DT2H2X---1.9.5.THM-Ref.s

%------------------------------------------------------------------------------
% File     : DT2H2X---1.9.5
% Problem  : ITP238^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm  : none
% Format   : tptp:raw
% Command  : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300

% Computer : n009.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 09:09:03 AM UTC 2026

% Result   : Theorem 59.76s 13.06s
% Output   : Refutation 59.76s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    9
%            Number of leaves      :    5
% Syntax   : Number of formulae    :   23 (  11 unt;   0 typ;   0 def)
%            Number of atoms       :   47 (  29 equ;   0 cnn)
%            Maximal formula atoms :    2 (   2 avg)
%            Number of connectives :  135 (  17   ~;   6   |;   0   &; 106   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   3 avg)
%            Number of types       :  110 ( 109 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1754 (1751 usr;  64 con; 0-8 aty)
%            Number of variables   :   17 (   0   ^;  15   !;   2   ?;  17   :)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(type_def_53,type,
    set_Pr958786334691620121nt_int: $tType ).

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

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

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

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

thf(type_def_58,type,
    list_P3795440434834930179_o_int: $tType ).

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

thf(type_def_60,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(type_def_61,type,
    produc2504756804600209347T_VEBT: $tType ).

thf(type_def_62,type,
    list_P5087981734274514673_int_o: $tType ).

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

thf(type_def_64,type,
    list_P5707943133018811711nt_int: $tType ).

thf(type_def_65,type,
    list_P8198026277950538467nt_nat: $tType ).

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

thf(type_def_67,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(type_def_68,type,
    produc1531783533982839933T_VEBT: $tType ).

thf(type_def_69,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(type_def_70,type,
    produc334124729049499915VEBT_o: $tType ).

thf(type_def_71,type,
    list_P4547456442757143711BT_int: $tType ).

thf(type_def_72,type,
    produc4894624898956917775BT_int: $tType ).

thf(type_def_73,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(type_def_74,type,
    produc9072475918466114483BT_nat: $tType ).

thf(type_def_75,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(type_def_76,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(type_def_77,type,
    list_P4002435161011370285od_o_o: $tType ).

thf(type_def_78,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(type_def_79,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(type_def_80,type,
    list_P6011104703257516679at_nat: $tType ).

thf(type_def_81,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(type_def_82,type,
    option4927543243414619207at_nat: $tType ).

thf(type_def_83,type,
    option6357759511663192854e_term: $tType ).

thf(type_def_84,type,
    produc6241069584506657477e_term: $tType ).

thf(type_def_85,type,
    produc8551481072490612790e_term: $tType ).

thf(type_def_86,type,
    set_Pr8056137968301705908nteger: $tType ).

thf(type_def_87,type,
    set_Pr1872883991513573699nt_int: $tType ).

thf(type_def_88,type,
    set_Pr1281608226676607948nteger: $tType ).

thf(type_def_89,type,
    set_Pr9222295170931077689nt_int: $tType ).

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

thf(type_def_91,type,
    produc8763457246119570046nteger: $tType ).

thf(type_def_92,type,
    produc7773217078559923341nt_int: $tType ).

thf(type_def_93,type,
    produc4953844613479565601on_nat: $tType ).

thf(type_def_94,type,
    produc2233624965454879586on_nat: $tType ).

thf(type_def_95,type,
    produc7248412053542808358at_nat: $tType ).

thf(type_def_96,type,
    produc4471711990508489141at_nat: $tType ).

thf(type_def_97,type,
    produc8306885398267862888on_nat: $tType ).

thf(type_def_98,type,
    produc3447558737645232053on_num: $tType ).

thf(type_def_99,type,
    produc7036089656553540234on_num: $tType ).

thf(type_def_100,type,
    produc1193250871479095198on_num: $tType ).

thf(type_def_101,type,
    produc1908205239877642774nteger: $tType ).

thf(type_def_102,type,
    produc2285326912895808259nt_int: $tType ).

thf(type_def_103,type,
    produc6121120109295599847at_nat: $tType ).

thf(type_def_104,type,
    produc5491161045314408544at_nat: $tType ).

thf(type_def_105,type,
    produc5542196010084753463at_nat: $tType ).

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

thf(type_def_107,type,
    produc8064648209034914857omplex: $tType ).

thf(type_def_108,type,
    produc2115011035271226405et_int: $tType ).

thf(type_def_109,type,
    produc7819656566062154093et_nat: $tType ).

thf(type_def_110,type,
    produc4411394909380815293omplex: $tType ).

thf(type_def_111,type,
    produc2422161461964618553l_real: $tType ).

thf(type_def_112,type,
    set_Pr6308028481084910985omplex: $tType ).

thf(type_def_113,type,
    set_Pr2522554150109002629et_int: $tType ).

thf(type_def_114,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(func_def_0,type,
    produc5542196010084753463at_nat: $tType ).

thf(func_def_1,type,
    set_Pr1281608226676607948nteger: $tType ).

thf(func_def_2,type,
    produc1908205239877642774nteger: $tType ).

thf(func_def_3,type,
    produc5491161045314408544at_nat: $tType ).

thf(func_def_4,type,
    set_Pr9222295170931077689nt_int: $tType ).

thf(func_def_5,type,
    produc2285326912895808259nt_int: $tType ).

thf(func_def_6,type,
    set_Pr8056137968301705908nteger: $tType ).

thf(func_def_7,type,
    produc8763457246119570046nteger: $tType ).

thf(func_def_8,type,
    set_Pr1872883991513573699nt_int: $tType ).

thf(func_def_9,type,
    produc7773217078559923341nt_int: $tType ).

thf(func_def_10,type,
    produc1193250871479095198on_num: $tType ).

thf(func_def_11,type,
    produc8306885398267862888on_nat: $tType ).

thf(func_def_12,type,
    produc6121120109295599847at_nat: $tType ).

thf(func_def_13,type,
    produc4471711990508489141at_nat: $tType ).

thf(func_def_14,type,
    produc7036089656553540234on_num: $tType ).

thf(func_def_15,type,
    produc2233624965454879586on_nat: $tType ).

thf(func_def_16,type,
    produc6241069584506657477e_term: $tType ).

thf(func_def_17,type,
    set_Pr6308028481084910985omplex: $tType ).

thf(func_def_18,type,
    set_fi4554929511873752355omplex: $tType ).

thf(func_def_19,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(func_def_20,type,
    produc8551481072490612790e_term: $tType ).

thf(func_def_21,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(func_def_22,type,
    set_Pr2522554150109002629et_int: $tType ).

thf(func_def_23,type,
    produc8064648209034914857omplex: $tType ).

thf(func_def_24,type,
    option6357759511663192854e_term: $tType ).

thf(func_def_25,type,
    produc3447558737645232053on_num: $tType ).

thf(func_def_26,type,
    produc4953844613479565601on_nat: $tType ).

thf(func_def_27,type,
    produc7248412053542808358at_nat: $tType ).

thf(func_def_28,type,
    set_fi7789364187291644575l_real: $tType ).

thf(func_def_29,type,
    filter6041513312241820739omplex: $tType ).

thf(func_def_30,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(func_def_31,type,
    list_P4547456442757143711BT_int: $tType ).

thf(func_def_32,type,
    list_P5647936690300460905T_VEBT: $tType ).

thf(func_def_33,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(func_def_34,type,
    produc8243902056947475879T_VEBT: $tType ).

thf(func_def_35,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(func_def_36,type,
    produc7819656566062154093et_nat: $tType ).

thf(func_def_37,type,
    produc2115011035271226405et_int: $tType ).

thf(func_def_38,type,
    produc8923325533196201883nteger: $tType ).

thf(func_def_39,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(func_def_40,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(func_def_41,type,
    filter2146258269922977983l_real: $tType ).

thf(func_def_42,type,
    option4927543243414619207at_nat: $tType ).

thf(func_def_43,type,
    filter1242075044329608583at_nat: $tType ).

thf(func_def_44,type,
    set_Pr6218003697084177305l_real: $tType ).

thf(func_def_45,type,
    list_P6011104703257516679at_nat: $tType ).

thf(func_def_46,type,
    list_P8198026277950538467nt_nat: $tType ).

thf(func_def_47,type,
    list_P5707943133018811711nt_int: $tType ).

thf(func_def_48,type,
    produc9072475918466114483BT_nat: $tType ).

thf(func_def_49,type,
    produc4894624898956917775BT_int: $tType ).

thf(func_def_50,type,
    produc1531783533982839933T_VEBT: $tType ).

thf(func_def_51,type,
    set_Pr1261947904930325089at_nat: $tType ).

thf(func_def_52,type,
    set_Pr958786334691620121nt_int: $tType ).

thf(func_def_53,type,
    produc4411394909380815293omplex: $tType ).

thf(func_def_54,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(func_def_55,type,
    list_P5087981734274514673_int_o: $tType ).

thf(func_def_56,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(func_def_57,type,
    list_P3795440434834930179_o_int: $tType ).

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

thf(func_def_59,type,
    produc334124729049499915VEBT_o: $tType ).

thf(func_def_60,type,
    produc2504756804600209347T_VEBT: $tType ).

thf(func_def_61,type,
    produc6271795597528267376eger_o: $tType ).

thf(func_def_62,type,
    produc2422161461964618553l_real: $tType ).

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

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

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

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

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

thf(func_def_68,type,
    list_P4002435161011370285od_o_o: $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(func_def_109,type,
    archim7802044766580827645g_real: real > int ).

thf(func_def_110,type,
    archim3151403230148437115or_rat: rat > int ).

thf(func_def_111,type,
    archim6058952711729229775r_real: real > int ).

thf(func_def_112,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

thf(func_def_113,type,
    bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(func_def_114,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

thf(func_def_115,type,
    bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).

thf(func_def_116,type,
    bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).

thf(func_def_117,type,
    bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_118,type,
    bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).

thf(func_def_119,type,
    bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).

thf(func_def_120,type,
    bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).

thf(func_def_121,type,
    bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_122,type,
    bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).

thf(func_def_123,type,
    bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_124,type,
    bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).

thf(func_def_125,type,
    bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).

thf(func_def_126,type,
    bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).

thf(func_def_127,type,
    bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).

thf(func_def_128,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

thf(func_def_129,type,
    bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(func_def_130,type,
    bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).

thf(func_def_131,type,
    bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).

thf(func_def_132,type,
    bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).

thf(func_def_133,type,
    bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).

thf(func_def_134,type,
    bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).

thf(func_def_135,type,
    bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).

thf(func_def_136,type,
    bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).

thf(func_def_137,type,
    bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).

thf(func_def_138,type,
    bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).

thf(func_def_139,type,
    bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(func_def_140,type,
    bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(func_def_141,type,
    bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_142,type,
    bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).

thf(func_def_143,type,
    bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).

thf(func_def_144,type,
    binomial: nat > nat > nat ).

thf(func_def_145,type,
    gbinomial_complex: complex > nat > complex ).

thf(func_def_146,type,
    gbinomial_int: int > nat > int ).

thf(func_def_147,type,
    gbinomial_nat: nat > nat > nat ).

thf(func_def_148,type,
    gbinomial_rat: rat > nat > rat ).

thf(func_def_149,type,
    gbinomial_real: real > nat > real ).

thf(func_def_150,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_151,type,
    bit_and_not_num: num > num > option_num ).

thf(func_def_152,type,
    bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_153,type,
    bit_concat_bit: nat > int > int > int ).

thf(func_def_154,type,
    bit_or_not_num_neg: num > num > num ).

thf(func_def_155,type,
    bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_156,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(func_def_157,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(func_def_158,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(func_def_159,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(func_def_160,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(func_def_161,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(func_def_162,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(func_def_163,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(func_def_164,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(func_def_165,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(func_def_166,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(func_def_167,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(func_def_168,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(func_def_169,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(func_def_170,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(func_def_171,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(func_def_172,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(func_def_173,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(func_def_174,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(func_def_175,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(func_def_176,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(func_def_177,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(func_def_178,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(func_def_179,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(func_def_180,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(func_def_181,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(func_def_182,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(func_def_183,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(func_def_184,type,
    bit_take_bit_num: nat > num > option_num ).

thf(func_def_185,type,
    bit_un1837492267222099188nd_num: num > num > option_num ).

thf(func_def_186,type,
    bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_187,type,
    bit_un6178654185764691216or_num: num > num > option_num ).

thf(func_def_188,type,
    bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_189,type,
    bit_un7362597486090784418nd_num: num > num > option_num ).

thf(func_def_190,type,
    bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_191,type,
    bit_un2480387367778600638or_num: num > num > option_num ).

thf(func_def_192,type,
    bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).

thf(func_def_193,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(func_def_194,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_195,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_196,type,
    code_int_of_integer: code_integer > int ).

thf(func_def_197,type,
    code_integer_of_int: int > code_integer ).

thf(func_def_198,type,
    code_integer_of_num: num > code_integer ).

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

thf(func_def_200,type,
    code_negative: num > code_integer ).

thf(func_def_201,type,
    code_num_of_integer: code_integer > num ).

thf(func_def_202,type,
    code_positive: num > code_integer ).

thf(func_def_203,type,
    code_Target_negative: num > int ).

thf(func_def_204,type,
    code_Target_positive: num > int ).

thf(func_def_205,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(func_def_206,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

thf(func_def_207,type,
    comple4887499456419720421f_real: set_real > real ).

thf(func_def_208,type,
    complete_Sup_Sup_int: set_int > int ).

thf(func_def_209,type,
    comple1385675409528146559p_real: set_real > real ).

thf(func_def_210,type,
    arg: complex > real ).

thf(func_def_211,type,
    cis: real > complex ).

thf(func_def_212,type,
    cnj: complex > complex ).

thf(func_def_213,type,
    complex2: real > real > complex ).

thf(func_def_214,type,
    im: complex > real ).

thf(func_def_215,type,
    re: complex > real ).

thf(func_def_216,type,
    csqrt: complex > complex ).

thf(func_def_217,type,
    imaginary_unit: complex ).

thf(func_def_218,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(func_def_219,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(func_def_220,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(func_def_221,type,
    adjust_div: product_prod_int_int > int ).

thf(func_def_222,type,
    adjust_mod: int > int > int ).

thf(func_def_223,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_224,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(func_def_225,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

thf(func_def_226,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(func_def_227,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(func_def_228,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(func_def_229,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(func_def_230,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(func_def_231,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_232,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(func_def_233,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_234,type,
    comm_s8582702949713902594nteger: code_integer > nat > code_integer ).

thf(func_def_235,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(func_def_236,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(func_def_237,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(func_def_238,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(func_def_239,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(func_def_240,type,
    semiri3624122377584611663nteger: nat > code_integer ).

thf(func_def_241,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(func_def_242,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(func_def_243,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(func_def_244,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(func_def_245,type,
    semiri2265585572941072030t_real: nat > real ).

thf(func_def_246,type,
    invers8013647133539491842omplex: complex > complex ).

thf(func_def_247,type,
    inverse_inverse_rat: rat > rat ).

thf(func_def_248,type,
    inverse_inverse_real: real > real ).

thf(func_def_249,type,
    at_bot_real: filter_real ).

thf(func_def_250,type,
    at_top_nat: filter_nat ).

thf(func_def_251,type,
    at_top_real: filter_real ).

thf(func_def_252,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(func_def_253,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(func_def_254,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(func_def_255,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(func_def_256,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(func_def_257,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(func_def_258,type,
    princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).

thf(func_def_259,type,
    princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).

thf(func_def_260,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

thf(func_def_261,type,
    finite_card_o: set_o > nat ).

thf(func_def_262,type,
    finite_card_complex: set_complex > nat ).

thf(func_def_263,type,
    finite_card_list_nat: set_list_nat > nat ).

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

thf(func_def_265,type,
    finite_card_char: set_char > nat ).

thf(func_def_266,type,
    finite_finite_o: set_o > $o ).

thf(func_def_267,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(func_def_268,type,
    finite_finite_int: set_int > $o ).

thf(func_def_269,type,
    finite_finite_list_o: set_list_o > $o ).

thf(func_def_270,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

thf(func_def_271,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(func_def_272,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(func_def_273,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

thf(func_def_274,type,
    finite_finite_nat: set_nat > $o ).

thf(func_def_275,type,
    finite_finite_num: set_num > $o ).

thf(func_def_276,type,
    finite_finite_rat: set_rat > $o ).

thf(func_def_277,type,
    finite_finite_real: set_real > $o ).

thf(func_def_278,type,
    finite6551019134538273531omplex: set_set_complex > $o ).

thf(func_def_279,type,
    finite6197958912794628473et_int: set_set_int > $o ).

thf(func_def_280,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(func_def_281,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

thf(func_def_282,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(func_def_283,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(func_def_284,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(func_def_285,type,
    comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_286,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_287,type,
    comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).

thf(func_def_288,type,
    comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).

thf(func_def_289,type,
    comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).

thf(func_def_290,type,
    id_o: $o > $o ).

thf(func_def_291,type,
    id_nat: nat > nat ).

thf(func_def_292,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_293,type,
    inj_on_nat_char: ( nat > char ) > set_nat > $o ).

thf(func_def_294,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(func_def_295,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_296,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_297,type,
    map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).

thf(func_def_298,type,
    map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).

thf(func_def_299,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_300,type,
    map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).

thf(func_def_301,type,
    map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).

thf(func_def_302,type,
    map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).

thf(func_def_303,type,
    strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_304,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(func_def_305,type,
    gcd_Gcd_int: set_int > int ).

thf(func_def_306,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(func_def_307,type,
    bezw: nat > nat > product_prod_int_int ).

thf(func_def_308,type,
    bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_309,type,
    gcd_gcd_int: int > int > int ).

thf(func_def_310,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(func_def_311,type,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_312,type,
    abs_abs_Code_integer: code_integer > code_integer ).

thf(func_def_313,type,
    abs_abs_complex: complex > complex ).

thf(func_def_314,type,
    abs_abs_int: int > int ).

thf(func_def_315,type,
    abs_abs_rat: rat > rat ).

thf(func_def_316,type,
    abs_abs_real: real > real ).

thf(func_def_317,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(func_def_318,type,
    minus_minus_complex: complex > complex > complex ).

thf(func_def_319,type,
    minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_320,type,
    minus_minus_int: int > int > int ).

thf(func_def_321,type,
    minus_minus_nat: nat > nat > nat ).

thf(func_def_322,type,
    minus_minus_rat: rat > rat > rat ).

thf(func_def_323,type,
    minus_minus_real: real > real > real ).

thf(func_def_324,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

thf(func_def_325,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(func_def_326,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_327,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(func_def_328,type,
    one_one_Code_integer: code_integer ).

thf(func_def_329,type,
    one_one_complex: complex ).

thf(func_def_330,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(func_def_331,type,
    one_one_int: int ).

thf(func_def_332,type,
    one_one_nat: nat ).

thf(func_def_333,type,
    one_one_rat: rat ).

thf(func_def_334,type,
    one_one_real: real ).

thf(func_def_335,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(func_def_336,type,
    plus_plus_complex: complex > complex > complex ).

thf(func_def_337,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_338,type,
    plus_plus_int: int > int > int ).

thf(func_def_339,type,
    plus_plus_nat: nat > nat > nat ).

thf(func_def_340,type,
    plus_plus_num: num > num > num ).

thf(func_def_341,type,
    plus_plus_rat: rat > rat > rat ).

thf(func_def_342,type,
    plus_plus_real: real > real > real ).

thf(func_def_343,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(func_def_344,type,
    sgn_sgn_complex: complex > complex ).

thf(func_def_345,type,
    sgn_sgn_int: int > int ).

thf(func_def_346,type,
    sgn_sgn_rat: rat > rat ).

thf(func_def_347,type,
    sgn_sgn_real: real > real ).

thf(func_def_348,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(func_def_349,type,
    times_times_complex: complex > complex > complex ).

thf(func_def_350,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_351,type,
    times_times_int: int > int > int ).

thf(func_def_352,type,
    times_times_nat: nat > nat > nat ).

thf(func_def_353,type,
    times_times_num: num > num > num ).

thf(func_def_354,type,
    times_times_rat: rat > rat > rat ).

thf(func_def_355,type,
    times_times_real: real > real > real ).

thf(func_def_356,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(func_def_357,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(func_def_358,type,
    uminus_uminus_int: int > int ).

thf(func_def_359,type,
    uminus_uminus_rat: rat > rat ).

thf(func_def_360,type,
    uminus_uminus_real: real > real ).

thf(func_def_361,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(func_def_362,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(func_def_363,type,
    uminus612125837232591019t_real: set_real > set_real ).

thf(func_def_364,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(func_def_365,type,
    zero_zero_complex: complex ).

thf(func_def_366,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(func_def_367,type,
    zero_zero_int: int ).

thf(func_def_368,type,
    zero_zero_nat: nat ).

thf(func_def_369,type,
    zero_zero_rat: rat ).

thf(func_def_370,type,
    zero_zero_real: real ).

thf(func_def_371,type,
    groups6621422865394947399nteger: ( complex > code_integer ) > set_complex > code_integer ).

thf(func_def_372,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_373,type,
    groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_374,type,
    groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_375,type,
    groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_376,type,
    groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).

thf(func_def_377,type,
    groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).

thf(func_def_378,type,
    groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).

thf(func_def_379,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(func_def_380,type,
    groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_381,type,
    groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_382,type,
    groups8778361861064173332t_real: ( int > real ) > set_int > real ).

thf(func_def_383,type,
    groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(func_def_384,type,
    groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_385,type,
    groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).

thf(func_def_386,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_387,type,
    groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_388,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(func_def_389,type,
    groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(func_def_390,type,
    groups4567486121110086003t_real: ( product_prod_nat_nat > real ) > set_Pr1261947904930325089at_nat > real ).

thf(func_def_391,type,
    groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).

thf(func_def_392,type,
    groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).

thf(func_def_393,type,
    groups1932886352136224148al_int: ( real > int ) > set_real > int ).

thf(func_def_394,type,
    groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_395,type,
    groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_396,type,
    groups8097168146408367636l_real: ( real > real ) > set_real > real ).

thf(func_def_397,type,
    groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).

thf(func_def_398,type,
    groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_399,type,
    groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_400,type,
    groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_401,type,
    groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_402,type,
    groups766887009212190081x_real: ( complex > real ) > set_complex > real ).

thf(func_def_403,type,
    groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).

thf(func_def_404,type,
    groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).

thf(func_def_405,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(func_def_406,type,
    groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_407,type,
    groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_408,type,
    groups2316167850115554303t_real: ( int > real ) > set_int > real ).

thf(func_def_409,type,
    groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(func_def_410,type,
    groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_411,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(func_def_412,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_413,type,
    groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_414,type,
    groups129246275422532515t_real: ( nat > real ) > set_nat > real ).

thf(func_def_415,type,
    groups4075276357253098568at_int: ( product_prod_nat_nat > int ) > set_Pr1261947904930325089at_nat > int ).

thf(func_def_416,type,
    groups4077766827762148844at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(func_def_417,type,
    groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).

thf(func_def_418,type,
    groups713298508707869441omplex: ( real > complex ) > set_real > complex ).

thf(func_def_419,type,
    groups4694064378042380927al_int: ( real > int ) > set_real > int ).

thf(func_def_420,type,
    groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_421,type,
    groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_422,type,
    groups1681761925125756287l_real: ( real > real ) > set_real > real ).

thf(func_def_423,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(func_def_424,type,
    groups4561878855575611511st_nat: list_nat > nat ).

thf(func_def_425,type,
    the_int: ( int > $o ) > int ).

thf(func_def_426,type,
    the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).

thf(func_def_427,type,
    the_real: ( real > $o ) > real ).

thf(func_def_428,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(func_def_429,type,
    if_complex: $o > complex > complex > complex ).

thf(func_def_430,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(func_def_431,type,
    if_int: $o > int > int > int ).

thf(func_def_432,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(func_def_433,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(func_def_434,type,
    if_nat: $o > nat > nat > nat ).

thf(func_def_435,type,
    if_num: $o > num > num > num ).

thf(func_def_436,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(func_def_437,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(func_def_438,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(func_def_439,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_440,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(func_def_441,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_442,type,
    if_rat: $o > rat > rat > rat ).

thf(func_def_443,type,
    if_real: $o > real > real > real ).

thf(func_def_444,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(func_def_445,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_446,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(func_def_447,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(func_def_448,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(func_def_449,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(func_def_450,type,
    intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_451,type,
    nat2: int > nat ).

thf(func_def_452,type,
    pcr_int: product_prod_nat_nat > int > $o ).

thf(func_def_453,type,
    power_int_real: real > int > real ).

thf(func_def_454,type,
    ring_1_Ints_real: set_real ).

thf(func_def_455,type,
    ring_17405671764205052669omplex: int > complex ).

thf(func_def_456,type,
    ring_1_of_int_int: int > int ).

thf(func_def_457,type,
    ring_1_of_int_rat: int > rat ).

thf(func_def_458,type,
    ring_1_of_int_real: int > real ).

thf(func_def_459,type,
    inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_460,type,
    inf_inf_nat: nat > nat > nat ).

thf(func_def_461,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_462,type,
    sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_463,type,
    sup_sup_nat: nat > nat > nat ).

thf(func_def_464,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_465,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(func_def_466,type,
    bfun_nat_real: ( nat > real ) > filter_nat > $o ).

thf(func_def_467,type,
    at_infinity_real: filter_real ).

thf(func_def_468,type,
    append_int: list_int > list_int > list_int ).

thf(func_def_469,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(func_def_470,type,
    drop_nat: nat > list_nat > list_nat ).

thf(func_def_471,type,
    fold_int_int: ( int > int > int ) > list_int > int > int ).

thf(func_def_472,type,
    fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(func_def_473,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(func_def_474,type,
    cons_int: int > list_int > list_int ).

thf(func_def_475,type,
    cons_nat: nat > list_nat > list_nat ).

thf(func_def_476,type,
    nil_int: list_int ).

thf(func_def_477,type,
    nil_nat: list_nat ).

thf(func_def_478,type,
    hd_nat: list_nat > nat ).

thf(func_def_479,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(func_def_480,type,
    set_o2: list_o > set_o ).

thf(func_def_481,type,
    set_complex2: list_complex > set_complex ).

thf(func_def_482,type,
    set_int2: list_int > set_int ).

thf(func_def_483,type,
    set_list_nat2: list_list_nat > set_list_nat ).

thf(func_def_484,type,
    set_nat2: list_nat > set_nat ).

thf(func_def_485,type,
    set_real2: list_real > set_real ).

thf(func_def_486,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_487,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

thf(func_def_488,type,
    list_update_o: list_o > nat > $o > list_o ).

thf(func_def_489,type,
    list_update_complex: list_complex > nat > complex > list_complex ).

thf(func_def_490,type,
    list_update_int: list_int > nat > int > list_int ).

thf(func_def_491,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(func_def_492,type,
    list_update_real: list_real > nat > real > list_real ).

thf(func_def_493,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_494,type,
    nth_o: list_o > nat > $o ).

thf(func_def_495,type,
    nth_complex: list_complex > nat > complex ).

thf(func_def_496,type,
    nth_int: list_int > nat > int ).

thf(func_def_497,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(func_def_498,type,
    nth_nat: list_nat > nat > nat ).

thf(func_def_499,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(func_def_500,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(func_def_501,type,
    nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).

thf(func_def_502,type,
    nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).

thf(func_def_503,type,
    nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).

thf(func_def_504,type,
    nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).

thf(func_def_505,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(func_def_506,type,
    nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).

thf(func_def_507,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(func_def_508,type,
    nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).

thf(func_def_509,type,
    nth_real: list_real > nat > real ).

thf(func_def_510,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_511,type,
    product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(func_def_512,type,
    product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(func_def_513,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(func_def_514,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(func_def_515,type,
    product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).

thf(func_def_516,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(func_def_517,type,
    product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

thf(func_def_518,type,
    produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(func_def_519,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_520,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(func_def_521,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_522,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(func_def_523,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(func_def_524,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(func_def_525,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_526,type,
    replicate_o: nat > $o > list_o ).

thf(func_def_527,type,
    replicate_complex: nat > complex > list_complex ).

thf(func_def_528,type,
    replicate_int: nat > int > list_int ).

thf(func_def_529,type,
    replicate_nat: nat > nat > list_nat ).

thf(func_def_530,type,
    replicate_real: nat > real > list_real ).

thf(func_def_531,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_532,type,
    sorted_wrt_int: ( int > int > $o ) > list_int > $o ).

thf(func_def_533,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(func_def_534,type,
    take_nat: nat > list_nat > list_nat ).

thf(func_def_535,type,
    upt: nat > nat > list_nat ).

thf(func_def_536,type,
    upto: int > int > list_int ).

thf(func_def_537,type,
    upto_aux: int > int > list_int > list_int ).

thf(func_def_538,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_539,type,
    suc: nat > nat ).

thf(func_def_540,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

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

thf(func_def_542,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_543,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

thf(func_def_544,type,
    pred: nat > nat ).

thf(func_def_545,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(func_def_546,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(func_def_547,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

thf(func_def_548,type,
    semiri1314217659103216013at_int: nat > int ).

thf(func_def_549,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(func_def_550,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(func_def_551,type,
    semiri5074537144036343181t_real: nat > real ).

thf(func_def_552,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

thf(func_def_553,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(func_def_554,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(func_def_555,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(func_def_556,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

thf(func_def_557,type,
    size_size_list_o: list_o > nat ).

thf(func_def_558,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(func_def_559,type,
    size_size_list_int: list_int > nat ).

thf(func_def_560,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(func_def_561,type,
    size_size_list_nat: list_nat > nat ).

thf(func_def_562,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

thf(func_def_563,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

thf(func_def_564,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(func_def_565,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

thf(func_def_566,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(func_def_567,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).

thf(func_def_568,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

thf(func_def_569,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

thf(func_def_570,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(func_def_571,type,
    size_size_list_real: list_real > nat ).

thf(func_def_572,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(func_def_573,type,
    size_size_num: num > nat ).

thf(func_def_574,type,
    size_size_option_nat: option_nat > nat ).

thf(func_def_575,type,
    size_size_option_num: option_num > nat ).

thf(func_def_576,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(func_def_577,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(func_def_578,type,
    nat_list_encode: list_nat > nat ).

thf(func_def_579,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(func_def_580,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(func_def_581,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_582,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(func_def_583,type,
    nat_set_decode: nat > set_nat ).

thf(func_def_584,type,
    nat_set_encode: set_nat > nat ).

thf(func_def_585,type,
    nat_triangle: nat > nat ).

thf(func_def_586,type,
    root: nat > real > real ).

thf(func_def_587,type,
    sqrt: real > real ).

thf(func_def_588,type,
    bitM: num > num ).

thf(func_def_589,type,
    inc: num > num ).

thf(func_def_590,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

thf(func_def_591,type,
    neg_nu7009210354673126013omplex: complex > complex ).

thf(func_def_592,type,
    neg_numeral_dbl_int: int > int ).

thf(func_def_593,type,
    neg_numeral_dbl_rat: rat > rat ).

thf(func_def_594,type,
    neg_numeral_dbl_real: real > real ).

thf(func_def_595,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(func_def_596,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(func_def_597,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(func_def_598,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(func_def_599,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(func_def_600,type,
    neg_nu5831290666863070958nteger: code_integer > code_integer ).

thf(func_def_601,type,
    neg_nu8557863876264182079omplex: complex > complex ).

thf(func_def_602,type,
    neg_nu5851722552734809277nc_int: int > int ).

thf(func_def_603,type,
    neg_nu5219082963157363817nc_rat: rat > rat ).

thf(func_def_604,type,
    neg_nu8295874005876285629c_real: real > real ).

thf(func_def_605,type,
    neg_numeral_sub_int: num > num > int ).

thf(func_def_606,type,
    bit0: num > num ).

thf(func_def_607,type,
    bit1: num > num ).

thf(func_def_608,type,
    one: num ).

thf(func_def_609,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(func_def_610,type,
    size_num: num > nat ).

thf(func_def_611,type,
    num_of_nat: nat > num ).

thf(func_def_612,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(func_def_613,type,
    numera6690914467698888265omplex: num > complex ).

thf(func_def_614,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(func_def_615,type,
    numeral_numeral_int: num > int ).

thf(func_def_616,type,
    numeral_numeral_nat: num > nat ).

thf(func_def_617,type,
    numeral_numeral_rat: num > rat ).

thf(func_def_618,type,
    numeral_numeral_real: num > real ).

thf(func_def_619,type,
    pow: num > num > num ).

thf(func_def_620,type,
    pred_numeral: num > nat ).

thf(func_def_621,type,
    sqr: num > num ).

thf(func_def_622,type,
    none_nat: option_nat ).

thf(func_def_623,type,
    none_num: option_num ).

thf(func_def_624,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(func_def_625,type,
    some_nat: nat > option_nat ).

thf(func_def_626,type,
    some_num: num > option_num ).

thf(func_def_627,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(func_def_628,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(func_def_629,type,
    case_option_int_num: int > ( num > int ) > option_num > int ).

thf(func_def_630,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(func_def_631,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(func_def_632,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

thf(func_def_633,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(func_def_634,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(func_def_635,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

thf(func_def_636,type,
    the_nat: option_nat > nat ).

thf(func_def_637,type,
    the_num: option_num > num ).

thf(func_def_638,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_639,type,
    bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(func_def_640,type,
    bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(func_def_641,type,
    bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(func_def_642,type,
    bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(func_def_643,type,
    bot_bot_int_int_o: int > int > $o ).

thf(func_def_644,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(func_def_645,type,
    bot_bot_nat: nat ).

thf(func_def_646,type,
    bot_bot_set_complex: set_complex ).

thf(func_def_647,type,
    bot_bot_set_int: set_int ).

thf(func_def_648,type,
    bot_bot_set_nat: set_nat ).

thf(func_def_649,type,
    bot_bot_set_num: set_num ).

thf(func_def_650,type,
    bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).

thf(func_def_651,type,
    bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).

thf(func_def_652,type,
    bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).

thf(func_def_653,type,
    bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).

thf(func_def_654,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(func_def_655,type,
    bot_bot_set_rat: set_rat ).

thf(func_def_656,type,
    bot_bot_set_real: set_real ).

thf(func_def_657,type,
    bot_bot_set_set_int: set_set_int ).

thf(func_def_658,type,
    ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).

thf(func_def_659,type,
    ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).

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

thf(func_def_661,type,
    ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_662,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(func_def_663,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(func_def_664,type,
    ord_less_int: int > int > $o ).

thf(func_def_665,type,
    ord_less_nat: nat > nat > $o ).

thf(func_def_666,type,
    ord_less_num: num > num > $o ).

thf(func_def_667,type,
    ord_less_rat: rat > rat > $o ).

thf(func_def_668,type,
    ord_less_real: real > real > $o ).

thf(func_def_669,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

thf(func_def_670,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

thf(func_def_671,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(func_def_672,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(func_def_673,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(func_def_674,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(func_def_675,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(func_def_676,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

thf(func_def_677,type,
    ord_le3636971675376928563eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(func_def_678,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_679,type,
    ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(func_def_680,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_681,type,
    ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).

thf(func_def_682,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(func_def_683,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_684,type,
    ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).

thf(func_def_685,type,
    ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).

thf(func_def_686,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_687,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_688,type,
    ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).

thf(func_def_689,type,
    ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).

thf(func_def_690,type,
    ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).

thf(func_def_691,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_692,type,
    ord_le2556027599737686990_num_o: ( product_prod_num_num > product_prod_num_num > $o ) > ( product_prod_num_num > product_prod_num_num > $o ) > $o ).

thf(func_def_693,type,
    ord_le2239182809043710856_num_o: ( product_prod_num_num > $o ) > ( product_prod_num_num > $o ) > $o ).

thf(func_def_694,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_695,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(func_def_696,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(func_def_697,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(func_def_698,type,
    ord_le4104064031414453916r_real: filter_real > filter_real > $o ).

thf(func_def_699,type,
    ord_less_eq_int: int > int > $o ).

thf(func_def_700,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(func_def_701,type,
    ord_less_eq_num: num > num > $o ).

thf(func_def_702,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(func_def_703,type,
    ord_less_eq_real: real > real > $o ).

thf(func_def_704,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(func_def_705,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(func_def_706,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

thf(func_def_707,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(func_def_708,type,
    ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).

thf(func_def_709,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(func_def_710,type,
    ord_less_eq_set_num: set_num > set_num > $o ).

thf(func_def_711,type,
    ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o ).

thf(func_def_712,type,
    ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(func_def_713,type,
    ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).

thf(func_def_714,type,
    ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(func_def_715,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(func_def_716,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(func_def_717,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(func_def_718,type,
    ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).

thf(func_def_719,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_720,type,
    ord_max_Code_integer: code_integer > code_integer > code_integer ).

thf(func_def_721,type,
    ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_722,type,
    ord_max_int: int > int > int ).

thf(func_def_723,type,
    ord_max_nat: nat > nat > nat ).

thf(func_def_724,type,
    ord_max_num: num > num > num ).

thf(func_def_725,type,
    ord_max_rat: rat > rat > rat ).

thf(func_def_726,type,
    ord_max_real: real > real > real ).

thf(func_def_727,type,
    ord_max_set_int: set_int > set_int > set_int ).

thf(func_def_728,type,
    ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_729,type,
    ord_min_nat: nat > nat > nat ).

thf(func_def_730,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(func_def_731,type,
    order_9091379641038594480t_real: ( nat > real ) > $o ).

thf(func_def_732,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(func_def_733,type,
    order_mono_nat_real: ( nat > real ) > $o ).

thf(func_def_734,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(func_def_735,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(func_def_736,type,
    top_top_set_o: set_o ).

thf(func_def_737,type,
    top_top_set_nat: set_nat ).

thf(func_def_738,type,
    top_top_set_real: set_real ).

thf(func_def_739,type,
    top_top_set_char: set_char ).

thf(func_def_740,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(func_def_741,type,
    power_power_complex: complex > nat > complex ).

thf(func_def_742,type,
    power_power_int: int > nat > int ).

thf(func_def_743,type,
    power_power_nat: nat > nat > nat ).

thf(func_def_744,type,
    power_power_rat: rat > nat > rat ).

thf(func_def_745,type,
    power_power_real: real > nat > real ).

thf(func_def_746,type,
    produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).

thf(func_def_747,type,
    produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).

thf(func_def_748,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(func_def_749,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(func_def_750,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(func_def_751,type,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

thf(func_def_752,type,
    produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).

thf(func_def_753,type,
    produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).

thf(func_def_754,type,
    produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).

thf(func_def_755,type,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(func_def_756,type,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

thf(func_def_757,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

thf(func_def_758,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

thf(func_def_759,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(func_def_760,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_761,type,
    product_Pair_int_o: int > $o > product_prod_int_o ).

thf(func_def_762,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(func_def_763,type,
    product_Pair_int_nat: int > nat > product_prod_int_nat ).

thf(func_def_764,type,
    produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).

thf(func_def_765,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_766,type,
    product_Pair_nat_num: nat > num > product_prod_nat_num ).

thf(func_def_767,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(func_def_768,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

thf(func_def_769,type,
    produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).

thf(func_def_770,type,
    produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).

thf(func_def_771,type,
    produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).

thf(func_def_772,type,
    produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex ).

thf(func_def_773,type,
    produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).

thf(func_def_774,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(func_def_775,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(func_def_776,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

thf(func_def_777,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(func_def_778,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(func_def_779,type,
    produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_780,type,
    produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).

thf(func_def_781,type,
    produc2592262431452330817omplex: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex ) > produc8763457246119570046nteger > set_complex ).

thf(func_def_782,type,
    produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).

thf(func_def_783,type,
    produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).

thf(func_def_784,type,
    produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).

thf(func_def_785,type,
    produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).

thf(func_def_786,type,
    produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).

thf(func_def_787,type,
    produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).

thf(func_def_788,type,
    produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).

thf(func_def_789,type,
    produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).

thf(func_def_790,type,
    produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).

thf(func_def_791,type,
    produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(func_def_792,type,
    produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).

thf(func_def_793,type,
    produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).

thf(func_def_794,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_795,type,
    produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).

thf(func_def_796,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(func_def_797,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(func_def_798,type,
    produc8580519160106071146omplex: ( int > int > set_complex ) > product_prod_int_int > set_complex ).

thf(func_def_799,type,
    produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).

thf(func_def_800,type,
    produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).

thf(func_def_801,type,
    produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).

thf(func_def_802,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_803,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_804,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_805,type,
    produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).

thf(func_def_806,type,
    produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).

thf(func_def_807,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(func_def_808,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_809,type,
    produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).

thf(func_def_810,type,
    produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).

thf(func_def_811,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

thf(func_def_812,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

thf(func_def_813,type,
    produc3914248068834153634plex_o: ( set_complex > set_complex > $o ) > produc8064648209034914857omplex > $o ).

thf(func_def_814,type,
    produc4109468873575309990_int_o: ( set_int > set_int > $o ) > produc2115011035271226405et_int > $o ).

thf(func_def_815,type,
    produc6247414631856714078_nat_o: ( set_nat > set_nat > $o ) > produc7819656566062154093et_nat > $o ).

thf(func_def_816,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(func_def_817,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_818,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(func_def_819,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_820,type,
    abs_Rat: product_prod_int_int > rat ).

thf(func_def_821,type,
    fract: int > int > rat ).

thf(func_def_822,type,
    frct: product_prod_int_int > rat ).

thf(func_def_823,type,
    rep_Rat: rat > product_prod_int_int ).

thf(func_def_824,type,
    field_5140801741446780682s_real: set_real ).

thf(func_def_825,type,
    field_7254667332652039916t_real: rat > real ).

thf(func_def_826,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(func_def_827,type,
    of_int: int > rat ).

thf(func_def_828,type,
    pcr_rat: product_prod_int_int > rat > $o ).

thf(func_def_829,type,
    positive: rat > $o ).

thf(func_def_830,type,
    quotient_of: rat > product_prod_int_int ).

thf(func_def_831,type,
    ratrel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_832,type,
    vanishes: ( nat > rat ) > $o ).

thf(func_def_833,type,
    real_V2521375963428798218omplex: set_complex ).

thf(func_def_834,type,
    real_V5970128139526366754l_real: ( real > real ) > $o ).

thf(func_def_835,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(func_def_836,type,
    real_V975177566351809787t_real: real > real > real ).

thf(func_def_837,type,
    real_V1022390504157884413omplex: complex > real ).

thf(func_def_838,type,
    real_V7735802525324610683m_real: real > real ).

thf(func_def_839,type,
    real_V4546457046886955230omplex: real > complex ).

thf(func_def_840,type,
    real_V2046097035970521341omplex: real > complex > complex ).

thf(func_def_841,type,
    real_V1485227260804924795R_real: real > real > real ).

thf(func_def_842,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

thf(func_def_843,type,
    algebr932160517623751201me_int: int > int > $o ).

thf(func_def_844,type,
    algebr934650988132801477me_nat: nat > nat > $o ).

thf(func_def_845,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(func_def_846,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(func_def_847,type,
    divide_divide_int: int > int > int ).

thf(func_def_848,type,
    divide_divide_nat: nat > nat > nat ).

thf(func_def_849,type,
    divide_divide_rat: rat > rat > rat ).

thf(func_def_850,type,
    divide_divide_real: real > real > real ).

thf(func_def_851,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(func_def_852,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(func_def_853,type,
    dvd_dvd_int: int > int > $o ).

thf(func_def_854,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(func_def_855,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(func_def_856,type,
    dvd_dvd_real: real > real > $o ).

thf(func_def_857,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

thf(func_def_858,type,
    modulo_modulo_int: int > int > int ).

thf(func_def_859,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(func_def_860,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(func_def_861,type,
    zero_n1201886186963655149omplex: $o > complex ).

thf(func_def_862,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(func_def_863,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(func_def_864,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

thf(func_def_865,type,
    zero_n3304061248610475627l_real: $o > real ).

thf(func_def_866,type,
    suminf_complex: ( nat > complex ) > complex ).

thf(func_def_867,type,
    suminf_int: ( nat > int ) > int ).

thf(func_def_868,type,
    suminf_nat: ( nat > nat ) > nat ).

thf(func_def_869,type,
    suminf_real: ( nat > real ) > real ).

thf(func_def_870,type,
    summable_complex: ( nat > complex ) > $o ).

thf(func_def_871,type,
    summable_int: ( nat > int ) > $o ).

thf(func_def_872,type,
    summable_nat: ( nat > nat ) > $o ).

thf(func_def_873,type,
    summable_real: ( nat > real ) > $o ).

thf(func_def_874,type,
    sums_complex: ( nat > complex ) > complex > $o ).

thf(func_def_875,type,
    sums_int: ( nat > int ) > int > $o ).

thf(func_def_876,type,
    sums_nat: ( nat > nat ) > nat > $o ).

thf(func_def_877,type,
    sums_real: ( nat > real ) > real > $o ).

thf(func_def_878,type,
    collect_o: ( $o > $o ) > set_o ).

thf(func_def_879,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

thf(func_def_880,type,
    collect_complex: ( complex > $o ) > set_complex ).

thf(func_def_881,type,
    collect_int: ( int > $o ) > set_int ).

thf(func_def_882,type,
    collect_list_o: ( list_o > $o ) > set_list_o ).

thf(func_def_883,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(func_def_884,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

thf(func_def_885,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(func_def_886,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

thf(func_def_887,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(func_def_888,type,
    collect_num: ( num > $o ) > set_num ).

thf(func_def_889,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(func_def_890,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(func_def_891,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(func_def_892,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(func_def_893,type,
    collec5108298041176329748omplex: ( produc8064648209034914857omplex > $o ) > set_Pr6308028481084910985omplex ).

thf(func_def_894,type,
    collec957716948307931664et_int: ( produc2115011035271226405et_int > $o ) > set_Pr2522554150109002629et_int ).

thf(func_def_895,type,
    collec6662362479098859352et_nat: ( produc7819656566062154093et_nat > $o ) > set_Pr5488025237498180813et_nat ).

thf(func_def_896,type,
    collect_rat: ( rat > $o ) > set_rat ).

thf(func_def_897,type,
    collect_real: ( real > $o ) > set_real ).

thf(func_def_898,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(func_def_899,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(func_def_900,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(func_def_901,type,
    collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).

thf(func_def_902,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(func_def_903,type,
    image_nat_int: ( nat > int ) > set_nat > set_int ).

thf(func_def_904,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(func_def_905,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

thf(func_def_906,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

thf(func_def_907,type,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(func_def_908,type,
    image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

thf(func_def_909,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(func_def_910,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

thf(func_def_911,type,
    insert_int: int > set_int > set_int ).

thf(func_def_912,type,
    insert_nat: nat > set_nat > set_nat ).

thf(func_def_913,type,
    insert_real: real > set_real > set_real ).

thf(func_def_914,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(func_def_915,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

thf(func_def_916,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(func_def_917,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(func_def_918,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(func_def_919,type,
    set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).

thf(func_def_920,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(func_def_921,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(func_def_922,type,
    set_or7049704709247886629st_num: num > num > set_num ).

thf(func_def_923,type,
    set_or633870826150836451st_rat: rat > rat > set_rat ).

thf(func_def_924,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(func_def_925,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

thf(func_def_926,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(func_def_927,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(func_def_928,type,
    set_ord_atLeast_real: real > set_real ).

thf(func_def_929,type,
    set_ord_atMost_int: int > set_int ).

thf(func_def_930,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(func_def_931,type,
    set_ord_atMost_num: num > set_num ).

thf(func_def_932,type,
    set_ord_atMost_rat: rat > set_rat ).

thf(func_def_933,type,
    set_ord_atMost_real: real > set_real ).

thf(func_def_934,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

thf(func_def_935,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(func_def_936,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(func_def_937,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(func_def_938,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(func_def_939,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(func_def_940,type,
    set_or5849166863359141190n_real: real > set_real ).

thf(func_def_941,type,
    set_ord_lessThan_int: int > set_int ).

thf(func_def_942,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(func_def_943,type,
    set_ord_lessThan_num: num > set_num ).

thf(func_def_944,type,
    set_ord_lessThan_rat: rat > set_rat ).

thf(func_def_945,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(func_def_946,type,
    ascii_of: char > char ).

thf(func_def_947,type,
    char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).

thf(func_def_948,type,
    comm_s629917340098488124ar_nat: char > nat ).

thf(func_def_949,type,
    integer_of_char: char > code_integer ).

thf(func_def_950,type,
    unique3096191561947761185of_nat: nat > char ).

thf(func_def_951,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(func_def_952,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(func_def_953,type,
    topolo4899668324122417113eq_int: ( nat > int ) > $o ).

thf(func_def_954,type,
    topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).

thf(func_def_955,type,
    topolo1459490580787246023eq_num: ( nat > num ) > $o ).

thf(func_def_956,type,
    topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).

thf(func_def_957,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(func_def_958,type,
    topolo3100542954746470799et_int: ( nat > set_int ) > $o ).

thf(func_def_959,type,
    topolo2177554685111907308n_real: real > set_real > filter_real ).

thf(func_def_960,type,
    topolo2815343760600316023s_real: real > filter_real ).

thf(func_def_961,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(func_def_962,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(func_def_963,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

thf(func_def_964,type,
    arccos: real > real ).

thf(func_def_965,type,
    arcosh_real: real > real ).

thf(func_def_966,type,
    arcsin: real > real ).

thf(func_def_967,type,
    arctan: real > real ).

thf(func_def_968,type,
    arsinh_real: real > real ).

thf(func_def_969,type,
    artanh_real: real > real ).

thf(func_def_970,type,
    cos_complex: complex > complex ).

thf(func_def_971,type,
    cos_real: real > real ).

thf(func_def_972,type,
    cos_coeff: nat > real ).

thf(func_def_973,type,
    cosh_real: real > real ).

thf(func_def_974,type,
    cot_real: real > real ).

thf(func_def_975,type,
    diffs_complex: ( nat > complex ) > nat > complex ).

thf(func_def_976,type,
    diffs_int: ( nat > int ) > nat > int ).

thf(func_def_977,type,
    diffs_rat: ( nat > rat ) > nat > rat ).

thf(func_def_978,type,
    diffs_real: ( nat > real ) > nat > real ).

thf(func_def_979,type,
    exp_complex: complex > complex ).

thf(func_def_980,type,
    exp_real: real > real ).

thf(func_def_981,type,
    ln_ln_real: real > real ).

thf(func_def_982,type,
    log: real > real > real ).

thf(func_def_983,type,
    pi: real ).

thf(func_def_984,type,
    powr_real: real > real > real ).

thf(func_def_985,type,
    sin_complex: complex > complex ).

thf(func_def_986,type,
    sin_real: real > real ).

thf(func_def_987,type,
    sin_coeff: nat > real ).

thf(func_def_988,type,
    sinh_real: real > real ).

thf(func_def_989,type,
    tan_complex: complex > complex ).

thf(func_def_990,type,
    tan_real: real > real ).

thf(func_def_991,type,
    tanh_complex: complex > complex ).

thf(func_def_992,type,
    tanh_real: real > real ).

thf(func_def_993,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

thf(func_def_994,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_995,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(func_def_996,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(func_def_997,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(func_def_998,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(func_def_999,type,
    vEBT_VEBT_low: nat > nat > nat ).

thf(func_def_1000,type,
    vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).

thf(func_def_1001,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1002,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(func_def_1003,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1004,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(func_def_1005,type,
    vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1006,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(func_def_1007,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_1008,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(func_def_1009,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(func_def_1010,type,
    vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1011,type,
    vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1012,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(func_def_1013,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(func_def_1014,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1015,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_1016,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(func_def_1017,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1018,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(func_def_1019,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(func_def_1020,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(func_def_1021,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(func_def_1022,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(func_def_1023,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(func_def_1024,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(func_def_1025,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(func_def_1026,type,
    vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(func_def_1027,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_1028,type,
    vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).

thf(func_def_1029,type,
    vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).

thf(func_def_1030,type,
    vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).

thf(func_def_1031,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(func_def_1032,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(func_def_1033,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1034,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(func_def_1035,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1036,type,
    vEBT_is_pred_in_set: set_nat > nat > nat > $o ).

thf(func_def_1037,type,
    vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).

thf(func_def_1038,type,
    vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1039,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

thf(func_def_1040,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(func_def_1041,type,
    accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).

thf(func_def_1042,type,
    accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).

thf(func_def_1043,type,
    accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).

thf(func_def_1044,type,
    accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).

thf(func_def_1045,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(func_def_1046,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_1047,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

thf(func_def_1048,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(func_def_1049,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(func_def_1050,type,
    finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).

thf(func_def_1051,type,
    finite_psubset_int: set_Pr2522554150109002629et_int ).

thf(func_def_1052,type,
    finite_psubset_nat: set_Pr5488025237498180813et_nat ).

thf(func_def_1053,type,
    measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).

thf(func_def_1054,type,
    fChoice_real: ( real > $o ) > real ).

thf(func_def_1055,type,
    member_o: $o > set_o > $o ).

thf(func_def_1056,type,
    member_complex: complex > set_complex > $o ).

thf(func_def_1057,type,
    member_int: int > set_int > $o ).

thf(func_def_1058,type,
    member_list_o: list_o > set_list_o > $o ).

thf(func_def_1059,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(func_def_1060,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).

thf(func_def_1061,type,
    member_nat: nat > set_nat > $o ).

thf(func_def_1062,type,
    member_num: num > set_num > $o ).

thf(func_def_1063,type,
    member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).

thf(func_def_1064,type,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(func_def_1065,type,
    member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).

thf(func_def_1066,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(func_def_1067,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(func_def_1068,type,
    member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).

thf(func_def_1069,type,
    member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).

thf(func_def_1070,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(func_def_1071,type,
    member_rat: rat > set_rat > $o ).

thf(func_def_1072,type,
    member_real: real > set_real > $o ).

thf(func_def_1073,type,
    member_set_int: set_int > set_set_int > $o ).

thf(func_def_1074,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_1075,type,
    deg: nat ).

thf(func_def_1076,type,
    m: nat ).

thf(func_def_1077,type,
    ma: nat ).

thf(func_def_1078,type,
    mi: nat ).

thf(func_def_1079,type,
    na: nat ).

thf(func_def_1080,type,
    summary: vEBT_VEBT ).

thf(func_def_1082,type,
    treeList: list_VEBT_VEBT ).

thf(func_def_1083,type,
    xa: nat ).

thf(func_def_1102,type,
    sP0: ( option4927543243414619207at_nat > option4927543243414619207at_nat > $o ) > option4927543243414619207at_nat > option4927543243414619207at_nat > $o ).

thf(func_def_1103,type,
    sP1: real > ( real > real > $o ) > real > $o ).

thf(func_def_1104,type,
    sP2: rat > rat > rat > $o ).

thf(func_def_1105,type,
    sP3: rat > rat > rat > $o ).

thf(func_def_1106,type,
    sP4: rat > rat > rat > $o ).

thf(func_def_1107,type,
    sP5: vEBT_VEBT > nat > $o ).

thf(func_def_1108,type,
    sP6: int > int > int > $o ).

thf(func_def_1109,type,
    sP7: int > int > int > $o ).

thf(func_def_1110,type,
    sP8: int > int > int > $o ).

thf(func_def_1111,type,
    sP9: ( option4927543243414619207at_nat > option_nat > $o ) > option4927543243414619207at_nat > option_nat > $o ).

thf(func_def_1112,type,
    sP10: int > ( int > $o ) > $o ).

thf(func_def_1113,type,
    sP11: vEBT_VEBT > nat > $o ).

thf(func_def_1114,type,
    sP12: ( option_nat > option4927543243414619207at_nat > $o ) > option_nat > option4927543243414619207at_nat > $o ).

thf(func_def_1115,type,
    sP13: ( int > $o ) > $o ).

thf(func_def_1116,type,
    sP14: ( option_nat > option_nat > $o ) > option_nat > option_nat > $o ).

thf(func_def_1117,type,
    sP15: nat > nat > nat > $o ).

thf(func_def_1118,type,
    sP16: nat > nat > nat > $o ).

thf(func_def_1119,type,
    sP17: nat > nat > nat > $o ).

thf(func_def_1120,type,
    sP18: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1121,type,
    sP19: nat > nat > list_VEBT_VEBT > $o ).

thf(func_def_1122,type,
    sP20: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1123,type,
    sP21: list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1124,type,
    sP22: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1125,type,
    sP23: nat > nat > nat > list_VEBT_VEBT > nat > $o ).

thf(func_def_1126,type,
    sP24: vEBT_VEBT > nat > $o ).

thf(func_def_1127,type,
    sP25: vEBT_VEBT > nat > $o ).

thf(func_def_1128,type,
    sP26: vEBT_VEBT > nat > $o ).

thf(func_def_1129,type,
    sP27: nat > vEBT_VEBT > $o ).

thf(func_def_1130,type,
    sP28: option_nat > vEBT_VEBT > $o ).

thf(func_def_1131,type,
    sP29: vEBT_VEBT > $o > $o ).

thf(func_def_1132,type,
    sP30: vEBT_VEBT > $o > $o ).

thf(func_def_1133,type,
    sK31: real > real > real > real ).

thf(func_def_1134,type,
    sK32: ( option4927543243414619207at_nat > option4927543243414619207at_nat > $o ) > option4927543243414619207at_nat > option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_1135,type,
    sK33: ( option4927543243414619207at_nat > option4927543243414619207at_nat > $o ) > option4927543243414619207at_nat > option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_1136,type,
    sK34: list_VEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1137,type,
    sK35: ( real > rat ) > real ).

thf(func_def_1138,type,
    sK36: ( real > rat ) > real ).

thf(func_def_1139,type,
    sK37: nat > nat > nat > nat ).

thf(func_def_1140,type,
    sK38: nat > nat > nat > nat ).

thf(func_def_1141,type,
    sK39: ( nat > rat ) > nat ).

thf(func_def_1142,type,
    sK40: ( nat > rat ) > nat ).

thf(func_def_1143,type,
    sK41: real > ( real > $o ) > real ).

thf(func_def_1144,type,
    sK42: real > ( real > $o ) > real ).

thf(func_def_1145,type,
    sK43: real > ( real > $o ) > real ).

thf(func_def_1146,type,
    sK44: real > ( real > $o ) > real ).

thf(func_def_1147,type,
    sK45: int > int ).

thf(func_def_1148,type,
    sK46: ( real > nat ) > real ).

thf(func_def_1149,type,
    sK47: ( real > nat ) > real ).

thf(func_def_1150,type,
    sK48: vEBT_VEBT > $o ).

thf(func_def_1151,type,
    sK49: vEBT_VEBT > $o ).

thf(func_def_1152,type,
    sK50: int > int > int ).

thf(func_def_1153,type,
    sK51: int > int ).

thf(func_def_1154,type,
    sK52: nat > list_VEBT_VEBT ).

thf(func_def_1155,type,
    sK53: int > int ).

thf(func_def_1156,type,
    sK54: real > ( real > real > $o ) > real > real ).

thf(func_def_1157,type,
    sK55: real > real > ( real > real > $o ) > real > real ).

thf(func_def_1158,type,
    sK56: real > real > ( real > real > $o ) > real > real ).

thf(func_def_1159,type,
    sK57: ( real > real > $o ) > real ).

thf(func_def_1160,type,
    sK58: ( real > real > $o ) > real ).

thf(func_def_1161,type,
    sK59: ( real > real > $o ) > real ).

thf(func_def_1162,type,
    sK60: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1163,type,
    sK61: int > ( int > $o ) > int ).

thf(func_def_1164,type,
    sK62: int > ( int > $o ) > int ).

thf(func_def_1165,type,
    sK63: ( int > $o ) > int ).

thf(func_def_1166,type,
    sK64: int > nat ).

thf(func_def_1167,type,
    sK65: ( real > real ) > real ).

thf(func_def_1168,type,
    sK66: ( real > real ) > real ).

thf(func_def_1169,type,
    sK67: nat > nat > nat ).

thf(func_def_1170,type,
    sK68: set_int > set_int > int ).

thf(func_def_1171,type,
    sK69: int > int ).

thf(func_def_1172,type,
    sK70: real > real > real ).

thf(func_def_1173,type,
    sK71: ( nat > $o ) > nat ).

thf(func_def_1174,type,
    sK72: ( rat > nat ) > rat ).

thf(func_def_1175,type,
    sK73: ( rat > nat ) > rat ).

thf(func_def_1176,type,
    sK74: int > int > int ).

thf(func_def_1177,type,
    sK75: nat > ( nat > $o ) > nat ).

thf(func_def_1178,type,
    sK76: ( nat > $o ) > nat ).

thf(func_def_1179,type,
    sK77: ( rat > real ) > rat ).

thf(func_def_1180,type,
    sK78: ( rat > real ) > rat ).

thf(func_def_1181,type,
    sK79: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1182,type,
    sK80: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1183,type,
    sK81: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1184,type,
    sK82: ( list_VEBT_VEBT > $o ) > list_VEBT_VEBT ).

thf(func_def_1185,type,
    sK83: real > nat > ( nat > real ) > nat > real ).

thf(func_def_1186,type,
    sK84: ( real > nat ) > real ).

thf(func_def_1187,type,
    sK85: ( real > nat ) > real ).

thf(func_def_1188,type,
    sK86: nat > nat ).

thf(func_def_1189,type,
    sK87: ( int > int > $o ) > int ).

thf(func_def_1190,type,
    sK88: ( int > int > $o ) > int ).

thf(func_def_1191,type,
    sK89: ( int > int > $o ) > int ).

thf(func_def_1192,type,
    sK90: ( int > int > $o ) > int ).

thf(func_def_1193,type,
    sK91: ( int > int > $o ) > int ).

thf(func_def_1194,type,
    sK92: ( nat > nat ) > nat ).

thf(func_def_1195,type,
    sK93: set_nat > nat > nat ).

thf(func_def_1196,type,
    sK94: real > real > real ).

thf(func_def_1197,type,
    sK95: int > nat ).

thf(func_def_1198,type,
    sK96: rat > rat ).

thf(func_def_1199,type,
    sK97: nat > nat > nat > nat ).

thf(func_def_1200,type,
    sK98: ( int > real ) > int ).

thf(func_def_1201,type,
    sK99: ( int > real ) > int ).

thf(func_def_1202,type,
    sK100: ( nat > $o ) > nat ).

thf(func_def_1203,type,
    sK101: ( nat > $o ) > nat ).

thf(func_def_1204,type,
    sK102: real > nat ).

thf(func_def_1205,type,
    sK103: nat > nat ).

thf(func_def_1206,type,
    sK104: int > nat ).

thf(func_def_1207,type,
    sK105: ( rat > nat ) > rat ).

thf(func_def_1208,type,
    sK106: ( rat > nat ) > rat ).

thf(func_def_1209,type,
    sK107: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1210,type,
    sK108: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1211,type,
    sK109: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1212,type,
    sK110: rat > rat ).

thf(func_def_1213,type,
    sK111: ( nat > $o ) > nat ).

thf(func_def_1214,type,
    sK112: code_integer > code_integer > code_integer ).

thf(func_def_1215,type,
    sK113: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1216,type,
    sK114: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1217,type,
    sK115: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1218,type,
    sK116: rat > rat ).

thf(func_def_1219,type,
    sK117: real > real > nat ).

thf(func_def_1220,type,
    sK118: ( nat > nat > $o ) > nat ).

thf(func_def_1221,type,
    sK119: ( nat > nat > $o ) > nat ).

thf(func_def_1222,type,
    sK120: ( nat > nat > $o ) > nat ).

thf(func_def_1223,type,
    sK121: ( nat > nat > $o ) > nat ).

thf(func_def_1224,type,
    sK122: ( nat > nat > $o ) > nat ).

thf(func_def_1225,type,
    sK123: ( real > rat ) > real ).

thf(func_def_1226,type,
    sK124: ( real > rat ) > real ).

thf(func_def_1227,type,
    sK125: nat > nat > nat ).

thf(func_def_1228,type,
    sK126: ( rat > nat ) > rat ).

thf(func_def_1229,type,
    sK127: ( rat > nat ) > rat ).

thf(func_def_1230,type,
    sK128: vEBT_VEBT > $o ).

thf(func_def_1231,type,
    sK129: vEBT_VEBT > $o ).

thf(func_def_1232,type,
    sK130: vEBT_VEBT > nat ).

thf(func_def_1233,type,
    sK131: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1234,type,
    sK132: vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1235,type,
    sK133: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1236,type,
    sK134: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1237,type,
    sK135: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1238,type,
    sK136: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1239,type,
    sK137: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1240,type,
    sK138: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1241,type,
    sK139: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1242,type,
    sK140: list_VEBT_VEBT > nat ).

thf(func_def_1243,type,
    sK141: vEBT_VEBT > nat ).

thf(func_def_1244,type,
    sK142: nat > nat ).

thf(func_def_1245,type,
    sK143: real > real > nat ).

thf(func_def_1246,type,
    sK144: rat > rat ).

thf(func_def_1247,type,
    sK145: ( nat > $o ) > nat ).

thf(func_def_1248,type,
    sK146: real > real > real ).

thf(func_def_1249,type,
    sK147: int > ( int > $o ) > int ).

thf(func_def_1250,type,
    sK148: int > ( int > $o ) > int ).

thf(func_def_1251,type,
    sK149: int > ( int > $o ) > int ).

thf(func_def_1252,type,
    sK150: int > ( int > $o ) > int ).

thf(func_def_1253,type,
    sK151: ( nat > nat > $o ) > nat ).

thf(func_def_1254,type,
    sK152: ( nat > nat > $o ) > nat ).

thf(func_def_1255,type,
    sK153: ( nat > nat > $o ) > nat ).

thf(func_def_1256,type,
    sK154: ( nat > nat > $o ) > nat ).

thf(func_def_1257,type,
    sK155: ( nat > nat > $o ) > nat ).

thf(func_def_1258,type,
    sK156: real > real ).

thf(func_def_1259,type,
    sK157: ( real > real > $o ) > real ).

thf(func_def_1260,type,
    sK158: ( real > real > $o ) > real ).

thf(func_def_1261,type,
    sK159: ( real > real > $o ) > real ).

thf(func_def_1262,type,
    sK160: ( real > real > $o ) > real ).

thf(func_def_1263,type,
    sK161: ( real > real > $o ) > real ).

thf(func_def_1264,type,
    sK162: ( real > real ) > real ).

thf(func_def_1265,type,
    sK163: ( real > real ) > real ).

thf(func_def_1266,type,
    sK164: ( nat > nat ) > nat ).

thf(func_def_1267,type,
    sK165: ( nat > nat ) > nat ).

thf(func_def_1268,type,
    sK166: real > real ).

thf(func_def_1269,type,
    sK167: rat > rat ).

thf(func_def_1270,type,
    sK168: ( rat > nat ) > rat ).

thf(func_def_1271,type,
    sK169: ( rat > nat ) > rat ).

thf(func_def_1272,type,
    sK170: ( rat > nat ) > rat ).

thf(func_def_1273,type,
    sK171: ( rat > nat ) > rat ).

thf(func_def_1274,type,
    sK172: int > int ).

thf(func_def_1275,type,
    sK173: real > nat > real ).

thf(func_def_1276,type,
    sK174: ( nat > nat > $o ) > nat ).

thf(func_def_1277,type,
    sK175: ( nat > nat > $o ) > nat ).

thf(func_def_1278,type,
    sK176: ( nat > nat > $o ) > nat ).

thf(func_def_1279,type,
    sK177: nat > ( nat > real ) > nat ).

thf(func_def_1280,type,
    sK178: ( nat > real ) > nat > nat ).

thf(func_def_1281,type,
    sK179: nat > nat > nat ).

thf(func_def_1282,type,
    sK180: real > real ).

thf(func_def_1283,type,
    sK181: ( int > rat ) > int ).

thf(func_def_1284,type,
    sK182: ( int > rat ) > int ).

thf(func_def_1285,type,
    sK183: real > real ).

thf(func_def_1286,type,
    sK184: rat > rat ).

thf(func_def_1287,type,
    sK185: ( rat > nat ) > rat ).

thf(func_def_1288,type,
    sK186: ( rat > nat ) > rat ).

thf(func_def_1289,type,
    sK187: ( int > $o ) > int > int ).

thf(func_def_1290,type,
    sK188: rat > rat > rat ).

thf(func_def_1291,type,
    sK189: nat > nat ).

thf(func_def_1292,type,
    sK190: nat > nat ).

thf(func_def_1293,type,
    sK191: nat > nat > nat > nat ).

thf(func_def_1294,type,
    sK192: real > nat ).

thf(func_def_1295,type,
    sK193: nat > nat > nat ).

thf(func_def_1296,type,
    sK194: real > real > real > real ).

thf(func_def_1297,type,
    sK195: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1298,type,
    sK196: int > int > int ).

thf(func_def_1299,type,
    sK197: real > nat ).

thf(func_def_1300,type,
    sK198: int > int > int ).

thf(func_def_1301,type,
    sK199: ( nat > int ) > nat ).

thf(func_def_1302,type,
    sK200: nat > nat > nat > nat ).

thf(func_def_1303,type,
    sK201: nat > nat > nat > nat ).

thf(func_def_1304,type,
    sK202: rat > rat > rat ).

thf(func_def_1305,type,
    sK203: ( rat > rat ) > rat ).

thf(func_def_1306,type,
    sK204: ( rat > rat ) > rat ).

thf(func_def_1307,type,
    sK205: vEBT_VEBT > nat > nat ).

thf(func_def_1308,type,
    sK206: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1309,type,
    sK207: vEBT_VEBT > nat > nat ).

thf(func_def_1310,type,
    sK208: vEBT_VEBT > nat > nat ).

thf(func_def_1311,type,
    sK209: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1312,type,
    sK210: nat > vEBT_VEBT > $o ).

thf(func_def_1313,type,
    sK211: nat > vEBT_VEBT > $o ).

thf(func_def_1314,type,
    sK212: ( rat > rat > $o ) > rat ).

thf(func_def_1315,type,
    sK213: ( rat > rat > $o ) > rat ).

thf(func_def_1316,type,
    sK214: ( rat > rat > $o ) > rat ).

thf(func_def_1317,type,
    sK215: ( rat > rat > $o ) > rat ).

thf(func_def_1318,type,
    sK216: ( rat > rat > $o ) > rat ).

thf(func_def_1319,type,
    sK217: ( nat > $o ) > nat > nat ).

thf(func_def_1320,type,
    sK218: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1321,type,
    sK219: list_VEBT_VEBT > nat ).

thf(func_def_1322,type,
    sK220: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1323,type,
    sK221: vEBT_VEBT > nat ).

thf(func_def_1324,type,
    sK222: nat > ( nat > real ) > real ).

thf(func_def_1325,type,
    sK223: ( nat > nat > $o ) > nat ).

thf(func_def_1326,type,
    sK224: ( nat > nat > $o ) > nat ).

thf(func_def_1327,type,
    sK225: ( nat > nat > $o ) > nat ).

thf(func_def_1328,type,
    sK226: ( nat > nat > $o ) > nat ).

thf(func_def_1329,type,
    sK227: nat > ( nat > $o ) > nat ).

thf(func_def_1330,type,
    sK228: nat > ( nat > $o ) > nat ).

thf(func_def_1331,type,
    sK229: ( rat > int ) > rat ).

thf(func_def_1332,type,
    sK230: ( rat > int ) > rat ).

thf(func_def_1333,type,
    sK231: ( nat > real ) > nat ).

thf(func_def_1334,type,
    sK232: ( nat > real ) > nat ).

thf(func_def_1335,type,
    sK233: ( rat > real ) > rat ).

thf(func_def_1336,type,
    sK234: ( rat > real ) > rat ).

thf(func_def_1337,type,
    sK235: nat > nat ).

thf(func_def_1338,type,
    sK236: ( rat > int ) > rat ).

thf(func_def_1339,type,
    sK237: ( rat > int ) > rat ).

thf(func_def_1340,type,
    sK238: int > int ).

thf(func_def_1341,type,
    sK239: ( real > int ) > real ).

thf(func_def_1342,type,
    sK240: ( real > int ) > real ).

thf(func_def_1343,type,
    sK241: ( nat > rat ) > nat ).

thf(func_def_1344,type,
    sK242: int > int ).

thf(func_def_1345,type,
    sK243: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1346,type,
    sK244: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1347,type,
    sK245: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1348,type,
    sK246: nat > nat ).

thf(func_def_1349,type,
    sK247: ( rat > rat ) > rat ).

thf(func_def_1350,type,
    sK248: ( rat > rat ) > rat ).

thf(func_def_1351,type,
    sK249: nat > nat > nat ).

thf(func_def_1352,type,
    sK250: ( rat > rat ) > rat ).

thf(func_def_1353,type,
    sK251: ( rat > rat ) > rat ).

thf(func_def_1354,type,
    sK252: real > real ).

thf(func_def_1355,type,
    sK253: ( nat > $o ) > nat ).

thf(func_def_1356,type,
    sK254: ( real > real ) > real ).

thf(func_def_1357,type,
    sK255: ( real > real ) > real ).

thf(func_def_1358,type,
    sK256: ( nat > rat ) > nat ).

thf(func_def_1359,type,
    sK257: ( nat > rat ) > nat ).

thf(func_def_1360,type,
    sK258: rat > rat ).

thf(func_def_1361,type,
    sK259: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1362,type,
    sK260: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1363,type,
    sK261: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1364,type,
    sK262: rat > rat ).

thf(func_def_1365,type,
    sK263: ( rat > nat ) > rat ).

thf(func_def_1366,type,
    sK264: ( rat > nat ) > rat ).

thf(func_def_1367,type,
    sK265: ( nat > rat ) > nat ).

thf(func_def_1368,type,
    sK266: ( nat > rat ) > nat ).

thf(func_def_1369,type,
    sK267: ( real > rat ) > real ).

thf(func_def_1370,type,
    sK268: ( real > rat ) > real ).

thf(func_def_1371,type,
    sK269: nat > nat ).

thf(func_def_1372,type,
    sK270: ( option4927543243414619207at_nat > option_nat > $o ) > option4927543243414619207at_nat > option_nat > product_prod_nat_nat ).

thf(func_def_1373,type,
    sK271: ( option4927543243414619207at_nat > option_nat > $o ) > option4927543243414619207at_nat > option_nat > nat ).

thf(func_def_1374,type,
    sK272: ( real > int ) > real ).

thf(func_def_1375,type,
    sK273: ( real > int ) > real ).

thf(func_def_1376,type,
    sK274: real > real > nat ).

thf(func_def_1377,type,
    sK275: nat > nat > nat > nat ).

thf(func_def_1378,type,
    sK276: real > real ).

thf(func_def_1379,type,
    sK277: ( int > real ) > int ).

thf(func_def_1380,type,
    sK278: ( int > real ) > int ).

thf(func_def_1381,type,
    sK279: int > int ).

thf(func_def_1382,type,
    sK280: ( nat > $o ) > nat ).

thf(func_def_1383,type,
    sK281: ( nat > $o ) > nat > nat ).

thf(func_def_1384,type,
    sK282: real > nat > real ).

thf(func_def_1385,type,
    sK283: real > real ).

thf(func_def_1386,type,
    sK284: rat > rat > rat > rat ).

thf(func_def_1387,type,
    sK285: ( nat > nat ) > nat ).

thf(func_def_1388,type,
    sK286: ( nat > nat ) > nat ).

thf(func_def_1389,type,
    sK287: nat > vEBT_VEBT > nat ).

thf(func_def_1390,type,
    sK288: int > ( int > $o ) > int ).

thf(func_def_1391,type,
    sK289: int > ( int > $o ) > int ).

thf(func_def_1392,type,
    sK290: ( nat > nat ) > nat ).

thf(func_def_1393,type,
    sK291: ( rat > rat ) > rat ).

thf(func_def_1394,type,
    sK292: ( rat > rat ) > rat ).

thf(func_def_1395,type,
    sK293: code_integer > code_integer > code_integer ).

thf(func_def_1396,type,
    sK294: ( rat > nat ) > rat ).

thf(func_def_1397,type,
    sK295: ( rat > nat ) > rat ).

thf(func_def_1398,type,
    sK296: int > int ).

thf(func_def_1399,type,
    sK297: ( nat > int ) > nat ).

thf(func_def_1400,type,
    sK298: nat > nat > nat ).

thf(func_def_1401,type,
    sK299: nat > nat > nat ).

thf(func_def_1402,type,
    sK300: nat > nat > nat ).

thf(func_def_1403,type,
    sK301: int > int ).

thf(func_def_1404,type,
    sK302: nat > nat > nat ).

thf(func_def_1405,type,
    sK303: ( real > real ) > real ).

thf(func_def_1406,type,
    sK304: ( real > real ) > real ).

thf(func_def_1407,type,
    sK305: nat > nat > nat ).

thf(func_def_1408,type,
    sK306: rat > rat > rat > rat ).

thf(func_def_1409,type,
    sK307: ( rat > nat ) > rat ).

thf(func_def_1410,type,
    sK308: ( rat > nat ) > rat ).

thf(func_def_1411,type,
    sK309: nat > nat > nat ).

thf(func_def_1412,type,
    sK310: nat > nat > nat ).

thf(func_def_1413,type,
    sK311: nat > nat > nat ).

thf(func_def_1414,type,
    sK312: ( nat > $o ) > nat ).

thf(func_def_1415,type,
    sK313: real > real ).

thf(func_def_1416,type,
    sK314: ( nat > real ) > nat > real > nat ).

thf(func_def_1417,type,
    sK315: ( real > rat ) > real ).

thf(func_def_1418,type,
    sK316: ( real > rat ) > real ).

thf(func_def_1419,type,
    sK317: ( rat > real ) > rat ).

thf(func_def_1420,type,
    sK318: ( rat > real ) > rat ).

thf(func_def_1421,type,
    sK319: int > nat ).

thf(func_def_1422,type,
    sK320: ( rat > real ) > rat ).

thf(func_def_1423,type,
    sK321: ( rat > real ) > rat ).

thf(func_def_1424,type,
    sK322: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1425,type,
    sK323: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1426,type,
    sK324: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1427,type,
    sK325: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1428,type,
    sK326: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1429,type,
    sK327: nat > vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1430,type,
    sK328: real > real > real > real ).

thf(func_def_1431,type,
    sK329: real > real ).

thf(func_def_1432,type,
    sK330: ( rat > rat ) > rat ).

thf(func_def_1433,type,
    sK331: ( rat > rat ) > rat ).

thf(func_def_1434,type,
    sK332: ( nat > int ) > nat ).

thf(func_def_1435,type,
    sK333: ( rat > int ) > rat ).

thf(func_def_1436,type,
    sK334: ( rat > int ) > rat ).

thf(func_def_1437,type,
    sK335: int > int ).

thf(func_def_1438,type,
    sK336: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1439,type,
    sK337: vEBT_VEBT > nat ).

thf(func_def_1440,type,
    sK338: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1441,type,
    sK339: vEBT_VEBT > nat ).

thf(func_def_1442,type,
    sK340: vEBT_VEBT > nat ).

thf(func_def_1443,type,
    sK341: vEBT_VEBT > nat ).

thf(func_def_1444,type,
    sK342: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1445,type,
    sK343: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1446,type,
    sK344: vEBT_VEBT > $o ).

thf(func_def_1447,type,
    sK345: vEBT_VEBT > $o ).

thf(func_def_1448,type,
    sK346: ( int > rat ) > int ).

thf(func_def_1449,type,
    sK347: ( int > rat ) > int ).

thf(func_def_1450,type,
    sK348: ( real > real ) > real ).

thf(func_def_1451,type,
    sK349: ( real > real ) > real ).

thf(func_def_1452,type,
    sK350: ( nat > set_int ) > nat ).

thf(func_def_1453,type,
    sK351: rat > rat > rat ).

thf(func_def_1454,type,
    sK352: nat > set_nat > nat ).

thf(func_def_1455,type,
    sK353: real > real > real > real ).

thf(func_def_1456,type,
    sK354: int > int ).

thf(func_def_1457,type,
    sK355: ( nat > $o ) > nat ).

thf(func_def_1458,type,
    sK356: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1459,type,
    sK357: vEBT_VEBT > nat > nat ).

thf(func_def_1460,type,
    sK358: vEBT_VEBT > nat > nat ).

thf(func_def_1461,type,
    sK359: vEBT_VEBT > nat > nat ).

thf(func_def_1462,type,
    sK360: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1463,type,
    sK361: nat > vEBT_VEBT > $o ).

thf(func_def_1464,type,
    sK362: nat > vEBT_VEBT > $o ).

thf(func_def_1465,type,
    sK363: vEBT_VEBT > nat ).

thf(func_def_1466,type,
    sK364: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1467,type,
    sK365: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1468,type,
    sK366: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1469,type,
    sK367: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1470,type,
    sK368: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1471,type,
    sK369: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1472,type,
    sK370: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1473,type,
    sK371: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1474,type,
    sK372: nat > nat ).

thf(func_def_1475,type,
    sK373: ( rat > real ) > rat ).

thf(func_def_1476,type,
    sK374: ( rat > real ) > rat ).

thf(func_def_1477,type,
    sK375: real > real > real ).

thf(func_def_1478,type,
    sK376: ( option_nat > option4927543243414619207at_nat > $o ) > option_nat > option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_1479,type,
    sK377: ( option_nat > option4927543243414619207at_nat > $o ) > option_nat > option4927543243414619207at_nat > nat ).

thf(func_def_1480,type,
    sK378: ( nat > nat > $o ) > nat ).

thf(func_def_1481,type,
    sK379: ( nat > nat > $o ) > nat ).

thf(func_def_1482,type,
    sK380: ( nat > nat > $o ) > nat ).

thf(func_def_1483,type,
    sK381: ( nat > nat > $o ) > nat ).

thf(func_def_1484,type,
    sK382: ( nat > rat ) > nat ).

thf(func_def_1485,type,
    sK383: ( nat > rat ) > nat ).

thf(func_def_1486,type,
    sK384: nat > nat ).

thf(func_def_1487,type,
    sK385: nat > nat > nat ).

thf(func_def_1488,type,
    sK386: ( int > $o ) > int ).

thf(func_def_1489,type,
    sK387: ( int > $o ) > int ).

thf(func_def_1490,type,
    sK388: ( nat > $o ) > nat ).

thf(func_def_1491,type,
    sK389: ( nat > real ) > nat ).

thf(func_def_1492,type,
    sK390: ( nat > real ) > nat ).

thf(func_def_1493,type,
    sK391: ( nat > rat ) > nat ).

thf(func_def_1494,type,
    sK392: ( nat > rat ) > nat ).

thf(func_def_1495,type,
    sK393: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1496,type,
    sK394: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1497,type,
    sK395: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1498,type,
    sK396: int > int > int ).

thf(func_def_1499,type,
    sK397: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_1500,type,
    sK398: real > real > nat ).

thf(func_def_1501,type,
    sK399: rat > rat > rat ).

thf(func_def_1502,type,
    sK400: nat > nat ).

thf(func_def_1503,type,
    sK401: rat > rat ).

thf(func_def_1504,type,
    sK402: real > real ).

thf(func_def_1505,type,
    sK403: int > nat ).

thf(func_def_1506,type,
    sK404: int > nat ).

thf(func_def_1507,type,
    sK405: num > num ).

thf(func_def_1508,type,
    sK406: num > num ).

thf(func_def_1509,type,
    sK407: ( rat > real ) > rat ).

thf(func_def_1510,type,
    sK408: ( rat > real ) > rat ).

thf(func_def_1511,type,
    sK409: ( int > $o ) > int > int ).

thf(func_def_1512,type,
    sK410: int > ( int > $o ) > int ).

thf(func_def_1513,type,
    sK411: nat > ( nat > real ) > real > nat > real ).

thf(func_def_1514,type,
    sK412: ( rat > rat > $o ) > rat ).

thf(func_def_1515,type,
    sK413: ( rat > rat > $o ) > rat ).

thf(func_def_1516,type,
    sK414: ( rat > rat > $o ) > rat ).

thf(func_def_1517,type,
    sK415: ( rat > rat > $o ) > rat ).

thf(func_def_1518,type,
    sK416: nat > nat ).

thf(func_def_1519,type,
    sK417: nat > ( nat > $o ) > nat ).

thf(func_def_1520,type,
    sK418: int > ( int > $o ) > int ).

thf(func_def_1521,type,
    sK419: ( real > real ) > real ).

thf(func_def_1522,type,
    sK420: ( real > real ) > real ).

thf(func_def_1523,type,
    sK421: ( option_nat > option_nat > $o ) > option_nat > option_nat > nat ).

thf(func_def_1524,type,
    sK422: ( option_nat > option_nat > $o ) > option_nat > option_nat > nat ).

thf(func_def_1525,type,
    sK423: ( rat > rat ) > rat ).

thf(func_def_1526,type,
    sK424: ( rat > rat ) > rat ).

thf(func_def_1527,type,
    sK425: ( nat > rat ) > nat ).

thf(func_def_1528,type,
    sK426: ( nat > rat ) > nat ).

thf(func_def_1529,type,
    sK427: int > int ).

thf(func_def_1530,type,
    sK428: ( nat > rat ) > nat ).

thf(func_def_1531,type,
    sK429: rat > rat ).

thf(func_def_1532,type,
    sK430: real > real ).

thf(func_def_1533,type,
    sK431: real > real > real ).

thf(func_def_1534,type,
    sK432: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1535,type,
    sK433: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1536,type,
    sK434: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1537,type,
    sK435: int > int > int ).

thf(func_def_1538,type,
    sK436: nat > nat > nat ).

thf(func_def_1539,type,
    sK437: nat > nat > nat ).

thf(func_def_1540,type,
    sK438: ( nat > $o ) > nat > nat ).

thf(func_def_1541,type,
    sK439: ( nat > $o ) > nat > nat ).

thf(func_def_1542,type,
    sK440: nat > nat ).

thf(func_def_1543,type,
    sK441: option_nat > nat ).

thf(func_def_1544,type,
    sK442: nat > nat > nat ).

thf(func_def_1545,type,
    sK443: ( rat > rat ) > rat ).

thf(func_def_1546,type,
    sK444: ( rat > rat ) > rat ).

thf(func_def_1547,type,
    sK445: ( nat > $o ) > nat ).

thf(func_def_1548,type,
    sK446: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1549,type,
    sK447: ( real > $o ) > ( real > $o ) > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1550,type,
    sK448: real > ( real > $o ) > ( real > $o ) > real ).

thf(func_def_1551,type,
    sK449: ( int > $o ) > int > int ).

thf(func_def_1552,type,
    sK450: nat > nat ).

thf(func_def_1553,type,
    sK451: nat > nat ).

thf(func_def_1554,type,
    sK452: nat > nat > nat ).

thf(func_def_1555,type,
    sK453: nat > nat > nat ).

thf(func_def_1556,type,
    sK454: nat > nat > nat ).

thf(func_def_1557,type,
    sK455: nat > nat ).

thf(func_def_1558,type,
    sK456: ( rat > int ) > rat ).

thf(func_def_1559,type,
    sK457: ( rat > int ) > rat ).

thf(func_def_1560,type,
    sK458: ( rat > rat ) > rat ).

thf(func_def_1561,type,
    sK459: ( rat > rat ) > rat ).

thf(func_def_1562,type,
    sK460: rat > rat ).

thf(func_def_1563,type,
    sK461: ( nat > $o ) > nat ).

thf(func_def_1564,type,
    sK462: nat > nat > nat ).

thf(func_def_1565,type,
    sK463: real > ( real > $o ) > real ).

thf(func_def_1566,type,
    sK464: real > ( real > $o ) > real ).

thf(func_def_1567,type,
    sK465: real > ( real > $o ) > real ).

thf(func_def_1568,type,
    sK466: real > ( real > $o ) > real ).

thf(func_def_1569,type,
    sK467: real > real ).

thf(func_def_1570,type,
    sK468: ( nat > rat ) > nat ).

thf(func_def_1571,type,
    sK469: int > nat ).

thf(func_def_1572,type,
    sK470: int > nat ).

thf(func_def_1573,type,
    sK471: ( rat > rat ) > rat ).

thf(func_def_1574,type,
    sK472: ( rat > rat ) > rat ).

thf(func_def_1575,type,
    sK473: ( nat > rat ) > nat ).

thf(func_def_1576,type,
    sK474: ( nat > nat > $o ) > nat ).

thf(func_def_1577,type,
    sK475: ( nat > nat > $o ) > nat ).

thf(func_def_1578,type,
    sK476: ( nat > nat > $o ) > nat ).

thf(func_def_1579,type,
    sK477: ( nat > nat > $o ) > nat ).

thf(func_def_1580,type,
    sK478: ( nat > nat > $o ) > nat ).

thf(func_def_1581,type,
    sK479: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1582,type,
    sK480: list_VEBT_VEBT > nat ).

thf(func_def_1583,type,
    sK481: list_VEBT_VEBT > nat > nat > nat > nat > nat ).

thf(func_def_1584,type,
    sK482: list_VEBT_VEBT > nat > nat > nat > nat > nat ).

thf(func_def_1585,type,
    sK483: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1586,type,
    sK484: vEBT_VEBT > nat > list_VEBT_VEBT > nat ).

thf(func_def_1587,type,
    sK485: code_integer > code_integer ).

thf(func_def_1588,type,
    sK486: rat > rat ).

thf(func_def_1589,type,
    sK487: ( nat > real ) > nat ).

thf(func_def_1590,type,
    sK488: nat > nat ).

thf(func_def_1591,type,
    sK489: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1592,type,
    sK490: int > int > int > int ).

thf(func_def_1593,type,
    sK491: vEBT_VEBT > $o ).

thf(func_def_1594,type,
    sK492: vEBT_VEBT > $o ).

thf(func_def_1595,type,
    sK493: ( real > real > $o ) > real ).

thf(func_def_1596,type,
    sK494: real > real ).

thf(func_def_1597,type,
    sK495: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1598,type,
    sK496: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1599,type,
    sK497: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1600,type,
    sK498: nat > nat ).

thf(func_def_1601,type,
    sK499: int > int ).

thf(func_def_1602,type,
    sK500: rat > rat > rat ).

thf(func_def_1603,type,
    sK501: real > nat > real ).

thf(func_def_1604,type,
    sK502: ( rat > real ) > rat ).

thf(func_def_1605,type,
    sK503: ( rat > real ) > rat ).

thf(func_def_1606,type,
    sK504: int > int > int > int ).

thf(func_def_1607,type,
    sK505: int > int > int > int ).

thf(func_def_1608,type,
    sK506: ( real > rat ) > real ).

thf(func_def_1609,type,
    sK507: ( real > rat ) > real ).

thf(func_def_1610,type,
    sK508: int > nat ).

thf(func_def_1611,type,
    sK509: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1612,type,
    sK510: list_VEBT_VEBT > nat ).

thf(func_def_1613,type,
    sK511: list_VEBT_VEBT > nat > nat > nat > nat > nat ).

thf(func_def_1614,type,
    sK512: list_VEBT_VEBT > nat > nat > nat > nat > nat ).

thf(func_def_1615,type,
    sK513: list_VEBT_VEBT > vEBT_VEBT > nat > nat ).

thf(func_def_1616,type,
    sK514: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1617,type,
    sK515: ( nat > int ) > nat ).

thf(func_def_1618,type,
    sK516: nat > nat > nat ).

thf(func_def_1619,type,
    sK517: ( rat > real ) > rat ).

thf(func_def_1620,type,
    sK518: ( rat > real ) > rat ).

thf(func_def_1621,type,
    sK519: rat > rat ).

thf(func_def_1622,type,
    sK520: nat > vEBT_VEBT > nat ).

thf(func_def_1623,type,
    sK521: nat > vEBT_VEBT > nat ).

thf(func_def_1624,type,
    sK522: nat > vEBT_VEBT > nat ).

thf(func_def_1625,type,
    sK523: nat > vEBT_VEBT > nat ).

thf(func_def_1626,type,
    sK524: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1627,type,
    sK525: nat > vEBT_VEBT > nat ).

thf(func_def_1628,type,
    sK526: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1629,type,
    sK527: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1630,type,
    sK528: vEBT_VEBT > nat > nat ).

thf(func_def_1631,type,
    sK529: vEBT_VEBT > nat > nat ).

thf(func_def_1632,type,
    sK530: vEBT_VEBT > nat > nat ).

thf(func_def_1633,type,
    sK531: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1634,type,
    sK532: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1635,type,
    sK533: vEBT_VEBT > nat > nat ).

thf(func_def_1636,type,
    sK534: vEBT_VEBT > nat > nat ).

thf(func_def_1637,type,
    sK535: vEBT_VEBT > nat > nat ).

thf(func_def_1638,type,
    sK536: vEBT_VEBT > nat > nat ).

thf(func_def_1639,type,
    sK537: vEBT_VEBT > nat > nat ).

thf(func_def_1640,type,
    sK538: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1641,type,
    sK539: vEBT_VEBT > nat > nat ).

thf(func_def_1642,type,
    sK540: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1643,type,
    sK541: vEBT_VEBT > nat > nat ).

thf(func_def_1644,type,
    sK542: vEBT_VEBT > nat > nat ).

thf(func_def_1645,type,
    sK543: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1646,type,
    sK544: vEBT_VEBT > $o ).

thf(func_def_1647,type,
    sK545: vEBT_VEBT > $o ).

thf(func_def_1648,type,
    sK546: nat > nat ).

thf(func_def_1649,type,
    sK547: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > nat ).

thf(func_def_1650,type,
    sK548: ( int > real ) > int ).

thf(func_def_1651,type,
    sK549: ( int > real ) > int ).

thf(func_def_1652,type,
    sK550: ( int > $o ) > int > int ).

thf(func_def_1653,type,
    sK551: ( int > $o ) > int > int ).

thf(func_def_1654,type,
    sK552: ( int > $o ) > int > int ).

thf(func_def_1655,type,
    sK553: ( int > $o ) > int > int ).

thf(func_def_1656,type,
    sK554: ( nat > $o ) > nat > nat ).

thf(func_def_1657,type,
    sK555: real > real ).

thf(func_def_1658,type,
    sK556: ( nat > real ) > nat ).

thf(func_def_1659,type,
    sK557: ( real > rat ) > real ).

thf(func_def_1660,type,
    sK558: ( real > rat ) > real ).

thf(func_def_1661,type,
    sK559: nat > nat ).

thf(func_def_1662,type,
    sK560: ( rat > rat ) > rat ).

thf(func_def_1663,type,
    sK561: ( rat > rat ) > rat ).

thf(func_def_1664,type,
    sK562: ( int > rat ) > int ).

thf(func_def_1665,type,
    sK563: ( int > rat ) > int ).

thf(func_def_1666,type,
    sK564: product_prod_nat_nat > nat ).

thf(func_def_1667,type,
    sK565: product_prod_nat_nat > nat ).

thf(func_def_1668,type,
    sK566: option_nat > vEBT_VEBT > $o ).

thf(func_def_1669,type,
    sK567: option_nat > vEBT_VEBT > $o ).

thf(func_def_1670,type,
    sK568: option_nat > vEBT_VEBT > nat ).

thf(func_def_1671,type,
    sK569: option_nat > vEBT_VEBT > nat ).

thf(func_def_1672,type,
    sK570: option_nat > vEBT_VEBT > nat ).

thf(func_def_1673,type,
    sK571: option_nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1674,type,
    sK572: option_nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1675,type,
    sK573: vEBT_VEBT > nat ).

thf(func_def_1676,type,
    sK574: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1677,type,
    sK575: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1678,type,
    sK576: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1679,type,
    sK577: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1680,type,
    sK578: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1681,type,
    sK579: int > nat ).

thf(func_def_1682,type,
    sK580: nat > nat > nat ).

thf(func_def_1683,type,
    sK581: nat > nat ).

thf(func_def_1684,type,
    sK582: ( int > $o ) > ( int > $o ) > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1685,type,
    sK583: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1686,type,
    sK584: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1687,type,
    sK585: real > real ).

thf(func_def_1688,type,
    sK586: int > int > int > int ).

thf(func_def_1689,type,
    sK587: int > int > int > int ).

thf(func_def_1690,type,
    sK588: real > real > nat ).

thf(func_def_1691,type,
    sK589: rat > rat ).

thf(func_def_1692,type,
    sK590: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1693,type,
    sK591: ( real > int ) > real ).

thf(func_def_1694,type,
    sK592: ( real > int ) > real ).

thf(func_def_1695,type,
    sK593: ( nat > set_int ) > nat ).

thf(func_def_1696,type,
    sK594: int > int ).

thf(func_def_1697,type,
    sK595: ( int > int > $o ) > int ).

thf(func_def_1698,type,
    sK596: ( int > int > $o ) > int ).

thf(func_def_1699,type,
    sK597: ( int > int > $o ) > int ).

thf(func_def_1700,type,
    sK598: ( int > int > $o ) > int ).

thf(func_def_1701,type,
    sK599: nat > nat > nat ).

thf(func_def_1702,type,
    sK600: int > int > int ).

thf(func_def_1703,type,
    sK601: int > int > int ).

thf(func_def_1704,type,
    sK602: ( nat > nat ) > nat ).

thf(func_def_1705,type,
    sK603: set_nat > nat > nat ).

thf(func_def_1706,type,
    sK604: set_nat > nat > nat ).

thf(func_def_1707,type,
    sK605: nat > nat > nat > nat ).

thf(func_def_1708,type,
    sK606: nat > nat > nat > nat ).

thf(func_def_1709,type,
    sK607: nat > nat > nat ).

thf(func_def_1710,type,
    sK608: ( int > $o ) > int ).

thf(func_def_1711,type,
    sK609: int > ( int > $o ) > ( int > $o ) > int ).

thf(func_def_1712,type,
    sK610: int > ( int > $o ) > int ).

thf(func_def_1713,type,
    sK611: int > ( int > $o ) > int ).

thf(func_def_1714,type,
    sK612: int > int ).

thf(func_def_1715,type,
    sK613: ( rat > int ) > rat ).

thf(func_def_1716,type,
    sK614: ( rat > int ) > rat ).

thf(func_def_1717,type,
    sK615: rat > rat ).

thf(func_def_1718,type,
    sK616: real > real ).

thf(func_def_1719,type,
    sK617: ( int > rat ) > int ).

thf(func_def_1720,type,
    sK618: ( int > rat ) > int ).

thf(func_def_1721,type,
    sK619: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1722,type,
    sK620: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1723,type,
    sK621: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1724,type,
    sK622: nat > nat > nat ).

thf(func_def_1725,type,
    sK623: nat > ( nat > $o ) > nat ).

thf(func_def_1726,type,
    sK624: ( nat > $o ) > nat ).

thf(func_def_1727,type,
    sK625: real > real > real ).

thf(func_def_1728,type,
    sK626: ( nat > rat ) > nat ).

thf(func_def_1729,type,
    sK627: ( nat > rat ) > nat ).

thf(func_def_1730,type,
    sK628: ( rat > rat ) > rat ).

thf(func_def_1731,type,
    sK629: ( rat > rat ) > rat ).

thf(func_def_1732,type,
    sK630: ( nat > real ) > nat ).

thf(func_def_1733,type,
    sK631: ( nat > real ) > nat ).

thf(func_def_1734,type,
    sK632: nat > nat > nat ).

thf(func_def_1735,type,
    sK633: vEBT_VEBT > nat ).

thf(func_def_1736,type,
    sK634: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1737,type,
    sK635: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1738,type,
    sK636: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1739,type,
    sK637: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1740,type,
    sK638: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1741,type,
    sK639: nat > $o > vEBT_VEBT > $o ).

thf(func_def_1742,type,
    sK640: nat > $o > vEBT_VEBT > $o ).

thf(func_def_1743,type,
    sK641: nat > $o > vEBT_VEBT > nat ).

thf(func_def_1744,type,
    sK642: nat > $o > vEBT_VEBT > nat ).

thf(func_def_1745,type,
    sK643: nat > $o > vEBT_VEBT > nat ).

thf(func_def_1746,type,
    sK644: nat > $o > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1747,type,
    sK645: nat > $o > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1748,type,
    sK646: vEBT_VEBT > product_prod_nat_nat ).

thf(func_def_1749,type,
    sK647: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1750,type,
    sK648: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1751,type,
    sK649: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1752,type,
    sK650: int > nat ).

thf(func_def_1753,type,
    sK651: ( rat > rat ) > rat ).

thf(func_def_1754,type,
    sK652: ( rat > rat ) > rat ).

thf(func_def_1755,type,
    sK653: int > ( int > $o ) > int ).

thf(func_def_1756,type,
    sK654: ( nat > nat ) > nat ).

thf(func_def_1757,type,
    sK655: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1758,type,
    sK656: nat > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1759,type,
    sK657: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).

thf(func_def_1760,type,
    sK658: rat > rat ).

thf(func_def_1761,type,
    sK659: ( real > nat ) > real ).

thf(func_def_1762,type,
    sK660: ( real > nat ) > real ).

thf(func_def_1763,type,
    sK661: ( nat > nat > $o ) > nat ).

thf(func_def_1764,type,
    sK662: ( nat > nat > $o ) > nat ).

thf(func_def_1765,type,
    sK663: ( nat > nat > $o ) > nat ).

thf(func_def_1766,type,
    sK664: ( nat > nat > $o ) > nat ).

thf(func_def_1767,type,
    sK665: ( rat > int ) > rat ).

thf(func_def_1768,type,
    sK666: ( rat > int ) > rat ).

thf(func_def_1769,type,
    sK667: real > real ).

thf(func_def_1770,type,
    sK668: ( rat > int ) > rat ).

thf(func_def_1771,type,
    sK669: ( rat > int ) > rat ).

thf(func_def_1772,type,
    sK670: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1773,type,
    sK671: ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1774,type,
    sK672: rat > ( rat > $o ) > ( rat > $o ) > rat ).

thf(func_def_1775,type,
    sK673: ( rat > int ) > rat ).

thf(func_def_1776,type,
    sK674: ( rat > int ) > rat ).

thf(func_def_1777,type,
    sK675: nat > set_nat > nat ).

thf(func_def_1778,type,
    sK676: set_nat > nat ).

thf(func_def_1779,type,
    sK677: ( nat > $o ) > nat > nat > nat ).

thf(func_def_1780,type,
    sK678: vEBT_VEBT > nat > nat > nat ).

thf(func_def_1781,type,
    sK679: ( int > $o ) > int > int ).

thf(func_def_1782,type,
    sK680: ( int > $o ) > int > int ).

thf(func_def_1783,type,
    sK681: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1784,type,
    sK682: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > nat ).

thf(func_def_1785,type,
    sK683: option_nat > nat ).

thf(func_def_1786,type,
    sK684: int > int > int ).

thf(func_def_1787,type,
    sK685: nat > ( nat > $o ) > nat > nat ).

thf(func_def_1788,type,
    sK686: nat > ( nat > $o ) > nat > nat ).

thf(func_def_1789,type,
    sK687: list_VEBT_VEBT > nat > vEBT_VEBT > nat ).

thf(func_def_1790,type,
    sK688: vEBT_VEBT > nat > list_VEBT_VEBT > nat ).

thf(func_def_1791,type,
    sK689: ( nat > vEBT_VEBT > $o ) > nat > nat ).

thf(func_def_1792,type,
    sK690: ( nat > vEBT_VEBT > $o ) > nat > list_VEBT_VEBT ).

thf(func_def_1793,type,
    sK691: ( nat > vEBT_VEBT > $o ) > nat > list_VEBT_VEBT ).

thf(func_def_1794,type,
    sK692: ( nat > vEBT_VEBT > $o ) > nat > list_VEBT_VEBT > nat ).

thf(func_def_1795,type,
    sK693: nat > ( nat > vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1796,type,
    sK694: nat > ( nat > $o ) > nat ).

thf(func_def_1797,type,
    sK695: nat > ( nat > $o ) > nat ).

thf(func_def_1798,type,
    sK696: nat > ( nat > $o ) > nat ).

thf(func_def_1799,type,
    sK697: nat > ( nat > $o ) > nat ).

thf(func_def_1800,type,
    sK698: list_VEBT_VEBT > nat > vEBT_VEBT > nat ).

thf(func_def_1801,type,
    sK699: nat > nat > nat > nat ).

thf(func_def_1802,type,
    sK700: nat > nat > nat > nat ).

thf(func_def_1803,type,
    sK701: nat > nat > nat ).

thf(func_def_1804,type,
    sK702: option_nat > nat ).

thf(func_def_1805,type,
    sK703: ( nat > real ) > real > nat > real ).

thf(func_def_1806,type,
    sK704: ( nat > real ) > real > nat > real ).

thf(func_def_1807,type,
    sK705: ( nat > real ) > nat > nat ).

thf(func_def_1808,type,
    sK706: ( int > $o ) > int > int > int ).

thf(func_def_1809,type,
    sK707: ( int > $o ) > int > int > int ).

thf(func_def_1810,type,
    sK708: ( int > $o ) > int > int > int ).

thf(func_def_1811,type,
    sK709: ( int > $o ) > int > int > int ).

thf(func_def_1812,type,
    sK710: ( int > $o ) > int > int > int ).

thf(func_def_1813,type,
    sK711: ( int > $o ) > int > int > int ).

thf(func_def_1814,type,
    sK712: ( int > $o ) > int > int > int ).

thf(func_def_1815,type,
    sK713: ( int > $o ) > int > int > int ).

thf(func_def_1816,type,
    sK714: ( int > $o ) > int > int > int ).

thf(func_def_1817,type,
    sK715: ( int > $o ) > int > int > int ).

thf(func_def_1818,type,
    sK716: ( int > $o ) > int > int > int ).

thf(func_def_1819,type,
    sK717: ( int > $o ) > int > int > int ).

thf(func_def_1820,type,
    sK718: ( int > $o ) > int > int > int ).

thf(func_def_1821,type,
    sK719: ( int > $o ) > int > int > int ).

thf(func_def_1822,type,
    sK720: ( int > $o ) > int > int > int ).

thf(func_def_1823,type,
    sK721: ( int > $o ) > int > int > int ).

thf(func_def_1824,type,
    sK722: ( int > $o ) > int > int > int ).

thf(func_def_1825,type,
    sK723: ( int > $o ) > int > int > int ).

thf(func_def_1826,type,
    sK724: ( nat > $o ) > nat > nat > nat ).

thf(func_def_1827,type,
    sK725: ( int > $o ) > int > int > int ).

thf(func_def_1828,type,
    sK726: ( int > $o ) > int > int > int ).

thf(func_def_1829,type,
    sK727: ( int > $o ) > int > int > int ).

thf(func_def_1830,type,
    sK728: ( int > $o ) > int > int > int ).

thf(func_def_1831,type,
    sK729: ( int > $o ) > int > int > int ).

thf(func_def_1832,type,
    sK730: ( int > $o ) > int > int > int ).

thf(func_def_1833,type,
    sK731: ( int > $o ) > int > int > int ).

thf(func_def_1834,type,
    sK732: ( int > $o ) > int > int > int ).

thf(func_def_1835,type,
    sK733: ( int > $o ) > int > int > int ).

thf(func_def_1836,type,
    sK734: ( int > $o ) > int > int > int ).

thf(func_def_1837,type,
    sK735: ( int > $o ) > int > int > int ).

thf(func_def_1838,type,
    sK736: ( int > $o ) > int > int > int ).

thf(func_def_1839,type,
    sK737: ( int > $o ) > int > int > int ).

thf(func_def_1840,type,
    sK738: ( int > $o ) > int > int > int ).

thf(func_def_1841,type,
    sK739: nat > nat ).

thf(func_def_1842,type,
    sK740: vEBT_VEBT > nat > list_VEBT_VEBT > nat ).

thf(func_def_1843,type,
    sK741: ( int > int > $o ) > int > int > int ).

thf(func_def_1844,type,
    sK742: ( int > int > $o ) > int > int > int ).

thf(func_def_1845,type,
    sK743: nat > vEBT_VEBT > nat > nat ).

thf(func_def_1846,type,
    sK744: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_1847,type,
    sK745: int > int > ( int > int > $o ) > int ).

thf(func_def_1848,type,
    sK746: int > int > ( int > int > $o ) > int ).

thf(func_def_1849,type,
    sK747: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_1850,type,
    sK748: nat > ( nat > $o ) > nat ).

thf(func_def_1851,type,
    sK749: nat > ( nat > $o ) > nat ).

thf(func_def_1852,type,
    sK750: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1853,type,
    sK751: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1854,type,
    sK752: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1855,type,
    sK753: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1856,type,
    sK754: nat > ( nat > $o ) > nat ).

thf(func_def_1857,type,
    sK755: nat > ( nat > $o ) > nat ).

thf(func_def_1858,type,
    sK756: set_nat > nat ).

thf(func_def_1859,type,
    sK757: nat > set_nat > nat ).

thf(func_def_1860,type,
    sK758: ( nat > $o ) > nat > nat ).

thf(func_def_1861,type,
    sK759: ( nat > $o ) > nat > nat ).

thf(func_def_1862,type,
    sK760: ( nat > real ) > nat > ( nat > real ) > real ).

thf(func_def_1863,type,
    sK761: ( nat > real ) > nat > ( nat > real ) > nat ).

thf(func_def_1864,type,
    sK762: nat > ( nat > real ) > real ).

thf(func_def_1865,type,
    sK763: nat > ( nat > real ) > nat ).

thf(func_def_1866,type,
    sK764: nat > nat > nat ).

thf(func_def_1867,type,
    sK765: nat > ( nat > $o ) > nat ).

thf(func_def_1868,type,
    sK766: nat > ( nat > $o ) > nat ).

thf(func_def_1869,type,
    sK767: vEBT_VEBT > $o ).

thf(func_def_1870,type,
    sK768: vEBT_VEBT > $o ).

thf(func_def_1871,type,
    sK769: list_VEBT_VEBT > vEBT_VEBT > nat ).

thf(func_def_1872,type,
    sK770: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1873,type,
    sK771: nat > nat > ( nat > $o ) > nat ).

thf(func_def_1874,type,
    sK772: ( nat > $o ) > nat > nat ).

thf(func_def_1875,type,
    sK773: ( nat > $o ) > nat > nat ).

thf(func_def_1876,type,
    sK774: ( nat > $o ) > nat > nat ).

thf(func_def_1877,type,
    ph775: 
      !>[X0: $tType] : X0 ).

thf(f43982,plain,
    $false,
    inference(subsumption_resolution,[status(thm)],[f43981,f30560]) ).

thf(f30560,plain,
    ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
   != none_nat ),
    inference(cnf_transformation,[status(thm)],[f1]) ).

thf(f1,axiom,
    ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
   != none_nat ),
    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',fact_0_False) ).

thf(f43981,plain,
    ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
    = none_nat ),
    inference(equality_resolution,[status(thm)],[f43980]) ).

thf(f43980,plain,
    ! [X0: option_nat] :
      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
       != X0 )
      | ( none_nat = X0 ) ),
    inference(superposition,[status(thm)],[f42972,f31771]) ).

thf(f31771,plain,
    ! [X0: option_nat] :
      ( ( ( some_nat @ ( sK441 @ X0 ) )
        = X0 )
      | ( none_nat = X0 ) ),
    inference(cnf_transformation,[status(thm)],[f28764]) ).

thf(f28764,plain,
    ! [X0: option_nat] :
      ( ( ( some_nat @ ( sK441 @ X0 ) )
        = X0 )
      | ( none_nat = X0 ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK441])],[f26553,f28763]) ).

thf(f28763,plain,
    ! [X0: option_nat] :
      ( ? [X1: nat] :
          ( ( some_nat @ X1 )
          = X0 )
     => ( ( some_nat @ ( sK441 @ X0 ) )
        = X0 ) ),
    introduced(definition,[],[choice_axiom]) ).

thf(f26553,plain,
    ! [X0: option_nat] :
      ( ? [X1: nat] :
          ( ( some_nat @ X1 )
          = X0 )
      | ( none_nat = X0 ) ),
    inference(ennf_transformation,[status(thm)],[f24578]) ).

thf(f24578,plain,
    ! [X0: option_nat] :
      ( ( none_nat != X0 )
     => ~ ! [X1: nat] :
            ( ( some_nat @ X1 )
           != X0 ) ),
    inference(rectify,[status(thm)],[f86]) ).

thf(f86,axiom,
    ! [X10: option_nat] :
      ( ( none_nat != X10 )
     => ~ ! [X23: nat] :
            ( ( some_nat @ X23 )
           != X10 ) ),
    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',fact_85_option_Oexhaust) ).

thf(f42972,plain,
    ! [X0: nat] :
      ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
     != ( some_nat @ X0 ) ),
    inference(subsumption_resolution,[status(thm)],[f29704,f30824]) ).

thf(f30824,plain,
    thesis != $true,
    inference(cnf_transformation,[status(thm)],[f24320]) ).

thf(f24320,plain,
    thesis != $true,
    inference(flattening,[status(thm)],[f16769]) ).

thf(f16769,plain,
    thesis != $true,
    inference(fool_elimination,[status(thm)],[f16768]) ).

thf(f16768,plain,
    ~ thesis,
    inference(rectify,[status(thm)],[f10250]) ).

thf(f10250,negated_conjecture,
    ~ thesis,
    inference(negated_conjecture,[status(cth)],[f10249]) ).

thf(f10249,conjecture,
    thesis,
    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',conj_1) ).

thf(f29704,plain,
    ! [X0: nat] :
      ( ( thesis = $true )
      | ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
       != ( some_nat @ X0 ) ) ),
    inference(cnf_transformation,[status(thm)],[f26591]) ).

thf(f26591,plain,
    ! [X0: nat] :
      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
       != ( some_nat @ X0 ) )
      | ( thesis = $true ) ),
    inference(ennf_transformation,[status(thm)],[f21395]) ).

thf(f21395,plain,
    ! [X0: nat] :
      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        = ( some_nat @ X0 ) )
     => ( thesis = $true ) ),
    inference(fool_elimination,[status(thm)],[f21394]) ).

thf(f21394,plain,
    ! [X0: nat] :
      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        = ( some_nat @ X0 ) )
     => thesis ),
    inference(rectify,[status(thm)],[f10248]) ).

thf(f10248,axiom,
    ! [X292: nat] :
      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        = ( some_nat @ X292 ) )
     => thesis ),
    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',conj_0) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12  % Problem  : ITP238^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.11/0.14  % Command  : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.35  % Computer : n009.cluster.edu
% 0.16/0.35  % Model    : x86_64 x86_64
% 0.16/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35  % Memory   : 8042.1875MB
% 0.16/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35  % CPULimit : 300
% 0.16/0.35  % WCLimit  : 300
% 0.16/0.35  % DateTime : Thu Feb 26 16:58:10 EST 2026
% 0.16/0.35  % CPUTime  : 
% 0.16/0.35  Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.47  ---- Original DTF file ---
% 0.30/0.47  thf(spec,logic,$$dhol).
% 0.30/0.47  %------------------------------------------------------------------------------
% 0.30/0.47  % File     : ITP238^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.30/0.47  % Domain   : Interactive Theorem Proving
% 0.30/0.47  % Problem  : Sledgehammer problem VEBT_Pred 00333_016090
% 0.30/0.47  % Version  : [Des22] axioms.
% 0.30/0.47  % English  :
% 0.30/0.47  
% 0.30/0.47  % Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.47  %          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.30/0.47  % Source   : [Des22]
% 0.30/0.47  % Names    : 0069_VEBT_Pred_00333_016090 [Des22]
% 0.30/0.47  
% 0.30/0.47  % Status   : Theorem
% 0.30/0.47  % Rating   : ? v9.2.1
% 0.30/0.47  % Syntax   : Number of formulae    : 11333 (5578 unt;1084 typ;   0 def)
% 0.30/0.47  %            Number of atoms       : 28549 (12091 equ;   0 cnn)
% 0.30/0.47  %            Maximal formula atoms :   71 (   2 avg)
% 0.30/0.47  %            Number of connectives : 115442 (2700   ~; 518   |;1769   &;99439   @)
% 0.30/0.47  %                                         (   0 <=>;11016  =>;   0  <=;   0 <~>)
% 0.30/0.47  %            Maximal formula depth :   39 (   6 avg)
% 0.30/0.47  %            Number of types       :  110 ( 109 usr)
% 0.30/0.47  %            Number of type conns  : 4888 (4888   >;   0   *;   0   +;   0  <<)
% 0.30/0.47  %            Number of symbols     :  978 ( 975 usr;  64 con; 0-8 aty)
% 0.30/0.47  %            Number of variables   : 26530 (2393   ^;23368   !; 769   ?;26530   :)
% 0.30/0.47  % SPC      : TH0_THM_EQU_NAR
% 0.30/0.47  
% 0.30/0.47  % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.47  %            from the van Emde Boas Trees session in the Archive of Formal
% 0.30/0.47  %            proofs - 
% 0.30/0.47  %            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.30/0.47  %            2022-02-17 22:49:49.367
% 0.30/0.47  % Bugfixes : Added ()s around negated sides of equations.
% 0.30/0.47  %------------------------------------------------------------------------------
% 0.30/0.47  % Could-be-implicit typings (109)
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% 0.30/0.47      bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.47      bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 0.30/0.47      bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.47      bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.30/0.47      bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
% 0.30/0.47      bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 0.30/0.47      bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.30/0.47      bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 0.30/0.47      bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.47      bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 0.30/0.47      bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 0.30/0.47      bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 0.30/0.47      bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 0.30/0.47      bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.47      bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.30/0.47      bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.47      bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 0.30/0.47      bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.30/0.47      bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.30/0.47      bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 0.30/0.47      bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.47      bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.47      bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Binomial_Obinomial,type,
% 0.30/0.47      binomial: nat > nat > nat ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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% 0.30/0.47  thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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% 0.30/0.47  thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
% 0.30/0.47      gbinomial_rat: rat > nat > rat ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
% 0.30/0.47      gbinomial_real: real > nat > real ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.30/0.47      bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.30/0.47      bit_and_not_num: num > num > option_num ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.30/0.47      bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.30/0.47      bit_concat_bit: nat > int > int > int ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.30/0.47      bit_or_not_num_neg: num > num > num ).
% 0.30/0.47  
% 0.30/0.47  thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 0.30/0.47      bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
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% 0.30/0.47  thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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% 0.30/0.48      ring_17405671764205052669omplex: int > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 0.30/0.48      ring_1_of_int_int: int > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 0.30/0.48      ring_1_of_int_rat: int > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 0.30/0.48      ring_1_of_int_real: int > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
% 0.30/0.48      inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 0.30/0.48      inf_inf_nat: nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 0.30/0.48      semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
% 0.30/0.48      sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 0.30/0.48      sup_sup_nat: nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.48      sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 0.30/0.48      lattic8265883725875713057ax_nat: set_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.30/0.48      bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
% 0.30/0.48      at_infinity_real: filter_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 0.30/0.48      append_int: list_int > list_int > list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 0.30/0.48      append_nat: list_nat > list_nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 0.30/0.48      drop_nat: nat > list_nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
% 0.30/0.48      fold_int_int: ( int > int > int ) > list_int > int > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.48      fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.30/0.48      linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.30/0.48      cons_int: int > list_int > list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.30/0.48      cons_nat: nat > list_nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 0.30/0.48      nil_int: list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.30/0.48      nil_nat: list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 0.30/0.48      hd_nat: list_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.48      map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 0.30/0.48      set_o2: list_o > set_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 0.30/0.48      set_complex2: list_complex > set_complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 0.30/0.48      set_int2: list_int > set_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.48      set_list_nat2: list_list_nat > set_list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.30/0.48      set_nat2: list_nat > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 0.30/0.48      set_real2: list_real > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist__update_001_Eo,type,
% 0.30/0.48      list_update_o: list_o > nat > $o > list_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
% 0.30/0.48      list_update_complex: list_complex > nat > complex > list_complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 0.30/0.48      list_update_int: list_int > nat > int > list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 0.30/0.48      list_update_nat: list_nat > nat > nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 0.30/0.48      list_update_real: list_real > nat > real > list_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001_Eo,type,
% 0.30/0.48      nth_o: list_o > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 0.30/0.48      nth_complex: list_complex > nat > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 0.30/0.48      nth_int: list_int > nat > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.48      nth_list_nat: list_list_nat > nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.30/0.48      nth_nat: list_nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.30/0.48      nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48      nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
% 0.30/0.48      nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.48      nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
% 0.30/0.48      nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48      nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.30/0.48      nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.30/0.48      nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.30/0.48      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.30/0.48      nth_real: list_real > nat > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 0.30/0.48      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 0.30/0.48      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 0.30/0.48      product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 0.30/0.48      product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 0.30/0.48      product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
% 0.30/0.48      product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 0.30/0.48      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.48      product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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% 0.30/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.30/0.48      product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.30/0.48      produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.30/0.48      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oreplicate_001_Eo,type,
% 0.30/0.48      replicate_o: nat > $o > list_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 0.30/0.48      replicate_complex: nat > complex > list_complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 0.30/0.48      replicate_int: nat > int > list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 0.30/0.48      replicate_nat: nat > nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 0.30/0.48      replicate_real: nat > real > list_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 0.30/0.48      sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 0.30/0.48      sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.30/0.48      take_nat: nat > list_nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oupt,type,
% 0.30/0.48      upt: nat > nat > list_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oupto,type,
% 0.30/0.48      upto: int > int > list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oupto__aux,type,
% 0.30/0.48      upto_aux: int > int > list_int > list_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_List_Oupto__rel,type,
% 0.30/0.48      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_OSuc,type,
% 0.30/0.48      suc: nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.30/0.48      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.30/0.48      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.48      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Onat_Opred,type,
% 0.30/0.48      pred: nat > nat ).
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% 0.30/0.48      semiri4939895301339042750nteger: nat > code_integer ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.30/0.48      semiri8010041392384452111omplex: nat > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.30/0.48      semiri1314217659103216013at_int: nat > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.30/0.48      semiri1316708129612266289at_nat: nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.30/0.48      semiri681578069525770553at_rat: nat > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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% 0.30/0.48  
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% 0.30/0.48      semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 0.30/0.48      semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 0.30/0.48  
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
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% 0.30/0.48  
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% 0.30/0.48  
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.30/0.48      size_size_list_real: list_real > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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% 0.30/0.48      size_size_num: num > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
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% 0.30/0.48  
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.30/0.48      nat_list_encode: list_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.30/0.48      nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.30/0.48      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.30/0.48      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.30/0.48      nat_prod_encode: product_prod_nat_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.30/0.48      nat_set_decode: nat > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.30/0.48      nat_set_encode: set_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.30/0.48      nat_triangle: nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_NthRoot_Oroot,type,
% 0.30/0.48      root: nat > real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_NthRoot_Osqrt,type,
% 0.30/0.48      sqrt: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_OBitM,type,
% 0.30/0.48      bitM: num > num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oinc,type,
% 0.30/0.48      inc: num > num ).
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 0.30/0.48      neg_nu7009210354673126013omplex: complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.30/0.48      neg_numeral_dbl_int: int > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.30/0.48      neg_numeral_dbl_rat: rat > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.30/0.48      neg_numeral_dbl_real: real > real ).
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.30/0.48      neg_nu3811975205180677377ec_int: int > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.30/0.48      neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 0.30/0.48      neg_nu6075765906172075777c_real: real > real ).
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% 0.30/0.48      neg_nu8557863876264182079omplex: complex > complex ).
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% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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% 0.30/0.48      neg_nu5219082963157363817nc_rat: rat > rat ).
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% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 0.30/0.48      neg_nu8295874005876285629c_real: real > real ).
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% 0.30/0.48  thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Onum_OBit0,type,
% 0.30/0.48      bit0: num > num ).
% 0.30/0.48  
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% 0.30/0.48      bit1: num > num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Onum_OOne,type,
% 0.30/0.48      one: num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.48      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
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% 0.30/0.48      num_of_nat: nat > num ).
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% 0.30/0.48  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.30/0.48      produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
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% 0.30/0.48  thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.30/0.48      produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 0.30/0.48      produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).
% 0.30/0.48  
% 0.30/0.48  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__Complex__Ocomplex_J,type,
% 0.30/0.48      produc2592262431452330817omplex: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex ) > produc8763457246119570046nteger > set_complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
% 0.30/0.48      produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.48      produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/0.48      produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
% 0.30/0.48      produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.48      produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/0.48      produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 0.30/0.48      produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo,type,
% 0.30/0.48      produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 0.30/0.48      produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 0.30/0.48      produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 0.30/0.48      produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.30/0.48      produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.30/0.48      produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 0.30/0.48      produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 0.30/0.48      produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.48      produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.30/0.48      produc8580519160106071146omplex: ( int > int > set_complex ) > product_prod_int_int > set_complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
% 0.30/0.48      produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.48      produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
% 0.30/0.48      produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.30/0.48      produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.30/0.48      produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 0.30/0.48      produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.30/0.48      produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 0.30/0.48      produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.48      produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48      produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.30/0.48      produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.30/0.48      produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.48      produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 0.30/0.48      produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J_001_Eo,type,
% 0.30/0.48      produc3914248068834153634plex_o: ( set_complex > set_complex > $o ) > produc8064648209034914857omplex > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
% 0.30/0.48      produc4109468873575309990_int_o: ( set_int > set_int > $o ) > produc2115011035271226405et_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
% 0.30/0.48      produc6247414631856714078_nat_o: ( set_nat > set_nat > $o ) > produc7819656566062154093et_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 0.30/0.48      product_fst_int_int: product_prod_int_int > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.48      product_fst_nat_nat: product_prod_nat_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 0.30/0.48      product_snd_int_int: product_prod_int_int > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.48      product_snd_nat_nat: product_prod_nat_nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_OAbs__Rat,type,
% 0.30/0.48      abs_Rat: product_prod_int_int > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_OFract,type,
% 0.30/0.48      fract: int > int > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_OFrct,type,
% 0.30/0.48      frct: product_prod_int_int > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_ORep__Rat,type,
% 0.30/0.48      rep_Rat: rat > product_prod_int_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 0.30/0.48      field_5140801741446780682s_real: set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 0.30/0.48      field_7254667332652039916t_real: rat > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Onormalize,type,
% 0.30/0.48      normalize: product_prod_int_int > product_prod_int_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Oof__int,type,
% 0.30/0.48      of_int: int > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Opcr__rat,type,
% 0.30/0.48      pcr_rat: product_prod_int_int > rat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Opositive,type,
% 0.30/0.48      positive: rat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Oquotient__of,type,
% 0.30/0.48      quotient_of: rat > product_prod_int_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Rat_Oratrel,type,
% 0.30/0.48      ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real_Ovanishes,type,
% 0.30/0.48      vanishes: ( nat > rat ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 0.30/0.48      real_V2521375963428798218omplex: set_complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.30/0.48      real_V5970128139526366754l_real: ( real > real ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 0.30/0.48      real_V3694042436643373181omplex: complex > complex > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.30/0.48      real_V975177566351809787t_real: real > real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 0.30/0.48      real_V1022390504157884413omplex: complex > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 0.30/0.48      real_V7735802525324610683m_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 0.30/0.48      real_V4546457046886955230omplex: real > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 0.30/0.48      real_V2046097035970521341omplex: real > complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 0.30/0.48      real_V1485227260804924795R_real: real > real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Relation_OField_001t__Nat__Onat,type,
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% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 0.30/0.48      set_or5832277885323065728an_int: int > int > set_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 0.30/0.48      set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 0.30/0.48      set_or1633881224788618240n_real: real > real > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 0.30/0.48      set_or5849166863359141190n_real: real > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 0.30/0.48      set_ord_lessThan_int: int > set_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 0.30/0.48      set_ord_lessThan_nat: nat > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 0.30/0.48      set_ord_lessThan_num: num > set_num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 0.30/0.48      set_ord_lessThan_rat: rat > set_rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 0.30/0.48      set_or5984915006950818249n_real: real > set_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_String_Oascii__of,type,
% 0.30/0.48      ascii_of: char > char ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_String_Ochar_OChar,type,
% 0.30/0.48      char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 0.30/0.48      comm_s629917340098488124ar_nat: char > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_String_Ointeger__of__char,type,
% 0.30/0.48      integer_of_char: char > code_integer ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 0.30/0.48      unique3096191561947761185of_nat: nat > char ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.30/0.48      topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.30/0.48      topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 0.30/0.48      topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 0.30/0.48      topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 0.30/0.48      topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 0.30/0.48      topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.30/0.48      topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 0.30/0.48      topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 0.30/0.48      topolo2177554685111907308n_real: real > set_real > filter_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.30/0.48      topolo2815343760600316023s_real: real > filter_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 0.30/0.48      topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 0.30/0.48      topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 0.30/0.48      topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oarccos,type,
% 0.30/0.48      arccos: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.30/0.48      arcosh_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oarcsin,type,
% 0.30/0.48      arcsin: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oarctan,type,
% 0.30/0.48      arctan: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.30/0.48      arsinh_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.30/0.48      artanh_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 0.30/0.48      cos_complex: complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 0.30/0.48      cos_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.30/0.48      cos_coeff: nat > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.30/0.48      cosh_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 0.30/0.48      cot_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 0.30/0.48      diffs_complex: ( nat > complex ) > nat > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 0.30/0.48      diffs_int: ( nat > int ) > nat > int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 0.30/0.48      diffs_rat: ( nat > rat ) > nat > rat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 0.30/0.48      diffs_real: ( nat > real ) > nat > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 0.30/0.48      exp_complex: complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 0.30/0.48      exp_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.30/0.48      ln_ln_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Olog,type,
% 0.30/0.48      log: real > real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Opi,type,
% 0.30/0.48      pi: real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.30/0.48      powr_real: real > real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.30/0.48      sin_complex: complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.30/0.48      sin_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Osin__coeff,type,
% 0.30/0.48      sin_coeff: nat > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.30/0.48      sinh_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 0.30/0.48      tan_complex: complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.30/0.48      tan_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 0.30/0.48      tanh_complex: complex > complex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.30/0.48      tanh_real: real > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.30/0.48      vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.30/0.48      vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.30/0.48      vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.30/0.48      vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.30/0.48      vEBT_VEBT_high: nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.30/0.48      vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.30/0.48      vEBT_VEBT_low: nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.30/0.48      vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.30/0.48      vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.30/0.48      vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.30/0.48      vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.30/0.48      vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.30/0.48      vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.30/0.48      vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.30/0.48      vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.30/0.48      vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.30/0.48      vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 0.30/0.48      vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 0.30/0.48      vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 0.30/0.48      vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 0.30/0.48      vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 0.30/0.48      vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 0.30/0.48      vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Member_Ovebt__member,type,
% 0.30/0.48      vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 0.30/0.48      vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 0.30/0.48      vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 0.30/0.48      vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 0.30/0.48      vEBT_VEBT_less: option_nat > option_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 0.30/0.48      vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 0.30/0.48      vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 0.30/0.48      vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 0.30/0.48      vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 0.30/0.48      vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 0.30/0.48      vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48      vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
% 0.30/0.48      vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
% 0.30/0.48      vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48      vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 0.30/0.48      vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 0.30/0.48      vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 0.30/0.48      vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 0.30/0.48      vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 0.30/0.48      vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 0.30/0.48      vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 0.30/0.48      vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 0.30/0.48      vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.48      accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 0.30/0.48      accp_nat: ( nat > nat > $o ) > nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.30/0.48      accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 0.30/0.48      accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 0.30/0.48      accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.30/0.48      accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.48      accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48      accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.30/0.48      accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.30/0.48      accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
% 0.30/0.48      finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
% 0.30/0.48      finite_psubset_int: set_Pr2522554150109002629et_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
% 0.30/0.48      finite_psubset_nat: set_Pr5488025237498180813et_nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
% 0.30/0.48      measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_fChoice_001t__Real__Oreal,type,
% 0.30/0.48      fChoice_real: ( real > $o ) > real ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001_Eo,type,
% 0.30/0.48      member_o: $o > set_o > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.30/0.48      member_complex: complex > set_complex > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Int__Oint,type,
% 0.30/0.48      member_int: int > set_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 0.30/0.48      member_list_o: list_o > set_list_o > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.48      member_list_nat: list_nat > set_list_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48      member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Nat__Onat,type,
% 0.30/0.48      member_nat: nat > set_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Num__Onum,type,
% 0.30/0.48      member_num: num > set_num > $o ).
% 0.30/0.48  
% 0.30/0.48  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,
% 0.30/0.48      member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
% 0.30/0.48  
% 0.30/0.48  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,
% 0.30/0.48      member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
% 0.30/0.48  
% 0.30/0.48  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,
% 0.30/0.48      member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
% 0.30/0.48  
% 0.30/0.48  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,
% 0.30/0.48      member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.48      member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 0.30/0.48      member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
% 0.30/0.48      member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.30/0.48      member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Rat__Orat,type,
% 0.30/0.48      member_rat: rat > set_rat > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Real__Oreal,type,
% 0.30/0.48      member_real: real > set_real > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.30/0.48      member_set_int: set_int > set_set_int > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_deg____,type,
% 0.30/0.48      deg: nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_m____,type,
% 0.30/0.48      m: nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_ma____,type,
% 0.30/0.48      ma: nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_mi____,type,
% 0.30/0.48      mi: nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_na____,type,
% 0.30/0.48      na: nat ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_summary____,type,
% 0.30/0.48      summary: vEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_thesis____,type,
% 0.30/0.48      thesis: $o ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_treeList____,type,
% 0.30/0.48      treeList: list_VEBT_VEBT ).
% 0.30/0.48  
% 0.30/0.48  thf(sy_v_xa____,type,
% 0.30/0.48      xa: nat ).
% 0.30/0.48  
% 0.30/0.48  % Relevant facts (10209)
% 0.30/0.48  thf(fact_0_False,axiom,
% 0.30/0.48      ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.48     != none_nat ) ).
% 0.30/0.48  
% 0.30/0.48  % False
% 0.30/0.48  thf(fact_1__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 0.30/0.48      ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48      = na ) ).
% 0.30/0.48  
% 0.30/0.48  % \<open>deg div 2 = n\<close>
% 0.30/0.48  thf(fact_2_semiring__norm_I85_J,axiom,
% 0.30/0.48      ! [M: num] :
% 0.30/0.48        ( ( bit0 @ M )
% 0.30/0.48       != one ) ).
% 0.30/0.48  
% 0.30/0.48  % semiring_norm(85)
% 0.30/0.48  thf(fact_3_semiring__norm_I83_J,axiom,
% 0.30/0.48      ! [N: num] :
% 0.30/0.48        ( one
% 0.30/0.48       != ( bit0 @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % semiring_norm(83)
% 0.30/0.48  thf(fact_4__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 0.30/0.48      ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.30/0.48  
% 0.30/0.48  % \<open>2 \<le> deg\<close>
% 0.30/0.48  thf(fact_5_numeral__Bit0__div__2,axiom,
% 0.30/0.48      ! [N: num] :
% 0.30/0.48        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48        = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_Bit0_div_2
% 0.30/0.48  thf(fact_6_numeral__Bit0__div__2,axiom,
% 0.30/0.48      ! [N: num] :
% 0.30/0.48        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.48        = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_Bit0_div_2
% 0.30/0.48  thf(fact_7_high__def,axiom,
% 0.30/0.48      ( vEBT_VEBT_high
% 0.30/0.48      = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % high_def
% 0.30/0.48  thf(fact_8_True,axiom,
% 0.30/0.48      ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 0.30/0.48  
% 0.30/0.48  % True
% 0.30/0.48  thf(fact_9_i1,axiom,
% 0.30/0.48      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.48        = none_nat )
% 0.30/0.48      | ~ ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % i1
% 0.30/0.48  thf(fact_10_divide__numeral__1,axiom,
% 0.30/0.48      ! [A: complex] :
% 0.30/0.48        ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.48        = A ) ).
% 0.30/0.48  
% 0.30/0.48  % divide_numeral_1
% 0.30/0.48  thf(fact_11_divide__numeral__1,axiom,
% 0.30/0.48      ! [A: real] :
% 0.30/0.48        ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.30/0.48        = A ) ).
% 0.30/0.48  
% 0.30/0.48  % divide_numeral_1
% 0.30/0.48  thf(fact_12_divide__numeral__1,axiom,
% 0.30/0.48      ! [A: rat] :
% 0.30/0.48        ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.30/0.48        = A ) ).
% 0.30/0.48  
% 0.30/0.48  % divide_numeral_1
% 0.30/0.48  thf(fact_13_option_Oinject,axiom,
% 0.30/0.48      ! [X2: nat,Y2: nat] :
% 0.30/0.48        ( ( ( some_nat @ X2 )
% 0.30/0.48          = ( some_nat @ Y2 ) )
% 0.30/0.48        = ( X2 = Y2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.inject
% 0.30/0.48  thf(fact_14_option_Oinject,axiom,
% 0.30/0.48      ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 0.30/0.48        ( ( ( some_P7363390416028606310at_nat @ X2 )
% 0.30/0.48          = ( some_P7363390416028606310at_nat @ Y2 ) )
% 0.30/0.48        = ( X2 = Y2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.inject
% 0.30/0.48  thf(fact_15_option_Oinject,axiom,
% 0.30/0.48      ! [X2: num,Y2: num] :
% 0.30/0.48        ( ( ( some_num @ X2 )
% 0.30/0.48          = ( some_num @ Y2 ) )
% 0.30/0.48        = ( X2 = Y2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.inject
% 0.30/0.48  thf(fact_16_verit__eq__simplify_I8_J,axiom,
% 0.30/0.48      ! [X2: num,Y2: num] :
% 0.30/0.48        ( ( ( bit0 @ X2 )
% 0.30/0.48          = ( bit0 @ Y2 ) )
% 0.30/0.48        = ( X2 = Y2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_eq_simplify(8)
% 0.30/0.48  thf(fact_17_semiring__norm_I87_J,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ( bit0 @ M )
% 0.30/0.48          = ( bit0 @ N ) )
% 0.30/0.48        = ( M = N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % semiring_norm(87)
% 0.30/0.48  thf(fact_18__C4_Ohyps_C_I5_J,axiom,
% 0.30/0.48      m = na ).
% 0.30/0.48  
% 0.30/0.48  % "4.hyps"(5)
% 0.30/0.48  thf(fact_19_max__in__set__def,axiom,
% 0.30/0.48      ( vEBT_VEBT_max_in_set
% 0.30/0.48      = ( ^ [Xs: set_nat,X: nat] :
% 0.30/0.48            ( ( member_nat @ X @ Xs )
% 0.30/0.48            & ! [Y: nat] :
% 0.30/0.48                ( ( member_nat @ Y @ Xs )
% 0.30/0.48               => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % max_in_set_def
% 0.30/0.48  thf(fact_20_greater__shift,axiom,
% 0.30/0.48      ( ord_less_nat
% 0.30/0.48      = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % greater_shift
% 0.30/0.48  thf(fact_21_min__in__set__def,axiom,
% 0.30/0.48      ( vEBT_VEBT_min_in_set
% 0.30/0.48      = ( ^ [Xs: set_nat,X: nat] :
% 0.30/0.48            ( ( member_nat @ X @ Xs )
% 0.30/0.48            & ! [Y: nat] :
% 0.30/0.48                ( ( member_nat @ Y @ Xs )
% 0.30/0.48               => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % min_in_set_def
% 0.30/0.48  thf(fact_22_power__shift,axiom,
% 0.30/0.48      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.48        ( ( ( power_power_nat @ X3 @ Y3 )
% 0.30/0.48          = Z )
% 0.30/0.48        = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 0.30/0.48          = ( some_nat @ Z ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % power_shift
% 0.30/0.48  thf(fact_23_bit__split__inv,axiom,
% 0.30/0.48      ! [X3: nat,D: nat] :
% 0.30/0.48        ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
% 0.30/0.48        = X3 ) ).
% 0.30/0.48  
% 0.30/0.48  % bit_split_inv
% 0.30/0.48  thf(fact_24_numeral__eq__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ( numera6690914467698888265omplex @ M )
% 0.30/0.48          = ( numera6690914467698888265omplex @ N ) )
% 0.30/0.48        = ( M = N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_eq_iff
% 0.30/0.48  thf(fact_25_numeral__eq__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ( numeral_numeral_real @ M )
% 0.30/0.48          = ( numeral_numeral_real @ N ) )
% 0.30/0.48        = ( M = N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_eq_iff
% 0.30/0.48  thf(fact_26_numeral__eq__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ( numeral_numeral_rat @ M )
% 0.30/0.48          = ( numeral_numeral_rat @ N ) )
% 0.30/0.48        = ( M = N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_eq_iff
% 0.30/0.48  thf(fact_27_numeral__eq__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ( numeral_numeral_nat @ M )
% 0.30/0.48          = ( numeral_numeral_nat @ N ) )
% 0.30/0.48        = ( M = N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_eq_iff
% 0.30/0.48  thf(fact_28_numeral__eq__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ( numeral_numeral_int @ M )
% 0.30/0.48          = ( numeral_numeral_int @ N ) )
% 0.30/0.48        = ( M = N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_eq_iff
% 0.30/0.48  thf(fact_29_numeral__le__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.48        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_le_iff
% 0.30/0.48  thf(fact_30_numeral__le__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.48        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_le_iff
% 0.30/0.48  thf(fact_31_numeral__le__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.48        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_le_iff
% 0.30/0.48  thf(fact_32_numeral__le__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.48        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_le_iff
% 0.30/0.48  thf(fact_33_numeral__less__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.48        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_less_iff
% 0.30/0.48  thf(fact_34_numeral__less__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.48        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_less_iff
% 0.30/0.48  thf(fact_35_numeral__less__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.48        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_less_iff
% 0.30/0.48  thf(fact_36_numeral__less__iff,axiom,
% 0.30/0.48      ! [M: num,N: num] :
% 0.30/0.48        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.48        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.48  
% 0.30/0.48  % numeral_less_iff
% 0.30/0.48  thf(fact_37_not__Some__eq,axiom,
% 0.30/0.48      ! [X3: option_nat] :
% 0.30/0.48        ( ( ! [Y: nat] :
% 0.30/0.48              ( X3
% 0.30/0.48             != ( some_nat @ Y ) ) )
% 0.30/0.48        = ( X3 = none_nat ) ) ).
% 0.30/0.48  
% 0.30/0.48  % not_Some_eq
% 0.30/0.48  thf(fact_38_not__Some__eq,axiom,
% 0.30/0.48      ! [X3: option4927543243414619207at_nat] :
% 0.30/0.48        ( ( ! [Y: product_prod_nat_nat] :
% 0.30/0.48              ( X3
% 0.30/0.48             != ( some_P7363390416028606310at_nat @ Y ) ) )
% 0.30/0.48        = ( X3 = none_P5556105721700978146at_nat ) ) ).
% 0.30/0.48  
% 0.30/0.48  % not_Some_eq
% 0.30/0.48  thf(fact_39_not__Some__eq,axiom,
% 0.30/0.48      ! [X3: option_num] :
% 0.30/0.48        ( ( ! [Y: num] :
% 0.30/0.48              ( X3
% 0.30/0.48             != ( some_num @ Y ) ) )
% 0.30/0.48        = ( X3 = none_num ) ) ).
% 0.30/0.48  
% 0.30/0.48  % not_Some_eq
% 0.30/0.48  thf(fact_40_not__None__eq,axiom,
% 0.30/0.48      ! [X3: option_nat] :
% 0.30/0.48        ( ( X3 != none_nat )
% 0.30/0.48        = ( ? [Y: nat] :
% 0.30/0.48              ( X3
% 0.30/0.48              = ( some_nat @ Y ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % not_None_eq
% 0.30/0.48  thf(fact_41_not__None__eq,axiom,
% 0.30/0.48      ! [X3: option4927543243414619207at_nat] :
% 0.30/0.48        ( ( X3 != none_P5556105721700978146at_nat )
% 0.30/0.48        = ( ? [Y: product_prod_nat_nat] :
% 0.30/0.48              ( X3
% 0.30/0.48              = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % not_None_eq
% 0.30/0.48  thf(fact_42_not__None__eq,axiom,
% 0.30/0.48      ! [X3: option_num] :
% 0.30/0.48        ( ( X3 != none_num )
% 0.30/0.48        = ( ? [Y: num] :
% 0.30/0.48              ( X3
% 0.30/0.48              = ( some_num @ Y ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % not_None_eq
% 0.30/0.48  thf(fact_43__C4_Ohyps_C_I4_J,axiom,
% 0.30/0.48      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.30/0.48      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.30/0.48  
% 0.30/0.48  % "4.hyps"(4)
% 0.30/0.48  thf(fact_44__092_060open_062length_AtreeList_A_061_A2_A_094_An_092_060close_062,axiom,
% 0.30/0.48      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.30/0.48      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ).
% 0.30/0.48  
% 0.30/0.48  % \<open>length treeList = 2 ^ n\<close>
% 0.30/0.48  thf(fact_45__092_060open_062high_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
% 0.30/0.48      ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
% 0.30/0.48      & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % \<open>high x n < 2 ^ n \<and> low x n < 2 ^ n\<close>
% 0.30/0.48  thf(fact_46__C4_Ohyps_C_I10_J,axiom,
% 0.30/0.48      ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.30/0.48  
% 0.30/0.48  % "4.hyps"(10)
% 0.30/0.48  thf(fact_47_less__shift,axiom,
% 0.30/0.48      ( ord_less_nat
% 0.30/0.48      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % less_shift
% 0.30/0.48  thf(fact_48_lesseq__shift,axiom,
% 0.30/0.48      ( ord_less_eq_nat
% 0.30/0.48      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % lesseq_shift
% 0.30/0.48  thf(fact_49_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.48      ! [B: real,A2: real] :
% 0.30/0.48        ( ( ~ ( ord_less_eq_real @ B @ A2 ) )
% 0.30/0.48        = ( ord_less_real @ A2 @ B ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(3)
% 0.30/0.48  thf(fact_50_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.48      ! [B: rat,A2: rat] :
% 0.30/0.48        ( ( ~ ( ord_less_eq_rat @ B @ A2 ) )
% 0.30/0.48        = ( ord_less_rat @ A2 @ B ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(3)
% 0.30/0.48  thf(fact_51_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.48      ! [B: num,A2: num] :
% 0.30/0.48        ( ( ~ ( ord_less_eq_num @ B @ A2 ) )
% 0.30/0.48        = ( ord_less_num @ A2 @ B ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(3)
% 0.30/0.48  thf(fact_52_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.48      ! [B: nat,A2: nat] :
% 0.30/0.48        ( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
% 0.30/0.48        = ( ord_less_nat @ A2 @ B ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(3)
% 0.30/0.48  thf(fact_53_verit__comp__simplify1_I3_J,axiom,
% 0.30/0.48      ! [B: int,A2: int] :
% 0.30/0.48        ( ( ~ ( ord_less_eq_int @ B @ A2 ) )
% 0.30/0.48        = ( ord_less_int @ A2 @ B ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(3)
% 0.30/0.48  thf(fact_54_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.48      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(2)
% 0.30/0.48  thf(fact_55_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.48      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(2)
% 0.30/0.48  thf(fact_56_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.48      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(2)
% 0.30/0.48  thf(fact_57_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.48      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(2)
% 0.30/0.48  thf(fact_58_verit__comp__simplify1_I2_J,axiom,
% 0.30/0.48      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(2)
% 0.30/0.48  thf(fact_59_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.48      ! [A: real] :
% 0.30/0.48        ~ ( ord_less_real @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(1)
% 0.30/0.48  thf(fact_60_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.48      ! [A: rat] :
% 0.30/0.48        ~ ( ord_less_rat @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(1)
% 0.30/0.48  thf(fact_61_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.48      ! [A: num] :
% 0.30/0.48        ~ ( ord_less_num @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(1)
% 0.30/0.48  thf(fact_62_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.48      ! [A: nat] :
% 0.30/0.48        ~ ( ord_less_nat @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(1)
% 0.30/0.48  thf(fact_63_verit__comp__simplify1_I1_J,axiom,
% 0.30/0.48      ! [A: int] :
% 0.30/0.48        ~ ( ord_less_int @ A @ A ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_comp_simplify1(1)
% 0.30/0.48  thf(fact_64_verit__la__disequality,axiom,
% 0.30/0.48      ! [A: rat,B2: rat] :
% 0.30/0.48        ( ( A = B2 )
% 0.30/0.48        | ~ ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.48        | ~ ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_la_disequality
% 0.30/0.48  thf(fact_65_verit__la__disequality,axiom,
% 0.30/0.48      ! [A: num,B2: num] :
% 0.30/0.48        ( ( A = B2 )
% 0.30/0.48        | ~ ( ord_less_eq_num @ A @ B2 )
% 0.30/0.48        | ~ ( ord_less_eq_num @ B2 @ A ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_la_disequality
% 0.30/0.48  thf(fact_66_verit__la__disequality,axiom,
% 0.30/0.48      ! [A: nat,B2: nat] :
% 0.30/0.48        ( ( A = B2 )
% 0.30/0.48        | ~ ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.48        | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_la_disequality
% 0.30/0.48  thf(fact_67_verit__la__disequality,axiom,
% 0.30/0.48      ! [A: int,B2: int] :
% 0.30/0.48        ( ( A = B2 )
% 0.30/0.48        | ~ ( ord_less_eq_int @ A @ B2 )
% 0.30/0.48        | ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.30/0.48  
% 0.30/0.48  % verit_la_disequality
% 0.30/0.48  thf(fact_68_div__le__dividend,axiom,
% 0.30/0.48      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.30/0.48  
% 0.30/0.48  % div_le_dividend
% 0.30/0.48  thf(fact_69_div__le__mono,axiom,
% 0.30/0.48      ! [M: nat,N: nat,K: nat] :
% 0.30/0.48        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.48       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % div_le_mono
% 0.30/0.48  thf(fact_70_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 0.30/0.48        ( ( ( X3 = none_nat )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_nat )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: nat,B3: nat] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_nat @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_nat @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_71_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.30/0.48        ( ( ( X3 = none_nat )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: nat,B3: product_prod_nat_nat] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_nat @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_72_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 0.30/0.48        ( ( ( X3 = none_nat )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_num )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: nat,B3: num] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_nat @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_num @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_73_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 0.30/0.48        ( ( ( X3 = none_P5556105721700978146at_nat )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_nat )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: product_prod_nat_nat,B3: nat] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_nat @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_74_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.30/0.48        ( ( ( X3 = none_P5556105721700978146at_nat )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_75_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 0.30/0.48        ( ( ( X3 = none_P5556105721700978146at_nat )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_num )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: product_prod_nat_nat,B3: num] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_num @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_76_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 0.30/0.48        ( ( ( X3 = none_num )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_nat )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: num,B3: nat] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_num @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_nat @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_77_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.30/0.48        ( ( ( X3 = none_num )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: num,B3: product_prod_nat_nat] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_num @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_78_combine__options__cases,axiom,
% 0.30/0.48      ! [X3: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 0.30/0.48        ( ( ( X3 = none_num )
% 0.30/0.48         => ( P @ X3 @ Y3 ) )
% 0.30/0.48       => ( ( ( Y3 = none_num )
% 0.30/0.48           => ( P @ X3 @ Y3 ) )
% 0.30/0.48         => ( ! [A3: num,B3: num] :
% 0.30/0.48                ( ( X3
% 0.30/0.48                  = ( some_num @ A3 ) )
% 0.30/0.48               => ( ( Y3
% 0.30/0.48                    = ( some_num @ B3 ) )
% 0.30/0.48                 => ( P @ X3 @ Y3 ) ) )
% 0.30/0.48           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % combine_options_cases
% 0.30/0.48  thf(fact_79_split__option__all,axiom,
% 0.30/0.48      ( ( ^ [P2: option_nat > $o] :
% 0.30/0.48          ! [X4: option_nat] : ( P2 @ X4 ) )
% 0.30/0.48      = ( ^ [P3: option_nat > $o] :
% 0.30/0.48            ( ( P3 @ none_nat )
% 0.30/0.48            & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % split_option_all
% 0.30/0.48  thf(fact_80_split__option__all,axiom,
% 0.30/0.48      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.30/0.48          ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 0.30/0.48      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.30/0.48            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.30/0.48            & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % split_option_all
% 0.30/0.48  thf(fact_81_split__option__all,axiom,
% 0.30/0.48      ( ( ^ [P2: option_num > $o] :
% 0.30/0.48          ! [X4: option_num] : ( P2 @ X4 ) )
% 0.30/0.48      = ( ^ [P3: option_num > $o] :
% 0.30/0.48            ( ( P3 @ none_num )
% 0.30/0.48            & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % split_option_all
% 0.30/0.48  thf(fact_82_split__option__ex,axiom,
% 0.30/0.48      ( ( ^ [P2: option_nat > $o] :
% 0.30/0.48          ? [X4: option_nat] : ( P2 @ X4 ) )
% 0.30/0.48      = ( ^ [P3: option_nat > $o] :
% 0.30/0.48            ( ( P3 @ none_nat )
% 0.30/0.48            | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % split_option_ex
% 0.30/0.48  thf(fact_83_split__option__ex,axiom,
% 0.30/0.48      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.30/0.48          ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 0.30/0.48      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.30/0.48            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.30/0.48            | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % split_option_ex
% 0.30/0.48  thf(fact_84_split__option__ex,axiom,
% 0.30/0.48      ( ( ^ [P2: option_num > $o] :
% 0.30/0.48          ? [X4: option_num] : ( P2 @ X4 ) )
% 0.30/0.48      = ( ^ [P3: option_num > $o] :
% 0.30/0.48            ( ( P3 @ none_num )
% 0.30/0.48            | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % split_option_ex
% 0.30/0.48  thf(fact_85_option_Oexhaust,axiom,
% 0.30/0.48      ! [Y3: option_nat] :
% 0.30/0.48        ( ( Y3 != none_nat )
% 0.30/0.48       => ~ ! [X22: nat] :
% 0.30/0.48              ( Y3
% 0.30/0.48             != ( some_nat @ X22 ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.exhaust
% 0.30/0.48  thf(fact_86_option_Oexhaust,axiom,
% 0.30/0.48      ! [Y3: option4927543243414619207at_nat] :
% 0.30/0.48        ( ( Y3 != none_P5556105721700978146at_nat )
% 0.30/0.48       => ~ ! [X22: product_prod_nat_nat] :
% 0.30/0.48              ( Y3
% 0.30/0.48             != ( some_P7363390416028606310at_nat @ X22 ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.exhaust
% 0.30/0.48  thf(fact_87_option_Oexhaust,axiom,
% 0.30/0.48      ! [Y3: option_num] :
% 0.30/0.48        ( ( Y3 != none_num )
% 0.30/0.48       => ~ ! [X22: num] :
% 0.30/0.48              ( Y3
% 0.30/0.48             != ( some_num @ X22 ) ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.exhaust
% 0.30/0.48  thf(fact_88_option_OdiscI,axiom,
% 0.30/0.48      ! [Option: option_nat,X2: nat] :
% 0.30/0.48        ( ( Option
% 0.30/0.48          = ( some_nat @ X2 ) )
% 0.30/0.48       => ( Option != none_nat ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.discI
% 0.30/0.48  thf(fact_89_option_OdiscI,axiom,
% 0.30/0.48      ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
% 0.30/0.48        ( ( Option
% 0.30/0.48          = ( some_P7363390416028606310at_nat @ X2 ) )
% 0.30/0.48       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.discI
% 0.30/0.48  thf(fact_90_option_OdiscI,axiom,
% 0.30/0.48      ! [Option: option_num,X2: num] :
% 0.30/0.48        ( ( Option
% 0.30/0.48          = ( some_num @ X2 ) )
% 0.30/0.48       => ( Option != none_num ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.discI
% 0.30/0.48  thf(fact_91_option_Odistinct_I1_J,axiom,
% 0.30/0.48      ! [X2: nat] :
% 0.30/0.48        ( none_nat
% 0.30/0.48       != ( some_nat @ X2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.distinct(1)
% 0.30/0.48  thf(fact_92_option_Odistinct_I1_J,axiom,
% 0.30/0.48      ! [X2: product_prod_nat_nat] :
% 0.30/0.48        ( none_P5556105721700978146at_nat
% 0.30/0.48       != ( some_P7363390416028606310at_nat @ X2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.distinct(1)
% 0.30/0.48  thf(fact_93_option_Odistinct_I1_J,axiom,
% 0.30/0.48      ! [X2: num] :
% 0.30/0.48        ( none_num
% 0.30/0.48       != ( some_num @ X2 ) ) ).
% 0.30/0.48  
% 0.30/0.48  % option.distinct(1)
% 0.30/0.48  thf(fact_94_mem__Collect__eq,axiom,
% 0.30/0.48      ! [A: complex,P: complex > $o] :
% 0.30/0.48        ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.30/0.48        = ( P @ A ) ) ).
% 0.30/0.48  
% 0.30/0.48  % mem_Collect_eq
% 0.30/0.48  thf(fact_95_mem__Collect__eq,axiom,
% 0.30/0.48      ! [A: real,P: real > $o] :
% 0.30/0.48        ( ( member_real @ A @ ( collect_real @ P ) )
% 0.30/0.48        = ( P @ A ) ) ).
% 0.30/0.48  
% 0.30/0.48  % mem_Collect_eq
% 0.30/0.48  thf(fact_96_mem__Collect__eq,axiom,
% 0.30/0.48      ! [A: list_nat,P: list_nat > $o] :
% 0.30/0.49        ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.30/0.49        = ( P @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mem_Collect_eq
% 0.30/0.49  thf(fact_97_mem__Collect__eq,axiom,
% 0.30/0.49      ! [A: nat,P: nat > $o] :
% 0.30/0.49        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.30/0.49        = ( P @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mem_Collect_eq
% 0.30/0.49  thf(fact_98_mem__Collect__eq,axiom,
% 0.30/0.49      ! [A: int,P: int > $o] :
% 0.30/0.49        ( ( member_int @ A @ ( collect_int @ P ) )
% 0.30/0.49        = ( P @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mem_Collect_eq
% 0.30/0.49  thf(fact_99_Collect__mem__eq,axiom,
% 0.30/0.49      ! [A4: set_complex] :
% 0.30/0.49        ( ( collect_complex
% 0.30/0.49          @ ^ [X: complex] : ( member_complex @ X @ A4 ) )
% 0.30/0.49        = A4 ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_mem_eq
% 0.30/0.49  thf(fact_100_Collect__mem__eq,axiom,
% 0.30/0.49      ! [A4: set_real] :
% 0.30/0.49        ( ( collect_real
% 0.30/0.49          @ ^ [X: real] : ( member_real @ X @ A4 ) )
% 0.30/0.49        = A4 ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_mem_eq
% 0.30/0.49  thf(fact_101_Collect__mem__eq,axiom,
% 0.30/0.49      ! [A4: set_list_nat] :
% 0.30/0.49        ( ( collect_list_nat
% 0.30/0.49          @ ^ [X: list_nat] : ( member_list_nat @ X @ A4 ) )
% 0.30/0.49        = A4 ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_mem_eq
% 0.30/0.49  thf(fact_102_Collect__mem__eq,axiom,
% 0.30/0.49      ! [A4: set_nat] :
% 0.30/0.49        ( ( collect_nat
% 0.30/0.49          @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
% 0.30/0.49        = A4 ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_mem_eq
% 0.30/0.49  thf(fact_103_Collect__mem__eq,axiom,
% 0.30/0.49      ! [A4: set_int] :
% 0.30/0.49        ( ( collect_int
% 0.30/0.49          @ ^ [X: int] : ( member_int @ X @ A4 ) )
% 0.30/0.49        = A4 ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_mem_eq
% 0.30/0.49  thf(fact_104_Collect__cong,axiom,
% 0.30/0.49      ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.49        ( ! [X5: complex] :
% 0.30/0.49            ( ( P @ X5 )
% 0.30/0.49            = ( Q @ X5 ) )
% 0.30/0.49       => ( ( collect_complex @ P )
% 0.30/0.49          = ( collect_complex @ Q ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_cong
% 0.30/0.49  thf(fact_105_Collect__cong,axiom,
% 0.30/0.49      ! [P: real > $o,Q: real > $o] :
% 0.30/0.49        ( ! [X5: real] :
% 0.30/0.49            ( ( P @ X5 )
% 0.30/0.49            = ( Q @ X5 ) )
% 0.30/0.49       => ( ( collect_real @ P )
% 0.30/0.49          = ( collect_real @ Q ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_cong
% 0.30/0.49  thf(fact_106_Collect__cong,axiom,
% 0.30/0.49      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.49        ( ! [X5: list_nat] :
% 0.30/0.49            ( ( P @ X5 )
% 0.30/0.49            = ( Q @ X5 ) )
% 0.30/0.49       => ( ( collect_list_nat @ P )
% 0.30/0.49          = ( collect_list_nat @ Q ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_cong
% 0.30/0.49  thf(fact_107_Collect__cong,axiom,
% 0.30/0.49      ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.49        ( ! [X5: nat] :
% 0.30/0.49            ( ( P @ X5 )
% 0.30/0.49            = ( Q @ X5 ) )
% 0.30/0.49       => ( ( collect_nat @ P )
% 0.30/0.49          = ( collect_nat @ Q ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_cong
% 0.30/0.49  thf(fact_108_Collect__cong,axiom,
% 0.30/0.49      ! [P: int > $o,Q: int > $o] :
% 0.30/0.49        ( ! [X5: int] :
% 0.30/0.49            ( ( P @ X5 )
% 0.30/0.49            = ( Q @ X5 ) )
% 0.30/0.49       => ( ( collect_int @ P )
% 0.30/0.49          = ( collect_int @ Q ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Collect_cong
% 0.30/0.49  thf(fact_109_verit__eq__simplify_I10_J,axiom,
% 0.30/0.49      ! [X2: num] :
% 0.30/0.49        ( one
% 0.30/0.49       != ( bit0 @ X2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % verit_eq_simplify(10)
% 0.30/0.49  thf(fact_110__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
% 0.30/0.49      vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 0.30/0.49  
% 0.30/0.49  % \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
% 0.30/0.49  thf(fact_111_minminNull,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT] :
% 0.30/0.49        ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49          = none_nat )
% 0.30/0.49       => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.30/0.49  
% 0.30/0.49  % minminNull
% 0.30/0.49  thf(fact_112_minNullmin,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT] :
% 0.30/0.49        ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.49       => ( ( vEBT_vebt_mint @ T )
% 0.30/0.49          = none_nat ) ) ).
% 0.30/0.49  
% 0.30/0.49  % minNullmin
% 0.30/0.49  thf(fact_113_power2__nat__le__imp__le,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_nat_le_imp_le
% 0.30/0.49  thf(fact_114_power2__nat__le__eq__le,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_nat_le_eq_le
% 0.30/0.49  thf(fact_115_self__le__ge2__pow,axiom,
% 0.30/0.49      ! [K: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.49       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % self_le_ge2_pow
% 0.30/0.49  thf(fact_116_less__exp,axiom,
% 0.30/0.49      ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_exp
% 0.30/0.49  thf(fact_117_high__bound__aux,axiom,
% 0.30/0.49      ! [Ma: nat,N: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.30/0.49       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % high_bound_aux
% 0.30/0.49  thf(fact_118_power__minus__is__div,axiom,
% 0.30/0.49      ! [B2: nat,A: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.49       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.49          = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_minus_is_div
% 0.30/0.49  thf(fact_119_misiz,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,M: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( some_nat @ M )
% 0.30/0.49            = ( vEBT_vebt_mint @ T ) )
% 0.30/0.49         => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % misiz
% 0.30/0.49  thf(fact_120_pow__sum,axiom,
% 0.30/0.49      ! [A: nat,B2: nat] :
% 0.30/0.49        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.49        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % pow_sum
% 0.30/0.49  thf(fact_121__092_060open_062x_A_092_060le_062_Ama_092_060close_062,axiom,
% 0.30/0.49      ord_less_eq_nat @ xa @ ma ).
% 0.30/0.49  
% 0.30/0.49  % \<open>x \<le> ma\<close>
% 0.30/0.49  thf(fact_122__C4_Ohyps_C_I6_J,axiom,
% 0.30/0.49      ( deg
% 0.30/0.49      = ( plus_plus_nat @ na @ m ) ) ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(6)
% 0.30/0.49  thf(fact_123__C4_Ohyps_C_I9_J,axiom,
% 0.30/0.49      ord_less_eq_nat @ mi @ ma ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(9)
% 0.30/0.49  thf(fact_124__C4_Ohyps_C_I2_J,axiom,
% 0.30/0.49      vEBT_invar_vebt @ summary @ m ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(2)
% 0.30/0.49  thf(fact_125_add__numeral__left,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: complex] :
% 0.30/0.49        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.30/0.49        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_numeral_left
% 0.30/0.49  thf(fact_126_add__numeral__left,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: real] :
% 0.30/0.49        ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.30/0.49        = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_numeral_left
% 0.30/0.49  thf(fact_127_add__numeral__left,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: rat] :
% 0.30/0.49        ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.30/0.49        = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_numeral_left
% 0.30/0.49  thf(fact_128_add__numeral__left,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: nat] :
% 0.30/0.49        ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.30/0.49        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_numeral_left
% 0.30/0.49  thf(fact_129_add__numeral__left,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: int] :
% 0.30/0.49        ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.30/0.49        = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_numeral_left
% 0.30/0.49  thf(fact_130_numeral__plus__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.49        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_plus_numeral
% 0.30/0.49  thf(fact_131_numeral__plus__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.49        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_plus_numeral
% 0.30/0.49  thf(fact_132_numeral__plus__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_plus_numeral
% 0.30/0.49  thf(fact_133_numeral__plus__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49        = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_plus_numeral
% 0.30/0.49  thf(fact_134_numeral__plus__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.49        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_plus_numeral
% 0.30/0.49  thf(fact_135_semiring__norm_I78_J,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.49        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(78)
% 0.30/0.49  thf(fact_136_semiring__norm_I71_J,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.49        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(71)
% 0.30/0.49  thf(fact_137_semiring__norm_I75_J,axiom,
% 0.30/0.49      ! [M: num] :
% 0.30/0.49        ~ ( ord_less_num @ M @ one ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(75)
% 0.30/0.49  thf(fact_138_semiring__norm_I68_J,axiom,
% 0.30/0.49      ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(68)
% 0.30/0.49  thf(fact_139_semiring__norm_I76_J,axiom,
% 0.30/0.49      ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(76)
% 0.30/0.49  thf(fact_140_semiring__norm_I69_J,axiom,
% 0.30/0.49      ! [M: num] :
% 0.30/0.49        ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(69)
% 0.30/0.49  thf(fact_141_add__self__div__2,axiom,
% 0.30/0.49      ! [M: nat] :
% 0.30/0.49        ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = M ) ).
% 0.30/0.49  
% 0.30/0.49  % add_self_div_2
% 0.30/0.49  thf(fact_142_local_Opower__def,axiom,
% 0.30/0.49      ( vEBT_VEBT_power
% 0.30/0.49      = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.30/0.49  
% 0.30/0.49  % local.power_def
% 0.30/0.49  thf(fact_143_is__num__normalize_I1_J,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % is_num_normalize(1)
% 0.30/0.49  thf(fact_144_is__num__normalize_I1_J,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % is_num_normalize(1)
% 0.30/0.49  thf(fact_145_is__num__normalize_I1_J,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % is_num_normalize(1)
% 0.30/0.49  thf(fact_146_le__num__One__iff,axiom,
% 0.30/0.49      ! [X3: num] :
% 0.30/0.49        ( ( ord_less_eq_num @ X3 @ one )
% 0.30/0.49        = ( X3 = one ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_num_One_iff
% 0.30/0.49  thf(fact_147_numeral__Bit0,axiom,
% 0.30/0.49      ! [N: num] :
% 0.30/0.49        ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.30/0.49        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_Bit0
% 0.30/0.49  thf(fact_148_numeral__Bit0,axiom,
% 0.30/0.49      ! [N: num] :
% 0.30/0.49        ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.30/0.49        = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_Bit0
% 0.30/0.49  thf(fact_149_numeral__Bit0,axiom,
% 0.30/0.49      ! [N: num] :
% 0.30/0.49        ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.30/0.49        = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_Bit0
% 0.30/0.49  thf(fact_150_numeral__Bit0,axiom,
% 0.30/0.49      ! [N: num] :
% 0.30/0.49        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.30/0.49        = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_Bit0
% 0.30/0.49  thf(fact_151_numeral__Bit0,axiom,
% 0.30/0.49      ! [N: num] :
% 0.30/0.49        ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.30/0.49        = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_Bit0
% 0.30/0.49  thf(fact_152_power2__commute,axiom,
% 0.30/0.49      ! [X3: complex,Y3: complex] :
% 0.30/0.49        ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_commute
% 0.30/0.49  thf(fact_153_power2__commute,axiom,
% 0.30/0.49      ! [X3: real,Y3: real] :
% 0.30/0.49        ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( power_power_real @ ( minus_minus_real @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_commute
% 0.30/0.49  thf(fact_154_power2__commute,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat] :
% 0.30/0.49        ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_commute
% 0.30/0.49  thf(fact_155_power2__commute,axiom,
% 0.30/0.49      ! [X3: int,Y3: int] :
% 0.30/0.49        ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( power_power_int @ ( minus_minus_int @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_commute
% 0.30/0.49  thf(fact_156_diff__le__diff__pow,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.49       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_le_diff_pow
% 0.30/0.49  thf(fact_157_power__divide,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,N: nat] :
% 0.30/0.49        ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B2 ) @ N )
% 0.30/0.49        = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_divide
% 0.30/0.49  thf(fact_158_power__divide,axiom,
% 0.30/0.49      ! [A: real,B2: real,N: nat] :
% 0.30/0.49        ( ( power_power_real @ ( divide_divide_real @ A @ B2 ) @ N )
% 0.30/0.49        = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_divide
% 0.30/0.49  thf(fact_159_power__divide,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.49        ( ( power_power_rat @ ( divide_divide_rat @ A @ B2 ) @ N )
% 0.30/0.49        = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_divide
% 0.30/0.49  thf(fact_160_mint__corr,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49            = ( some_nat @ X3 ) )
% 0.30/0.49         => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mint_corr
% 0.30/0.49  thf(fact_161_mint__sound,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 0.30/0.49         => ( ( vEBT_vebt_mint @ T )
% 0.30/0.49            = ( some_nat @ X3 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mint_sound
% 0.30/0.49  thf(fact_162_add__shift,axiom,
% 0.30/0.49      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.49        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.30/0.49          = Z )
% 0.30/0.49        = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 0.30/0.49          = ( some_nat @ Z ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_shift
% 0.30/0.49  thf(fact_163_Nat_Oadd__diff__assoc,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.49          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.add_diff_assoc
% 0.30/0.49  thf(fact_164_Nat_Oadd__diff__assoc2,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/0.49          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.add_diff_assoc2
% 0.30/0.49  thf(fact_165_Nat_Odiff__diff__right,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.49          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.diff_diff_right
% 0.30/0.49  thf(fact_166_le__add__diff__inverse,axiom,
% 0.30/0.49      ! [B2: real,A: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse
% 0.30/0.49  thf(fact_167_le__add__diff__inverse,axiom,
% 0.30/0.49      ! [B2: rat,A: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse
% 0.30/0.49  thf(fact_168_le__add__diff__inverse,axiom,
% 0.30/0.49      ! [B2: nat,A: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse
% 0.30/0.49  thf(fact_169_le__add__diff__inverse,axiom,
% 0.30/0.49      ! [B2: int,A: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse
% 0.30/0.49  thf(fact_170_le__add__diff__inverse2,axiom,
% 0.30/0.49      ! [B2: real,A: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse2
% 0.30/0.49  thf(fact_171_le__add__diff__inverse2,axiom,
% 0.30/0.49      ! [B2: rat,A: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse2
% 0.30/0.49  thf(fact_172_le__add__diff__inverse2,axiom,
% 0.30/0.49      ! [B2: nat,A: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse2
% 0.30/0.49  thf(fact_173_le__add__diff__inverse2,axiom,
% 0.30/0.49      ! [B2: int,A: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.49       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add_diff_inverse2
% 0.30/0.49  thf(fact_174_member__bound,axiom,
% 0.30/0.49      ! [Tree: vEBT_VEBT,X3: nat,N: nat] :
% 0.30/0.49        ( ( vEBT_vebt_member @ Tree @ X3 )
% 0.30/0.49       => ( ( vEBT_invar_vebt @ Tree @ N )
% 0.30/0.49         => ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % member_bound
% 0.30/0.49  thf(fact_175__C4_Ohyps_C_I11_J,axiom,
% 0.30/0.49      ( ( mi != ma )
% 0.30/0.49     => ! [I2: nat] :
% 0.30/0.49          ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.30/0.49         => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.30/0.49                = I2 )
% 0.30/0.49             => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.30/0.49            & ! [X6: nat] :
% 0.30/0.49                ( ( ( ( vEBT_VEBT_high @ X6 @ na )
% 0.30/0.49                    = I2 )
% 0.30/0.49                  & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ X6 @ na ) ) )
% 0.30/0.49               => ( ( ord_less_nat @ mi @ X6 )
% 0.30/0.49                  & ( ord_less_eq_nat @ X6 @ ma ) ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(11)
% 0.30/0.49  thf(fact_176_div__exp__eq,axiom,
% 0.30/0.49      ! [A: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49        = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % div_exp_eq
% 0.30/0.49  thf(fact_177_div__exp__eq,axiom,
% 0.30/0.49      ! [A: int,M: nat,N: nat] :
% 0.30/0.49        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49        = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % div_exp_eq
% 0.30/0.49  thf(fact_178_field__less__half__sum,axiom,
% 0.30/0.49      ! [X3: real,Y3: real] :
% 0.30/0.49        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.49       => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % field_less_half_sum
% 0.30/0.49  thf(fact_179_field__less__half__sum,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat] :
% 0.30/0.49        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.49       => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % field_less_half_sum
% 0.30/0.49  thf(fact_180_not__min__Null__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT] :
% 0.30/0.49        ( ~ ( vEBT_VEBT_minNull @ T )
% 0.30/0.49       => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % not_min_Null_member
% 0.30/0.49  thf(fact_181_min__Null__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,X3: nat] :
% 0.30/0.49        ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.49       => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % min_Null_member
% 0.30/0.49  thf(fact_182_both__member__options__equiv__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 0.30/0.49          = ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % both_member_options_equiv_member
% 0.30/0.49  thf(fact_183_valid__member__both__member__options,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 0.30/0.49         => ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % valid_member_both_member_options
% 0.30/0.49  thf(fact_184_add__def,axiom,
% 0.30/0.49      ( vEBT_VEBT_add
% 0.30/0.49      = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_def
% 0.30/0.49  thf(fact_185_mint__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49            = ( some_nat @ Maxi ) )
% 0.30/0.49         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mint_member
% 0.30/0.49  thf(fact_186_member__correct,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( vEBT_vebt_member @ T @ X3 )
% 0.30/0.49          = ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % member_correct
% 0.30/0.49  thf(fact_187_mint__corr__help,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,Mini: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49            = ( some_nat @ Mini ) )
% 0.30/0.49         => ( ( vEBT_vebt_member @ T @ X3 )
% 0.30/0.49           => ( ord_less_eq_nat @ Mini @ X3 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mint_corr_help
% 0.30/0.49  thf(fact_188_set__vebt__set__vebt_H__valid,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( vEBT_set_vebt @ T )
% 0.30/0.49          = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % set_vebt_set_vebt'_valid
% 0.30/0.49  thf(fact_189_nat__add__left__cancel__less,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.49        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_add_left_cancel_less
% 0.30/0.49  thf(fact_190_nat__add__left__cancel__le,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.49        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_add_left_cancel_le
% 0.30/0.49  thf(fact_191_diff__diff__cancel,axiom,
% 0.30/0.49      ! [I: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ N )
% 0.30/0.49       => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.30/0.49          = I ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_diff_cancel
% 0.30/0.49  thf(fact_192_diff__diff__left,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/0.49        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_diff_left
% 0.30/0.49  thf(fact_193_semiring__norm_I6_J,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.49        = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(6)
% 0.30/0.49  thf(fact_194_semiring__norm_I2_J,axiom,
% 0.30/0.49      ( ( plus_plus_num @ one @ one )
% 0.30/0.49      = ( bit0 @ one ) ) ).
% 0.30/0.49  
% 0.30/0.49  % semiring_norm(2)
% 0.30/0.49  thf(fact_195_pred__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,X3: nat,Y3: nat] :
% 0.30/0.49        ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Y3 )
% 0.30/0.49        = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.30/0.49          & ( ord_less_nat @ Y3 @ X3 )
% 0.30/0.49          & ! [Z2: nat] :
% 0.30/0.49              ( ( ( vEBT_vebt_member @ T @ Z2 )
% 0.30/0.49                & ( ord_less_nat @ Z2 @ X3 ) )
% 0.30/0.49             => ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % pred_member
% 0.30/0.49  thf(fact_196__C4_Ohyps_C_I7_J,axiom,
% 0.30/0.49      ! [I2: nat] :
% 0.30/0.49        ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.30/0.49       => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ X7 ) )
% 0.30/0.49          = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(7)
% 0.30/0.49  thf(fact_197_add__One__commute,axiom,
% 0.30/0.49      ! [N: num] :
% 0.30/0.49        ( ( plus_plus_num @ one @ N )
% 0.30/0.49        = ( plus_plus_num @ N @ one ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_One_commute
% 0.30/0.49  thf(fact_198_linorder__neqE__linordered__idom,axiom,
% 0.30/0.49      ! [X3: real,Y3: real] :
% 0.30/0.49        ( ( X3 != Y3 )
% 0.30/0.49       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.30/0.49         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linorder_neqE_linordered_idom
% 0.30/0.49  thf(fact_199_linorder__neqE__linordered__idom,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat] :
% 0.30/0.49        ( ( X3 != Y3 )
% 0.30/0.49       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.49         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linorder_neqE_linordered_idom
% 0.30/0.49  thf(fact_200_linorder__neqE__linordered__idom,axiom,
% 0.30/0.49      ! [X3: int,Y3: int] :
% 0.30/0.49        ( ( X3 != Y3 )
% 0.30/0.49       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.30/0.49         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linorder_neqE_linordered_idom
% 0.30/0.49  thf(fact_201_nat__neq__iff,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( M != N )
% 0.30/0.49        = ( ( ord_less_nat @ M @ N )
% 0.30/0.49          | ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_neq_iff
% 0.30/0.49  thf(fact_202_less__not__refl,axiom,
% 0.30/0.49      ! [N: nat] :
% 0.30/0.49        ~ ( ord_less_nat @ N @ N ) ).
% 0.30/0.49  
% 0.30/0.49  % less_not_refl
% 0.30/0.49  thf(fact_203_less__not__refl2,axiom,
% 0.30/0.49      ! [N: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_nat @ N @ M )
% 0.30/0.49       => ( M != N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_not_refl2
% 0.30/0.49  thf(fact_204_less__not__refl3,axiom,
% 0.30/0.49      ! [S: nat,T: nat] :
% 0.30/0.49        ( ( ord_less_nat @ S @ T )
% 0.30/0.49       => ( S != T ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_not_refl3
% 0.30/0.49  thf(fact_205_less__irrefl__nat,axiom,
% 0.30/0.49      ! [N: nat] :
% 0.30/0.49        ~ ( ord_less_nat @ N @ N ) ).
% 0.30/0.49  
% 0.30/0.49  % less_irrefl_nat
% 0.30/0.49  thf(fact_206_nat__less__induct,axiom,
% 0.30/0.49      ! [P: nat > $o,N: nat] :
% 0.30/0.49        ( ! [N3: nat] :
% 0.30/0.49            ( ! [M2: nat] :
% 0.30/0.49                ( ( ord_less_nat @ M2 @ N3 )
% 0.30/0.49               => ( P @ M2 ) )
% 0.30/0.49           => ( P @ N3 ) )
% 0.30/0.49       => ( P @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_less_induct
% 0.30/0.49  thf(fact_207_infinite__descent,axiom,
% 0.30/0.49      ! [P: nat > $o,N: nat] :
% 0.30/0.49        ( ! [N3: nat] :
% 0.30/0.49            ( ~ ( P @ N3 )
% 0.30/0.49           => ? [M2: nat] :
% 0.30/0.49                ( ( ord_less_nat @ M2 @ N3 )
% 0.30/0.49                & ~ ( P @ M2 ) ) )
% 0.30/0.49       => ( P @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % infinite_descent
% 0.30/0.49  thf(fact_208_linorder__neqE__nat,axiom,
% 0.30/0.49      ! [X3: nat,Y3: nat] :
% 0.30/0.49        ( ( X3 != Y3 )
% 0.30/0.49       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.49         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linorder_neqE_nat
% 0.30/0.49  thf(fact_209_le__refl,axiom,
% 0.30/0.49      ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.30/0.49  
% 0.30/0.49  % le_refl
% 0.30/0.49  thf(fact_210_le__trans,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ord_less_eq_nat @ J @ K )
% 0.30/0.49         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_trans
% 0.30/0.49  thf(fact_211_eq__imp__le,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( M = N )
% 0.30/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_imp_le
% 0.30/0.49  thf(fact_212_le__antisym,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.49         => ( M = N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_antisym
% 0.30/0.49  thf(fact_213_nat__le__linear,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49        | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_le_linear
% 0.30/0.49  thf(fact_214_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.49      ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.49        ( ( P @ K )
% 0.30/0.49       => ( ! [Y4: nat] :
% 0.30/0.49              ( ( P @ Y4 )
% 0.30/0.49             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.30/0.49         => ? [X5: nat] :
% 0.30/0.49              ( ( P @ X5 )
% 0.30/0.49              & ! [Y5: nat] :
% 0.30/0.49                  ( ( P @ Y5 )
% 0.30/0.49                 => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.ex_has_greatest_nat
% 0.30/0.49  thf(fact_215_size__neq__size__imp__neq,axiom,
% 0.30/0.49      ! [X3: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 0.30/0.49        ( ( ( size_s6755466524823107622T_VEBT @ X3 )
% 0.30/0.49         != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 0.30/0.49       => ( X3 != Y3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % size_neq_size_imp_neq
% 0.30/0.49  thf(fact_216_size__neq__size__imp__neq,axiom,
% 0.30/0.49      ! [X3: num,Y3: num] :
% 0.30/0.49        ( ( ( size_size_num @ X3 )
% 0.30/0.49         != ( size_size_num @ Y3 ) )
% 0.30/0.49       => ( X3 != Y3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % size_neq_size_imp_neq
% 0.30/0.49  thf(fact_217_size__neq__size__imp__neq,axiom,
% 0.30/0.49      ! [X3: list_o,Y3: list_o] :
% 0.30/0.49        ( ( ( size_size_list_o @ X3 )
% 0.30/0.49         != ( size_size_list_o @ Y3 ) )
% 0.30/0.49       => ( X3 != Y3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % size_neq_size_imp_neq
% 0.30/0.49  thf(fact_218_size__neq__size__imp__neq,axiom,
% 0.30/0.49      ! [X3: list_nat,Y3: list_nat] :
% 0.30/0.49        ( ( ( size_size_list_nat @ X3 )
% 0.30/0.49         != ( size_size_list_nat @ Y3 ) )
% 0.30/0.49       => ( X3 != Y3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % size_neq_size_imp_neq
% 0.30/0.49  thf(fact_219_size__neq__size__imp__neq,axiom,
% 0.30/0.49      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.30/0.49        ( ( ( size_size_VEBT_VEBT @ X3 )
% 0.30/0.49         != ( size_size_VEBT_VEBT @ Y3 ) )
% 0.30/0.49       => ( X3 != Y3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % size_neq_size_imp_neq
% 0.30/0.49  thf(fact_220_diff__commute,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/0.49        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_commute
% 0.30/0.49  thf(fact_221_add__diff__add,axiom,
% 0.30/0.49      ! [A: complex,C: complex,B2: complex,D: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B2 @ D ) )
% 0.30/0.49        = ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ ( minus_minus_complex @ C @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_add
% 0.30/0.49  thf(fact_222_add__diff__add,axiom,
% 0.30/0.49      ! [A: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
% 0.30/0.49        = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_add
% 0.30/0.49  thf(fact_223_add__diff__add,axiom,
% 0.30/0.49      ! [A: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) )
% 0.30/0.49        = ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_add
% 0.30/0.49  thf(fact_224_add__diff__add,axiom,
% 0.30/0.49      ! [A: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
% 0.30/0.49        = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_add
% 0.30/0.49  thf(fact_225_nat__less__le,axiom,
% 0.30/0.49      ( ord_less_nat
% 0.30/0.49      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.49            ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.49            & ( M3 != N2 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_less_le
% 0.30/0.49  thf(fact_226_less__imp__le__nat,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_nat @ M @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_imp_le_nat
% 0.30/0.49  thf(fact_227_le__eq__less__or__eq,axiom,
% 0.30/0.49      ( ord_less_eq_nat
% 0.30/0.49      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.49            ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.49            | ( M3 = N2 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_eq_less_or_eq
% 0.30/0.49  thf(fact_228_less__or__eq__imp__le,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ( ord_less_nat @ M @ N )
% 0.30/0.49          | ( M = N ) )
% 0.30/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_or_eq_imp_le
% 0.30/0.49  thf(fact_229_le__neq__implies__less,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49       => ( ( M != N )
% 0.30/0.49         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_neq_implies_less
% 0.30/0.49  thf(fact_230_less__mono__imp__le__mono,axiom,
% 0.30/0.49      ! [F: nat > nat,I: nat,J: nat] :
% 0.30/0.49        ( ! [I3: nat,J2: nat] :
% 0.30/0.49            ( ( ord_less_nat @ I3 @ J2 )
% 0.30/0.49           => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.30/0.49       => ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_mono_imp_le_mono
% 0.30/0.49  thf(fact_231_add__lessD1,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.30/0.49       => ( ord_less_nat @ I @ K ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_lessD1
% 0.30/0.49  thf(fact_232_add__less__mono,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_nat @ I @ J )
% 0.30/0.49       => ( ( ord_less_nat @ K @ L )
% 0.30/0.49         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_mono
% 0.30/0.49  thf(fact_233_not__add__less1,axiom,
% 0.30/0.49      ! [I: nat,J: nat] :
% 0.30/0.49        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.30/0.49  
% 0.30/0.49  % not_add_less1
% 0.30/0.49  thf(fact_234_not__add__less2,axiom,
% 0.30/0.49      ! [J: nat,I: nat] :
% 0.30/0.49        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.30/0.49  
% 0.30/0.49  % not_add_less2
% 0.30/0.49  thf(fact_235_add__less__mono1,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_nat @ I @ J )
% 0.30/0.49       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_mono1
% 0.30/0.49  thf(fact_236_trans__less__add1,axiom,
% 0.30/0.49      ! [I: nat,J: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_nat @ I @ J )
% 0.30/0.49       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % trans_less_add1
% 0.30/0.49  thf(fact_237_trans__less__add2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_nat @ I @ J )
% 0.30/0.49       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % trans_less_add2
% 0.30/0.49  thf(fact_238_less__add__eq__less,axiom,
% 0.30/0.49      ! [K: nat,L: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_nat @ K @ L )
% 0.30/0.49       => ( ( ( plus_plus_nat @ M @ L )
% 0.30/0.49            = ( plus_plus_nat @ K @ N ) )
% 0.30/0.49         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_eq_less
% 0.30/0.49  thf(fact_239_add__leE,axiom,
% 0.30/0.49      ! [M: nat,K: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/0.49       => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49           => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_leE
% 0.30/0.49  thf(fact_240_le__add1,axiom,
% 0.30/0.49      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add1
% 0.30/0.49  thf(fact_241_le__add2,axiom,
% 0.30/0.49      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_add2
% 0.30/0.49  thf(fact_242_add__leD1,axiom,
% 0.30/0.49      ! [M: nat,K: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_leD1
% 0.30/0.49  thf(fact_243_add__leD2,axiom,
% 0.30/0.49      ! [M: nat,K: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_leD2
% 0.30/0.49  thf(fact_244_le__Suc__ex,axiom,
% 0.30/0.49      ! [K: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.49       => ? [N3: nat] :
% 0.30/0.49            ( L
% 0.30/0.49            = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_Suc_ex
% 0.30/0.49  thf(fact_245_add__le__mono,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_mono
% 0.30/0.49  thf(fact_246_add__le__mono1,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_mono1
% 0.30/0.49  thf(fact_247_trans__le__add1,axiom,
% 0.30/0.49      ! [I: nat,J: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % trans_le_add1
% 0.30/0.49  thf(fact_248_trans__le__add2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,M: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % trans_le_add2
% 0.30/0.49  thf(fact_249_nat__le__iff__add,axiom,
% 0.30/0.49      ( ord_less_eq_nat
% 0.30/0.49      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.49          ? [K2: nat] :
% 0.30/0.49            ( N2
% 0.30/0.49            = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_le_iff_add
% 0.30/0.49  thf(fact_250_diff__less__mono2,axiom,
% 0.30/0.49      ! [M: nat,N: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_nat @ M @ N )
% 0.30/0.49       => ( ( ord_less_nat @ M @ L )
% 0.30/0.49         => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_less_mono2
% 0.30/0.49  thf(fact_251_less__imp__diff__less,axiom,
% 0.30/0.49      ! [J: nat,K: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_nat @ J @ K )
% 0.30/0.49       => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_imp_diff_less
% 0.30/0.49  thf(fact_252_eq__diff__iff,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.49       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.49         => ( ( ( minus_minus_nat @ M @ K )
% 0.30/0.49              = ( minus_minus_nat @ N @ K ) )
% 0.30/0.49            = ( M = N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_diff_iff
% 0.30/0.49  thf(fact_253_le__diff__iff,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.49       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.49         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/0.49            = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_diff_iff
% 0.30/0.49  thf(fact_254_Nat_Odiff__diff__eq,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.49       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.49         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/0.49            = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.diff_diff_eq
% 0.30/0.49  thf(fact_255_diff__le__mono,axiom,
% 0.30/0.49      ! [M: nat,N: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_le_mono
% 0.30/0.49  thf(fact_256_diff__le__self,axiom,
% 0.30/0.49      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_le_self
% 0.30/0.49  thf(fact_257_le__diff__iff_H,axiom,
% 0.30/0.49      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ A @ C )
% 0.30/0.49       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.49         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
% 0.30/0.49            = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_diff_iff'
% 0.30/0.49  thf(fact_258_diff__le__mono2,axiom,
% 0.30/0.49      ! [M: nat,N: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_le_mono2
% 0.30/0.49  thf(fact_259_Nat_Odiff__cancel,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/0.49        = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.diff_cancel
% 0.30/0.49  thf(fact_260_diff__cancel2,axiom,
% 0.30/0.49      ! [M: nat,K: nat,N: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.49        = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_cancel2
% 0.30/0.49  thf(fact_261_diff__add__inverse,axiom,
% 0.30/0.49      ! [N: nat,M: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.30/0.49        = M ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_add_inverse
% 0.30/0.49  thf(fact_262_diff__add__inverse2,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.30/0.49        = M ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_add_inverse2
% 0.30/0.49  thf(fact_263_add__le__add__imp__diff__le,axiom,
% 0.30/0.49      ! [I: real,K: real,N: real,J: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.30/0.49       => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.30/0.49         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.30/0.49           => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.30/0.49             => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_add_imp_diff_le
% 0.30/0.49  thf(fact_264_add__le__add__imp__diff__le,axiom,
% 0.30/0.49      ! [I: rat,K: rat,N: rat,J: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.30/0.49       => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.49         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.30/0.49           => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.49             => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_add_imp_diff_le
% 0.30/0.49  thf(fact_265_add__le__add__imp__diff__le,axiom,
% 0.30/0.49      ! [I: nat,K: nat,N: nat,J: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.30/0.49       => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.49         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.30/0.49           => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.49             => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_add_imp_diff_le
% 0.30/0.49  thf(fact_266_add__le__add__imp__diff__le,axiom,
% 0.30/0.49      ! [I: int,K: int,N: int,J: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.30/0.49       => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.30/0.49         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.30/0.49           => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.30/0.49             => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_add_imp_diff_le
% 0.30/0.49  thf(fact_267_add__le__imp__le__diff,axiom,
% 0.30/0.49      ! [I: real,K: real,N: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.30/0.49       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_imp_le_diff
% 0.30/0.49  thf(fact_268_add__le__imp__le__diff,axiom,
% 0.30/0.49      ! [I: rat,K: rat,N: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.30/0.49       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_imp_le_diff
% 0.30/0.49  thf(fact_269_add__le__imp__le__diff,axiom,
% 0.30/0.49      ! [I: nat,K: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.30/0.49       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_imp_le_diff
% 0.30/0.49  thf(fact_270_add__le__imp__le__diff,axiom,
% 0.30/0.49      ! [I: int,K: int,N: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.30/0.49       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_imp_le_diff
% 0.30/0.49  thf(fact_271_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.49      ! [A: real,B2: real] :
% 0.30/0.49        ( ~ ( ord_less_real @ A @ B2 )
% 0.30/0.49       => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linordered_semidom_class.add_diff_inverse
% 0.30/0.49  thf(fact_272_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.49      ! [A: rat,B2: rat] :
% 0.30/0.49        ( ~ ( ord_less_rat @ A @ B2 )
% 0.30/0.49       => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linordered_semidom_class.add_diff_inverse
% 0.30/0.49  thf(fact_273_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.49      ! [A: nat,B2: nat] :
% 0.30/0.49        ( ~ ( ord_less_nat @ A @ B2 )
% 0.30/0.49       => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linordered_semidom_class.add_diff_inverse
% 0.30/0.49  thf(fact_274_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.49      ! [A: int,B2: int] :
% 0.30/0.49        ( ~ ( ord_less_int @ A @ B2 )
% 0.30/0.49       => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.49          = A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % linordered_semidom_class.add_diff_inverse
% 0.30/0.49  thf(fact_275_mono__nat__linear__lb,axiom,
% 0.30/0.49      ! [F: nat > nat,M: nat,K: nat] :
% 0.30/0.49        ( ! [M4: nat,N3: nat] :
% 0.30/0.49            ( ( ord_less_nat @ M4 @ N3 )
% 0.30/0.49           => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.30/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mono_nat_linear_lb
% 0.30/0.49  thf(fact_276_less__diff__iff,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.49       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.49         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/0.49            = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_diff_iff
% 0.30/0.49  thf(fact_277_diff__less__mono,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.49       => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.49         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_less_mono
% 0.30/0.49  thf(fact_278_add__diff__inverse__nat,axiom,
% 0.30/0.49      ! [M: nat,N: nat] :
% 0.30/0.49        ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.49       => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.49          = M ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_inverse_nat
% 0.30/0.49  thf(fact_279_less__diff__conv,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.49        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_diff_conv
% 0.30/0.49  thf(fact_280_Nat_Ole__imp__diff__is__add,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ( minus_minus_nat @ J @ I )
% 0.30/0.49            = K )
% 0.30/0.49          = ( J
% 0.30/0.49            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.le_imp_diff_is_add
% 0.30/0.49  thf(fact_281_Nat_Odiff__add__assoc2,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.30/0.49          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.diff_add_assoc2
% 0.30/0.49  thf(fact_282_Nat_Odiff__add__assoc,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.30/0.49          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.diff_add_assoc
% 0.30/0.49  thf(fact_283_Nat_Ole__diff__conv2,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.49          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % Nat.le_diff_conv2
% 0.30/0.49  thf(fact_284_le__diff__conv,axiom,
% 0.30/0.49      ! [J: nat,K: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/0.49        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_diff_conv
% 0.30/0.49  thf(fact_285_less__diff__conv2,axiom,
% 0.30/0.49      ! [K: nat,J: nat,I: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.49       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/0.49          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_diff_conv2
% 0.30/0.49  thf(fact_286_field__sum__of__halves,axiom,
% 0.30/0.49      ! [X3: real] :
% 0.30/0.49        ( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = X3 ) ).
% 0.30/0.49  
% 0.30/0.49  % field_sum_of_halves
% 0.30/0.49  thf(fact_287_field__sum__of__halves,axiom,
% 0.30/0.49      ! [X3: rat] :
% 0.30/0.49        ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = X3 ) ).
% 0.30/0.49  
% 0.30/0.49  % field_sum_of_halves
% 0.30/0.49  thf(fact_288_in__children__def,axiom,
% 0.30/0.49      ( vEBT_V5917875025757280293ildren
% 0.30/0.49      = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % in_children_def
% 0.30/0.49  thf(fact_289_maxt__sound,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 0.30/0.49         => ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49            = ( some_nat @ X3 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % maxt_sound
% 0.30/0.49  thf(fact_290_maxt__corr,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49            = ( some_nat @ X3 ) )
% 0.30/0.49         => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % maxt_corr
% 0.30/0.49  thf(fact_291_post__member__pre__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49         => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49           => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X3 ) @ Y3 )
% 0.30/0.49             => ( ( vEBT_vebt_member @ T @ Y3 )
% 0.30/0.49                | ( X3 = Y3 ) ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % post_member_pre_member
% 0.30/0.49  thf(fact_292_valid__insert__both__member__options__pres,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49         => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49           => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 0.30/0.49             => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X3 ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % valid_insert_both_member_options_pres
% 0.30/0.49  thf(fact_293_valid__insert__both__member__options__add,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49         => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X3 ) @ X3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % valid_insert_both_member_options_add
% 0.30/0.49  thf(fact_294_maxt__corr__help,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,Maxi: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49            = ( some_nat @ Maxi ) )
% 0.30/0.49         => ( ( vEBT_vebt_member @ T @ X3 )
% 0.30/0.49           => ( ord_less_eq_nat @ X3 @ Maxi ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % maxt_corr_help
% 0.30/0.49  thf(fact_295_maxt__member,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.49       => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49            = ( some_nat @ Maxi ) )
% 0.30/0.49         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % maxt_member
% 0.30/0.49  thf(fact_296_both__member__options__ding,axiom,
% 0.30/0.49      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.30/0.49       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.49         => ( ( 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 ) ) ) ) )
% 0.30/0.49           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % both_member_options_ding
% 0.30/0.49  thf(fact_297_maxbmo,axiom,
% 0.30/0.49      ! [T: vEBT_VEBT,X3: nat] :
% 0.30/0.49        ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49          = ( some_nat @ X3 ) )
% 0.30/0.49       => ( vEBT_V8194947554948674370ptions @ T @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % maxbmo
% 0.30/0.49  thf(fact_298_bit__concat__def,axiom,
% 0.30/0.49      ( vEBT_VEBT_bit_concat
% 0.30/0.49      = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % bit_concat_def
% 0.30/0.49  thf(fact_299_low__inv,axiom,
% 0.30/0.49      ! [X3: nat,N: nat,Y3: nat] :
% 0.30/0.49        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
% 0.30/0.49          = X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % low_inv
% 0.30/0.49  thf(fact_300_deg__deg__n,axiom,
% 0.30/0.49      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.30/0.49       => ( Deg = N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % deg_deg_n
% 0.30/0.49  thf(fact_301_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.30/0.49        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_left_semiring_numeral
% 0.30/0.49  thf(fact_302_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: real] :
% 0.30/0.49        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.30/0.49        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_left_semiring_numeral
% 0.30/0.49  thf(fact_303_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.30/0.49        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_left_semiring_numeral
% 0.30/0.49  thf(fact_304_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.30/0.49        = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_left_semiring_numeral
% 0.30/0.49  thf(fact_305_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.49      ! [V: num,W: num,Z: int] :
% 0.30/0.49        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.30/0.49        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_left_semiring_numeral
% 0.30/0.49  thf(fact_306_numeral__times__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.49        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_times_numeral
% 0.30/0.49  thf(fact_307_numeral__times__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.49        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_times_numeral
% 0.30/0.49  thf(fact_308_numeral__times__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.49        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_times_numeral
% 0.30/0.49  thf(fact_309_numeral__times__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.49        = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_times_numeral
% 0.30/0.49  thf(fact_310_numeral__times__numeral,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.49        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % numeral_times_numeral
% 0.30/0.49  thf(fact_311_high__inv,axiom,
% 0.30/0.49      ! [X3: nat,N: nat,Y3: nat] :
% 0.30/0.49        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
% 0.30/0.49          = Y3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % high_inv
% 0.30/0.49  thf(fact_312_distrib__right__numeral,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,V: num] :
% 0.30/0.49        ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.49        = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right_numeral
% 0.30/0.49  thf(fact_313_distrib__right__numeral,axiom,
% 0.30/0.49      ! [A: real,B2: real,V: num] :
% 0.30/0.49        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right_numeral
% 0.30/0.49  thf(fact_314_distrib__right__numeral,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,V: num] :
% 0.30/0.49        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right_numeral
% 0.30/0.49  thf(fact_315_distrib__right__numeral,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,V: num] :
% 0.30/0.49        ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right_numeral
% 0.30/0.49  thf(fact_316_distrib__right__numeral,axiom,
% 0.30/0.49      ! [A: int,B2: int,V: num] :
% 0.30/0.49        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right_numeral
% 0.30/0.49  thf(fact_317_distrib__left__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: complex,C: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left_numeral
% 0.30/0.49  thf(fact_318_distrib__left__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left_numeral
% 0.30/0.49  thf(fact_319_distrib__left__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left_numeral
% 0.30/0.49  thf(fact_320_distrib__left__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: nat,C: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left_numeral
% 0.30/0.49  thf(fact_321_distrib__left__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left_numeral
% 0.30/0.49  thf(fact_322_right__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: complex,C: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib_numeral
% 0.30/0.49  thf(fact_323_right__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib_numeral
% 0.30/0.49  thf(fact_324_right__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib_numeral
% 0.30/0.49  thf(fact_325_right__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [V: num,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib_numeral
% 0.30/0.49  thf(fact_326_left__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,V: num] :
% 0.30/0.49        ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.49        = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib_numeral
% 0.30/0.49  thf(fact_327_left__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [A: real,B2: real,V: num] :
% 0.30/0.49        ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.49        = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib_numeral
% 0.30/0.49  thf(fact_328_left__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,V: num] :
% 0.30/0.49        ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.49        = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib_numeral
% 0.30/0.49  thf(fact_329_left__diff__distrib__numeral,axiom,
% 0.30/0.49      ! [A: int,B2: int,V: num] :
% 0.30/0.49        ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.49        = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib_numeral
% 0.30/0.49  thf(fact_330_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [B2: real,W: num,A: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 0.30/0.49        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % divide_le_eq_numeral1(1)
% 0.30/0.49  thf(fact_331_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [B2: rat,W: num,A: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.30/0.49        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % divide_le_eq_numeral1(1)
% 0.30/0.49  thf(fact_332_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [A: real,B2: real,W: num] :
% 0.30/0.49        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.49        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_divide_eq_numeral1(1)
% 0.30/0.49  thf(fact_333_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,W: num] :
% 0.30/0.49        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.49        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_divide_eq_numeral1(1)
% 0.30/0.49  thf(fact_334_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [B2: real,W: num,A: real] :
% 0.30/0.49        ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 0.30/0.49        = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % divide_less_eq_numeral1(1)
% 0.30/0.49  thf(fact_335_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [B2: rat,W: num,A: rat] :
% 0.30/0.49        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.30/0.49        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % divide_less_eq_numeral1(1)
% 0.30/0.49  thf(fact_336_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [A: real,B2: real,W: num] :
% 0.30/0.49        ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.49        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_divide_eq_numeral1(1)
% 0.30/0.49  thf(fact_337_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,W: num] :
% 0.30/0.49        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.49        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_divide_eq_numeral1(1)
% 0.30/0.49  thf(fact_338_power__add__numeral2,axiom,
% 0.30/0.49      ! [A: complex,M: num,N: num,B2: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.30/0.49        = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral2
% 0.30/0.49  thf(fact_339_power__add__numeral2,axiom,
% 0.30/0.49      ! [A: real,M: num,N: num,B2: real] :
% 0.30/0.49        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.30/0.49        = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral2
% 0.30/0.49  thf(fact_340_power__add__numeral2,axiom,
% 0.30/0.49      ! [A: rat,M: num,N: num,B2: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.30/0.49        = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral2
% 0.30/0.49  thf(fact_341_power__add__numeral2,axiom,
% 0.30/0.49      ! [A: nat,M: num,N: num,B2: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.30/0.49        = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral2
% 0.30/0.49  thf(fact_342_power__add__numeral2,axiom,
% 0.30/0.49      ! [A: int,M: num,N: num,B2: int] :
% 0.30/0.49        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 0.30/0.49        = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral2
% 0.30/0.49  thf(fact_343_power__add__numeral,axiom,
% 0.30/0.49      ! [A: complex,M: num,N: num] :
% 0.30/0.49        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.49        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral
% 0.30/0.49  thf(fact_344_power__add__numeral,axiom,
% 0.30/0.49      ! [A: real,M: num,N: num] :
% 0.30/0.49        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.49        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral
% 0.30/0.49  thf(fact_345_power__add__numeral,axiom,
% 0.30/0.49      ! [A: rat,M: num,N: num] :
% 0.30/0.49        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.49        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral
% 0.30/0.49  thf(fact_346_power__add__numeral,axiom,
% 0.30/0.49      ! [A: nat,M: num,N: num] :
% 0.30/0.49        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.49        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral
% 0.30/0.49  thf(fact_347_power__add__numeral,axiom,
% 0.30/0.49      ! [A: int,M: num,N: num] :
% 0.30/0.49        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.30/0.49        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add_numeral
% 0.30/0.49  thf(fact_348__C4_Ohyps_C_I8_J,axiom,
% 0.30/0.49      ( ( mi = ma )
% 0.30/0.49     => ! [X6: vEBT_VEBT] :
% 0.30/0.49          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.30/0.49         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(8)
% 0.30/0.49  thf(fact_349_combine__common__factor,axiom,
% 0.30/0.49      ! [A: real,E: real,B2: real,C: real] :
% 0.30/0.49        ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % combine_common_factor
% 0.30/0.49  thf(fact_350_combine__common__factor,axiom,
% 0.30/0.49      ! [A: rat,E: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ E ) @ C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % combine_common_factor
% 0.30/0.49  thf(fact_351_combine__common__factor,axiom,
% 0.30/0.49      ! [A: nat,E: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % combine_common_factor
% 0.30/0.49  thf(fact_352_combine__common__factor,axiom,
% 0.30/0.49      ! [A: int,E: int,B2: int,C: int] :
% 0.30/0.49        ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % combine_common_factor
% 0.30/0.49  thf(fact_353_distrib__right,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right
% 0.30/0.49  thf(fact_354_distrib__right,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right
% 0.30/0.49  thf(fact_355_distrib__right,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right
% 0.30/0.49  thf(fact_356_distrib__right,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_right
% 0.30/0.49  thf(fact_357_distrib__left,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left
% 0.30/0.49  thf(fact_358_distrib__left,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left
% 0.30/0.49  thf(fact_359_distrib__left,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left
% 0.30/0.49  thf(fact_360_distrib__left,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % distrib_left
% 0.30/0.49  thf(fact_361_comm__semiring__class_Odistrib,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % comm_semiring_class.distrib
% 0.30/0.49  thf(fact_362_comm__semiring__class_Odistrib,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % comm_semiring_class.distrib
% 0.30/0.49  thf(fact_363_comm__semiring__class_Odistrib,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % comm_semiring_class.distrib
% 0.30/0.49  thf(fact_364_comm__semiring__class_Odistrib,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % comm_semiring_class.distrib
% 0.30/0.49  thf(fact_365_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ring_class.ring_distribs(1)
% 0.30/0.49  thf(fact_366_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ring_class.ring_distribs(1)
% 0.30/0.49  thf(fact_367_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ring_class.ring_distribs(1)
% 0.30/0.49  thf(fact_368_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ring_class.ring_distribs(2)
% 0.30/0.49  thf(fact_369_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ring_class.ring_distribs(2)
% 0.30/0.49  thf(fact_370_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ring_class.ring_distribs(2)
% 0.30/0.49  thf(fact_371_right__diff__distrib_H,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.49        ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib'
% 0.30/0.49  thf(fact_372_right__diff__distrib_H,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib'
% 0.30/0.49  thf(fact_373_right__diff__distrib_H,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib'
% 0.30/0.49  thf(fact_374_right__diff__distrib_H,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib'
% 0.30/0.49  thf(fact_375_right__diff__distrib_H,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib'
% 0.30/0.49  thf(fact_376_left__diff__distrib_H,axiom,
% 0.30/0.49      ! [B2: complex,C: complex,A: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( minus_minus_complex @ B2 @ C ) @ A )
% 0.30/0.49        = ( minus_minus_complex @ ( times_times_complex @ B2 @ A ) @ ( times_times_complex @ C @ A ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib'
% 0.30/0.49  thf(fact_377_left__diff__distrib_H,axiom,
% 0.30/0.49      ! [B2: real,C: real,A: real] :
% 0.30/0.49        ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
% 0.30/0.49        = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib'
% 0.30/0.49  thf(fact_378_left__diff__distrib_H,axiom,
% 0.30/0.49      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A )
% 0.30/0.49        = ( minus_minus_rat @ ( times_times_rat @ B2 @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib'
% 0.30/0.49  thf(fact_379_left__diff__distrib_H,axiom,
% 0.30/0.49      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
% 0.30/0.49        = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib'
% 0.30/0.49  thf(fact_380_left__diff__distrib_H,axiom,
% 0.30/0.49      ! [B2: int,C: int,A: int] :
% 0.30/0.49        ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
% 0.30/0.49        = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib'
% 0.30/0.49  thf(fact_381_right__diff__distrib,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.49        ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib
% 0.30/0.49  thf(fact_382_right__diff__distrib,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib
% 0.30/0.49  thf(fact_383_right__diff__distrib,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib
% 0.30/0.49  thf(fact_384_right__diff__distrib,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.30/0.49        = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % right_diff_distrib
% 0.30/0.49  thf(fact_385_left__diff__distrib,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 0.30/0.49        = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib
% 0.30/0.49  thf(fact_386_left__diff__distrib,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.30/0.49        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib
% 0.30/0.49  thf(fact_387_left__diff__distrib,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.30/0.49        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib
% 0.30/0.49  thf(fact_388_left__diff__distrib,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.49        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_diff_distrib
% 0.30/0.49  thf(fact_389_power__commuting__commutes,axiom,
% 0.30/0.49      ! [X3: complex,Y3: complex,N: nat] :
% 0.30/0.49        ( ( ( times_times_complex @ X3 @ Y3 )
% 0.30/0.49          = ( times_times_complex @ Y3 @ X3 ) )
% 0.30/0.49       => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ Y3 )
% 0.30/0.49          = ( times_times_complex @ Y3 @ ( power_power_complex @ X3 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commuting_commutes
% 0.30/0.49  thf(fact_390_power__commuting__commutes,axiom,
% 0.30/0.49      ! [X3: real,Y3: real,N: nat] :
% 0.30/0.49        ( ( ( times_times_real @ X3 @ Y3 )
% 0.30/0.49          = ( times_times_real @ Y3 @ X3 ) )
% 0.30/0.49       => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ Y3 )
% 0.30/0.49          = ( times_times_real @ Y3 @ ( power_power_real @ X3 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commuting_commutes
% 0.30/0.49  thf(fact_391_power__commuting__commutes,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat,N: nat] :
% 0.30/0.49        ( ( ( times_times_rat @ X3 @ Y3 )
% 0.30/0.49          = ( times_times_rat @ Y3 @ X3 ) )
% 0.30/0.49       => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ Y3 )
% 0.30/0.49          = ( times_times_rat @ Y3 @ ( power_power_rat @ X3 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commuting_commutes
% 0.30/0.49  thf(fact_392_power__commuting__commutes,axiom,
% 0.30/0.49      ! [X3: nat,Y3: nat,N: nat] :
% 0.30/0.49        ( ( ( times_times_nat @ X3 @ Y3 )
% 0.30/0.49          = ( times_times_nat @ Y3 @ X3 ) )
% 0.30/0.49       => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ Y3 )
% 0.30/0.49          = ( times_times_nat @ Y3 @ ( power_power_nat @ X3 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commuting_commutes
% 0.30/0.49  thf(fact_393_power__commuting__commutes,axiom,
% 0.30/0.49      ! [X3: int,Y3: int,N: nat] :
% 0.30/0.49        ( ( ( times_times_int @ X3 @ Y3 )
% 0.30/0.49          = ( times_times_int @ Y3 @ X3 ) )
% 0.30/0.49       => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ Y3 )
% 0.30/0.49          = ( times_times_int @ Y3 @ ( power_power_int @ X3 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commuting_commutes
% 0.30/0.49  thf(fact_394_power__mult__distrib,axiom,
% 0.30/0.49      ! [A: complex,B2: complex,N: nat] :
% 0.30/0.49        ( ( power_power_complex @ ( times_times_complex @ A @ B2 ) @ N )
% 0.30/0.49        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult_distrib
% 0.30/0.49  thf(fact_395_power__mult__distrib,axiom,
% 0.30/0.49      ! [A: real,B2: real,N: nat] :
% 0.30/0.49        ( ( power_power_real @ ( times_times_real @ A @ B2 ) @ N )
% 0.30/0.49        = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult_distrib
% 0.30/0.49  thf(fact_396_power__mult__distrib,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.49        ( ( power_power_rat @ ( times_times_rat @ A @ B2 ) @ N )
% 0.30/0.49        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult_distrib
% 0.30/0.49  thf(fact_397_power__mult__distrib,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.49        ( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
% 0.30/0.49        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult_distrib
% 0.30/0.49  thf(fact_398_power__mult__distrib,axiom,
% 0.30/0.49      ! [A: int,B2: int,N: nat] :
% 0.30/0.49        ( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
% 0.30/0.49        = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult_distrib
% 0.30/0.49  thf(fact_399_power__commutes,axiom,
% 0.30/0.49      ! [A: complex,N: nat] :
% 0.30/0.49        ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.30/0.49        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commutes
% 0.30/0.49  thf(fact_400_power__commutes,axiom,
% 0.30/0.49      ! [A: real,N: nat] :
% 0.30/0.49        ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.30/0.49        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commutes
% 0.30/0.49  thf(fact_401_power__commutes,axiom,
% 0.30/0.49      ! [A: rat,N: nat] :
% 0.30/0.49        ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.30/0.49        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commutes
% 0.30/0.49  thf(fact_402_power__commutes,axiom,
% 0.30/0.49      ! [A: nat,N: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.30/0.49        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commutes
% 0.30/0.49  thf(fact_403_power__commutes,axiom,
% 0.30/0.49      ! [A: int,N: nat] :
% 0.30/0.49        ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.30/0.49        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_commutes
% 0.30/0.49  thf(fact_404_power__mult,axiom,
% 0.30/0.49      ! [A: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.49        = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult
% 0.30/0.49  thf(fact_405_power__mult,axiom,
% 0.30/0.49      ! [A: real,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.49        = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult
% 0.30/0.49  thf(fact_406_power__mult,axiom,
% 0.30/0.49      ! [A: int,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.49        = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult
% 0.30/0.49  thf(fact_407_power__mult,axiom,
% 0.30/0.49      ! [A: complex,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.30/0.49        = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_mult
% 0.30/0.49  thf(fact_408_mult__le__mono2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_le_mono2
% 0.30/0.49  thf(fact_409_mult__le__mono1,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_le_mono1
% 0.30/0.49  thf(fact_410_mult__le__mono,axiom,
% 0.30/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ord_less_eq_nat @ K @ L )
% 0.30/0.49         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_le_mono
% 0.30/0.49  thf(fact_411_le__square,axiom,
% 0.30/0.49      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_square
% 0.30/0.49  thf(fact_412_le__cube,axiom,
% 0.30/0.49      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % le_cube
% 0.30/0.49  thf(fact_413_left__add__mult__distrib,axiom,
% 0.30/0.49      ! [I: nat,U: nat,J: nat,K: nat] :
% 0.30/0.49        ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_add_mult_distrib
% 0.30/0.49  thf(fact_414_add__mult__distrib2,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_mult_distrib2
% 0.30/0.49  thf(fact_415_add__mult__distrib,axiom,
% 0.30/0.49      ! [M: nat,N: nat,K: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_mult_distrib
% 0.30/0.49  thf(fact_416_diff__mult__distrib2,axiom,
% 0.30/0.49      ! [K: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.49        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_mult_distrib2
% 0.30/0.49  thf(fact_417_diff__mult__distrib,axiom,
% 0.30/0.49      ! [M: nat,N: nat,K: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.30/0.49        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_mult_distrib
% 0.30/0.49  thf(fact_418_div__mult2__eq,axiom,
% 0.30/0.49      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.49        ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.30/0.49        = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % div_mult2_eq
% 0.30/0.49  thf(fact_419_mult__numeral__1,axiom,
% 0.30/0.49      ! [A: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1
% 0.30/0.49  thf(fact_420_mult__numeral__1,axiom,
% 0.30/0.49      ! [A: real] :
% 0.30/0.49        ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1
% 0.30/0.49  thf(fact_421_mult__numeral__1,axiom,
% 0.30/0.49      ! [A: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1
% 0.30/0.49  thf(fact_422_mult__numeral__1,axiom,
% 0.30/0.49      ! [A: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1
% 0.30/0.49  thf(fact_423_mult__numeral__1,axiom,
% 0.30/0.49      ! [A: int] :
% 0.30/0.49        ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1
% 0.30/0.49  thf(fact_424_mult__numeral__1__right,axiom,
% 0.30/0.49      ! [A: complex] :
% 0.30/0.49        ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1_right
% 0.30/0.49  thf(fact_425_mult__numeral__1__right,axiom,
% 0.30/0.49      ! [A: real] :
% 0.30/0.49        ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1_right
% 0.30/0.49  thf(fact_426_mult__numeral__1__right,axiom,
% 0.30/0.49      ! [A: rat] :
% 0.30/0.49        ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1_right
% 0.30/0.49  thf(fact_427_mult__numeral__1__right,axiom,
% 0.30/0.49      ! [A: nat] :
% 0.30/0.49        ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1_right
% 0.30/0.49  thf(fact_428_mult__numeral__1__right,axiom,
% 0.30/0.49      ! [A: int] :
% 0.30/0.49        ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_numeral_1_right
% 0.30/0.49  thf(fact_429_mult__diff__mult,axiom,
% 0.30/0.49      ! [X3: complex,Y3: complex,A: complex,B2: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ Y3 ) @ ( times_times_complex @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_complex @ ( times_times_complex @ X3 @ ( minus_minus_complex @ Y3 @ B2 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X3 @ A ) @ B2 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_diff_mult
% 0.30/0.49  thf(fact_430_mult__diff__mult,axiom,
% 0.30/0.49      ! [X3: real,Y3: real,A: real,B2: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( times_times_real @ X3 @ Y3 ) @ ( times_times_real @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y3 @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B2 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_diff_mult
% 0.30/0.49  thf(fact_431_mult__diff__mult,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat,A: rat,B2: rat] :
% 0.30/0.49        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y3 ) @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y3 @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B2 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_diff_mult
% 0.30/0.49  thf(fact_432_mult__diff__mult,axiom,
% 0.30/0.49      ! [X3: int,Y3: int,A: int,B2: int] :
% 0.30/0.49        ( ( minus_minus_int @ ( times_times_int @ X3 @ Y3 ) @ ( times_times_int @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y3 @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B2 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_diff_mult
% 0.30/0.49  thf(fact_433_square__diff__square__factored,axiom,
% 0.30/0.49      ! [X3: complex,Y3: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ ( times_times_complex @ Y3 @ Y3 ) )
% 0.30/0.49        = ( times_times_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % square_diff_square_factored
% 0.30/0.49  thf(fact_434_square__diff__square__factored,axiom,
% 0.30/0.49      ! [X3: real,Y3: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.30/0.49        = ( times_times_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % square_diff_square_factored
% 0.30/0.49  thf(fact_435_square__diff__square__factored,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat] :
% 0.30/0.49        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.30/0.49        = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % square_diff_square_factored
% 0.30/0.49  thf(fact_436_square__diff__square__factored,axiom,
% 0.30/0.49      ! [X3: int,Y3: int] :
% 0.30/0.49        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.30/0.49        = ( times_times_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( minus_minus_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % square_diff_square_factored
% 0.30/0.49  thf(fact_437_eq__add__iff2,axiom,
% 0.30/0.49      ! [A: complex,E: complex,C: complex,B2: complex,D: complex] :
% 0.30/0.49        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( C
% 0.30/0.49          = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff2
% 0.30/0.49  thf(fact_438_eq__add__iff2,axiom,
% 0.30/0.49      ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( C
% 0.30/0.49          = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff2
% 0.30/0.49  thf(fact_439_eq__add__iff2,axiom,
% 0.30/0.49      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( C
% 0.30/0.49          = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff2
% 0.30/0.49  thf(fact_440_eq__add__iff2,axiom,
% 0.30/0.49      ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( C
% 0.30/0.49          = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff2
% 0.30/0.49  thf(fact_441_eq__add__iff1,axiom,
% 0.30/0.49      ! [A: complex,E: complex,C: complex,B2: complex,D: complex] :
% 0.30/0.49        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ E ) @ C )
% 0.30/0.49          = D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff1
% 0.30/0.49  thf(fact_442_eq__add__iff1,axiom,
% 0.30/0.49      ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
% 0.30/0.49          = D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff1
% 0.30/0.49  thf(fact_443_eq__add__iff1,axiom,
% 0.30/0.49      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C )
% 0.30/0.49          = D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff1
% 0.30/0.49  thf(fact_444_eq__add__iff1,axiom,
% 0.30/0.49      ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.30/0.49          = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
% 0.30/0.49          = D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % eq_add_iff1
% 0.30/0.49  thf(fact_445_power__add,axiom,
% 0.30/0.49      ! [A: complex,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.49        = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add
% 0.30/0.49  thf(fact_446_power__add,axiom,
% 0.30/0.49      ! [A: real,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.49        = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add
% 0.30/0.49  thf(fact_447_power__add,axiom,
% 0.30/0.49      ! [A: rat,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.49        = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add
% 0.30/0.49  thf(fact_448_power__add,axiom,
% 0.30/0.49      ! [A: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.49        = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add
% 0.30/0.49  thf(fact_449_power__add,axiom,
% 0.30/0.49      ! [A: int,M: nat,N: nat] :
% 0.30/0.49        ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.49        = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_add
% 0.30/0.49  thf(fact_450_less__mult__imp__div__less,axiom,
% 0.30/0.49      ! [M: nat,I: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.30/0.49       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_mult_imp_div_less
% 0.30/0.49  thf(fact_451_times__div__less__eq__dividend,axiom,
% 0.30/0.49      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.30/0.49  
% 0.30/0.49  % times_div_less_eq_dividend
% 0.30/0.49  thf(fact_452_div__times__less__eq__dividend,axiom,
% 0.30/0.49      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.30/0.49  
% 0.30/0.49  % div_times_less_eq_dividend
% 0.30/0.49  thf(fact_453_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.49      ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ordered_ring_class.le_add_iff1
% 0.30/0.49  thf(fact_454_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.49      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ordered_ring_class.le_add_iff1
% 0.30/0.49  thf(fact_455_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.49      ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ordered_ring_class.le_add_iff1
% 0.30/0.49  thf(fact_456_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.49      ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ordered_ring_class.le_add_iff2
% 0.30/0.49  thf(fact_457_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.49      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ordered_ring_class.le_add_iff2
% 0.30/0.49  thf(fact_458_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.49      ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % ordered_ring_class.le_add_iff2
% 0.30/0.49  thf(fact_459_less__add__iff1,axiom,
% 0.30/0.49      ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_iff1
% 0.30/0.49  thf(fact_460_less__add__iff1,axiom,
% 0.30/0.49      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_iff1
% 0.30/0.49  thf(fact_461_less__add__iff1,axiom,
% 0.30/0.49      ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_iff1
% 0.30/0.49  thf(fact_462_less__add__iff2,axiom,
% 0.30/0.49      ! [A: real,E: real,C: real,B2: real,D: real] :
% 0.30/0.49        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_iff2
% 0.30/0.49  thf(fact_463_less__add__iff2,axiom,
% 0.30/0.49      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_iff2
% 0.30/0.49  thf(fact_464_less__add__iff2,axiom,
% 0.30/0.49      ! [A: int,E: int,C: int,B2: int,D: int] :
% 0.30/0.49        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 0.30/0.49        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % less_add_iff2
% 0.30/0.49  thf(fact_465_nat__eq__add__iff1,axiom,
% 0.30/0.49      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.49       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.30/0.49            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.30/0.49            = N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_eq_add_iff1
% 0.30/0.49  thf(fact_466_nat__eq__add__iff2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.30/0.49            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( M
% 0.30/0.49            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_eq_add_iff2
% 0.30/0.49  thf(fact_467_nat__le__add__iff1,axiom,
% 0.30/0.49      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.49       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_le_add_iff1
% 0.30/0.49  thf(fact_468_nat__le__add__iff2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_le_add_iff2
% 0.30/0.49  thf(fact_469_nat__diff__add__eq1,axiom,
% 0.30/0.49      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.49       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_diff_add_eq1
% 0.30/0.49  thf(fact_470_nat__diff__add__eq2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_diff_add_eq2
% 0.30/0.49  thf(fact_471_mult__2,axiom,
% 0.30/0.49      ! [Z: complex] :
% 0.30/0.49        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.30/0.49        = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2
% 0.30/0.49  thf(fact_472_mult__2,axiom,
% 0.30/0.49      ! [Z: real] :
% 0.30/0.49        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.30/0.49        = ( plus_plus_real @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2
% 0.30/0.49  thf(fact_473_mult__2,axiom,
% 0.30/0.49      ! [Z: rat] :
% 0.30/0.49        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.30/0.49        = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2
% 0.30/0.49  thf(fact_474_mult__2,axiom,
% 0.30/0.49      ! [Z: nat] :
% 0.30/0.49        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.30/0.49        = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2
% 0.30/0.49  thf(fact_475_mult__2,axiom,
% 0.30/0.49      ! [Z: int] :
% 0.30/0.49        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.30/0.49        = ( plus_plus_int @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2
% 0.30/0.49  thf(fact_476_mult__2__right,axiom,
% 0.30/0.49      ! [Z: complex] :
% 0.30/0.49        ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2_right
% 0.30/0.49  thf(fact_477_mult__2__right,axiom,
% 0.30/0.49      ! [Z: real] :
% 0.30/0.49        ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( plus_plus_real @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2_right
% 0.30/0.49  thf(fact_478_mult__2__right,axiom,
% 0.30/0.49      ! [Z: rat] :
% 0.30/0.49        ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2_right
% 0.30/0.49  thf(fact_479_mult__2__right,axiom,
% 0.30/0.49      ! [Z: nat] :
% 0.30/0.49        ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2_right
% 0.30/0.49  thf(fact_480_mult__2__right,axiom,
% 0.30/0.49      ! [Z: int] :
% 0.30/0.49        ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( plus_plus_int @ Z @ Z ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mult_2_right
% 0.30/0.49  thf(fact_481_left__add__twice,axiom,
% 0.30/0.49      ! [A: complex,B2: complex] :
% 0.30/0.49        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_add_twice
% 0.30/0.49  thf(fact_482_left__add__twice,axiom,
% 0.30/0.49      ! [A: real,B2: real] :
% 0.30/0.49        ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_add_twice
% 0.30/0.49  thf(fact_483_left__add__twice,axiom,
% 0.30/0.49      ! [A: rat,B2: rat] :
% 0.30/0.49        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_add_twice
% 0.30/0.49  thf(fact_484_left__add__twice,axiom,
% 0.30/0.49      ! [A: nat,B2: nat] :
% 0.30/0.49        ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_add_twice
% 0.30/0.49  thf(fact_485_left__add__twice,axiom,
% 0.30/0.49      ! [A: int,B2: int] :
% 0.30/0.49        ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.49        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % left_add_twice
% 0.30/0.49  thf(fact_486_power4__eq__xxxx,axiom,
% 0.30/0.49      ! [X3: complex] :
% 0.30/0.49        ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power4_eq_xxxx
% 0.30/0.49  thf(fact_487_power4__eq__xxxx,axiom,
% 0.30/0.49      ! [X3: real] :
% 0.30/0.49        ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = ( times_times_real @ ( times_times_real @ ( times_times_real @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power4_eq_xxxx
% 0.30/0.49  thf(fact_488_power4__eq__xxxx,axiom,
% 0.30/0.49      ! [X3: rat] :
% 0.30/0.49        ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power4_eq_xxxx
% 0.30/0.49  thf(fact_489_power4__eq__xxxx,axiom,
% 0.30/0.49      ! [X3: nat] :
% 0.30/0.49        ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power4_eq_xxxx
% 0.30/0.49  thf(fact_490_power4__eq__xxxx,axiom,
% 0.30/0.49      ! [X3: int] :
% 0.30/0.49        ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.49        = ( times_times_int @ ( times_times_int @ ( times_times_int @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power4_eq_xxxx
% 0.30/0.49  thf(fact_491_power2__eq__square,axiom,
% 0.30/0.49      ! [A: complex] :
% 0.30/0.49        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( times_times_complex @ A @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_eq_square
% 0.30/0.49  thf(fact_492_power2__eq__square,axiom,
% 0.30/0.49      ! [A: real] :
% 0.30/0.49        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( times_times_real @ A @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_eq_square
% 0.30/0.49  thf(fact_493_power2__eq__square,axiom,
% 0.30/0.49      ! [A: rat] :
% 0.30/0.49        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( times_times_rat @ A @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_eq_square
% 0.30/0.49  thf(fact_494_power2__eq__square,axiom,
% 0.30/0.49      ! [A: nat] :
% 0.30/0.49        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( times_times_nat @ A @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_eq_square
% 0.30/0.49  thf(fact_495_power2__eq__square,axiom,
% 0.30/0.49      ! [A: int] :
% 0.30/0.49        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( times_times_int @ A @ A ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_eq_square
% 0.30/0.49  thf(fact_496_power__even__eq,axiom,
% 0.30/0.49      ! [A: nat,N: nat] :
% 0.30/0.49        ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49        = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_even_eq
% 0.30/0.49  thf(fact_497_power__even__eq,axiom,
% 0.30/0.49      ! [A: real,N: nat] :
% 0.30/0.49        ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49        = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_even_eq
% 0.30/0.49  thf(fact_498_power__even__eq,axiom,
% 0.30/0.49      ! [A: int,N: nat] :
% 0.30/0.49        ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49        = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_even_eq
% 0.30/0.49  thf(fact_499_power__even__eq,axiom,
% 0.30/0.49      ! [A: complex,N: nat] :
% 0.30/0.49        ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.49        = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power_even_eq
% 0.30/0.49  thf(fact_500_nat__less__add__iff2,axiom,
% 0.30/0.49      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.49       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_less_add_iff2
% 0.30/0.49  thf(fact_501_nat__less__add__iff1,axiom,
% 0.30/0.49      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.49       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.30/0.49          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % nat_less_add_iff1
% 0.30/0.49  thf(fact_502_is__pred__in__set__def,axiom,
% 0.30/0.49      ( vEBT_is_pred_in_set
% 0.30/0.49      = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 0.30/0.49            ( ( member_nat @ Y @ Xs )
% 0.30/0.49            & ( ord_less_nat @ Y @ X )
% 0.30/0.49            & ! [Z2: nat] :
% 0.30/0.49                ( ( member_nat @ Z2 @ Xs )
% 0.30/0.49               => ( ( ord_less_nat @ Z2 @ X )
% 0.30/0.49                 => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % is_pred_in_set_def
% 0.30/0.49  thf(fact_503_power2__sum,axiom,
% 0.30/0.49      ! [X3: complex,Y3: complex] :
% 0.30/0.49        ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_sum
% 0.30/0.49  thf(fact_504_power2__sum,axiom,
% 0.30/0.49      ! [X3: real,Y3: real] :
% 0.30/0.49        ( ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_sum
% 0.30/0.49  thf(fact_505_power2__sum,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat] :
% 0.30/0.49        ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_sum
% 0.30/0.49  thf(fact_506_power2__sum,axiom,
% 0.30/0.49      ! [X3: nat,Y3: nat] :
% 0.30/0.49        ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_sum
% 0.30/0.49  thf(fact_507_power2__sum,axiom,
% 0.30/0.49      ! [X3: int,Y3: int] :
% 0.30/0.49        ( ( power_power_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_sum
% 0.30/0.49  thf(fact_508_power2__diff,axiom,
% 0.30/0.49      ! [X3: complex,Y3: complex] :
% 0.30/0.49        ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_diff
% 0.30/0.49  thf(fact_509_power2__diff,axiom,
% 0.30/0.49      ! [X3: real,Y3: real] :
% 0.30/0.49        ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_diff
% 0.30/0.49  thf(fact_510_power2__diff,axiom,
% 0.30/0.49      ! [X3: rat,Y3: rat] :
% 0.30/0.49        ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_diff
% 0.30/0.49  thf(fact_511_power2__diff,axiom,
% 0.30/0.49      ! [X3: int,Y3: int] :
% 0.30/0.49        ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49        = ( 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 ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % power2_diff
% 0.30/0.49  thf(fact_512_mul__def,axiom,
% 0.30/0.49      ( vEBT_VEBT_mul
% 0.30/0.49      = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mul_def
% 0.30/0.49  thf(fact_513_mul__shift,axiom,
% 0.30/0.49      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.49        ( ( ( times_times_nat @ X3 @ Y3 )
% 0.30/0.49          = Z )
% 0.30/0.49        = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 0.30/0.49          = ( some_nat @ Z ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mul_shift
% 0.30/0.49  thf(fact_514_member__inv,axiom,
% 0.30/0.49      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.30/0.49        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.49       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49          & ( ( X3 = Mi )
% 0.30/0.49            | ( X3 = Ma )
% 0.30/0.49            | ( ( ord_less_nat @ X3 @ Ma )
% 0.30/0.49              & ( ord_less_nat @ Mi @ X3 )
% 0.30/0.49              & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.49              & ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % member_inv
% 0.30/0.49  thf(fact_515_thisvalid,axiom,
% 0.30/0.49      vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 0.30/0.49  
% 0.30/0.49  % thisvalid
% 0.30/0.49  thf(fact_516_mintlistlength,axiom,
% 0.30/0.49      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.30/0.49       => ( ( Mi != Ma )
% 0.30/0.49         => ( ( ord_less_nat @ Mi @ Ma )
% 0.30/0.49            & ? [M4: nat] :
% 0.30/0.49                ( ( ( some_nat @ M4 )
% 0.30/0.49                  = ( vEBT_vebt_mint @ Summary ) )
% 0.30/0.49                & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mintlistlength
% 0.30/0.49  thf(fact_517_sum__squares__bound,axiom,
% 0.30/0.49      ! [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 ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % sum_squares_bound
% 0.30/0.49  thf(fact_518_sum__squares__bound,axiom,
% 0.30/0.49      ! [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 ) ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % sum_squares_bound
% 0.30/0.49  thf(fact_519__C4_Ohyps_C_I3_J,axiom,
% 0.30/0.49      ! [X3: nat,Px: nat] :
% 0.30/0.49        ( ( ( vEBT_vebt_pred @ summary @ X3 )
% 0.30/0.49          = ( some_nat @ Px ) )
% 0.30/0.49        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X3 @ Px ) ) ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(3)
% 0.30/0.49  thf(fact_520_mi__ma__2__deg,axiom,
% 0.30/0.49      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.30/0.49       => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.30/0.49          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mi_ma_2_deg
% 0.30/0.49  thf(fact_521_enat__ord__number_I1_J,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.30/0.49        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % enat_ord_number(1)
% 0.30/0.49  thf(fact_522_enat__ord__number_I2_J,axiom,
% 0.30/0.49      ! [M: num,N: num] :
% 0.30/0.49        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.30/0.49        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % enat_ord_number(2)
% 0.30/0.49  thf(fact_523_add__diff__cancel__right_H,axiom,
% 0.30/0.49      ! [A: complex,B2: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_right'
% 0.30/0.49  thf(fact_524_add__diff__cancel__right_H,axiom,
% 0.30/0.49      ! [A: real,B2: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_right'
% 0.30/0.49  thf(fact_525_add__diff__cancel__right_H,axiom,
% 0.30/0.49      ! [A: rat,B2: rat] :
% 0.30/0.49        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_right'
% 0.30/0.49  thf(fact_526_add__diff__cancel__right_H,axiom,
% 0.30/0.49      ! [A: nat,B2: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_right'
% 0.30/0.49  thf(fact_527_add__diff__cancel__right_H,axiom,
% 0.30/0.49      ! [A: int,B2: int] :
% 0.30/0.49        ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_right'
% 0.30/0.49  thf(fact_528_inthall,axiom,
% 0.30/0.49      ! [Xs2: list_real,P: real > $o,N: nat] :
% 0.30/0.49        ( ! [X5: real] :
% 0.30/0.49            ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
% 0.30/0.49           => ( P @ X5 ) )
% 0.30/0.49       => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.49         => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % inthall
% 0.30/0.49  thf(fact_529_inthall,axiom,
% 0.30/0.49      ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 0.30/0.49        ( ! [X5: complex] :
% 0.30/0.49            ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.49           => ( P @ X5 ) )
% 0.30/0.49       => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.49         => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % inthall
% 0.30/0.49  thf(fact_530_inthall,axiom,
% 0.30/0.49      ! [Xs2: list_int,P: int > $o,N: nat] :
% 0.30/0.49        ( ! [X5: int] :
% 0.30/0.49            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 0.30/0.49           => ( P @ X5 ) )
% 0.30/0.49       => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.49         => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % inthall
% 0.30/0.49  thf(fact_531_inthall,axiom,
% 0.30/0.49      ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.30/0.49        ( ! [X5: vEBT_VEBT] :
% 0.30/0.49            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.49           => ( P @ X5 ) )
% 0.30/0.49       => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.49         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % inthall
% 0.30/0.49  thf(fact_532_inthall,axiom,
% 0.30/0.49      ! [Xs2: list_o,P: $o > $o,N: nat] :
% 0.30/0.49        ( ! [X5: $o] :
% 0.30/0.49            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 0.30/0.49           => ( P @ X5 ) )
% 0.30/0.49       => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.49         => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % inthall
% 0.30/0.49  thf(fact_533_inthall,axiom,
% 0.30/0.49      ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 0.30/0.49        ( ! [X5: nat] :
% 0.30/0.49            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.49           => ( P @ X5 ) )
% 0.30/0.49       => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.49         => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % inthall
% 0.30/0.49  thf(fact_534_add__left__cancel,axiom,
% 0.30/0.49      ! [A: real,B2: real,C: real] :
% 0.30/0.49        ( ( ( plus_plus_real @ A @ B2 )
% 0.30/0.49          = ( plus_plus_real @ A @ C ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_left_cancel
% 0.30/0.49  thf(fact_535_add__left__cancel,axiom,
% 0.30/0.49      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.49        ( ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.49          = ( plus_plus_rat @ A @ C ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_left_cancel
% 0.30/0.49  thf(fact_536_add__left__cancel,axiom,
% 0.30/0.49      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.49        ( ( ( plus_plus_nat @ A @ B2 )
% 0.30/0.49          = ( plus_plus_nat @ A @ C ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_left_cancel
% 0.30/0.49  thf(fact_537_add__left__cancel,axiom,
% 0.30/0.49      ! [A: int,B2: int,C: int] :
% 0.30/0.49        ( ( ( plus_plus_int @ A @ B2 )
% 0.30/0.49          = ( plus_plus_int @ A @ C ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_left_cancel
% 0.30/0.49  thf(fact_538_add__right__cancel,axiom,
% 0.30/0.49      ! [B2: real,A: real,C: real] :
% 0.30/0.49        ( ( ( plus_plus_real @ B2 @ A )
% 0.30/0.49          = ( plus_plus_real @ C @ A ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_right_cancel
% 0.30/0.49  thf(fact_539_add__right__cancel,axiom,
% 0.30/0.49      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.49        ( ( ( plus_plus_rat @ B2 @ A )
% 0.30/0.49          = ( plus_plus_rat @ C @ A ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_right_cancel
% 0.30/0.49  thf(fact_540_add__right__cancel,axiom,
% 0.30/0.49      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.49        ( ( ( plus_plus_nat @ B2 @ A )
% 0.30/0.49          = ( plus_plus_nat @ C @ A ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_right_cancel
% 0.30/0.49  thf(fact_541_add__right__cancel,axiom,
% 0.30/0.49      ! [B2: int,A: int,C: int] :
% 0.30/0.49        ( ( ( plus_plus_int @ B2 @ A )
% 0.30/0.49          = ( plus_plus_int @ C @ A ) )
% 0.30/0.49        = ( B2 = C ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_right_cancel
% 0.30/0.49  thf(fact_542__C4_Ohyps_C_I1_J,axiom,
% 0.30/0.49      ! [X6: vEBT_VEBT] :
% 0.30/0.49        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.30/0.49       => ( ( vEBT_invar_vebt @ X6 @ na )
% 0.30/0.49          & ! [Xa: nat,Xb: nat] :
% 0.30/0.49              ( ( ( vEBT_vebt_pred @ X6 @ Xa )
% 0.30/0.49                = ( some_nat @ Xb ) )
% 0.30/0.49              = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ X6 ) @ Xa @ Xb ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % "4.hyps"(1)
% 0.30/0.49  thf(fact_543_mi__eq__ma__no__ch,axiom,
% 0.30/0.49      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.30/0.49       => ( ( Mi = Ma )
% 0.30/0.49         => ( ! [X6: vEBT_VEBT] :
% 0.30/0.49                ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49               => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) )
% 0.30/0.49            & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.30/0.49  
% 0.30/0.49  % mi_eq_ma_no_ch
% 0.30/0.49  thf(fact_544_add__le__cancel__left,axiom,
% 0.30/0.49      ! [C: real,A: real,B2: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_left
% 0.30/0.49  thf(fact_545_add__le__cancel__left,axiom,
% 0.30/0.49      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_left
% 0.30/0.49  thf(fact_546_add__le__cancel__left,axiom,
% 0.30/0.49      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_left
% 0.30/0.49  thf(fact_547_add__le__cancel__left,axiom,
% 0.30/0.49      ! [C: int,A: int,B2: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_left
% 0.30/0.49  thf(fact_548_add__le__cancel__right,axiom,
% 0.30/0.49      ! [A: real,C: real,B2: real] :
% 0.30/0.49        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_right
% 0.30/0.49  thf(fact_549_add__le__cancel__right,axiom,
% 0.30/0.49      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.49        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_right
% 0.30/0.49  thf(fact_550_add__le__cancel__right,axiom,
% 0.30/0.49      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_right
% 0.30/0.49  thf(fact_551_add__le__cancel__right,axiom,
% 0.30/0.49      ! [A: int,C: int,B2: int] :
% 0.30/0.49        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_le_cancel_right
% 0.30/0.49  thf(fact_552_add__less__cancel__left,axiom,
% 0.30/0.49      ! [C: real,A: real,B2: real] :
% 0.30/0.49        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_left
% 0.30/0.49  thf(fact_553_add__less__cancel__left,axiom,
% 0.30/0.49      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_left
% 0.30/0.49  thf(fact_554_add__less__cancel__left,axiom,
% 0.30/0.49      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_nat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_left
% 0.30/0.49  thf(fact_555_add__less__cancel__left,axiom,
% 0.30/0.49      ! [C: int,A: int,B2: int] :
% 0.30/0.49        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.30/0.49        = ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_left
% 0.30/0.49  thf(fact_556_add__less__cancel__right,axiom,
% 0.30/0.49      ! [A: real,C: real,B2: real] :
% 0.30/0.49        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_right
% 0.30/0.49  thf(fact_557_add__less__cancel__right,axiom,
% 0.30/0.49      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_right
% 0.30/0.49  thf(fact_558_add__less__cancel__right,axiom,
% 0.30/0.49      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_nat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_right
% 0.30/0.49  thf(fact_559_add__less__cancel__right,axiom,
% 0.30/0.49      ! [A: int,C: int,B2: int] :
% 0.30/0.49        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.49        = ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_less_cancel_right
% 0.30/0.49  thf(fact_560_add__diff__cancel,axiom,
% 0.30/0.49      ! [A: complex,B2: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel
% 0.30/0.49  thf(fact_561_add__diff__cancel,axiom,
% 0.30/0.49      ! [A: real,B2: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel
% 0.30/0.49  thf(fact_562_add__diff__cancel,axiom,
% 0.30/0.49      ! [A: rat,B2: rat] :
% 0.30/0.49        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel
% 0.30/0.49  thf(fact_563_add__diff__cancel,axiom,
% 0.30/0.49      ! [A: int,B2: int] :
% 0.30/0.49        ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel
% 0.30/0.49  thf(fact_564_diff__add__cancel,axiom,
% 0.30/0.49      ! [A: complex,B2: complex] :
% 0.30/0.49        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_add_cancel
% 0.30/0.49  thf(fact_565_diff__add__cancel,axiom,
% 0.30/0.49      ! [A: real,B2: real] :
% 0.30/0.49        ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_add_cancel
% 0.30/0.49  thf(fact_566_diff__add__cancel,axiom,
% 0.30/0.49      ! [A: rat,B2: rat] :
% 0.30/0.49        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_add_cancel
% 0.30/0.49  thf(fact_567_diff__add__cancel,axiom,
% 0.30/0.49      ! [A: int,B2: int] :
% 0.30/0.49        ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 0.30/0.49        = A ) ).
% 0.30/0.49  
% 0.30/0.49  % diff_add_cancel
% 0.30/0.49  thf(fact_568_add__diff__cancel__left,axiom,
% 0.30/0.49      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B2 ) )
% 0.30/0.49        = ( minus_minus_complex @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left
% 0.30/0.49  thf(fact_569_add__diff__cancel__left,axiom,
% 0.30/0.49      ! [C: real,A: real,B2: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.30/0.49        = ( minus_minus_real @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left
% 0.30/0.49  thf(fact_570_add__diff__cancel__left,axiom,
% 0.30/0.49      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.49        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.30/0.49        = ( minus_minus_rat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left
% 0.30/0.49  thf(fact_571_add__diff__cancel__left,axiom,
% 0.30/0.49      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.49        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.30/0.49        = ( minus_minus_nat @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left
% 0.30/0.49  thf(fact_572_add__diff__cancel__left,axiom,
% 0.30/0.49      ! [C: int,A: int,B2: int] :
% 0.30/0.49        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.30/0.49        = ( minus_minus_int @ A @ B2 ) ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left
% 0.30/0.49  thf(fact_573_add__diff__cancel__left_H,axiom,
% 0.30/0.49      ! [A: complex,B2: complex] :
% 0.30/0.49        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ A )
% 0.30/0.49        = B2 ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left'
% 0.30/0.49  thf(fact_574_add__diff__cancel__left_H,axiom,
% 0.30/0.49      ! [A: real,B2: real] :
% 0.30/0.49        ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
% 0.30/0.49        = B2 ) ).
% 0.30/0.49  
% 0.30/0.49  % add_diff_cancel_left'
% 0.30/0.49  thf(fact_575_add__diff__cancel__left_H,axiom,
% 0.30/0.49      ! [A: rat,B2: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ A )
% 0.30/0.50        = B2 ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_left'
% 0.30/0.50  thf(fact_576_add__diff__cancel__left_H,axiom,
% 0.30/0.50      ! [A: nat,B2: nat] :
% 0.30/0.50        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
% 0.30/0.50        = B2 ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_left'
% 0.30/0.50  thf(fact_577_add__diff__cancel__left_H,axiom,
% 0.30/0.50      ! [A: int,B2: int] :
% 0.30/0.50        ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
% 0.30/0.50        = B2 ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_left'
% 0.30/0.50  thf(fact_578_add__diff__cancel__right,axiom,
% 0.30/0.50      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.50        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_complex @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_right
% 0.30/0.50  thf(fact_579_add__diff__cancel__right,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_real @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_right
% 0.30/0.50  thf(fact_580_add__diff__cancel__right,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_rat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_right
% 0.30/0.50  thf(fact_581_add__diff__cancel__right,axiom,
% 0.30/0.50      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.50        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_nat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_right
% 0.30/0.50  thf(fact_582_add__diff__cancel__right,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_int @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_cancel_right
% 0.30/0.50  thf(fact_583_insert__simp__mima,axiom,
% 0.30/0.50      ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50        ( ( ( X3 = Mi )
% 0.30/0.50          | ( X3 = Ma ) )
% 0.30/0.50       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50         => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.50            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % insert_simp_mima
% 0.30/0.50  thf(fact_584_semiring__norm_I13_J,axiom,
% 0.30/0.50      ! [M: num,N: num] :
% 0.30/0.50        ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.50        = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % semiring_norm(13)
% 0.30/0.50  thf(fact_585_semiring__norm_I11_J,axiom,
% 0.30/0.50      ! [M: num] :
% 0.30/0.50        ( ( times_times_num @ M @ one )
% 0.30/0.50        = M ) ).
% 0.30/0.50  
% 0.30/0.50  % semiring_norm(11)
% 0.30/0.50  thf(fact_586_semiring__norm_I12_J,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( times_times_num @ one @ N )
% 0.30/0.50        = N ) ).
% 0.30/0.50  
% 0.30/0.50  % semiring_norm(12)
% 0.30/0.50  thf(fact_587_pred__max,axiom,
% 0.30/0.50      ! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50       => ( ( ord_less_nat @ Ma @ X3 )
% 0.30/0.50         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.50            = ( some_nat @ Ma ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % pred_max
% 0.30/0.50  thf(fact_588_num__double,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.30/0.50        = ( bit0 @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % num_double
% 0.30/0.50  thf(fact_589_power__mult__numeral,axiom,
% 0.30/0.50      ! [A: nat,M: num,N: num] :
% 0.30/0.50        ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_mult_numeral
% 0.30/0.50  thf(fact_590_power__mult__numeral,axiom,
% 0.30/0.50      ! [A: real,M: num,N: num] :
% 0.30/0.50        ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_mult_numeral
% 0.30/0.50  thf(fact_591_power__mult__numeral,axiom,
% 0.30/0.50      ! [A: int,M: num,N: num] :
% 0.30/0.50        ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_mult_numeral
% 0.30/0.50  thf(fact_592_power__mult__numeral,axiom,
% 0.30/0.50      ! [A: complex,M: num,N: num] :
% 0.30/0.50        ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_mult_numeral
% 0.30/0.50  thf(fact_593_pred__list__to__short,axiom,
% 0.30/0.50      ! [Deg: nat,X3: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.30/0.50        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50       => ( ( ord_less_eq_nat @ X3 @ Ma )
% 0.30/0.50         => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.50              = none_nat ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % pred_list_to_short
% 0.30/0.50  thf(fact_594_enat__less__induct,axiom,
% 0.30/0.50      ! [P: extended_enat > $o,N: extended_enat] :
% 0.30/0.50        ( ! [N3: extended_enat] :
% 0.30/0.50            ( ! [M2: extended_enat] :
% 0.30/0.50                ( ( ord_le72135733267957522d_enat @ M2 @ N3 )
% 0.30/0.50               => ( P @ M2 ) )
% 0.30/0.50           => ( P @ N3 ) )
% 0.30/0.50       => ( P @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % enat_less_induct
% 0.30/0.50  thf(fact_595_add__diff__assoc__enat,axiom,
% 0.30/0.50      ! [Z: extended_enat,Y3: extended_enat,X3: extended_enat] :
% 0.30/0.50        ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 0.30/0.50       => ( ( plus_p3455044024723400733d_enat @ X3 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 0.30/0.50          = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_assoc_enat
% 0.30/0.50  thf(fact_596_L2__set__mult__ineq__lemma,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % L2_set_mult_ineq_lemma
% 0.30/0.50  thf(fact_597_four__x__squared,axiom,
% 0.30/0.50      ! [X3: real] :
% 0.30/0.50        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50        = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % four_x_squared
% 0.30/0.50  thf(fact_598_div__mult2__numeral__eq,axiom,
% 0.30/0.50      ! [A: nat,K: num,L: num] :
% 0.30/0.50        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 0.30/0.50        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % div_mult2_numeral_eq
% 0.30/0.50  thf(fact_599_div__mult2__numeral__eq,axiom,
% 0.30/0.50      ! [A: int,K: num,L: num] :
% 0.30/0.50        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 0.30/0.50        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % div_mult2_numeral_eq
% 0.30/0.50  thf(fact_600_vebt__pred_Osimps_I4_J,axiom,
% 0.30/0.50      ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 0.30/0.50        ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 0.30/0.50        = none_nat ) ).
% 0.30/0.50  
% 0.30/0.50  % vebt_pred.simps(4)
% 0.30/0.50  thf(fact_601_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50  thf(fact_602_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50  thf(fact_603_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50  thf(fact_604_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.50  thf(fact_605_mult_Oassoc,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.assoc
% 0.30/0.50  thf(fact_606_mult_Oassoc,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.assoc
% 0.30/0.50  thf(fact_607_mult_Oassoc,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.assoc
% 0.30/0.50  thf(fact_608_mult_Oassoc,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.assoc
% 0.30/0.50  thf(fact_609_mult_Ocommute,axiom,
% 0.30/0.50      ( times_times_real
% 0.30/0.50      = ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.commute
% 0.30/0.50  thf(fact_610_mult_Ocommute,axiom,
% 0.30/0.50      ( times_times_rat
% 0.30/0.50      = ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.commute
% 0.30/0.50  thf(fact_611_mult_Ocommute,axiom,
% 0.30/0.50      ( times_times_nat
% 0.30/0.50      = ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.commute
% 0.30/0.50  thf(fact_612_mult_Ocommute,axiom,
% 0.30/0.50      ( times_times_int
% 0.30/0.50      = ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.commute
% 0.30/0.50  thf(fact_613_mult_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: real,A: real,C: real] :
% 0.30/0.50        ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
% 0.30/0.50        = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.left_commute
% 0.30/0.50  thf(fact_614_mult_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.50        ( ( times_times_rat @ B2 @ ( times_times_rat @ A @ C ) )
% 0.30/0.50        = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.left_commute
% 0.30/0.50  thf(fact_615_mult_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.50        ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
% 0.30/0.50        = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.left_commute
% 0.30/0.50  thf(fact_616_mult_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: int,A: int,C: int] :
% 0.30/0.50        ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
% 0.30/0.50        = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.left_commute
% 0.30/0.50  thf(fact_617_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_add_class.add_ac(1)
% 0.30/0.50  thf(fact_618_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_add_class.add_ac(1)
% 0.30/0.50  thf(fact_619_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_add_class.add_ac(1)
% 0.30/0.50  thf(fact_620_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ab_semigroup_add_class.add_ac(1)
% 0.30/0.50  thf(fact_621_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ( plus_plus_real @ I @ K )
% 0.30/0.50          = ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(4)
% 0.30/0.50  thf(fact_622_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ( plus_plus_rat @ I @ K )
% 0.30/0.50          = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(4)
% 0.30/0.50  thf(fact_623_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ( plus_plus_nat @ I @ K )
% 0.30/0.50          = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(4)
% 0.30/0.50  thf(fact_624_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ( plus_plus_int @ I @ K )
% 0.30/0.50          = ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(4)
% 0.30/0.50  thf(fact_625_group__cancel_Oadd1,axiom,
% 0.30/0.50      ! [A4: real,K: real,A: real,B2: real] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_real @ K @ A ) )
% 0.30/0.50       => ( ( plus_plus_real @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add1
% 0.30/0.50  thf(fact_626_group__cancel_Oadd1,axiom,
% 0.30/0.50      ! [A4: rat,K: rat,A: rat,B2: rat] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_rat @ K @ A ) )
% 0.30/0.50       => ( ( plus_plus_rat @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add1
% 0.30/0.50  thf(fact_627_group__cancel_Oadd1,axiom,
% 0.30/0.50      ! [A4: nat,K: nat,A: nat,B2: nat] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_nat @ K @ A ) )
% 0.30/0.50       => ( ( plus_plus_nat @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add1
% 0.30/0.50  thf(fact_628_group__cancel_Oadd1,axiom,
% 0.30/0.50      ! [A4: int,K: int,A: int,B2: int] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_int @ K @ A ) )
% 0.30/0.50       => ( ( plus_plus_int @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add1
% 0.30/0.50  thf(fact_629_group__cancel_Oadd2,axiom,
% 0.30/0.50      ! [B5: real,K: real,B2: real,A: real] :
% 0.30/0.50        ( ( B5
% 0.30/0.50          = ( plus_plus_real @ K @ B2 ) )
% 0.30/0.50       => ( ( plus_plus_real @ A @ B5 )
% 0.30/0.50          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add2
% 0.30/0.50  thf(fact_630_group__cancel_Oadd2,axiom,
% 0.30/0.50      ! [B5: rat,K: rat,B2: rat,A: rat] :
% 0.30/0.50        ( ( B5
% 0.30/0.50          = ( plus_plus_rat @ K @ B2 ) )
% 0.30/0.50       => ( ( plus_plus_rat @ A @ B5 )
% 0.30/0.50          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add2
% 0.30/0.50  thf(fact_631_group__cancel_Oadd2,axiom,
% 0.30/0.50      ! [B5: nat,K: nat,B2: nat,A: nat] :
% 0.30/0.50        ( ( B5
% 0.30/0.50          = ( plus_plus_nat @ K @ B2 ) )
% 0.30/0.50       => ( ( plus_plus_nat @ A @ B5 )
% 0.30/0.50          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add2
% 0.30/0.50  thf(fact_632_group__cancel_Oadd2,axiom,
% 0.30/0.50      ! [B5: int,K: int,B2: int,A: int] :
% 0.30/0.50        ( ( B5
% 0.30/0.50          = ( plus_plus_int @ K @ B2 ) )
% 0.30/0.50       => ( ( plus_plus_int @ A @ B5 )
% 0.30/0.50          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.add2
% 0.30/0.50  thf(fact_633_add_Oassoc,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.assoc
% 0.30/0.50  thf(fact_634_add_Oassoc,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.assoc
% 0.30/0.50  thf(fact_635_add_Oassoc,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.assoc
% 0.30/0.50  thf(fact_636_add_Oassoc,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.assoc
% 0.30/0.50  thf(fact_637_add_Oleft__cancel,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ( plus_plus_real @ A @ B2 )
% 0.30/0.50          = ( plus_plus_real @ A @ C ) )
% 0.30/0.50        = ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_cancel
% 0.30/0.50  thf(fact_638_add_Oleft__cancel,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.50          = ( plus_plus_rat @ A @ C ) )
% 0.30/0.50        = ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_cancel
% 0.30/0.50  thf(fact_639_add_Oleft__cancel,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ( plus_plus_int @ A @ B2 )
% 0.30/0.50          = ( plus_plus_int @ A @ C ) )
% 0.30/0.50        = ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_cancel
% 0.30/0.50  thf(fact_640_add_Oright__cancel,axiom,
% 0.30/0.50      ! [B2: real,A: real,C: real] :
% 0.30/0.50        ( ( ( plus_plus_real @ B2 @ A )
% 0.30/0.50          = ( plus_plus_real @ C @ A ) )
% 0.30/0.50        = ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.right_cancel
% 0.30/0.50  thf(fact_641_add_Oright__cancel,axiom,
% 0.30/0.50      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.50        ( ( ( plus_plus_rat @ B2 @ A )
% 0.30/0.50          = ( plus_plus_rat @ C @ A ) )
% 0.30/0.50        = ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.right_cancel
% 0.30/0.50  thf(fact_642_add_Oright__cancel,axiom,
% 0.30/0.50      ! [B2: int,A: int,C: int] :
% 0.30/0.50        ( ( ( plus_plus_int @ B2 @ A )
% 0.30/0.50          = ( plus_plus_int @ C @ A ) )
% 0.30/0.50        = ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.right_cancel
% 0.30/0.50  thf(fact_643_add_Ocommute,axiom,
% 0.30/0.50      ( plus_plus_real
% 0.30/0.50      = ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.commute
% 0.30/0.50  thf(fact_644_add_Ocommute,axiom,
% 0.30/0.50      ( plus_plus_rat
% 0.30/0.50      = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.commute
% 0.30/0.50  thf(fact_645_add_Ocommute,axiom,
% 0.30/0.50      ( plus_plus_nat
% 0.30/0.50      = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.commute
% 0.30/0.50  thf(fact_646_add_Ocommute,axiom,
% 0.30/0.50      ( plus_plus_int
% 0.30/0.50      = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.commute
% 0.30/0.50  thf(fact_647_add_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: real,A: real,C: real] :
% 0.30/0.50        ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
% 0.30/0.50        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_commute
% 0.30/0.50  thf(fact_648_add_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.50        ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A @ C ) )
% 0.30/0.50        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_commute
% 0.30/0.50  thf(fact_649_add_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.50        ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
% 0.30/0.50        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_commute
% 0.30/0.50  thf(fact_650_add_Oleft__commute,axiom,
% 0.30/0.50      ! [B2: int,A: int,C: int] :
% 0.30/0.50        ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
% 0.30/0.50        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add.left_commute
% 0.30/0.50  thf(fact_651_add__left__imp__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ( plus_plus_real @ A @ B2 )
% 0.30/0.50          = ( plus_plus_real @ A @ C ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_imp_eq
% 0.30/0.50  thf(fact_652_add__left__imp__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.50          = ( plus_plus_rat @ A @ C ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_imp_eq
% 0.30/0.50  thf(fact_653_add__left__imp__eq,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ( plus_plus_nat @ A @ B2 )
% 0.30/0.50          = ( plus_plus_nat @ A @ C ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_imp_eq
% 0.30/0.50  thf(fact_654_add__left__imp__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ( plus_plus_int @ A @ B2 )
% 0.30/0.50          = ( plus_plus_int @ A @ C ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_imp_eq
% 0.30/0.50  thf(fact_655_add__right__imp__eq,axiom,
% 0.30/0.50      ! [B2: real,A: real,C: real] :
% 0.30/0.50        ( ( ( plus_plus_real @ B2 @ A )
% 0.30/0.50          = ( plus_plus_real @ C @ A ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_imp_eq
% 0.30/0.50  thf(fact_656_add__right__imp__eq,axiom,
% 0.30/0.50      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.50        ( ( ( plus_plus_rat @ B2 @ A )
% 0.30/0.50          = ( plus_plus_rat @ C @ A ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_imp_eq
% 0.30/0.50  thf(fact_657_add__right__imp__eq,axiom,
% 0.30/0.50      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.50        ( ( ( plus_plus_nat @ B2 @ A )
% 0.30/0.50          = ( plus_plus_nat @ C @ A ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_imp_eq
% 0.30/0.50  thf(fact_658_add__right__imp__eq,axiom,
% 0.30/0.50      ! [B2: int,A: int,C: int] :
% 0.30/0.50        ( ( ( plus_plus_int @ B2 @ A )
% 0.30/0.50          = ( plus_plus_int @ C @ A ) )
% 0.30/0.50       => ( B2 = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_imp_eq
% 0.30/0.50  thf(fact_659_diff__eq__diff__eq,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex,D: complex] :
% 0.30/0.50        ( ( ( minus_minus_complex @ A @ B2 )
% 0.30/0.50          = ( minus_minus_complex @ C @ D ) )
% 0.30/0.50       => ( ( A = B2 )
% 0.30/0.50          = ( C = D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_eq
% 0.30/0.50  thf(fact_660_diff__eq__diff__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ( minus_minus_real @ A @ B2 )
% 0.30/0.50          = ( minus_minus_real @ C @ D ) )
% 0.30/0.50       => ( ( A = B2 )
% 0.30/0.50          = ( C = D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_eq
% 0.30/0.50  thf(fact_661_diff__eq__diff__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ( minus_minus_rat @ A @ B2 )
% 0.30/0.50          = ( minus_minus_rat @ C @ D ) )
% 0.30/0.50       => ( ( A = B2 )
% 0.30/0.50          = ( C = D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_eq
% 0.30/0.50  thf(fact_662_diff__eq__diff__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ( minus_minus_int @ A @ B2 )
% 0.30/0.50          = ( minus_minus_int @ C @ D ) )
% 0.30/0.50       => ( ( A = B2 )
% 0.30/0.50          = ( C = D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_eq
% 0.30/0.50  thf(fact_663_diff__right__commute,axiom,
% 0.30/0.50      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.50        ( ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B2 )
% 0.30/0.50        = ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_commute
% 0.30/0.50  thf(fact_664_diff__right__commute,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
% 0.30/0.50        = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_commute
% 0.30/0.50  thf(fact_665_diff__right__commute,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 )
% 0.30/0.50        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_commute
% 0.30/0.50  thf(fact_666_diff__right__commute,axiom,
% 0.30/0.50      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.50        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
% 0.30/0.50        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_commute
% 0.30/0.50  thf(fact_667_diff__right__commute,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
% 0.30/0.50        = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_commute
% 0.30/0.50  thf(fact_668_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( ord_less_eq_real @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(3)
% 0.30/0.50  thf(fact_669_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( ord_less_eq_rat @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(3)
% 0.30/0.50  thf(fact_670_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(3)
% 0.30/0.50  thf(fact_671_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( ord_less_eq_int @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(3)
% 0.30/0.50  thf(fact_672_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(2)
% 0.30/0.50  thf(fact_673_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_eq_rat @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(2)
% 0.30/0.50  thf(fact_674_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_eq_nat @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(2)
% 0.30/0.50  thf(fact_675_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_eq_int @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(2)
% 0.30/0.50  thf(fact_676_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( ord_less_eq_real @ I @ J )
% 0.30/0.50          & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(1)
% 0.30/0.50  thf(fact_677_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( ord_less_eq_rat @ I @ J )
% 0.30/0.50          & ( ord_less_eq_rat @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(1)
% 0.30/0.50  thf(fact_678_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.50          & ( ord_less_eq_nat @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(1)
% 0.30/0.50  thf(fact_679_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( ord_less_eq_int @ I @ J )
% 0.30/0.50          & ( ord_less_eq_int @ K @ L ) )
% 0.30/0.50       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_semiring(1)
% 0.30/0.50  thf(fact_680_add__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.50         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono
% 0.30/0.50  thf(fact_681_add__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.50         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono
% 0.30/0.50  thf(fact_682_add__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.50         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono
% 0.30/0.50  thf(fact_683_add__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.50         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono
% 0.30/0.50  thf(fact_684_add__left__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_mono
% 0.30/0.50  thf(fact_685_add__left__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_mono
% 0.30/0.50  thf(fact_686_add__left__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_mono
% 0.30/0.50  thf(fact_687_add__left__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_left_mono
% 0.30/0.50  thf(fact_688_less__eqE,axiom,
% 0.30/0.50      ! [A: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ~ ! [C2: nat] :
% 0.30/0.50              ( B2
% 0.30/0.50             != ( plus_plus_nat @ A @ C2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_eqE
% 0.30/0.50  thf(fact_689_add__right__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_mono
% 0.30/0.50  thf(fact_690_add__right__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_mono
% 0.30/0.50  thf(fact_691_add__right__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_mono
% 0.30/0.50  thf(fact_692_add__right__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_right_mono
% 0.30/0.50  thf(fact_693_le__iff__add,axiom,
% 0.30/0.50      ( ord_less_eq_nat
% 0.30/0.50      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.50          ? [C3: nat] :
% 0.30/0.50            ( B4
% 0.30/0.50            = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % le_iff_add
% 0.30/0.50  thf(fact_694_add__le__imp__le__left,axiom,
% 0.30/0.50      ! [C: real,A: real,B2: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_left
% 0.30/0.50  thf(fact_695_add__le__imp__le__left,axiom,
% 0.30/0.50      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_left
% 0.30/0.50  thf(fact_696_add__le__imp__le__left,axiom,
% 0.30/0.50      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_left
% 0.30/0.50  thf(fact_697_add__le__imp__le__left,axiom,
% 0.30/0.50      ! [C: int,A: int,B2: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_left
% 0.30/0.50  thf(fact_698_add__le__imp__le__right,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_right
% 0.30/0.50  thf(fact_699_add__le__imp__le__right,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_right
% 0.30/0.50  thf(fact_700_add__le__imp__le__right,axiom,
% 0.30/0.50      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_right
% 0.30/0.50  thf(fact_701_add__le__imp__le__right,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_imp_le_right
% 0.30/0.50  thf(fact_702_diff__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,D: real,C: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_real @ D @ C )
% 0.30/0.50         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_mono
% 0.30/0.50  thf(fact_703_diff__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,D: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_rat @ D @ C )
% 0.30/0.50         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_mono
% 0.30/0.50  thf(fact_704_diff__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,D: int,C: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_int @ D @ C )
% 0.30/0.50         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_mono
% 0.30/0.50  thf(fact_705_diff__left__mono,axiom,
% 0.30/0.50      ! [B2: real,A: real,C: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.50       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_left_mono
% 0.30/0.50  thf(fact_706_diff__left__mono,axiom,
% 0.30/0.50      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.50       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_left_mono
% 0.30/0.50  thf(fact_707_diff__left__mono,axiom,
% 0.30/0.50      ! [B2: int,A: int,C: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.50       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_left_mono
% 0.30/0.50  thf(fact_708_diff__right__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_mono
% 0.30/0.50  thf(fact_709_diff__right__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_mono
% 0.30/0.50  thf(fact_710_diff__right__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_right_mono
% 0.30/0.50  thf(fact_711_diff__eq__diff__less__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ( minus_minus_real @ A @ B2 )
% 0.30/0.50          = ( minus_minus_real @ C @ D ) )
% 0.30/0.50       => ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_less_eq
% 0.30/0.50  thf(fact_712_diff__eq__diff__less__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ( minus_minus_rat @ A @ B2 )
% 0.30/0.50          = ( minus_minus_rat @ C @ D ) )
% 0.30/0.50       => ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_less_eq
% 0.30/0.50  thf(fact_713_diff__eq__diff__less__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ( minus_minus_int @ A @ B2 )
% 0.30/0.50          = ( minus_minus_int @ C @ D ) )
% 0.30/0.50       => ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_less_eq
% 0.30/0.50  thf(fact_714_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( ord_less_real @ I @ J )
% 0.30/0.50          & ( ord_less_real @ K @ L ) )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(5)
% 0.30/0.50  thf(fact_715_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( ord_less_rat @ I @ J )
% 0.30/0.50          & ( ord_less_rat @ K @ L ) )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(5)
% 0.30/0.50  thf(fact_716_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( ord_less_nat @ I @ J )
% 0.30/0.50          & ( ord_less_nat @ K @ L ) )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(5)
% 0.30/0.50  thf(fact_717_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( ord_less_int @ I @ J )
% 0.30/0.50          & ( ord_less_int @ K @ L ) )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(5)
% 0.30/0.50  thf(fact_718_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_real @ K @ L ) )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(2)
% 0.30/0.50  thf(fact_719_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_rat @ K @ L ) )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(2)
% 0.30/0.50  thf(fact_720_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_nat @ K @ L ) )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(2)
% 0.30/0.50  thf(fact_721_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( I = J )
% 0.30/0.50          & ( ord_less_int @ K @ L ) )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(2)
% 0.30/0.50  thf(fact_722_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( ord_less_real @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(1)
% 0.30/0.50  thf(fact_723_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( ord_less_rat @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(1)
% 0.30/0.50  thf(fact_724_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( ord_less_nat @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(1)
% 0.30/0.50  thf(fact_725_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( ord_less_int @ I @ J )
% 0.30/0.50          & ( K = L ) )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(1)
% 0.30/0.50  thf(fact_726_add__strict__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_real @ C @ D )
% 0.30/0.50         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_mono
% 0.30/0.50  thf(fact_727_add__strict__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_rat @ C @ D )
% 0.30/0.50         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_mono
% 0.30/0.50  thf(fact_728_add__strict__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.50        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_nat @ C @ D )
% 0.30/0.50         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_mono
% 0.30/0.50  thf(fact_729_add__strict__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_int @ C @ D )
% 0.30/0.50         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_mono
% 0.30/0.50  thf(fact_730_add__strict__left__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_left_mono
% 0.30/0.50  thf(fact_731_add__strict__left__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_left_mono
% 0.30/0.50  thf(fact_732_add__strict__left__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_left_mono
% 0.30/0.50  thf(fact_733_add__strict__left__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_left_mono
% 0.30/0.50  thf(fact_734_add__strict__right__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_right_mono
% 0.30/0.50  thf(fact_735_add__strict__right__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_right_mono
% 0.30/0.50  thf(fact_736_add__strict__right__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_right_mono
% 0.30/0.50  thf(fact_737_add__strict__right__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_strict_right_mono
% 0.30/0.50  thf(fact_738_add__less__imp__less__left,axiom,
% 0.30/0.50      ! [C: real,A: real,B2: real] :
% 0.30/0.50        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_left
% 0.30/0.50  thf(fact_739_add__less__imp__less__left,axiom,
% 0.30/0.50      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_left
% 0.30/0.50  thf(fact_740_add__less__imp__less__left,axiom,
% 0.30/0.50      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_nat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_left
% 0.30/0.50  thf(fact_741_add__less__imp__less__left,axiom,
% 0.30/0.50      ! [C: int,A: int,B2: int] :
% 0.30/0.50        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 0.30/0.50       => ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_left
% 0.30/0.50  thf(fact_742_add__less__imp__less__right,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_right
% 0.30/0.50  thf(fact_743_add__less__imp__less__right,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_right
% 0.30/0.50  thf(fact_744_add__less__imp__less__right,axiom,
% 0.30/0.50      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_nat @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_right
% 0.30/0.50  thf(fact_745_add__less__imp__less__right,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.50       => ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_imp_less_right
% 0.30/0.50  thf(fact_746_diff__strict__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,D: real,C: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_real @ D @ C )
% 0.30/0.50         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_mono
% 0.30/0.50  thf(fact_747_diff__strict__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,D: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_rat @ D @ C )
% 0.30/0.50         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_mono
% 0.30/0.50  thf(fact_748_diff__strict__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,D: int,C: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_int @ D @ C )
% 0.30/0.50         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_mono
% 0.30/0.50  thf(fact_749_diff__eq__diff__less,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ( minus_minus_real @ A @ B2 )
% 0.30/0.50          = ( minus_minus_real @ C @ D ) )
% 0.30/0.50       => ( ( ord_less_real @ A @ B2 )
% 0.30/0.50          = ( ord_less_real @ C @ D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_less
% 0.30/0.50  thf(fact_750_diff__eq__diff__less,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ( minus_minus_rat @ A @ B2 )
% 0.30/0.50          = ( minus_minus_rat @ C @ D ) )
% 0.30/0.50       => ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50          = ( ord_less_rat @ C @ D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_less
% 0.30/0.50  thf(fact_751_diff__eq__diff__less,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ( minus_minus_int @ A @ B2 )
% 0.30/0.50          = ( minus_minus_int @ C @ D ) )
% 0.30/0.50       => ( ( ord_less_int @ A @ B2 )
% 0.30/0.50          = ( ord_less_int @ C @ D ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_diff_less
% 0.30/0.50  thf(fact_752_diff__strict__left__mono,axiom,
% 0.30/0.50      ! [B2: real,A: real,C: real] :
% 0.30/0.50        ( ( ord_less_real @ B2 @ A )
% 0.30/0.50       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_left_mono
% 0.30/0.50  thf(fact_753_diff__strict__left__mono,axiom,
% 0.30/0.50      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.50       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_left_mono
% 0.30/0.50  thf(fact_754_diff__strict__left__mono,axiom,
% 0.30/0.50      ! [B2: int,A: int,C: int] :
% 0.30/0.50        ( ( ord_less_int @ B2 @ A )
% 0.30/0.50       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_left_mono
% 0.30/0.50  thf(fact_755_diff__strict__right__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_right_mono
% 0.30/0.50  thf(fact_756_diff__strict__right__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_right_mono
% 0.30/0.50  thf(fact_757_diff__strict__right__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_strict_right_mono
% 0.30/0.50  thf(fact_758_group__cancel_Osub1,axiom,
% 0.30/0.50      ! [A4: complex,K: complex,A: complex,B2: complex] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_complex @ K @ A ) )
% 0.30/0.50       => ( ( minus_minus_complex @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.sub1
% 0.30/0.50  thf(fact_759_group__cancel_Osub1,axiom,
% 0.30/0.50      ! [A4: real,K: real,A: real,B2: real] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_real @ K @ A ) )
% 0.30/0.50       => ( ( minus_minus_real @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.sub1
% 0.30/0.50  thf(fact_760_group__cancel_Osub1,axiom,
% 0.30/0.50      ! [A4: rat,K: rat,A: rat,B2: rat] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_rat @ K @ A ) )
% 0.30/0.50       => ( ( minus_minus_rat @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.sub1
% 0.30/0.50  thf(fact_761_group__cancel_Osub1,axiom,
% 0.30/0.50      ! [A4: int,K: int,A: int,B2: int] :
% 0.30/0.50        ( ( A4
% 0.30/0.50          = ( plus_plus_int @ K @ A ) )
% 0.30/0.50       => ( ( minus_minus_int @ A4 @ B2 )
% 0.30/0.50          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % group_cancel.sub1
% 0.30/0.50  thf(fact_762_diff__eq__eq,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( ( minus_minus_complex @ A @ B2 )
% 0.30/0.50          = C )
% 0.30/0.50        = ( A
% 0.30/0.50          = ( plus_plus_complex @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_eq
% 0.30/0.50  thf(fact_763_diff__eq__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ( minus_minus_real @ A @ B2 )
% 0.30/0.50          = C )
% 0.30/0.50        = ( A
% 0.30/0.50          = ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_eq
% 0.30/0.50  thf(fact_764_diff__eq__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ( minus_minus_rat @ A @ B2 )
% 0.30/0.50          = C )
% 0.30/0.50        = ( A
% 0.30/0.50          = ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_eq
% 0.30/0.50  thf(fact_765_diff__eq__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ( minus_minus_int @ A @ B2 )
% 0.30/0.50          = C )
% 0.30/0.50        = ( A
% 0.30/0.50          = ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_eq_eq
% 0.30/0.50  thf(fact_766_eq__diff__eq,axiom,
% 0.30/0.50      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.50        ( ( A
% 0.30/0.50          = ( minus_minus_complex @ C @ B2 ) )
% 0.30/0.50        = ( ( plus_plus_complex @ A @ B2 )
% 0.30/0.50          = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % eq_diff_eq
% 0.30/0.50  thf(fact_767_eq__diff__eq,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( A
% 0.30/0.50          = ( minus_minus_real @ C @ B2 ) )
% 0.30/0.50        = ( ( plus_plus_real @ A @ B2 )
% 0.30/0.50          = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % eq_diff_eq
% 0.30/0.50  thf(fact_768_eq__diff__eq,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( A
% 0.30/0.50          = ( minus_minus_rat @ C @ B2 ) )
% 0.30/0.50        = ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.50          = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % eq_diff_eq
% 0.30/0.50  thf(fact_769_eq__diff__eq,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( A
% 0.30/0.50          = ( minus_minus_int @ C @ B2 ) )
% 0.30/0.50        = ( ( plus_plus_int @ A @ B2 )
% 0.30/0.50          = C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % eq_diff_eq
% 0.30/0.50  thf(fact_770_add__diff__eq,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( plus_plus_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_eq
% 0.30/0.50  thf(fact_771_add__diff__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_eq
% 0.30/0.50  thf(fact_772_add__diff__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_eq
% 0.30/0.50  thf(fact_773_add__diff__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_diff_eq
% 0.30/0.50  thf(fact_774_diff__diff__eq2,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( minus_minus_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq2
% 0.30/0.50  thf(fact_775_diff__diff__eq2,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq2
% 0.30/0.50  thf(fact_776_diff__diff__eq2,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq2
% 0.30/0.50  thf(fact_777_diff__diff__eq2,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq2
% 0.30/0.50  thf(fact_778_diff__add__eq,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq
% 0.30/0.50  thf(fact_779_diff__add__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq
% 0.30/0.50  thf(fact_780_diff__add__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq
% 0.30/0.50  thf(fact_781_diff__add__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq
% 0.30/0.50  thf(fact_782_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( minus_minus_complex @ A @ ( plus_plus_complex @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq_diff_diff_swap
% 0.30/0.50  thf(fact_783_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq_diff_diff_swap
% 0.30/0.50  thf(fact_784_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq_diff_diff_swap
% 0.30/0.50  thf(fact_785_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.50        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add_eq_diff_diff_swap
% 0.30/0.50  thf(fact_786_add__implies__diff,axiom,
% 0.30/0.50      ! [C: complex,B2: complex,A: complex] :
% 0.30/0.50        ( ( ( plus_plus_complex @ C @ B2 )
% 0.30/0.50          = A )
% 0.30/0.50       => ( C
% 0.30/0.50          = ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_implies_diff
% 0.30/0.50  thf(fact_787_add__implies__diff,axiom,
% 0.30/0.50      ! [C: real,B2: real,A: real] :
% 0.30/0.50        ( ( ( plus_plus_real @ C @ B2 )
% 0.30/0.50          = A )
% 0.30/0.50       => ( C
% 0.30/0.50          = ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_implies_diff
% 0.30/0.50  thf(fact_788_add__implies__diff,axiom,
% 0.30/0.50      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.50        ( ( ( plus_plus_rat @ C @ B2 )
% 0.30/0.50          = A )
% 0.30/0.50       => ( C
% 0.30/0.50          = ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_implies_diff
% 0.30/0.50  thf(fact_789_add__implies__diff,axiom,
% 0.30/0.50      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.50        ( ( ( plus_plus_nat @ C @ B2 )
% 0.30/0.50          = A )
% 0.30/0.50       => ( C
% 0.30/0.50          = ( minus_minus_nat @ A @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_implies_diff
% 0.30/0.50  thf(fact_790_add__implies__diff,axiom,
% 0.30/0.50      ! [C: int,B2: int,A: int] :
% 0.30/0.50        ( ( ( plus_plus_int @ C @ B2 )
% 0.30/0.50          = A )
% 0.30/0.50       => ( C
% 0.30/0.50          = ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_implies_diff
% 0.30/0.50  thf(fact_791_diff__diff__eq,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_complex @ A @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq
% 0.30/0.50  thf(fact_792_diff__diff__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq
% 0.30/0.50  thf(fact_793_diff__diff__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq
% 0.30/0.50  thf(fact_794_diff__diff__eq,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq
% 0.30/0.50  thf(fact_795_diff__diff__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_diff_eq
% 0.30/0.50  thf(fact_796_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( ord_less_eq_real @ I @ J )
% 0.30/0.50          & ( ord_less_real @ K @ L ) )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(4)
% 0.30/0.50  thf(fact_797_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( ord_less_eq_rat @ I @ J )
% 0.30/0.50          & ( ord_less_rat @ K @ L ) )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(4)
% 0.30/0.50  thf(fact_798_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.50          & ( ord_less_nat @ K @ L ) )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(4)
% 0.30/0.50  thf(fact_799_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( ord_less_eq_int @ I @ J )
% 0.30/0.50          & ( ord_less_int @ K @ L ) )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(4)
% 0.30/0.50  thf(fact_800_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.50      ! [I: real,J: real,K: real,L: real] :
% 0.30/0.50        ( ( ( ord_less_real @ I @ J )
% 0.30/0.50          & ( ord_less_eq_real @ K @ L ) )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(3)
% 0.30/0.50  thf(fact_801_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.50      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.30/0.50        ( ( ( ord_less_rat @ I @ J )
% 0.30/0.50          & ( ord_less_eq_rat @ K @ L ) )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(3)
% 0.30/0.50  thf(fact_802_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.50      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/0.50        ( ( ( ord_less_nat @ I @ J )
% 0.30/0.50          & ( ord_less_eq_nat @ K @ L ) )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(3)
% 0.30/0.50  thf(fact_803_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.50      ! [I: int,J: int,K: int,L: int] :
% 0.30/0.50        ( ( ( ord_less_int @ I @ J )
% 0.30/0.50          & ( ord_less_eq_int @ K @ L ) )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono_thms_linordered_field(3)
% 0.30/0.50  thf(fact_804_add__le__less__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_real @ C @ D )
% 0.30/0.50         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_less_mono
% 0.30/0.50  thf(fact_805_add__le__less__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_rat @ C @ D )
% 0.30/0.50         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_less_mono
% 0.30/0.50  thf(fact_806_add__le__less__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_nat @ C @ D )
% 0.30/0.50         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_less_mono
% 0.30/0.50  thf(fact_807_add__le__less__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_int @ C @ D )
% 0.30/0.50         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_le_less_mono
% 0.30/0.50  thf(fact_808_add__less__le__mono,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.50         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_le_mono
% 0.30/0.50  thf(fact_809_add__less__le__mono,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.50         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_le_mono
% 0.30/0.50  thf(fact_810_add__less__le__mono,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.50        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.50         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_le_mono
% 0.30/0.50  thf(fact_811_add__less__le__mono,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.50         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_less_le_mono
% 0.30/0.50  thf(fact_812_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50         => ( ( ( minus_minus_nat @ B2 @ A )
% 0.30/0.50              = C )
% 0.30/0.50            = ( B2
% 0.30/0.50              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.30/0.50  thf(fact_813_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/0.50      ! [A: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
% 0.30/0.50          = B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.30/0.50  thf(fact_814_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 0.30/0.50          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.30/0.50  thf(fact_815_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
% 0.30/0.50          = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.30/0.50  thf(fact_816_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
% 0.30/0.50          = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.30/0.50  thf(fact_817_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
% 0.30/0.50          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.30/0.50  thf(fact_818_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 0.30/0.50          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.30/0.50  thf(fact_819_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 0.30/0.50          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.30/0.50  thf(fact_820_le__add__diff,axiom,
% 0.30/0.50      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % le_add_diff
% 0.30/0.50  thf(fact_821_diff__add,axiom,
% 0.30/0.50      ! [A: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.50       => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
% 0.30/0.50          = B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_add
% 0.30/0.50  thf(fact_822_le__diff__eq,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 0.30/0.50        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % le_diff_eq
% 0.30/0.50  thf(fact_823_le__diff__eq,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 0.30/0.50        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % le_diff_eq
% 0.30/0.50  thf(fact_824_le__diff__eq,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 0.30/0.50        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % le_diff_eq
% 0.30/0.50  thf(fact_825_diff__le__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_le_eq
% 0.30/0.50  thf(fact_826_diff__le__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_le_eq
% 0.30/0.50  thf(fact_827_diff__le__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_le_eq
% 0.30/0.50  thf(fact_828_less__diff__eq,axiom,
% 0.30/0.50      ! [A: real,C: real,B2: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 0.30/0.50        = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_diff_eq
% 0.30/0.50  thf(fact_829_less__diff__eq,axiom,
% 0.30/0.50      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 0.30/0.50        = ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_diff_eq
% 0.30/0.50  thf(fact_830_less__diff__eq,axiom,
% 0.30/0.50      ! [A: int,C: int,B2: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 0.30/0.50        = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_diff_eq
% 0.30/0.50  thf(fact_831_diff__less__eq,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_less_eq
% 0.30/0.50  thf(fact_832_diff__less__eq,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_less_eq
% 0.30/0.50  thf(fact_833_diff__less__eq,axiom,
% 0.30/0.50      ! [A: int,B2: int,C: int] :
% 0.30/0.50        ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.50        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % diff_less_eq
% 0.30/0.50  thf(fact_834__C2_C,axiom,
% 0.30/0.50      ( ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50          = none_nat )
% 0.30/0.50       => ( ( ( ord_less_nat @ mi @ xa )
% 0.30/0.50           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.50              = ( some_nat @ mi ) ) )
% 0.30/0.50          & ( ~ ( ord_less_nat @ mi @ xa )
% 0.30/0.50           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.50              = none_nat ) ) ) )
% 0.30/0.50      & ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50         != none_nat )
% 0.30/0.50       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.50          = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % "2"
% 0.30/0.50  thf(fact_835_invar__vebt_Ointros_I4_J,axiom,
% 0.30/0.50      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.30/0.50        ( ! [X5: vEBT_VEBT] :
% 0.30/0.50            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.50           => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.50       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.50         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.50              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50           => ( ( M = N )
% 0.30/0.50             => ( ( Deg
% 0.30/0.50                  = ( plus_plus_nat @ N @ M ) )
% 0.30/0.50               => ( ! [I3: nat] :
% 0.30/0.50                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
% 0.30/0.50                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.30/0.50                 => ( ( ( Mi = Ma )
% 0.30/0.50                     => ! [X5: vEBT_VEBT] :
% 0.30/0.50                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.50                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 0.30/0.50                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.30/0.50                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.50                       => ( ( ( Mi != Ma )
% 0.30/0.50                           => ! [I3: nat] :
% 0.30/0.50                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.30/0.50                                      = I3 )
% 0.30/0.50                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.30/0.50                                  & ! [X5: nat] :
% 0.30/0.50                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 0.30/0.50                                          = I3 )
% 0.30/0.50                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 0.30/0.50                                     => ( ( ord_less_nat @ Mi @ X5 )
% 0.30/0.50                                        & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 0.30/0.50                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % invar_vebt.intros(4)
% 0.30/0.50  thf(fact_836_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.30/0.50      ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.50        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.30/0.50         => ( ( 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 ) ) ) ) )
% 0.30/0.50           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % both_member_options_from_chilf_to_complete_tree
% 0.30/0.50  thf(fact_837_both__member__options__from__complete__tree__to__child,axiom,
% 0.30/0.50      ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.30/0.50       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.50         => ( ( 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 ) ) ) ) )
% 0.30/0.50            | ( X3 = Mi )
% 0.30/0.50            | ( X3 = Ma ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % both_member_options_from_complete_tree_to_child
% 0.30/0.50  thf(fact_838_invar__vebt_Ointros_I2_J,axiom,
% 0.30/0.50      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.30/0.50        ( ! [X5: vEBT_VEBT] :
% 0.30/0.50            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.50           => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.50       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.50         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.50              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50           => ( ( M = N )
% 0.30/0.50             => ( ( Deg
% 0.30/0.50                  = ( plus_plus_nat @ N @ M ) )
% 0.30/0.50               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.30/0.50                 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.50                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.50                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 0.30/0.50                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % invar_vebt.intros(2)
% 0.30/0.50  thf(fact_839_summaxma,axiom,
% 0.30/0.50      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.30/0.50       => ( ( Mi != Ma )
% 0.30/0.50         => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 0.30/0.50            = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % summaxma
% 0.30/0.50  thf(fact_840_set__n__deg__not__0,axiom,
% 0.30/0.50      ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 0.30/0.50        ( ! [X5: vEBT_VEBT] :
% 0.30/0.50            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.50           => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.50       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.50            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.50         => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % set_n_deg_not_0
% 0.30/0.50  thf(fact_841_vebt__maxt_Osimps_I3_J,axiom,
% 0.30/0.50      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.50        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.50        = ( some_nat @ Ma ) ) ).
% 0.30/0.50  
% 0.30/0.50  % vebt_maxt.simps(3)
% 0.30/0.50  thf(fact_842_vebt__mint_Osimps_I3_J,axiom,
% 0.30/0.50      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.50        ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.50        = ( some_nat @ Mi ) ) ).
% 0.30/0.50  
% 0.30/0.50  % vebt_mint.simps(3)
% 0.30/0.50  thf(fact_843_times__divide__eq__right,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.50        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % times_divide_eq_right
% 0.30/0.50  thf(fact_844_times__divide__eq__right,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.50        = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % times_divide_eq_right
% 0.30/0.50  thf(fact_845_times__divide__eq__right,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( times_times_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.50        = ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % times_divide_eq_right
% 0.30/0.50  thf(fact_846_divide__divide__eq__right,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.50        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % divide_divide_eq_right
% 0.30/0.50  thf(fact_847_divide__divide__eq__right,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.50        = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % divide_divide_eq_right
% 0.30/0.50  thf(fact_848_divide__divide__eq__right,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.50        = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % divide_divide_eq_right
% 0.30/0.50  thf(fact_849__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
% 0.30/0.50      ord_less_eq_nat @ one_one_nat @ na ).
% 0.30/0.50  
% 0.30/0.50  % \<open>1 \<le> n\<close>
% 0.30/0.50  thf(fact_850_real__divide__square__eq,axiom,
% 0.30/0.50      ! [R: real,A: real] :
% 0.30/0.50        ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 0.30/0.50        = ( divide_divide_real @ A @ R ) ) ).
% 0.30/0.50  
% 0.30/0.50  % real_divide_square_eq
% 0.30/0.50  thf(fact_851_VEBT_Oinject_I1_J,axiom,
% 0.30/0.50      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.30/0.50        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.30/0.50          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.30/0.50        = ( ( X11 = Y11 )
% 0.30/0.50          & ( X12 = Y12 )
% 0.30/0.50          & ( X13 = Y13 )
% 0.30/0.50          & ( X14 = Y14 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % VEBT.inject(1)
% 0.30/0.50  thf(fact_852_mult__1,axiom,
% 0.30/0.50      ! [A: complex] :
% 0.30/0.50        ( ( times_times_complex @ one_one_complex @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult_1
% 0.30/0.50  thf(fact_853_mult__1,axiom,
% 0.30/0.50      ! [A: real] :
% 0.30/0.50        ( ( times_times_real @ one_one_real @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult_1
% 0.30/0.50  thf(fact_854_mult__1,axiom,
% 0.30/0.50      ! [A: rat] :
% 0.30/0.50        ( ( times_times_rat @ one_one_rat @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult_1
% 0.30/0.50  thf(fact_855_mult__1,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( times_times_nat @ one_one_nat @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult_1
% 0.30/0.50  thf(fact_856_mult__1,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( times_times_int @ one_one_int @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult_1
% 0.30/0.50  thf(fact_857_mult_Oright__neutral,axiom,
% 0.30/0.50      ! [A: complex] :
% 0.30/0.50        ( ( times_times_complex @ A @ one_one_complex )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.right_neutral
% 0.30/0.50  thf(fact_858_mult_Oright__neutral,axiom,
% 0.30/0.50      ! [A: real] :
% 0.30/0.50        ( ( times_times_real @ A @ one_one_real )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.right_neutral
% 0.30/0.50  thf(fact_859_mult_Oright__neutral,axiom,
% 0.30/0.50      ! [A: rat] :
% 0.30/0.50        ( ( times_times_rat @ A @ one_one_rat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.right_neutral
% 0.30/0.50  thf(fact_860_mult_Oright__neutral,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( times_times_nat @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.right_neutral
% 0.30/0.50  thf(fact_861_mult_Oright__neutral,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( times_times_int @ A @ one_one_int )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.right_neutral
% 0.30/0.50  thf(fact_862_times__divide__eq__left,axiom,
% 0.30/0.50      ! [B2: complex,C: complex,A: complex] :
% 0.30/0.50        ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
% 0.30/0.50        = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % times_divide_eq_left
% 0.30/0.50  thf(fact_863_times__divide__eq__left,axiom,
% 0.30/0.50      ! [B2: real,C: real,A: real] :
% 0.30/0.50        ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.50        = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % times_divide_eq_left
% 0.30/0.50  thf(fact_864_times__divide__eq__left,axiom,
% 0.30/0.50      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.50        ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.50        = ( divide_divide_rat @ ( times_times_rat @ B2 @ A ) @ C ) ) ).
% 0.30/0.50  
% 0.30/0.50  % times_divide_eq_left
% 0.30/0.50  thf(fact_865_divide__divide__eq__left,axiom,
% 0.30/0.50      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.50        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 0.30/0.50        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % divide_divide_eq_left
% 0.30/0.50  thf(fact_866_divide__divide__eq__left,axiom,
% 0.30/0.50      ! [A: real,B2: real,C: real] :
% 0.30/0.50        ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 0.30/0.50        = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % divide_divide_eq_left
% 0.30/0.50  thf(fact_867_divide__divide__eq__left,axiom,
% 0.30/0.50      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.50        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 0.30/0.50        = ( divide_divide_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % divide_divide_eq_left
% 0.30/0.50  thf(fact_868_div__by__1,axiom,
% 0.30/0.50      ! [A: complex] :
% 0.30/0.50        ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % div_by_1
% 0.30/0.50  thf(fact_869_div__by__1,axiom,
% 0.30/0.50      ! [A: real] :
% 0.30/0.50        ( ( divide_divide_real @ A @ one_one_real )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % div_by_1
% 0.30/0.50  thf(fact_870_div__by__1,axiom,
% 0.30/0.50      ! [A: rat] :
% 0.30/0.50        ( ( divide_divide_rat @ A @ one_one_rat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % div_by_1
% 0.30/0.50  thf(fact_871_div__by__1,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( divide_divide_nat @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % div_by_1
% 0.30/0.50  thf(fact_872_div__by__1,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( divide_divide_int @ A @ one_one_int )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % div_by_1
% 0.30/0.50  thf(fact_873_bits__div__by__1,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( divide_divide_nat @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % bits_div_by_1
% 0.30/0.50  thf(fact_874_bits__div__by__1,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( divide_divide_int @ A @ one_one_int )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % bits_div_by_1
% 0.30/0.50  thf(fact_875_power__one,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( power_power_rat @ one_one_rat @ N )
% 0.30/0.50        = one_one_rat ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one
% 0.30/0.50  thf(fact_876_power__one,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( power_power_nat @ one_one_nat @ N )
% 0.30/0.50        = one_one_nat ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one
% 0.30/0.50  thf(fact_877_power__one,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( power_power_real @ one_one_real @ N )
% 0.30/0.50        = one_one_real ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one
% 0.30/0.50  thf(fact_878_power__one,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( power_power_int @ one_one_int @ N )
% 0.30/0.50        = one_one_int ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one
% 0.30/0.50  thf(fact_879_power__one,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( power_power_complex @ one_one_complex @ N )
% 0.30/0.50        = one_one_complex ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one
% 0.30/0.50  thf(fact_880_power__one__right,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( power_power_nat @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_right
% 0.30/0.50  thf(fact_881_power__one__right,axiom,
% 0.30/0.50      ! [A: real] :
% 0.30/0.50        ( ( power_power_real @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_right
% 0.30/0.50  thf(fact_882_power__one__right,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( power_power_int @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_right
% 0.30/0.50  thf(fact_883_power__one__right,axiom,
% 0.30/0.50      ! [A: complex] :
% 0.30/0.50        ( ( power_power_complex @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_right
% 0.30/0.50  thf(fact_884_nat__1__eq__mult__iff,axiom,
% 0.30/0.50      ! [M: nat,N: nat] :
% 0.30/0.50        ( ( one_one_nat
% 0.30/0.50          = ( times_times_nat @ M @ N ) )
% 0.30/0.50        = ( ( M = one_one_nat )
% 0.30/0.50          & ( N = one_one_nat ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % nat_1_eq_mult_iff
% 0.30/0.50  thf(fact_885_nat__mult__eq__1__iff,axiom,
% 0.30/0.50      ! [M: nat,N: nat] :
% 0.30/0.50        ( ( ( times_times_nat @ M @ N )
% 0.30/0.50          = one_one_nat )
% 0.30/0.50        = ( ( M = one_one_nat )
% 0.30/0.50          & ( N = one_one_nat ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % nat_mult_eq_1_iff
% 0.30/0.50  thf(fact_886_numeral__eq__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ( numera6690914467698888265omplex @ N )
% 0.30/0.50          = one_one_complex )
% 0.30/0.50        = ( N = one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_eq_one_iff
% 0.30/0.50  thf(fact_887_numeral__eq__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ( numeral_numeral_real @ N )
% 0.30/0.50          = one_one_real )
% 0.30/0.50        = ( N = one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_eq_one_iff
% 0.30/0.50  thf(fact_888_numeral__eq__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ( numeral_numeral_rat @ N )
% 0.30/0.50          = one_one_rat )
% 0.30/0.50        = ( N = one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_eq_one_iff
% 0.30/0.50  thf(fact_889_numeral__eq__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ( numeral_numeral_nat @ N )
% 0.30/0.50          = one_one_nat )
% 0.30/0.50        = ( N = one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_eq_one_iff
% 0.30/0.50  thf(fact_890_numeral__eq__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ( numeral_numeral_int @ N )
% 0.30/0.50          = one_one_int )
% 0.30/0.50        = ( N = one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_eq_one_iff
% 0.30/0.50  thf(fact_891_one__eq__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( one_one_complex
% 0.30/0.50          = ( numera6690914467698888265omplex @ N ) )
% 0.30/0.50        = ( one = N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_eq_numeral_iff
% 0.30/0.50  thf(fact_892_one__eq__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( one_one_real
% 0.30/0.50          = ( numeral_numeral_real @ N ) )
% 0.30/0.50        = ( one = N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_eq_numeral_iff
% 0.30/0.50  thf(fact_893_one__eq__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( one_one_rat
% 0.30/0.50          = ( numeral_numeral_rat @ N ) )
% 0.30/0.50        = ( one = N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_eq_numeral_iff
% 0.30/0.50  thf(fact_894_one__eq__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( one_one_nat
% 0.30/0.50          = ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( one = N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_eq_numeral_iff
% 0.30/0.50  thf(fact_895_one__eq__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( one_one_int
% 0.30/0.50          = ( numeral_numeral_int @ N ) )
% 0.30/0.50        = ( one = N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_eq_numeral_iff
% 0.30/0.50  thf(fact_896_power__inject__exp,axiom,
% 0.30/0.50      ! [A: real,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50       => ( ( ( power_power_real @ A @ M )
% 0.30/0.50            = ( power_power_real @ A @ N ) )
% 0.30/0.50          = ( M = N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_inject_exp
% 0.30/0.50  thf(fact_897_power__inject__exp,axiom,
% 0.30/0.50      ! [A: rat,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50       => ( ( ( power_power_rat @ A @ M )
% 0.30/0.50            = ( power_power_rat @ A @ N ) )
% 0.30/0.50          = ( M = N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_inject_exp
% 0.30/0.50  thf(fact_898_power__inject__exp,axiom,
% 0.30/0.50      ! [A: nat,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50       => ( ( ( power_power_nat @ A @ M )
% 0.30/0.50            = ( power_power_nat @ A @ N ) )
% 0.30/0.50          = ( M = N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_inject_exp
% 0.30/0.50  thf(fact_899_power__inject__exp,axiom,
% 0.30/0.50      ! [A: int,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50       => ( ( ( power_power_int @ A @ M )
% 0.30/0.50            = ( power_power_int @ A @ N ) )
% 0.30/0.50          = ( M = N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_inject_exp
% 0.30/0.50  thf(fact_900_option_Ocollapse,axiom,
% 0.30/0.50      ! [Option: option_nat] :
% 0.30/0.50        ( ( Option != none_nat )
% 0.30/0.50       => ( ( some_nat @ ( the_nat @ Option ) )
% 0.30/0.50          = Option ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.collapse
% 0.30/0.50  thf(fact_901_option_Ocollapse,axiom,
% 0.30/0.50      ! [Option: option4927543243414619207at_nat] :
% 0.30/0.50        ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.50       => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.30/0.50          = Option ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.collapse
% 0.30/0.50  thf(fact_902_option_Ocollapse,axiom,
% 0.30/0.50      ! [Option: option_num] :
% 0.30/0.50        ( ( Option != none_num )
% 0.30/0.50       => ( ( some_num @ ( the_num @ Option ) )
% 0.30/0.50          = Option ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.collapse
% 0.30/0.50  thf(fact_903_power__strict__increasing__iff,axiom,
% 0.30/0.50      ! [B2: real,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.50       => ( ( ord_less_real @ ( power_power_real @ B2 @ X3 ) @ ( power_power_real @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing_iff
% 0.30/0.50  thf(fact_904_power__strict__increasing__iff,axiom,
% 0.30/0.50      ! [B2: rat,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ B2 )
% 0.30/0.50       => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X3 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing_iff
% 0.30/0.50  thf(fact_905_power__strict__increasing__iff,axiom,
% 0.30/0.50      ! [B2: nat,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ B2 )
% 0.30/0.50       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X3 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing_iff
% 0.30/0.50  thf(fact_906_power__strict__increasing__iff,axiom,
% 0.30/0.50      ! [B2: int,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ B2 )
% 0.30/0.50       => ( ( ord_less_int @ ( power_power_int @ B2 @ X3 ) @ ( power_power_int @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing_iff
% 0.30/0.50  thf(fact_907_one__add__one,axiom,
% 0.30/0.50      ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.30/0.50      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_add_one
% 0.30/0.50  thf(fact_908_one__add__one,axiom,
% 0.30/0.50      ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.30/0.50      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_add_one
% 0.30/0.50  thf(fact_909_one__add__one,axiom,
% 0.30/0.50      ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.30/0.50      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_add_one
% 0.30/0.50  thf(fact_910_one__add__one,axiom,
% 0.30/0.50      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.30/0.50      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_add_one
% 0.30/0.50  thf(fact_911_one__add__one,axiom,
% 0.30/0.50      ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.30/0.50      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_add_one
% 0.30/0.50  thf(fact_912_power__increasing__iff,axiom,
% 0.30/0.50      ! [B2: real,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X3 ) @ ( power_power_real @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing_iff
% 0.30/0.50  thf(fact_913_power__increasing__iff,axiom,
% 0.30/0.50      ! [B2: rat,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X3 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing_iff
% 0.30/0.50  thf(fact_914_power__increasing__iff,axiom,
% 0.30/0.50      ! [B2: nat,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X3 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing_iff
% 0.30/0.50  thf(fact_915_power__increasing__iff,axiom,
% 0.30/0.50      ! [B2: int,X3: nat,Y3: nat] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ B2 )
% 0.30/0.50       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X3 ) @ ( power_power_int @ B2 @ Y3 ) )
% 0.30/0.50          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing_iff
% 0.30/0.50  thf(fact_916_numeral__plus__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.30/0.50        = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_plus_one
% 0.30/0.50  thf(fact_917_numeral__plus__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.30/0.50        = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_plus_one
% 0.30/0.50  thf(fact_918_numeral__plus__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.30/0.50        = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_plus_one
% 0.30/0.50  thf(fact_919_numeral__plus__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.30/0.50        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_plus_one
% 0.30/0.50  thf(fact_920_numeral__plus__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.30/0.50        = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_plus_one
% 0.30/0.50  thf(fact_921_one__plus__numeral,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.50        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral
% 0.30/0.50  thf(fact_922_one__plus__numeral,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.50        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral
% 0.30/0.50  thf(fact_923_one__plus__numeral,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.50        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral
% 0.30/0.50  thf(fact_924_one__plus__numeral,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral
% 0.30/0.50  thf(fact_925_one__plus__numeral,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.50        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral
% 0.30/0.50  thf(fact_926_numeral__le__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.30/0.50        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_le_one_iff
% 0.30/0.50  thf(fact_927_numeral__le__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.30/0.50        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_le_one_iff
% 0.30/0.50  thf(fact_928_numeral__le__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.30/0.50        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_le_one_iff
% 0.30/0.50  thf(fact_929_numeral__le__one__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.30/0.50        = ( ord_less_eq_num @ N @ one ) ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_le_one_iff
% 0.30/0.50  thf(fact_930_one__less__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_less_numeral_iff
% 0.30/0.50  thf(fact_931_one__less__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_less_numeral_iff
% 0.30/0.50  thf(fact_932_one__less__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_less_numeral_iff
% 0.30/0.50  thf(fact_933_one__less__numeral__iff,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.50        = ( ord_less_num @ one @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_less_numeral_iff
% 0.30/0.50  thf(fact_934_one__reorient,axiom,
% 0.30/0.50      ! [X3: complex] :
% 0.30/0.50        ( ( one_one_complex = X3 )
% 0.30/0.50        = ( X3 = one_one_complex ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_reorient
% 0.30/0.50  thf(fact_935_one__reorient,axiom,
% 0.30/0.50      ! [X3: real] :
% 0.30/0.50        ( ( one_one_real = X3 )
% 0.30/0.50        = ( X3 = one_one_real ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_reorient
% 0.30/0.50  thf(fact_936_one__reorient,axiom,
% 0.30/0.50      ! [X3: rat] :
% 0.30/0.50        ( ( one_one_rat = X3 )
% 0.30/0.50        = ( X3 = one_one_rat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_reorient
% 0.30/0.50  thf(fact_937_one__reorient,axiom,
% 0.30/0.50      ! [X3: nat] :
% 0.30/0.50        ( ( one_one_nat = X3 )
% 0.30/0.50        = ( X3 = one_one_nat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_reorient
% 0.30/0.50  thf(fact_938_one__reorient,axiom,
% 0.30/0.50      ! [X3: int] :
% 0.30/0.50        ( ( one_one_int = X3 )
% 0.30/0.50        = ( X3 = one_one_int ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_reorient
% 0.30/0.50  thf(fact_939_less__eq__real__def,axiom,
% 0.30/0.50      ( ord_less_eq_real
% 0.30/0.50      = ( ^ [X: real,Y: real] :
% 0.30/0.50            ( ( ord_less_real @ X @ Y )
% 0.30/0.50            | ( X = Y ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_eq_real_def
% 0.30/0.50  thf(fact_940_complete__real,axiom,
% 0.30/0.50      ! [S2: set_real] :
% 0.30/0.50        ( ? [X6: real] : ( member_real @ X6 @ S2 )
% 0.30/0.50       => ( ? [Z3: real] :
% 0.30/0.50            ! [X5: real] :
% 0.30/0.50              ( ( member_real @ X5 @ S2 )
% 0.30/0.50             => ( ord_less_eq_real @ X5 @ Z3 ) )
% 0.30/0.50         => ? [Y4: real] :
% 0.30/0.50              ( ! [X6: real] :
% 0.30/0.50                  ( ( member_real @ X6 @ S2 )
% 0.30/0.50                 => ( ord_less_eq_real @ X6 @ Y4 ) )
% 0.30/0.50              & ! [Z3: real] :
% 0.30/0.50                  ( ! [X5: real] :
% 0.30/0.50                      ( ( member_real @ X5 @ S2 )
% 0.30/0.50                     => ( ord_less_eq_real @ X5 @ Z3 ) )
% 0.30/0.50                 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % complete_real
% 0.30/0.50  thf(fact_941_le__numeral__extra_I4_J,axiom,
% 0.30/0.50      ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.30/0.50  
% 0.30/0.50  % le_numeral_extra(4)
% 0.30/0.50  thf(fact_942_le__numeral__extra_I4_J,axiom,
% 0.30/0.50      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.30/0.50  
% 0.30/0.50  % le_numeral_extra(4)
% 0.30/0.50  thf(fact_943_le__numeral__extra_I4_J,axiom,
% 0.30/0.50      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.30/0.50  
% 0.30/0.50  % le_numeral_extra(4)
% 0.30/0.50  thf(fact_944_le__numeral__extra_I4_J,axiom,
% 0.30/0.50      ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.30/0.50  
% 0.30/0.50  % le_numeral_extra(4)
% 0.30/0.50  thf(fact_945_less__numeral__extra_I4_J,axiom,
% 0.30/0.50      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.30/0.50  
% 0.30/0.50  % less_numeral_extra(4)
% 0.30/0.50  thf(fact_946_less__numeral__extra_I4_J,axiom,
% 0.30/0.50      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.30/0.50  
% 0.30/0.50  % less_numeral_extra(4)
% 0.30/0.50  thf(fact_947_less__numeral__extra_I4_J,axiom,
% 0.30/0.50      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/0.50  
% 0.30/0.50  % less_numeral_extra(4)
% 0.30/0.50  thf(fact_948_less__numeral__extra_I4_J,axiom,
% 0.30/0.50      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.30/0.50  
% 0.30/0.50  % less_numeral_extra(4)
% 0.30/0.50  thf(fact_949_mult_Ocomm__neutral,axiom,
% 0.30/0.50      ! [A: complex] :
% 0.30/0.50        ( ( times_times_complex @ A @ one_one_complex )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.comm_neutral
% 0.30/0.50  thf(fact_950_mult_Ocomm__neutral,axiom,
% 0.30/0.50      ! [A: real] :
% 0.30/0.50        ( ( times_times_real @ A @ one_one_real )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.comm_neutral
% 0.30/0.50  thf(fact_951_mult_Ocomm__neutral,axiom,
% 0.30/0.50      ! [A: rat] :
% 0.30/0.50        ( ( times_times_rat @ A @ one_one_rat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.comm_neutral
% 0.30/0.50  thf(fact_952_mult_Ocomm__neutral,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( times_times_nat @ A @ one_one_nat )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.comm_neutral
% 0.30/0.50  thf(fact_953_mult_Ocomm__neutral,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( times_times_int @ A @ one_one_int )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % mult.comm_neutral
% 0.30/0.50  thf(fact_954_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50      ! [A: complex] :
% 0.30/0.50        ( ( times_times_complex @ one_one_complex @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % comm_monoid_mult_class.mult_1
% 0.30/0.50  thf(fact_955_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50      ! [A: real] :
% 0.30/0.50        ( ( times_times_real @ one_one_real @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % comm_monoid_mult_class.mult_1
% 0.30/0.50  thf(fact_956_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50      ! [A: rat] :
% 0.30/0.50        ( ( times_times_rat @ one_one_rat @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % comm_monoid_mult_class.mult_1
% 0.30/0.50  thf(fact_957_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50      ! [A: nat] :
% 0.30/0.50        ( ( times_times_nat @ one_one_nat @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % comm_monoid_mult_class.mult_1
% 0.30/0.50  thf(fact_958_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.50      ! [A: int] :
% 0.30/0.50        ( ( times_times_int @ one_one_int @ A )
% 0.30/0.50        = A ) ).
% 0.30/0.50  
% 0.30/0.50  % comm_monoid_mult_class.mult_1
% 0.30/0.50  thf(fact_959_nat__mult__1,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( times_times_nat @ one_one_nat @ N )
% 0.30/0.50        = N ) ).
% 0.30/0.50  
% 0.30/0.50  % nat_mult_1
% 0.30/0.50  thf(fact_960_nat__mult__1__right,axiom,
% 0.30/0.50      ! [N: nat] :
% 0.30/0.50        ( ( times_times_nat @ N @ one_one_nat )
% 0.30/0.50        = N ) ).
% 0.30/0.50  
% 0.30/0.50  % nat_mult_1_right
% 0.30/0.50  thf(fact_961_option_Osel,axiom,
% 0.30/0.50      ! [X2: nat] :
% 0.30/0.50        ( ( the_nat @ ( some_nat @ X2 ) )
% 0.30/0.50        = X2 ) ).
% 0.30/0.50  
% 0.30/0.50  % option.sel
% 0.30/0.50  thf(fact_962_option_Osel,axiom,
% 0.30/0.50      ! [X2: product_prod_nat_nat] :
% 0.30/0.50        ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
% 0.30/0.50        = X2 ) ).
% 0.30/0.50  
% 0.30/0.50  % option.sel
% 0.30/0.50  thf(fact_963_option_Osel,axiom,
% 0.30/0.50      ! [X2: num] :
% 0.30/0.50        ( ( the_num @ ( some_num @ X2 ) )
% 0.30/0.50        = X2 ) ).
% 0.30/0.50  
% 0.30/0.50  % option.sel
% 0.30/0.50  thf(fact_964_option_Oexpand,axiom,
% 0.30/0.50      ! [Option: option_nat,Option2: option_nat] :
% 0.30/0.50        ( ( ( Option = none_nat )
% 0.30/0.50          = ( Option2 = none_nat ) )
% 0.30/0.50       => ( ( ( Option != none_nat )
% 0.30/0.50           => ( ( Option2 != none_nat )
% 0.30/0.50             => ( ( the_nat @ Option )
% 0.30/0.50                = ( the_nat @ Option2 ) ) ) )
% 0.30/0.50         => ( Option = Option2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.expand
% 0.30/0.50  thf(fact_965_option_Oexpand,axiom,
% 0.30/0.50      ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.30/0.50        ( ( ( Option = none_P5556105721700978146at_nat )
% 0.30/0.50          = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.30/0.50       => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.50           => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.30/0.50             => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.30/0.50                = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.30/0.50         => ( Option = Option2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.expand
% 0.30/0.50  thf(fact_966_option_Oexpand,axiom,
% 0.30/0.50      ! [Option: option_num,Option2: option_num] :
% 0.30/0.50        ( ( ( Option = none_num )
% 0.30/0.50          = ( Option2 = none_num ) )
% 0.30/0.50       => ( ( ( Option != none_num )
% 0.30/0.50           => ( ( Option2 != none_num )
% 0.30/0.50             => ( ( the_num @ Option )
% 0.30/0.50                = ( the_num @ Option2 ) ) ) )
% 0.30/0.50         => ( Option = Option2 ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.expand
% 0.30/0.50  thf(fact_967_less__half__sum,axiom,
% 0.30/0.50      ! [A: real,B2: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_half_sum
% 0.30/0.50  thf(fact_968_less__half__sum,axiom,
% 0.30/0.50      ! [A: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_half_sum
% 0.30/0.50  thf(fact_969_gt__half__sum,axiom,
% 0.30/0.50      ! [A: real,B2: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % gt_half_sum
% 0.30/0.50  thf(fact_970_gt__half__sum,axiom,
% 0.30/0.50      ! [A: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).
% 0.30/0.50  
% 0.30/0.50  % gt_half_sum
% 0.30/0.50  thf(fact_971_one__le__numeral,axiom,
% 0.30/0.50      ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_numeral
% 0.30/0.50  thf(fact_972_one__le__numeral,axiom,
% 0.30/0.50      ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_numeral
% 0.30/0.50  thf(fact_973_one__le__numeral,axiom,
% 0.30/0.50      ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_numeral
% 0.30/0.50  thf(fact_974_one__le__numeral,axiom,
% 0.30/0.50      ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_numeral
% 0.30/0.50  thf(fact_975_not__numeral__less__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.30/0.50  
% 0.30/0.50  % not_numeral_less_one
% 0.30/0.50  thf(fact_976_not__numeral__less__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.30/0.50  
% 0.30/0.50  % not_numeral_less_one
% 0.30/0.50  thf(fact_977_not__numeral__less__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.30/0.50  
% 0.30/0.50  % not_numeral_less_one
% 0.30/0.50  thf(fact_978_not__numeral__less__one,axiom,
% 0.30/0.50      ! [N: num] :
% 0.30/0.50        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.30/0.50  
% 0.30/0.50  % not_numeral_less_one
% 0.30/0.50  thf(fact_979_less__1__mult,axiom,
% 0.30/0.50      ! [M: real,N: real] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ M )
% 0.30/0.50       => ( ( ord_less_real @ one_one_real @ N )
% 0.30/0.50         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_1_mult
% 0.30/0.50  thf(fact_980_less__1__mult,axiom,
% 0.30/0.50      ! [M: rat,N: rat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ M )
% 0.30/0.50       => ( ( ord_less_rat @ one_one_rat @ N )
% 0.30/0.50         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_1_mult
% 0.30/0.50  thf(fact_981_less__1__mult,axiom,
% 0.30/0.50      ! [M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ M )
% 0.30/0.50       => ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.50         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_1_mult
% 0.30/0.50  thf(fact_982_less__1__mult,axiom,
% 0.30/0.50      ! [M: int,N: int] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ M )
% 0.30/0.50       => ( ( ord_less_int @ one_one_int @ N )
% 0.30/0.50         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_1_mult
% 0.30/0.50  thf(fact_983_less__add__one,axiom,
% 0.30/0.50      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_add_one
% 0.30/0.50  thf(fact_984_less__add__one,axiom,
% 0.30/0.50      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_add_one
% 0.30/0.50  thf(fact_985_less__add__one,axiom,
% 0.30/0.50      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_add_one
% 0.30/0.50  thf(fact_986_less__add__one,axiom,
% 0.30/0.50      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.30/0.50  
% 0.30/0.50  % less_add_one
% 0.30/0.50  thf(fact_987_add__mono1,axiom,
% 0.30/0.50      ! [A: real,B2: real] :
% 0.30/0.50        ( ( ord_less_real @ A @ B2 )
% 0.30/0.50       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono1
% 0.30/0.50  thf(fact_988_add__mono1,axiom,
% 0.30/0.50      ! [A: rat,B2: rat] :
% 0.30/0.50        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.50       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono1
% 0.30/0.50  thf(fact_989_add__mono1,axiom,
% 0.30/0.50      ! [A: nat,B2: nat] :
% 0.30/0.50        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.50       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono1
% 0.30/0.50  thf(fact_990_add__mono1,axiom,
% 0.30/0.50      ! [A: int,B2: int] :
% 0.30/0.50        ( ( ord_less_int @ A @ B2 )
% 0.30/0.50       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % add_mono1
% 0.30/0.50  thf(fact_991_one__plus__numeral__commute,axiom,
% 0.30/0.50      ! [X3: num] :
% 0.30/0.50        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
% 0.30/0.50        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral_commute
% 0.30/0.50  thf(fact_992_one__plus__numeral__commute,axiom,
% 0.30/0.50      ! [X3: num] :
% 0.30/0.50        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
% 0.30/0.50        = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral_commute
% 0.30/0.50  thf(fact_993_one__plus__numeral__commute,axiom,
% 0.30/0.50      ! [X3: num] :
% 0.30/0.50        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
% 0.30/0.50        = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral_commute
% 0.30/0.50  thf(fact_994_one__plus__numeral__commute,axiom,
% 0.30/0.50      ! [X3: num] :
% 0.30/0.50        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
% 0.30/0.50        = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral_commute
% 0.30/0.50  thf(fact_995_one__plus__numeral__commute,axiom,
% 0.30/0.50      ! [X3: num] :
% 0.30/0.50        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
% 0.30/0.50        = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_plus_numeral_commute
% 0.30/0.50  thf(fact_996_numeral__One,axiom,
% 0.30/0.50      ( ( numera6690914467698888265omplex @ one )
% 0.30/0.50      = one_one_complex ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_One
% 0.30/0.50  thf(fact_997_numeral__One,axiom,
% 0.30/0.50      ( ( numeral_numeral_real @ one )
% 0.30/0.50      = one_one_real ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_One
% 0.30/0.50  thf(fact_998_numeral__One,axiom,
% 0.30/0.50      ( ( numeral_numeral_rat @ one )
% 0.30/0.50      = one_one_rat ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_One
% 0.30/0.50  thf(fact_999_numeral__One,axiom,
% 0.30/0.50      ( ( numeral_numeral_nat @ one )
% 0.30/0.50      = one_one_nat ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_One
% 0.30/0.50  thf(fact_1000_numeral__One,axiom,
% 0.30/0.50      ( ( numeral_numeral_int @ one )
% 0.30/0.50      = one_one_int ) ).
% 0.30/0.50  
% 0.30/0.50  % numeral_One
% 0.30/0.50  thf(fact_1001_one__le__power,axiom,
% 0.30/0.50      ! [A: real,N: nat] :
% 0.30/0.50        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.50       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_power
% 0.30/0.50  thf(fact_1002_one__le__power,axiom,
% 0.30/0.50      ! [A: rat,N: nat] :
% 0.30/0.50        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.50       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_power
% 0.30/0.50  thf(fact_1003_one__le__power,axiom,
% 0.30/0.50      ! [A: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.50       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_power
% 0.30/0.50  thf(fact_1004_one__le__power,axiom,
% 0.30/0.50      ! [A: int,N: nat] :
% 0.30/0.50        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.50       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % one_le_power
% 0.30/0.50  thf(fact_1005_left__right__inverse__power,axiom,
% 0.30/0.50      ! [X3: complex,Y3: complex,N: nat] :
% 0.30/0.50        ( ( ( times_times_complex @ X3 @ Y3 )
% 0.30/0.50          = one_one_complex )
% 0.30/0.50       => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.30/0.50          = one_one_complex ) ) ).
% 0.30/0.50  
% 0.30/0.50  % left_right_inverse_power
% 0.30/0.50  thf(fact_1006_left__right__inverse__power,axiom,
% 0.30/0.50      ! [X3: real,Y3: real,N: nat] :
% 0.30/0.50        ( ( ( times_times_real @ X3 @ Y3 )
% 0.30/0.50          = one_one_real )
% 0.30/0.50       => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.30/0.50          = one_one_real ) ) ).
% 0.30/0.50  
% 0.30/0.50  % left_right_inverse_power
% 0.30/0.50  thf(fact_1007_left__right__inverse__power,axiom,
% 0.30/0.50      ! [X3: rat,Y3: rat,N: nat] :
% 0.30/0.50        ( ( ( times_times_rat @ X3 @ Y3 )
% 0.30/0.50          = one_one_rat )
% 0.30/0.50       => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.30/0.50          = one_one_rat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % left_right_inverse_power
% 0.30/0.50  thf(fact_1008_left__right__inverse__power,axiom,
% 0.30/0.50      ! [X3: nat,Y3: nat,N: nat] :
% 0.30/0.50        ( ( ( times_times_nat @ X3 @ Y3 )
% 0.30/0.50          = one_one_nat )
% 0.30/0.50       => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 0.30/0.50          = one_one_nat ) ) ).
% 0.30/0.50  
% 0.30/0.50  % left_right_inverse_power
% 0.30/0.50  thf(fact_1009_left__right__inverse__power,axiom,
% 0.30/0.50      ! [X3: int,Y3: int,N: nat] :
% 0.30/0.50        ( ( ( times_times_int @ X3 @ Y3 )
% 0.30/0.50          = one_one_int )
% 0.30/0.50       => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.30/0.50          = one_one_int ) ) ).
% 0.30/0.50  
% 0.30/0.50  % left_right_inverse_power
% 0.30/0.50  thf(fact_1010_power__one__over,axiom,
% 0.30/0.50      ! [A: complex,N: nat] :
% 0.30/0.50        ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.30/0.50        = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_over
% 0.30/0.50  thf(fact_1011_power__one__over,axiom,
% 0.30/0.50      ! [A: real,N: nat] :
% 0.30/0.50        ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.30/0.50        = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_over
% 0.30/0.50  thf(fact_1012_power__one__over,axiom,
% 0.30/0.50      ! [A: rat,N: nat] :
% 0.30/0.50        ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.30/0.50        = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_one_over
% 0.30/0.50  thf(fact_1013_numerals_I1_J,axiom,
% 0.30/0.50      ( ( numeral_numeral_nat @ one )
% 0.30/0.50      = one_one_nat ) ).
% 0.30/0.50  
% 0.30/0.50  % numerals(1)
% 0.30/0.50  thf(fact_1014_option_Oexhaust__sel,axiom,
% 0.30/0.50      ! [Option: option_nat] :
% 0.30/0.50        ( ( Option != none_nat )
% 0.30/0.50       => ( Option
% 0.30/0.50          = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.exhaust_sel
% 0.30/0.50  thf(fact_1015_option_Oexhaust__sel,axiom,
% 0.30/0.50      ! [Option: option4927543243414619207at_nat] :
% 0.30/0.50        ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.50       => ( Option
% 0.30/0.50          = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.exhaust_sel
% 0.30/0.50  thf(fact_1016_option_Oexhaust__sel,axiom,
% 0.30/0.50      ! [Option: option_num] :
% 0.30/0.50        ( ( Option != none_num )
% 0.30/0.50       => ( Option
% 0.30/0.50          = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % option.exhaust_sel
% 0.30/0.50  thf(fact_1017_linordered__field__no__ub,axiom,
% 0.30/0.50      ! [X6: real] :
% 0.30/0.50      ? [X_1: real] : ( ord_less_real @ X6 @ X_1 ) ).
% 0.30/0.50  
% 0.30/0.50  % linordered_field_no_ub
% 0.30/0.50  thf(fact_1018_linordered__field__no__ub,axiom,
% 0.30/0.50      ! [X6: rat] :
% 0.30/0.50      ? [X_1: rat] : ( ord_less_rat @ X6 @ X_1 ) ).
% 0.30/0.50  
% 0.30/0.50  % linordered_field_no_ub
% 0.30/0.50  thf(fact_1019_linordered__field__no__lb,axiom,
% 0.30/0.50      ! [X6: real] :
% 0.30/0.50      ? [Y4: real] : ( ord_less_real @ Y4 @ X6 ) ).
% 0.30/0.50  
% 0.30/0.50  % linordered_field_no_lb
% 0.30/0.50  thf(fact_1020_linordered__field__no__lb,axiom,
% 0.30/0.50      ! [X6: rat] :
% 0.30/0.50      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X6 ) ).
% 0.30/0.50  
% 0.30/0.50  % linordered_field_no_lb
% 0.30/0.50  thf(fact_1021_power__less__power__Suc,axiom,
% 0.30/0.50      ! [A: real,N: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_power_Suc
% 0.30/0.50  thf(fact_1022_power__less__power__Suc,axiom,
% 0.30/0.50      ! [A: rat,N: nat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_power_Suc
% 0.30/0.50  thf(fact_1023_power__less__power__Suc,axiom,
% 0.30/0.50      ! [A: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_power_Suc
% 0.30/0.50  thf(fact_1024_power__less__power__Suc,axiom,
% 0.30/0.50      ! [A: int,N: nat] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_power_Suc
% 0.30/0.50  thf(fact_1025_power__gt1__lemma,axiom,
% 0.30/0.50      ! [A: real,N: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_gt1_lemma
% 0.30/0.50  thf(fact_1026_power__gt1__lemma,axiom,
% 0.30/0.50      ! [A: rat,N: nat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_gt1_lemma
% 0.30/0.50  thf(fact_1027_power__gt1__lemma,axiom,
% 0.30/0.50      ! [A: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_gt1_lemma
% 0.30/0.50  thf(fact_1028_power__gt1__lemma,axiom,
% 0.30/0.50      ! [A: int,N: nat] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_gt1_lemma
% 0.30/0.50  thf(fact_1029_square__diff__one__factored,axiom,
% 0.30/0.50      ! [X3: complex] :
% 0.30/0.50        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
% 0.30/0.50        = ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % square_diff_one_factored
% 0.30/0.50  thf(fact_1030_square__diff__one__factored,axiom,
% 0.30/0.50      ! [X3: real] :
% 0.30/0.50        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
% 0.30/0.50        = ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % square_diff_one_factored
% 0.30/0.50  thf(fact_1031_square__diff__one__factored,axiom,
% 0.30/0.50      ! [X3: rat] :
% 0.30/0.50        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
% 0.30/0.50        = ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % square_diff_one_factored
% 0.30/0.50  thf(fact_1032_square__diff__one__factored,axiom,
% 0.30/0.50      ! [X3: int] :
% 0.30/0.50        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
% 0.30/0.50        = ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % square_diff_one_factored
% 0.30/0.50  thf(fact_1033_power__strict__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: real] :
% 0.30/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing
% 0.30/0.50  thf(fact_1034_power__strict__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: rat] :
% 0.30/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing
% 0.30/0.50  thf(fact_1035_power__strict__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: nat] :
% 0.30/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing
% 0.30/0.50  thf(fact_1036_power__strict__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: int] :
% 0.30/0.50        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_strict_increasing
% 0.30/0.50  thf(fact_1037_power__less__imp__less__exp,axiom,
% 0.30/0.50      ! [A: real,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.30/0.50         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_imp_less_exp
% 0.30/0.50  thf(fact_1038_power__less__imp__less__exp,axiom,
% 0.30/0.50      ! [A: rat,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.30/0.50         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_imp_less_exp
% 0.30/0.50  thf(fact_1039_power__less__imp__less__exp,axiom,
% 0.30/0.50      ! [A: nat,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.50         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_imp_less_exp
% 0.30/0.50  thf(fact_1040_power__less__imp__less__exp,axiom,
% 0.30/0.50      ! [A: int,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.50         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_less_imp_less_exp
% 0.30/0.50  thf(fact_1041_power__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: real] :
% 0.30/0.50        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.50         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing
% 0.30/0.50  thf(fact_1042_power__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: rat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.50         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing
% 0.30/0.50  thf(fact_1043_power__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: nat] :
% 0.30/0.50        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.50         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing
% 0.30/0.50  thf(fact_1044_power__increasing,axiom,
% 0.30/0.50      ! [N: nat,N4: nat,A: int] :
% 0.30/0.50        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.50       => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.50         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 0.30/0.50  
% 0.30/0.50  % power_increasing
% 0.30/0.50  thf(fact_1045_power__le__imp__le__exp,axiom,
% 0.30/0.50      ! [A: real,M: nat,N: nat] :
% 0.30/0.50        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.30/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_le_imp_le_exp
% 0.30/0.51  thf(fact_1046_power__le__imp__le__exp,axiom,
% 0.30/0.51      ! [A: rat,M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.51       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.30/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_le_imp_le_exp
% 0.30/0.51  thf(fact_1047_power__le__imp__le__exp,axiom,
% 0.30/0.51      ! [A: nat,M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.51       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_le_imp_le_exp
% 0.30/0.51  thf(fact_1048_power__le__imp__le__exp,axiom,
% 0.30/0.51      ! [A: int,M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.51       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.51         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_le_imp_le_exp
% 0.30/0.51  thf(fact_1049_one__power2,axiom,
% 0.30/0.51      ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51      = one_one_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % one_power2
% 0.30/0.51  thf(fact_1050_one__power2,axiom,
% 0.30/0.51      ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51      = one_one_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % one_power2
% 0.30/0.51  thf(fact_1051_one__power2,axiom,
% 0.30/0.51      ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51      = one_one_real ) ).
% 0.30/0.51  
% 0.30/0.51  % one_power2
% 0.30/0.51  thf(fact_1052_one__power2,axiom,
% 0.30/0.51      ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51      = one_one_int ) ).
% 0.30/0.51  
% 0.30/0.51  % one_power2
% 0.30/0.51  thf(fact_1053_one__power2,axiom,
% 0.30/0.51      ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51      = one_one_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % one_power2
% 0.30/0.51  thf(fact_1054_nat__1__add__1,axiom,
% 0.30/0.51      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.30/0.51      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nat_1_add_1
% 0.30/0.51  thf(fact_1055_times__divide__times__eq,axiom,
% 0.30/0.51      ! [X3: complex,Y3: complex,Z: complex,W: complex] :
% 0.30/0.51        ( ( times_times_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.30/0.51        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % times_divide_times_eq
% 0.30/0.51  thf(fact_1056_times__divide__times__eq,axiom,
% 0.30/0.51      ! [X3: real,Y3: real,Z: real,W: real] :
% 0.30/0.51        ( ( times_times_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.30/0.51        = ( divide_divide_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % times_divide_times_eq
% 0.30/0.51  thf(fact_1057_times__divide__times__eq,axiom,
% 0.30/0.51      ! [X3: rat,Y3: rat,Z: rat,W: rat] :
% 0.30/0.51        ( ( times_times_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.30/0.51        = ( divide_divide_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % times_divide_times_eq
% 0.30/0.51  thf(fact_1058_divide__divide__times__eq,axiom,
% 0.30/0.51      ! [X3: complex,Y3: complex,Z: complex,W: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.30/0.51        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_divide_times_eq
% 0.30/0.51  thf(fact_1059_divide__divide__times__eq,axiom,
% 0.30/0.51      ! [X3: real,Y3: real,Z: real,W: real] :
% 0.30/0.51        ( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.30/0.51        = ( divide_divide_real @ ( times_times_real @ X3 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_divide_times_eq
% 0.30/0.51  thf(fact_1060_divide__divide__times__eq,axiom,
% 0.30/0.51      ! [X3: rat,Y3: rat,Z: rat,W: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.30/0.51        = ( divide_divide_rat @ ( times_times_rat @ X3 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_divide_times_eq
% 0.30/0.51  thf(fact_1061_divide__divide__eq__left_H,axiom,
% 0.30/0.51      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 0.30/0.51        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_divide_eq_left'
% 0.30/0.51  thf(fact_1062_divide__divide__eq__left_H,axiom,
% 0.30/0.51      ! [A: real,B2: real,C: real] :
% 0.30/0.51        ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 0.30/0.51        = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_divide_eq_left'
% 0.30/0.51  thf(fact_1063_divide__divide__eq__left_H,axiom,
% 0.30/0.51      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 0.30/0.51        = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_divide_eq_left'
% 0.30/0.51  thf(fact_1064_add__divide__distrib,axiom,
% 0.30/0.51      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B2 ) @ C )
% 0.30/0.51        = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_divide_distrib
% 0.30/0.51  thf(fact_1065_add__divide__distrib,axiom,
% 0.30/0.51      ! [A: real,B2: real,C: real] :
% 0.30/0.51        ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 0.30/0.51        = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_divide_distrib
% 0.30/0.51  thf(fact_1066_add__divide__distrib,axiom,
% 0.30/0.51      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 0.30/0.51        = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_divide_distrib
% 0.30/0.51  thf(fact_1067_diff__divide__distrib,axiom,
% 0.30/0.51      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 0.30/0.51        = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_divide_distrib
% 0.30/0.51  thf(fact_1068_diff__divide__distrib,axiom,
% 0.30/0.51      ! [A: real,B2: real,C: real] :
% 0.30/0.51        ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 0.30/0.51        = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_divide_distrib
% 0.30/0.51  thf(fact_1069_diff__divide__distrib,axiom,
% 0.30/0.51      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 0.30/0.51        = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_divide_distrib
% 0.30/0.51  thf(fact_1070_ex__power__ivl1,axiom,
% 0.30/0.51      ! [B2: nat,K: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.51       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.30/0.51         => ? [N3: nat] :
% 0.30/0.51              ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
% 0.30/0.51              & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % ex_power_ivl1
% 0.30/0.51  thf(fact_1071_ex__power__ivl2,axiom,
% 0.30/0.51      ! [B2: nat,K: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.51       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.51         => ? [N3: nat] :
% 0.30/0.51              ( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
% 0.30/0.51              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % ex_power_ivl2
% 0.30/0.51  thf(fact_1072_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.30/0.51      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
% 0.30/0.51        ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B2 ) )
% 0.30/0.51        = ( some_P7363390416028606310at_nat @ ( F @ A @ B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(3)
% 0.30/0.51  thf(fact_1073_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.30/0.51      ! [F: num > num > num,A: num,B2: num] :
% 0.30/0.51        ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B2 ) )
% 0.30/0.51        = ( some_num @ ( F @ A @ B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(3)
% 0.30/0.51  thf(fact_1074_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.30/0.51      ! [F: nat > nat > nat,A: nat,B2: nat] :
% 0.30/0.51        ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B2 ) )
% 0.30/0.51        = ( some_nat @ ( F @ A @ B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(3)
% 0.30/0.51  thf(fact_1075_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.30/0.51      ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 0.30/0.51        ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 0.30/0.51        = none_P5556105721700978146at_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(1)
% 0.30/0.51  thf(fact_1076_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.30/0.51      ! [Uu: num > num > num,Uv: option_num] :
% 0.30/0.51        ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 0.30/0.51        = none_num ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(1)
% 0.30/0.51  thf(fact_1077_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.30/0.51      ! [Uu: nat > nat > nat,Uv: option_nat] :
% 0.30/0.51        ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 0.30/0.51        = none_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(1)
% 0.30/0.51  thf(fact_1078_vebt__mint_Osimps_I2_J,axiom,
% 0.30/0.51      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.30/0.51        ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.30/0.51        = none_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % vebt_mint.simps(2)
% 0.30/0.51  thf(fact_1079_vebt__maxt_Osimps_I2_J,axiom,
% 0.30/0.51      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.30/0.51        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.30/0.51        = none_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % vebt_maxt.simps(2)
% 0.30/0.51  thf(fact_1080_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.30/0.51      ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 0.30/0.51        ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 0.30/0.51        = none_P5556105721700978146at_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(2)
% 0.30/0.51  thf(fact_1081_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.30/0.51      ! [Uw: num > num > num,V: num] :
% 0.30/0.51        ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 0.30/0.51        = none_num ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(2)
% 0.30/0.51  thf(fact_1082_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.30/0.51      ! [Uw: nat > nat > nat,V: nat] :
% 0.30/0.51        ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 0.30/0.51        = none_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.simps(2)
% 0.30/0.51  thf(fact_1083_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51      ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.30/0.51        ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb2 )
% 0.30/0.51          = Y3 )
% 0.30/0.51       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.30/0.51           => ( Y3 != none_P5556105721700978146at_nat ) )
% 0.30/0.51         => ( ( ? [V2: product_prod_nat_nat] :
% 0.30/0.51                  ( Xa2
% 0.30/0.51                  = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.30/0.51             => ( ( Xb2 = none_P5556105721700978146at_nat )
% 0.30/0.51               => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 0.30/0.51           => ~ ! [A3: product_prod_nat_nat] :
% 0.30/0.51                  ( ( Xa2
% 0.30/0.51                    = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.51                 => ! [B3: product_prod_nat_nat] :
% 0.30/0.51                      ( ( Xb2
% 0.30/0.51                        = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.30/0.51                     => ( Y3
% 0.30/0.51                       != ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.elims
% 0.30/0.51  thf(fact_1084_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51      ! [X3: num > num > num,Xa2: option_num,Xb2: option_num,Y3: option_num] :
% 0.30/0.51        ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb2 )
% 0.30/0.51          = Y3 )
% 0.30/0.51       => ( ( ( Xa2 = none_num )
% 0.30/0.51           => ( Y3 != none_num ) )
% 0.30/0.51         => ( ( ? [V2: num] :
% 0.30/0.51                  ( Xa2
% 0.30/0.51                  = ( some_num @ V2 ) )
% 0.30/0.51             => ( ( Xb2 = none_num )
% 0.30/0.51               => ( Y3 != none_num ) ) )
% 0.30/0.51           => ~ ! [A3: num] :
% 0.30/0.51                  ( ( Xa2
% 0.30/0.51                    = ( some_num @ A3 ) )
% 0.30/0.51                 => ! [B3: num] :
% 0.30/0.51                      ( ( Xb2
% 0.30/0.51                        = ( some_num @ B3 ) )
% 0.30/0.51                     => ( Y3
% 0.30/0.51                       != ( some_num @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.elims
% 0.30/0.51  thf(fact_1085_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51      ! [X3: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y3: option_nat] :
% 0.30/0.51        ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb2 )
% 0.30/0.51          = Y3 )
% 0.30/0.51       => ( ( ( Xa2 = none_nat )
% 0.30/0.51           => ( Y3 != none_nat ) )
% 0.30/0.51         => ( ( ? [V2: nat] :
% 0.30/0.51                  ( Xa2
% 0.30/0.51                  = ( some_nat @ V2 ) )
% 0.30/0.51             => ( ( Xb2 = none_nat )
% 0.30/0.51               => ( Y3 != none_nat ) ) )
% 0.30/0.51           => ~ ! [A3: nat] :
% 0.30/0.51                  ( ( Xa2
% 0.30/0.51                    = ( some_nat @ A3 ) )
% 0.30/0.51                 => ! [B3: nat] :
% 0.30/0.51                      ( ( Xb2
% 0.30/0.51                        = ( some_nat @ B3 ) )
% 0.30/0.51                     => ( Y3
% 0.30/0.51                       != ( some_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.elims
% 0.30/0.51  thf(fact_1086_nested__mint,axiom,
% 0.30/0.51      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.30/0.51       => ( ( N
% 0.30/0.51            = ( suc @ ( suc @ Va ) ) )
% 0.30/0.51         => ( ~ ( ord_less_nat @ Ma @ Mi )
% 0.30/0.51           => ( ( Ma != Mi )
% 0.30/0.51             => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nested_mint
% 0.30/0.51  thf(fact_1087_real__average__minus__second,axiom,
% 0.30/0.51      ! [B2: real,A: real] :
% 0.30/0.51        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.30/0.51        = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % real_average_minus_second
% 0.30/0.51  thf(fact_1088_real__average__minus__first,axiom,
% 0.30/0.51      ! [A: real,B2: real] :
% 0.30/0.51        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.30/0.51        = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % real_average_minus_first
% 0.30/0.51  thf(fact_1089__C1_C,axiom,
% 0.30/0.51      ( ( ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51           != none_nat )
% 0.30/0.51          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.51          = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51      & ( ~ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51             != none_nat )
% 0.30/0.51            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.51          = ( if_option_nat
% 0.30/0.51            @ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51              = none_nat )
% 0.30/0.51            @ ( if_option_nat @ ( ord_less_nat @ mi @ xa ) @ ( some_nat @ mi ) @ none_nat )
% 0.30/0.51            @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % "1"
% 0.30/0.51  thf(fact_1090_invar__vebt_Ointros_I5_J,axiom,
% 0.30/0.51      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.30/0.51        ( ! [X5: vEBT_VEBT] :
% 0.30/0.51            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51           => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.51       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.51         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.51              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51           => ( ( M
% 0.30/0.51                = ( suc @ N ) )
% 0.30/0.51             => ( ( Deg
% 0.30/0.51                  = ( plus_plus_nat @ N @ M ) )
% 0.30/0.51               => ( ! [I3: nat] :
% 0.30/0.51                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
% 0.30/0.51                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.30/0.51                 => ( ( ( Mi = Ma )
% 0.30/0.51                     => ! [X5: vEBT_VEBT] :
% 0.30/0.51                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 0.30/0.51                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.30/0.51                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.51                       => ( ( ( Mi != Ma )
% 0.30/0.51                           => ! [I3: nat] :
% 0.30/0.51                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.30/0.51                                      = I3 )
% 0.30/0.51                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.30/0.51                                  & ! [X5: nat] :
% 0.30/0.51                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 0.30/0.51                                          = I3 )
% 0.30/0.51                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 0.30/0.51                                     => ( ( ord_less_nat @ Mi @ X5 )
% 0.30/0.51                                        & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 0.30/0.51                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % invar_vebt.intros(5)
% 0.30/0.51  thf(fact_1091_pred__less__length__list,axiom,
% 0.30/0.51      ! [Deg: nat,X3: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51       => ( ( ord_less_eq_nat @ X3 @ Ma )
% 0.30/0.51         => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.51              = ( if_option_nat
% 0.30/0.51                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51                   != none_nat )
% 0.30/0.51                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( 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 ) ) ) ) ) )
% 0.30/0.51                @ ( if_option_nat
% 0.30/0.51                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51                    = none_nat )
% 0.30/0.51                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
% 0.30/0.51                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % pred_less_length_list
% 0.30/0.51  thf(fact_1092_pred__lesseq__max,axiom,
% 0.30/0.51      ! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51       => ( ( ord_less_eq_nat @ X3 @ Ma )
% 0.30/0.51         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.51            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51              @ ( if_option_nat
% 0.30/0.51                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51                   != none_nat )
% 0.30/0.51                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( 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 ) ) ) ) ) )
% 0.30/0.51                @ ( if_option_nat
% 0.30/0.51                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51                    = none_nat )
% 0.30/0.51                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
% 0.30/0.51                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51              @ none_nat ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % pred_lesseq_max
% 0.30/0.51  thf(fact_1093_divmod__step__eq,axiom,
% 0.30/0.51      ! [L: num,R: nat,Q2: nat] :
% 0.30/0.51        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
% 0.30/0.51         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.30/0.51            = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L ) ) ) ) )
% 0.30/0.51        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
% 0.30/0.51         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.30/0.51            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divmod_step_eq
% 0.30/0.51  thf(fact_1094_divmod__step__eq,axiom,
% 0.30/0.51      ! [L: num,R: int,Q2: int] :
% 0.30/0.51        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
% 0.30/0.51         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.51            = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L ) ) ) ) )
% 0.30/0.51        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
% 0.30/0.51         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.51            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divmod_step_eq
% 0.30/0.51  thf(fact_1095_divmod__step__eq,axiom,
% 0.30/0.51      ! [L: num,R: code_integer,Q2: code_integer] :
% 0.30/0.51        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
% 0.30/0.51         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 0.30/0.51            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 0.30/0.51        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
% 0.30/0.51         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 0.30/0.51            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divmod_step_eq
% 0.30/0.51  thf(fact_1096_maxt__corr__help__empty,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51       => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.51            = none_nat )
% 0.30/0.51         => ( ( vEBT_VEBT_set_vebt @ T )
% 0.30/0.51            = bot_bot_set_nat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % maxt_corr_help_empty
% 0.30/0.51  thf(fact_1097_mint__corr__help__empty,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51       => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.51            = none_nat )
% 0.30/0.51         => ( ( vEBT_VEBT_set_vebt @ T )
% 0.30/0.51            = bot_bot_set_nat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mint_corr_help_empty
% 0.30/0.51  thf(fact_1098_all__set__conv__all__nth,axiom,
% 0.30/0.51      ! [Xs2: list_int,P: int > $o] :
% 0.30/0.51        ( ( ! [X: int] :
% 0.30/0.51              ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X ) ) )
% 0.30/0.51        = ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51             => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_set_conv_all_nth
% 0.30/0.51  thf(fact_1099_all__set__conv__all__nth,axiom,
% 0.30/0.51      ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.51        ( ( ! [X: vEBT_VEBT] :
% 0.30/0.51              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X ) ) )
% 0.30/0.51        = ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51             => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_set_conv_all_nth
% 0.30/0.51  thf(fact_1100_all__set__conv__all__nth,axiom,
% 0.30/0.51      ! [Xs2: list_o,P: $o > $o] :
% 0.30/0.51        ( ( ! [X: $o] :
% 0.30/0.51              ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X ) ) )
% 0.30/0.51        = ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51             => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_set_conv_all_nth
% 0.30/0.51  thf(fact_1101_all__set__conv__all__nth,axiom,
% 0.30/0.51      ! [Xs2: list_nat,P: nat > $o] :
% 0.30/0.51        ( ( ! [X: nat] :
% 0.30/0.51              ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X ) ) )
% 0.30/0.51        = ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51             => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_set_conv_all_nth
% 0.30/0.51  thf(fact_1102_even__odd__cases,axiom,
% 0.30/0.51      ! [X3: nat] :
% 0.30/0.51        ( ! [N3: nat] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( plus_plus_nat @ N3 @ N3 ) )
% 0.30/0.51       => ~ ! [N3: nat] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % even_odd_cases
% 0.30/0.51  thf(fact_1103_set__vebt_H__def,axiom,
% 0.30/0.51      ( vEBT_VEBT_set_vebt
% 0.30/0.51      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_vebt'_def
% 0.30/0.51  thf(fact_1104_deg__SUcn__Node,axiom,
% 0.30/0.51      ! [Tree: vEBT_VEBT,N: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 0.30/0.51       => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.51            ( Tree
% 0.30/0.51            = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % deg_SUcn_Node
% 0.30/0.51  thf(fact_1105_zdiv__numeral__Bit0,axiom,
% 0.30/0.51      ! [V: num,W: num] :
% 0.30/0.51        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.51        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % zdiv_numeral_Bit0
% 0.30/0.51  thf(fact_1106_old_Onat_Oinject,axiom,
% 0.30/0.51      ! [Nat: nat,Nat2: nat] :
% 0.30/0.51        ( ( ( suc @ Nat )
% 0.30/0.51          = ( suc @ Nat2 ) )
% 0.30/0.51        = ( Nat = Nat2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % old.nat.inject
% 0.30/0.51  thf(fact_1107_nat_Oinject,axiom,
% 0.30/0.51      ! [X2: nat,Y2: nat] :
% 0.30/0.51        ( ( ( suc @ X2 )
% 0.30/0.51          = ( suc @ Y2 ) )
% 0.30/0.51        = ( X2 = Y2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nat.inject
% 0.30/0.51  thf(fact_1108_lessI,axiom,
% 0.30/0.51      ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lessI
% 0.30/0.51  thf(fact_1109_Suc__mono,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ N )
% 0.30/0.51       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_mono
% 0.30/0.51  thf(fact_1110_Suc__less__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.51        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_less_eq
% 0.30/0.51  thf(fact_1111_Suc__le__mono,axiom,
% 0.30/0.51      ! [N: nat,M: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.30/0.51        = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_le_mono
% 0.30/0.51  thf(fact_1112_add__Suc__right,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.30/0.51        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_Suc_right
% 0.30/0.51  thf(fact_1113_diff__Suc__Suc,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.51        = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_Suc_Suc
% 0.30/0.51  thf(fact_1114_Suc__diff__diff,axiom,
% 0.30/0.51      ! [M: nat,N: nat,K: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.30/0.51        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_diff_diff
% 0.30/0.51  thf(fact_1115_mult__Suc__right,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.30/0.51        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_Suc_right
% 0.30/0.51  thf(fact_1116_diff__Suc__1,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.30/0.51        = N ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_Suc_1
% 0.30/0.51  thf(fact_1117_diff__Suc__diff__eq2,axiom,
% 0.30/0.51      ! [K: nat,J: nat,I: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.51       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.30/0.51          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_Suc_diff_eq2
% 0.30/0.51  thf(fact_1118_diff__Suc__diff__eq1,axiom,
% 0.30/0.51      ! [K: nat,J: nat,I: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.51       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.30/0.51          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_Suc_diff_eq1
% 0.30/0.51  thf(fact_1119_Suc__numeral,axiom,
% 0.30/0.51      ! [N: num] :
% 0.30/0.51        ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.30/0.51        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_numeral
% 0.30/0.51  thf(fact_1120_add__2__eq__Suc_H,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51        = ( suc @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_2_eq_Suc'
% 0.30/0.51  thf(fact_1121_add__2__eq__Suc,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.51        = ( suc @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_2_eq_Suc
% 0.30/0.51  thf(fact_1122_div2__Suc__Suc,axiom,
% 0.30/0.51      ! [M: nat] :
% 0.30/0.51        ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.51        = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % div2_Suc_Suc
% 0.30/0.51  thf(fact_1123_Suc__1,axiom,
% 0.30/0.51      ( ( suc @ one_one_nat )
% 0.30/0.51      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_1
% 0.30/0.51  thf(fact_1124_mult__commute__abs,axiom,
% 0.30/0.51      ! [C: real] :
% 0.30/0.51        ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
% 0.30/0.51        = ( times_times_real @ C ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_commute_abs
% 0.30/0.51  thf(fact_1125_mult__commute__abs,axiom,
% 0.30/0.51      ! [C: rat] :
% 0.30/0.51        ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
% 0.30/0.51        = ( times_times_rat @ C ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_commute_abs
% 0.30/0.51  thf(fact_1126_mult__commute__abs,axiom,
% 0.30/0.51      ! [C: nat] :
% 0.30/0.51        ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
% 0.30/0.51        = ( times_times_nat @ C ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_commute_abs
% 0.30/0.51  thf(fact_1127_mult__commute__abs,axiom,
% 0.30/0.51      ! [C: int] :
% 0.30/0.51        ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
% 0.30/0.51        = ( times_times_int @ C ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_commute_abs
% 0.30/0.51  thf(fact_1128_n__not__Suc__n,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( N
% 0.30/0.51       != ( suc @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % n_not_Suc_n
% 0.30/0.51  thf(fact_1129_Suc__inject,axiom,
% 0.30/0.51      ! [X3: nat,Y3: nat] :
% 0.30/0.51        ( ( ( suc @ X3 )
% 0.30/0.51          = ( suc @ Y3 ) )
% 0.30/0.51       => ( X3 = Y3 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_inject
% 0.30/0.51  thf(fact_1130_lambda__one,axiom,
% 0.30/0.51      ( ( ^ [X: complex] : X )
% 0.30/0.51      = ( times_times_complex @ one_one_complex ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lambda_one
% 0.30/0.51  thf(fact_1131_lambda__one,axiom,
% 0.30/0.51      ( ( ^ [X: real] : X )
% 0.30/0.51      = ( times_times_real @ one_one_real ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lambda_one
% 0.30/0.51  thf(fact_1132_lambda__one,axiom,
% 0.30/0.51      ( ( ^ [X: rat] : X )
% 0.30/0.51      = ( times_times_rat @ one_one_rat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lambda_one
% 0.30/0.51  thf(fact_1133_lambda__one,axiom,
% 0.30/0.51      ( ( ^ [X: nat] : X )
% 0.30/0.51      = ( times_times_nat @ one_one_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lambda_one
% 0.30/0.51  thf(fact_1134_lambda__one,axiom,
% 0.30/0.51      ( ( ^ [X: int] : X )
% 0.30/0.51      = ( times_times_int @ one_one_int ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lambda_one
% 0.30/0.51  thf(fact_1135_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.51      ! [X3: produc8306885398267862888on_nat] :
% 0.30/0.51        ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.30/0.51       => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.30/0.51         => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
% 0.30/0.51                ( X3
% 0.30/0.51               != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.cases
% 0.30/0.51  thf(fact_1136_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.51      ! [X3: produc5542196010084753463at_nat] :
% 0.30/0.51        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.30/0.51       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.30/0.51         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.30/0.51                ( X3
% 0.30/0.51               != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.cases
% 0.30/0.51  thf(fact_1137_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.51      ! [X3: produc1193250871479095198on_num] :
% 0.30/0.51        ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.30/0.51       => ( ! [Uw2: num > num > num,V2: num] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.30/0.51         => ~ ! [F2: num > num > num,A3: num,B3: num] :
% 0.30/0.51                ( X3
% 0.30/0.51               != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_shift.cases
% 0.30/0.51  thf(fact_1138_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.51      ! [X3: produc2233624965454879586on_nat] :
% 0.30/0.51        ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.30/0.51       => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.30/0.51         => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
% 0.30/0.51                ( X3
% 0.30/0.51               != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_comp_shift.cases
% 0.30/0.51  thf(fact_1139_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.51      ! [X3: produc5491161045314408544at_nat] :
% 0.30/0.51        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.30/0.51       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.30/0.51         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.30/0.51                ( X3
% 0.30/0.51               != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_comp_shift.cases
% 0.30/0.51  thf(fact_1140_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.51      ! [X3: produc7036089656553540234on_num] :
% 0.30/0.51        ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.30/0.51            ( X3
% 0.30/0.51           != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.30/0.51       => ( ! [Uw2: num > num > $o,V2: num] :
% 0.30/0.51              ( X3
% 0.30/0.51             != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.30/0.51         => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
% 0.30/0.51                ( X3
% 0.30/0.51               != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.option_comp_shift.cases
% 0.30/0.51  thf(fact_1141_Nat_OlessE,axiom,
% 0.30/0.51      ! [I: nat,K: nat] :
% 0.30/0.51        ( ( ord_less_nat @ I @ K )
% 0.30/0.51       => ( ( K
% 0.30/0.51           != ( suc @ I ) )
% 0.30/0.51         => ~ ! [J2: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I @ J2 )
% 0.30/0.51               => ( K
% 0.30/0.51                 != ( suc @ J2 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Nat.lessE
% 0.30/0.51  thf(fact_1142_Suc__lessD,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.30/0.51       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_lessD
% 0.30/0.51  thf(fact_1143_Suc__lessE,axiom,
% 0.30/0.51      ! [I: nat,K: nat] :
% 0.30/0.51        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.30/0.51       => ~ ! [J2: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I @ J2 )
% 0.30/0.51             => ( K
% 0.30/0.51               != ( suc @ J2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_lessE
% 0.30/0.51  thf(fact_1144_Suc__lessI,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ N )
% 0.30/0.51       => ( ( ( suc @ M )
% 0.30/0.51           != N )
% 0.30/0.51         => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_lessI
% 0.30/0.51  thf(fact_1145_less__SucE,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.51       => ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.51         => ( M = N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_SucE
% 0.30/0.51  thf(fact_1146_less__SucI,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ N )
% 0.30/0.51       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_SucI
% 0.30/0.51  thf(fact_1147_Ex__less__Suc,axiom,
% 0.30/0.51      ! [N: nat,P: nat > $o] :
% 0.30/0.51        ( ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.51              & ( P @ I4 ) ) )
% 0.30/0.51        = ( ( P @ N )
% 0.30/0.51          | ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ N )
% 0.30/0.51              & ( P @ I4 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Ex_less_Suc
% 0.30/0.51  thf(fact_1148_less__Suc__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.51        = ( ( ord_less_nat @ M @ N )
% 0.30/0.51          | ( M = N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_Suc_eq
% 0.30/0.51  thf(fact_1149_not__less__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.30/0.51        = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % not_less_eq
% 0.30/0.51  thf(fact_1150_All__less__Suc,axiom,
% 0.30/0.51      ! [N: nat,P: nat > $o] :
% 0.30/0.51        ( ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.51             => ( P @ I4 ) ) )
% 0.30/0.51        = ( ( P @ N )
% 0.30/0.51          & ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ N )
% 0.30/0.51             => ( P @ I4 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % All_less_Suc
% 0.30/0.51  thf(fact_1151_Suc__less__eq2,axiom,
% 0.30/0.51      ! [N: nat,M: nat] :
% 0.30/0.51        ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.51        = ( ? [M5: nat] :
% 0.30/0.51              ( ( M
% 0.30/0.51                = ( suc @ M5 ) )
% 0.30/0.51              & ( ord_less_nat @ N @ M5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_less_eq2
% 0.30/0.51  thf(fact_1152_less__antisym,axiom,
% 0.30/0.51      ! [N: nat,M: nat] :
% 0.30/0.51        ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.51       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.51         => ( M = N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_antisym
% 0.30/0.51  thf(fact_1153_Suc__less__SucD,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.51       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_less_SucD
% 0.30/0.51  thf(fact_1154_less__trans__Suc,axiom,
% 0.30/0.51      ! [I: nat,J: nat,K: nat] :
% 0.30/0.51        ( ( ord_less_nat @ I @ J )
% 0.30/0.51       => ( ( ord_less_nat @ J @ K )
% 0.30/0.51         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_trans_Suc
% 0.30/0.51  thf(fact_1155_less__Suc__induct,axiom,
% 0.30/0.51      ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.30/0.51        ( ( ord_less_nat @ I @ J )
% 0.30/0.51       => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.30/0.51         => ( ! [I3: nat,J2: nat,K3: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I3 @ J2 )
% 0.30/0.51               => ( ( ord_less_nat @ J2 @ K3 )
% 0.30/0.51                 => ( ( P @ I3 @ J2 )
% 0.30/0.51                   => ( ( P @ J2 @ K3 )
% 0.30/0.51                     => ( P @ I3 @ K3 ) ) ) ) )
% 0.30/0.51           => ( P @ I @ J ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_Suc_induct
% 0.30/0.51  thf(fact_1156_strict__inc__induct,axiom,
% 0.30/0.51      ! [I: nat,J: nat,P: nat > $o] :
% 0.30/0.51        ( ( ord_less_nat @ I @ J )
% 0.30/0.51       => ( ! [I3: nat] :
% 0.30/0.51              ( ( J
% 0.30/0.51                = ( suc @ I3 ) )
% 0.30/0.51             => ( P @ I3 ) )
% 0.30/0.51         => ( ! [I3: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I3 @ J )
% 0.30/0.51               => ( ( P @ ( suc @ I3 ) )
% 0.30/0.51                 => ( P @ I3 ) ) )
% 0.30/0.51           => ( P @ I ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % strict_inc_induct
% 0.30/0.51  thf(fact_1157_not__less__less__Suc__eq,axiom,
% 0.30/0.51      ! [N: nat,M: nat] :
% 0.30/0.51        ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.51       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.51          = ( N = M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % not_less_less_Suc_eq
% 0.30/0.51  thf(fact_1158_Suc__leD,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.51       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_leD
% 0.30/0.51  thf(fact_1159_le__SucE,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.51       => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.51         => ( M
% 0.30/0.51            = ( suc @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % le_SucE
% 0.30/0.51  thf(fact_1160_le__SucI,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.51       => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % le_SucI
% 0.30/0.51  thf(fact_1161_Suc__le__D,axiom,
% 0.30/0.51      ! [N: nat,M6: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 0.30/0.51       => ? [M4: nat] :
% 0.30/0.51            ( M6
% 0.30/0.51            = ( suc @ M4 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_le_D
% 0.30/0.51  thf(fact_1162_le__Suc__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.51        = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.51          | ( M
% 0.30/0.51            = ( suc @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % le_Suc_eq
% 0.30/0.51  thf(fact_1163_Suc__n__not__le__n,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_n_not_le_n
% 0.30/0.51  thf(fact_1164_not__less__eq__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.30/0.51        = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.30/0.51  
% 0.30/0.51  % not_less_eq_eq
% 0.30/0.51  thf(fact_1165_full__nat__induct,axiom,
% 0.30/0.51      ! [P: nat > $o,N: nat] :
% 0.30/0.51        ( ! [N3: nat] :
% 0.30/0.51            ( ! [M2: nat] :
% 0.30/0.51                ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
% 0.30/0.51               => ( P @ M2 ) )
% 0.30/0.51           => ( P @ N3 ) )
% 0.30/0.51       => ( P @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % full_nat_induct
% 0.30/0.51  thf(fact_1166_nat__induct__at__least,axiom,
% 0.30/0.51      ! [M: nat,N: nat,P: nat > $o] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.51       => ( ( P @ M )
% 0.30/0.51         => ( ! [N3: nat] :
% 0.30/0.51                ( ( ord_less_eq_nat @ M @ N3 )
% 0.30/0.51               => ( ( P @ N3 )
% 0.30/0.51                 => ( P @ ( suc @ N3 ) ) ) )
% 0.30/0.51           => ( P @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nat_induct_at_least
% 0.30/0.51  thf(fact_1167_transitive__stepwise__le,axiom,
% 0.30/0.51      ! [M: nat,N: nat,R2: nat > nat > $o] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.51       => ( ! [X5: nat] : ( R2 @ X5 @ X5 )
% 0.30/0.51         => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 0.30/0.51                ( ( R2 @ X5 @ Y4 )
% 0.30/0.51               => ( ( R2 @ Y4 @ Z4 )
% 0.30/0.51                 => ( R2 @ X5 @ Z4 ) ) )
% 0.30/0.51           => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
% 0.30/0.51             => ( R2 @ M @ N ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % transitive_stepwise_le
% 0.30/0.51  thf(fact_1168_nat__arith_Osuc1,axiom,
% 0.30/0.51      ! [A4: nat,K: nat,A: nat] :
% 0.30/0.51        ( ( A4
% 0.30/0.51          = ( plus_plus_nat @ K @ A ) )
% 0.30/0.51       => ( ( suc @ A4 )
% 0.30/0.51          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nat_arith.suc1
% 0.30/0.51  thf(fact_1169_add__Suc,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.30/0.51        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_Suc
% 0.30/0.51  thf(fact_1170_add__Suc__shift,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.30/0.51        = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_Suc_shift
% 0.30/0.51  thf(fact_1171_zero__induct__lemma,axiom,
% 0.30/0.51      ! [P: nat > $o,K: nat,I: nat] :
% 0.30/0.51        ( ( P @ K )
% 0.30/0.51       => ( ! [N3: nat] :
% 0.30/0.51              ( ( P @ ( suc @ N3 ) )
% 0.30/0.51             => ( P @ N3 ) )
% 0.30/0.51         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % zero_induct_lemma
% 0.30/0.51  thf(fact_1172_Suc__mult__cancel1,axiom,
% 0.30/0.51      ! [K: nat,M: nat,N: nat] :
% 0.30/0.51        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.30/0.51          = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.30/0.51        = ( M = N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_mult_cancel1
% 0.30/0.51  thf(fact_1173_numeral__code_I2_J,axiom,
% 0.30/0.51      ! [N: num] :
% 0.30/0.51        ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.30/0.51        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % numeral_code(2)
% 0.30/0.51  thf(fact_1174_numeral__code_I2_J,axiom,
% 0.30/0.51      ! [N: num] :
% 0.30/0.51        ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.30/0.51        = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % numeral_code(2)
% 0.30/0.51  thf(fact_1175_numeral__code_I2_J,axiom,
% 0.30/0.51      ! [N: num] :
% 0.30/0.51        ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.30/0.51        = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % numeral_code(2)
% 0.30/0.51  thf(fact_1176_numeral__code_I2_J,axiom,
% 0.30/0.51      ! [N: num] :
% 0.30/0.51        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.30/0.51        = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % numeral_code(2)
% 0.30/0.51  thf(fact_1177_numeral__code_I2_J,axiom,
% 0.30/0.51      ! [N: num] :
% 0.30/0.51        ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.30/0.51        = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % numeral_code(2)
% 0.30/0.51  thf(fact_1178_set__vebt__def,axiom,
% 0.30/0.51      ( vEBT_set_vebt
% 0.30/0.51      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_vebt_def
% 0.30/0.51  thf(fact_1179_real__arch__pow,axiom,
% 0.30/0.51      ! [X3: real,Y3: real] :
% 0.30/0.51        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.51       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X3 @ N3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % real_arch_pow
% 0.30/0.51  thf(fact_1180_power__numeral__even,axiom,
% 0.30/0.51      ! [Z: complex,W: num] :
% 0.30/0.51        ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51        = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_numeral_even
% 0.30/0.51  thf(fact_1181_power__numeral__even,axiom,
% 0.30/0.51      ! [Z: real,W: num] :
% 0.30/0.51        ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51        = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_numeral_even
% 0.30/0.51  thf(fact_1182_power__numeral__even,axiom,
% 0.30/0.51      ! [Z: rat,W: num] :
% 0.30/0.51        ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51        = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_numeral_even
% 0.30/0.51  thf(fact_1183_power__numeral__even,axiom,
% 0.30/0.51      ! [Z: nat,W: num] :
% 0.30/0.51        ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51        = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_numeral_even
% 0.30/0.51  thf(fact_1184_power__numeral__even,axiom,
% 0.30/0.51      ! [Z: int,W: num] :
% 0.30/0.51        ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.51        = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_numeral_even
% 0.30/0.51  thf(fact_1185_set__conv__nth,axiom,
% 0.30/0.51      ( set_complex2
% 0.30/0.51      = ( ^ [Xs: list_complex] :
% 0.30/0.51            ( collect_complex
% 0.30/0.51            @ ^ [Uu3: complex] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_complex @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1186_set__conv__nth,axiom,
% 0.30/0.51      ( set_real2
% 0.30/0.51      = ( ^ [Xs: list_real] :
% 0.30/0.51            ( collect_real
% 0.30/0.51            @ ^ [Uu3: real] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_real @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1187_set__conv__nth,axiom,
% 0.30/0.51      ( set_list_nat2
% 0.30/0.51      = ( ^ [Xs: list_list_nat] :
% 0.30/0.51            ( collect_list_nat
% 0.30/0.51            @ ^ [Uu3: list_nat] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_list_nat @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1188_set__conv__nth,axiom,
% 0.30/0.51      ( set_int2
% 0.30/0.51      = ( ^ [Xs: list_int] :
% 0.30/0.51            ( collect_int
% 0.30/0.51            @ ^ [Uu3: int] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_int @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1189_set__conv__nth,axiom,
% 0.30/0.51      ( set_VEBT_VEBT2
% 0.30/0.51      = ( ^ [Xs: list_VEBT_VEBT] :
% 0.30/0.51            ( collect_VEBT_VEBT
% 0.30/0.51            @ ^ [Uu3: vEBT_VEBT] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1190_set__conv__nth,axiom,
% 0.30/0.51      ( set_o2
% 0.30/0.51      = ( ^ [Xs: list_o] :
% 0.30/0.51            ( collect_o
% 0.30/0.51            @ ^ [Uu3: $o] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_o @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1191_set__conv__nth,axiom,
% 0.30/0.51      ( set_nat2
% 0.30/0.51      = ( ^ [Xs: list_nat] :
% 0.30/0.51            ( collect_nat
% 0.30/0.51            @ ^ [Uu3: nat] :
% 0.30/0.51              ? [I4: nat] :
% 0.30/0.51                ( ( Uu3
% 0.30/0.51                  = ( nth_nat @ Xs @ I4 ) )
% 0.30/0.51                & ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % set_conv_nth
% 0.30/0.51  thf(fact_1192_lift__Suc__mono__less,axiom,
% 0.30/0.51      ! [F: nat > real,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less
% 0.30/0.51  thf(fact_1193_lift__Suc__mono__less,axiom,
% 0.30/0.51      ! [F: nat > rat,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less
% 0.30/0.51  thf(fact_1194_lift__Suc__mono__less,axiom,
% 0.30/0.51      ! [F: nat > num,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less
% 0.30/0.51  thf(fact_1195_lift__Suc__mono__less,axiom,
% 0.30/0.51      ! [F: nat > nat,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less
% 0.30/0.51  thf(fact_1196_lift__Suc__mono__less,axiom,
% 0.30/0.51      ! [F: nat > int,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less
% 0.30/0.51  thf(fact_1197_lift__Suc__mono__less__iff,axiom,
% 0.30/0.51      ! [F: nat > real,N: nat,M: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.51          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less_iff
% 0.30/0.51  thf(fact_1198_lift__Suc__mono__less__iff,axiom,
% 0.30/0.51      ! [F: nat > rat,N: nat,M: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.51          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less_iff
% 0.30/0.51  thf(fact_1199_lift__Suc__mono__less__iff,axiom,
% 0.30/0.51      ! [F: nat > num,N: nat,M: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.51          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less_iff
% 0.30/0.51  thf(fact_1200_lift__Suc__mono__less__iff,axiom,
% 0.30/0.51      ! [F: nat > nat,N: nat,M: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.51          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less_iff
% 0.30/0.51  thf(fact_1201_lift__Suc__mono__less__iff,axiom,
% 0.30/0.51      ! [F: nat > int,N: nat,M: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.30/0.51          = ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_less_iff
% 0.30/0.51  thf(fact_1202_power__Suc2,axiom,
% 0.30/0.51      ! [A: complex,N: nat] :
% 0.30/0.51        ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc2
% 0.30/0.51  thf(fact_1203_power__Suc2,axiom,
% 0.30/0.51      ! [A: real,N: nat] :
% 0.30/0.51        ( ( power_power_real @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc2
% 0.30/0.51  thf(fact_1204_power__Suc2,axiom,
% 0.30/0.51      ! [A: rat,N: nat] :
% 0.30/0.51        ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc2
% 0.30/0.51  thf(fact_1205_power__Suc2,axiom,
% 0.30/0.51      ! [A: nat,N: nat] :
% 0.30/0.51        ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc2
% 0.30/0.51  thf(fact_1206_power__Suc2,axiom,
% 0.30/0.51      ! [A: int,N: nat] :
% 0.30/0.51        ( ( power_power_int @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc2
% 0.30/0.51  thf(fact_1207_power__Suc,axiom,
% 0.30/0.51      ! [A: complex,N: nat] :
% 0.30/0.51        ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc
% 0.30/0.51  thf(fact_1208_power__Suc,axiom,
% 0.30/0.51      ! [A: real,N: nat] :
% 0.30/0.51        ( ( power_power_real @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc
% 0.30/0.51  thf(fact_1209_power__Suc,axiom,
% 0.30/0.51      ! [A: rat,N: nat] :
% 0.30/0.51        ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc
% 0.30/0.51  thf(fact_1210_power__Suc,axiom,
% 0.30/0.51      ! [A: nat,N: nat] :
% 0.30/0.51        ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc
% 0.30/0.51  thf(fact_1211_power__Suc,axiom,
% 0.30/0.51      ! [A: int,N: nat] :
% 0.30/0.51        ( ( power_power_int @ A @ ( suc @ N ) )
% 0.30/0.51        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_Suc
% 0.30/0.51  thf(fact_1212_lift__Suc__mono__le,axiom,
% 0.30/0.51      ! [F: nat > set_int,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_le
% 0.30/0.51  thf(fact_1213_lift__Suc__mono__le,axiom,
% 0.30/0.51      ! [F: nat > rat,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_le
% 0.30/0.51  thf(fact_1214_lift__Suc__mono__le,axiom,
% 0.30/0.51      ! [F: nat > num,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_le
% 0.30/0.51  thf(fact_1215_lift__Suc__mono__le,axiom,
% 0.30/0.51      ! [F: nat > nat,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_le
% 0.30/0.51  thf(fact_1216_lift__Suc__mono__le,axiom,
% 0.30/0.51      ! [F: nat > int,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_mono_le
% 0.30/0.51  thf(fact_1217_lift__Suc__antimono__le,axiom,
% 0.30/0.51      ! [F: nat > set_int,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_antimono_le
% 0.30/0.51  thf(fact_1218_lift__Suc__antimono__le,axiom,
% 0.30/0.51      ! [F: nat > rat,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_antimono_le
% 0.30/0.51  thf(fact_1219_lift__Suc__antimono__le,axiom,
% 0.30/0.51      ! [F: nat > num,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_antimono_le
% 0.30/0.51  thf(fact_1220_lift__Suc__antimono__le,axiom,
% 0.30/0.51      ! [F: nat > nat,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_antimono_le
% 0.30/0.51  thf(fact_1221_lift__Suc__antimono__le,axiom,
% 0.30/0.51      ! [F: nat > int,N: nat,N5: nat] :
% 0.30/0.51        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.30/0.51       => ( ( ord_less_eq_nat @ N @ N5 )
% 0.30/0.51         => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % lift_Suc_antimono_le
% 0.30/0.51  thf(fact_1222_Suc__leI,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ N )
% 0.30/0.51       => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_leI
% 0.30/0.51  thf(fact_1223_Suc__le__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.51        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_le_eq
% 0.30/0.51  thf(fact_1224_dec__induct,axiom,
% 0.30/0.51      ! [I: nat,J: nat,P: nat > $o] :
% 0.30/0.51        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.51       => ( ( P @ I )
% 0.30/0.51         => ( ! [N3: nat] :
% 0.30/0.51                ( ( ord_less_eq_nat @ I @ N3 )
% 0.30/0.51               => ( ( ord_less_nat @ N3 @ J )
% 0.30/0.51                 => ( ( P @ N3 )
% 0.30/0.51                   => ( P @ ( suc @ N3 ) ) ) ) )
% 0.30/0.51           => ( P @ J ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % dec_induct
% 0.30/0.51  thf(fact_1225_inc__induct,axiom,
% 0.30/0.51      ! [I: nat,J: nat,P: nat > $o] :
% 0.30/0.51        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.51       => ( ( P @ J )
% 0.30/0.51         => ( ! [N3: nat] :
% 0.30/0.51                ( ( ord_less_eq_nat @ I @ N3 )
% 0.30/0.51               => ( ( ord_less_nat @ N3 @ J )
% 0.30/0.51                 => ( ( P @ ( suc @ N3 ) )
% 0.30/0.51                   => ( P @ N3 ) ) ) )
% 0.30/0.51           => ( P @ I ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % inc_induct
% 0.30/0.51  thf(fact_1226_Suc__le__lessD,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.51       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_le_lessD
% 0.30/0.51  thf(fact_1227_le__less__Suc__eq,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.51       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.51          = ( N = M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % le_less_Suc_eq
% 0.30/0.51  thf(fact_1228_less__Suc__eq__le,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.51        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_Suc_eq_le
% 0.30/0.51  thf(fact_1229_less__eq__Suc__le,axiom,
% 0.30/0.51      ( ord_less_nat
% 0.30/0.51      = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_eq_Suc_le
% 0.30/0.51  thf(fact_1230_le__imp__less__Suc,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.51       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % le_imp_less_Suc
% 0.30/0.51  thf(fact_1231_less__imp__Suc__add,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ N )
% 0.30/0.51       => ? [K3: nat] :
% 0.30/0.51            ( N
% 0.30/0.51            = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_imp_Suc_add
% 0.30/0.51  thf(fact_1232_less__iff__Suc__add,axiom,
% 0.30/0.51      ( ord_less_nat
% 0.30/0.51      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.51          ? [K2: nat] :
% 0.30/0.51            ( N2
% 0.30/0.51            = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_iff_Suc_add
% 0.30/0.51  thf(fact_1233_less__add__Suc2,axiom,
% 0.30/0.51      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_add_Suc2
% 0.30/0.51  thf(fact_1234_less__add__Suc1,axiom,
% 0.30/0.51      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_add_Suc1
% 0.30/0.51  thf(fact_1235_less__natE,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ M @ N )
% 0.30/0.51       => ~ ! [Q3: nat] :
% 0.30/0.51              ( N
% 0.30/0.51             != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % less_natE
% 0.30/0.51  thf(fact_1236_diff__less__Suc,axiom,
% 0.30/0.51      ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_less_Suc
% 0.30/0.51  thf(fact_1237_Suc__diff__Suc,axiom,
% 0.30/0.51      ! [N: nat,M: nat] :
% 0.30/0.51        ( ( ord_less_nat @ N @ M )
% 0.30/0.51       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.30/0.51          = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_diff_Suc
% 0.30/0.51  thf(fact_1238_Suc__mult__less__cancel1,axiom,
% 0.30/0.51      ! [K: nat,M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.30/0.51        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_mult_less_cancel1
% 0.30/0.51  thf(fact_1239_Suc__diff__le,axiom,
% 0.30/0.51      ! [N: nat,M: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.51       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.30/0.51          = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_diff_le
% 0.30/0.51  thf(fact_1240_Suc__mult__le__cancel1,axiom,
% 0.30/0.51      ! [K: nat,M: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.30/0.51        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_mult_le_cancel1
% 0.30/0.51  thf(fact_1241_mult__Suc,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.30/0.51        = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_Suc
% 0.30/0.51  thf(fact_1242_Suc__div__le__mono,axiom,
% 0.30/0.51      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_div_le_mono
% 0.30/0.51  thf(fact_1243_Suc__eq__plus1__left,axiom,
% 0.30/0.51      ( suc
% 0.30/0.51      = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_eq_plus1_left
% 0.30/0.51  thf(fact_1244_plus__1__eq__Suc,axiom,
% 0.30/0.51      ( ( plus_plus_nat @ one_one_nat )
% 0.30/0.51      = suc ) ).
% 0.30/0.51  
% 0.30/0.51  % plus_1_eq_Suc
% 0.30/0.51  thf(fact_1245_Suc__eq__plus1,axiom,
% 0.30/0.51      ( suc
% 0.30/0.51      = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_eq_plus1
% 0.30/0.51  thf(fact_1246_diff__Suc__eq__diff__pred,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.30/0.51        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_Suc_eq_diff_pred
% 0.30/0.51  thf(fact_1247_power__gt1,axiom,
% 0.30/0.51      ! [A: real,N: nat] :
% 0.30/0.51        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.51       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_gt1
% 0.30/0.51  thf(fact_1248_power__gt1,axiom,
% 0.30/0.51      ! [A: rat,N: nat] :
% 0.30/0.51        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.51       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_gt1
% 0.30/0.51  thf(fact_1249_power__gt1,axiom,
% 0.30/0.51      ! [A: nat,N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.51       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_gt1
% 0.30/0.51  thf(fact_1250_power__gt1,axiom,
% 0.30/0.51      ! [A: int,N: nat] :
% 0.30/0.51        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.51       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_gt1
% 0.30/0.51  thf(fact_1251_subset__code_I1_J,axiom,
% 0.30/0.51      ! [Xs2: list_real,B5: set_real] :
% 0.30/0.51        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
% 0.30/0.51        = ( ! [X: real] :
% 0.30/0.51              ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 0.30/0.51             => ( member_real @ X @ B5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % subset_code(1)
% 0.30/0.51  thf(fact_1252_subset__code_I1_J,axiom,
% 0.30/0.51      ! [Xs2: list_complex,B5: set_complex] :
% 0.30/0.51        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B5 )
% 0.30/0.51        = ( ! [X: complex] :
% 0.30/0.51              ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 0.30/0.51             => ( member_complex @ X @ B5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % subset_code(1)
% 0.30/0.51  thf(fact_1253_subset__code_I1_J,axiom,
% 0.30/0.51      ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
% 0.30/0.51        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
% 0.30/0.51        = ( ! [X: vEBT_VEBT] :
% 0.30/0.51              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51             => ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % subset_code(1)
% 0.30/0.51  thf(fact_1254_subset__code_I1_J,axiom,
% 0.30/0.51      ! [Xs2: list_nat,B5: set_nat] :
% 0.30/0.51        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
% 0.30/0.51        = ( ! [X: nat] :
% 0.30/0.51              ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51             => ( member_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % subset_code(1)
% 0.30/0.51  thf(fact_1255_subset__code_I1_J,axiom,
% 0.30/0.51      ! [Xs2: list_int,B5: set_int] :
% 0.30/0.51        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
% 0.30/0.51        = ( ! [X: int] :
% 0.30/0.51              ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.30/0.51             => ( member_int @ X @ B5 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % subset_code(1)
% 0.30/0.51  thf(fact_1256_Ex__list__of__length,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51      ? [Xs3: list_VEBT_VEBT] :
% 0.30/0.51        ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.30/0.51        = N ) ).
% 0.30/0.51  
% 0.30/0.51  % Ex_list_of_length
% 0.30/0.51  thf(fact_1257_Ex__list__of__length,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51      ? [Xs3: list_o] :
% 0.30/0.51        ( ( size_size_list_o @ Xs3 )
% 0.30/0.51        = N ) ).
% 0.30/0.51  
% 0.30/0.51  % Ex_list_of_length
% 0.30/0.51  thf(fact_1258_Ex__list__of__length,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51      ? [Xs3: list_nat] :
% 0.30/0.51        ( ( size_size_list_nat @ Xs3 )
% 0.30/0.51        = N ) ).
% 0.30/0.51  
% 0.30/0.51  % Ex_list_of_length
% 0.30/0.51  thf(fact_1259_neq__if__length__neq,axiom,
% 0.30/0.51      ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.51        ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.51         != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.30/0.51       => ( Xs2 != Ys ) ) ).
% 0.30/0.51  
% 0.30/0.51  % neq_if_length_neq
% 0.30/0.51  thf(fact_1260_neq__if__length__neq,axiom,
% 0.30/0.51      ! [Xs2: list_o,Ys: list_o] :
% 0.30/0.51        ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.51         != ( size_size_list_o @ Ys ) )
% 0.30/0.51       => ( Xs2 != Ys ) ) ).
% 0.30/0.51  
% 0.30/0.51  % neq_if_length_neq
% 0.30/0.51  thf(fact_1261_neq__if__length__neq,axiom,
% 0.30/0.51      ! [Xs2: list_nat,Ys: list_nat] :
% 0.30/0.51        ( ( ( size_size_list_nat @ Xs2 )
% 0.30/0.51         != ( size_size_list_nat @ Ys ) )
% 0.30/0.51       => ( Xs2 != Ys ) ) ).
% 0.30/0.51  
% 0.30/0.51  % neq_if_length_neq
% 0.30/0.51  thf(fact_1262_div__nat__eqI,axiom,
% 0.30/0.51      ! [N: nat,Q2: nat,M: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 0.30/0.51       => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 0.30/0.51         => ( ( divide_divide_nat @ M @ N )
% 0.30/0.51            = Q2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % div_nat_eqI
% 0.30/0.51  thf(fact_1263_Suc__nat__number__of__add,axiom,
% 0.30/0.51      ! [V: num,N: nat] :
% 0.30/0.51        ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.30/0.51        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_nat_number_of_add
% 0.30/0.51  thf(fact_1264_two__realpow__ge__one,axiom,
% 0.30/0.51      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % two_realpow_ge_one
% 0.30/0.51  thf(fact_1265_power__odd__eq,axiom,
% 0.30/0.51      ! [A: complex,N: nat] :
% 0.30/0.51        ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.51        = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_odd_eq
% 0.30/0.51  thf(fact_1266_power__odd__eq,axiom,
% 0.30/0.51      ! [A: real,N: nat] :
% 0.30/0.51        ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.51        = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_odd_eq
% 0.30/0.51  thf(fact_1267_power__odd__eq,axiom,
% 0.30/0.51      ! [A: rat,N: nat] :
% 0.30/0.51        ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.51        = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_odd_eq
% 0.30/0.51  thf(fact_1268_power__odd__eq,axiom,
% 0.30/0.51      ! [A: nat,N: nat] :
% 0.30/0.51        ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.51        = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_odd_eq
% 0.30/0.51  thf(fact_1269_power__odd__eq,axiom,
% 0.30/0.51      ! [A: int,N: nat] :
% 0.30/0.51        ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.51        = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % power_odd_eq
% 0.30/0.51  thf(fact_1270_length__induct,axiom,
% 0.30/0.51      ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 0.30/0.51        ( ! [Xs3: list_VEBT_VEBT] :
% 0.30/0.51            ( ! [Ys2: list_VEBT_VEBT] :
% 0.30/0.51                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.30/0.51               => ( P @ Ys2 ) )
% 0.30/0.51           => ( P @ Xs3 ) )
% 0.30/0.51       => ( P @ Xs2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % length_induct
% 0.30/0.51  thf(fact_1271_length__induct,axiom,
% 0.30/0.51      ! [P: list_o > $o,Xs2: list_o] :
% 0.30/0.51        ( ! [Xs3: list_o] :
% 0.30/0.51            ( ! [Ys2: list_o] :
% 0.30/0.51                ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 0.30/0.51               => ( P @ Ys2 ) )
% 0.30/0.51           => ( P @ Xs3 ) )
% 0.30/0.51       => ( P @ Xs2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % length_induct
% 0.30/0.51  thf(fact_1272_length__induct,axiom,
% 0.30/0.51      ! [P: list_nat > $o,Xs2: list_nat] :
% 0.30/0.51        ( ! [Xs3: list_nat] :
% 0.30/0.51            ( ! [Ys2: list_nat] :
% 0.30/0.51                ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 0.30/0.51               => ( P @ Ys2 ) )
% 0.30/0.51           => ( P @ Xs3 ) )
% 0.30/0.51       => ( P @ Xs2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % length_induct
% 0.30/0.51  thf(fact_1273_vebt__pred_Osimps_I7_J,axiom,
% 0.30/0.51      ! [Ma: nat,X3: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51        ( ( ( ord_less_nat @ Ma @ X3 )
% 0.30/0.51         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.51            = ( some_nat @ Ma ) ) )
% 0.30/0.51        & ( ~ ( ord_less_nat @ Ma @ X3 )
% 0.30/0.51         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.51            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51              @ ( if_option_nat
% 0.30/0.51                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51                   != none_nat )
% 0.30/0.51                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( 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 ) ) ) ) ) )
% 0.30/0.51                @ ( if_option_nat
% 0.30/0.51                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51                    = none_nat )
% 0.30/0.51                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
% 0.30/0.51                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51              @ none_nat ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % vebt_pred.simps(7)
% 0.30/0.51  thf(fact_1274_nth__equalityI,axiom,
% 0.30/0.51      ! [Xs2: list_int,Ys: list_int] :
% 0.30/0.51        ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.51          = ( size_size_list_int @ Ys ) )
% 0.30/0.51       => ( ! [I3: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51             => ( ( nth_int @ Xs2 @ I3 )
% 0.30/0.51                = ( nth_int @ Ys @ I3 ) ) )
% 0.30/0.51         => ( Xs2 = Ys ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_equalityI
% 0.30/0.51  thf(fact_1275_nth__equalityI,axiom,
% 0.30/0.51      ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.51        ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.51          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.30/0.51       => ( ! [I3: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51             => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 0.30/0.51                = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
% 0.30/0.51         => ( Xs2 = Ys ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_equalityI
% 0.30/0.51  thf(fact_1276_nth__equalityI,axiom,
% 0.30/0.51      ! [Xs2: list_o,Ys: list_o] :
% 0.30/0.51        ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.51          = ( size_size_list_o @ Ys ) )
% 0.30/0.51       => ( ! [I3: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51             => ( ( nth_o @ Xs2 @ I3 )
% 0.30/0.51                = ( nth_o @ Ys @ I3 ) ) )
% 0.30/0.51         => ( Xs2 = Ys ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_equalityI
% 0.30/0.51  thf(fact_1277_nth__equalityI,axiom,
% 0.30/0.51      ! [Xs2: list_nat,Ys: list_nat] :
% 0.30/0.51        ( ( ( size_size_list_nat @ Xs2 )
% 0.30/0.51          = ( size_size_list_nat @ Ys ) )
% 0.30/0.51       => ( ! [I3: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51             => ( ( nth_nat @ Xs2 @ I3 )
% 0.30/0.51                = ( nth_nat @ Ys @ I3 ) ) )
% 0.30/0.51         => ( Xs2 = Ys ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_equalityI
% 0.30/0.51  thf(fact_1278_Skolem__list__nth,axiom,
% 0.30/0.51      ! [K: nat,P: nat > int > $o] :
% 0.30/0.51        ( ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51             => ? [X7: int] : ( P @ I4 @ X7 ) ) )
% 0.30/0.51        = ( ? [Xs: list_int] :
% 0.30/0.51              ( ( ( size_size_list_int @ Xs )
% 0.30/0.51                = K )
% 0.30/0.51              & ! [I4: nat] :
% 0.30/0.51                  ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51                 => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Skolem_list_nth
% 0.30/0.51  thf(fact_1279_Skolem__list__nth,axiom,
% 0.30/0.51      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.30/0.51        ( ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51             => ? [X7: vEBT_VEBT] : ( P @ I4 @ X7 ) ) )
% 0.30/0.51        = ( ? [Xs: list_VEBT_VEBT] :
% 0.30/0.51              ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.51                = K )
% 0.30/0.51              & ! [I4: nat] :
% 0.30/0.51                  ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51                 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Skolem_list_nth
% 0.30/0.51  thf(fact_1280_Skolem__list__nth,axiom,
% 0.30/0.51      ! [K: nat,P: nat > $o > $o] :
% 0.30/0.51        ( ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51             => ? [X7: $o] : ( P @ I4 @ X7 ) ) )
% 0.30/0.51        = ( ? [Xs: list_o] :
% 0.30/0.51              ( ( ( size_size_list_o @ Xs )
% 0.30/0.51                = K )
% 0.30/0.51              & ! [I4: nat] :
% 0.30/0.51                  ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51                 => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Skolem_list_nth
% 0.30/0.51  thf(fact_1281_Skolem__list__nth,axiom,
% 0.30/0.51      ! [K: nat,P: nat > nat > $o] :
% 0.30/0.51        ( ( ! [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51             => ? [X7: nat] : ( P @ I4 @ X7 ) ) )
% 0.30/0.51        = ( ? [Xs: list_nat] :
% 0.30/0.51              ( ( ( size_size_list_nat @ Xs )
% 0.30/0.51                = K )
% 0.30/0.51              & ! [I4: nat] :
% 0.30/0.51                  ( ( ord_less_nat @ I4 @ K )
% 0.30/0.51                 => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Skolem_list_nth
% 0.30/0.51  thf(fact_1282_list__eq__iff__nth__eq,axiom,
% 0.30/0.51      ( ( ^ [Y6: list_int,Z5: list_int] : ( Y6 = Z5 ) )
% 0.30/0.51      = ( ^ [Xs: list_int,Ys3: list_int] :
% 0.30/0.51            ( ( ( size_size_list_int @ Xs )
% 0.30/0.51              = ( size_size_list_int @ Ys3 ) )
% 0.30/0.51            & ! [I4: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.30/0.51               => ( ( nth_int @ Xs @ I4 )
% 0.30/0.51                  = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_eq_iff_nth_eq
% 0.30/0.51  thf(fact_1283_list__eq__iff__nth__eq,axiom,
% 0.30/0.51      ( ( ^ [Y6: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : ( Y6 = Z5 ) )
% 0.30/0.51      = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.30/0.51            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.51              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.30/0.51            & ! [I4: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.51               => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 0.30/0.51                  = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_eq_iff_nth_eq
% 0.30/0.51  thf(fact_1284_list__eq__iff__nth__eq,axiom,
% 0.30/0.51      ( ( ^ [Y6: list_o,Z5: list_o] : ( Y6 = Z5 ) )
% 0.30/0.51      = ( ^ [Xs: list_o,Ys3: list_o] :
% 0.30/0.51            ( ( ( size_size_list_o @ Xs )
% 0.30/0.51              = ( size_size_list_o @ Ys3 ) )
% 0.30/0.51            & ! [I4: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.30/0.51               => ( ( nth_o @ Xs @ I4 )
% 0.30/0.51                  = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_eq_iff_nth_eq
% 0.30/0.51  thf(fact_1285_list__eq__iff__nth__eq,axiom,
% 0.30/0.51      ( ( ^ [Y6: list_nat,Z5: list_nat] : ( Y6 = Z5 ) )
% 0.30/0.51      = ( ^ [Xs: list_nat,Ys3: list_nat] :
% 0.30/0.51            ( ( ( size_size_list_nat @ Xs )
% 0.30/0.51              = ( size_size_list_nat @ Ys3 ) )
% 0.30/0.51            & ! [I4: nat] :
% 0.30/0.51                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.51               => ( ( nth_nat @ Xs @ I4 )
% 0.30/0.51                  = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_eq_iff_nth_eq
% 0.30/0.51  thf(fact_1286_invar__vebt_Ointros_I3_J,axiom,
% 0.30/0.51      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.30/0.51        ( ! [X5: vEBT_VEBT] :
% 0.30/0.51            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51           => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.30/0.51       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.51         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.51              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51           => ( ( M
% 0.30/0.51                = ( suc @ N ) )
% 0.30/0.51             => ( ( Deg
% 0.30/0.51                  = ( plus_plus_nat @ N @ M ) )
% 0.30/0.51               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.30/0.51                 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.51                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 0.30/0.51                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % invar_vebt.intros(3)
% 0.30/0.51  thf(fact_1287_discrete,axiom,
% 0.30/0.51      ( ord_less_nat
% 0.30/0.51      = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % discrete
% 0.30/0.51  thf(fact_1288_discrete,axiom,
% 0.30/0.51      ( ord_less_int
% 0.30/0.51      = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % discrete
% 0.30/0.51  thf(fact_1289_nth__mem,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_real] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.51       => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_mem
% 0.30/0.51  thf(fact_1290_nth__mem,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_complex] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.51       => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_mem
% 0.30/0.51  thf(fact_1291_nth__mem,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_int] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51       => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_mem
% 0.30/0.51  thf(fact_1292_nth__mem,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_VEBT_VEBT] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_mem
% 0.30/0.51  thf(fact_1293_nth__mem,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_o] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51       => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_mem
% 0.30/0.51  thf(fact_1294_nth__mem,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_nat] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51       => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % nth_mem
% 0.30/0.51  thf(fact_1295_list__ball__nth,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_int,P: int > $o] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51       => ( ! [X5: int] :
% 0.30/0.51              ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X5 ) )
% 0.30/0.51         => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_ball_nth
% 0.30/0.51  thf(fact_1296_list__ball__nth,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51       => ( ! [X5: vEBT_VEBT] :
% 0.30/0.51              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X5 ) )
% 0.30/0.51         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_ball_nth
% 0.30/0.51  thf(fact_1297_list__ball__nth,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_o,P: $o > $o] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51       => ( ! [X5: $o] :
% 0.30/0.51              ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X5 ) )
% 0.30/0.51         => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_ball_nth
% 0.30/0.51  thf(fact_1298_list__ball__nth,axiom,
% 0.30/0.51      ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 0.30/0.51        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51       => ( ! [X5: nat] :
% 0.30/0.51              ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51             => ( P @ X5 ) )
% 0.30/0.51         => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % list_ball_nth
% 0.30/0.51  thf(fact_1299_in__set__conv__nth,axiom,
% 0.30/0.51      ! [X3: real,Xs2: list_real] :
% 0.30/0.51        ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 0.30/0.51        = ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.51              & ( ( nth_real @ Xs2 @ I4 )
% 0.30/0.51                = X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % in_set_conv_nth
% 0.30/0.51  thf(fact_1300_in__set__conv__nth,axiom,
% 0.30/0.51      ! [X3: complex,Xs2: list_complex] :
% 0.30/0.51        ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.51        = ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.51              & ( ( nth_complex @ Xs2 @ I4 )
% 0.30/0.51                = X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % in_set_conv_nth
% 0.30/0.51  thf(fact_1301_in__set__conv__nth,axiom,
% 0.30/0.51      ! [X3: int,Xs2: list_int] :
% 0.30/0.51        ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.30/0.51        = ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51              & ( ( nth_int @ Xs2 @ I4 )
% 0.30/0.51                = X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % in_set_conv_nth
% 0.30/0.51  thf(fact_1302_in__set__conv__nth,axiom,
% 0.30/0.51      ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.30/0.51        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51        = ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51              & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 0.30/0.51                = X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % in_set_conv_nth
% 0.30/0.51  thf(fact_1303_in__set__conv__nth,axiom,
% 0.30/0.51      ! [X3: $o,Xs2: list_o] :
% 0.30/0.51        ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.30/0.51        = ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51              & ( ( nth_o @ Xs2 @ I4 )
% 0.30/0.51                = X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % in_set_conv_nth
% 0.30/0.51  thf(fact_1304_in__set__conv__nth,axiom,
% 0.30/0.51      ! [X3: nat,Xs2: list_nat] :
% 0.30/0.51        ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51        = ( ? [I4: nat] :
% 0.30/0.51              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51              & ( ( nth_nat @ Xs2 @ I4 )
% 0.30/0.51                = X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % in_set_conv_nth
% 0.30/0.51  thf(fact_1305_all__nth__imp__all__set,axiom,
% 0.30/0.51      ! [Xs2: list_real,P: real > $o,X3: real] :
% 0.30/0.51        ( ! [I3: nat] :
% 0.30/0.51            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.51           => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
% 0.30/0.51       => ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 0.30/0.51         => ( P @ X3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_nth_imp_all_set
% 0.30/0.51  thf(fact_1306_all__nth__imp__all__set,axiom,
% 0.30/0.51      ! [Xs2: list_complex,P: complex > $o,X3: complex] :
% 0.30/0.51        ( ! [I3: nat] :
% 0.30/0.51            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.51           => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
% 0.30/0.51       => ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.51         => ( P @ X3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_nth_imp_all_set
% 0.30/0.51  thf(fact_1307_all__nth__imp__all__set,axiom,
% 0.30/0.51      ! [Xs2: list_int,P: int > $o,X3: int] :
% 0.30/0.51        ( ! [I3: nat] :
% 0.30/0.51            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.51           => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
% 0.30/0.51       => ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.30/0.51         => ( P @ X3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_nth_imp_all_set
% 0.30/0.51  thf(fact_1308_all__nth__imp__all__set,axiom,
% 0.30/0.51      ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 0.30/0.51        ( ! [I3: nat] :
% 0.30/0.51            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.51           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
% 0.30/0.51       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.51         => ( P @ X3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_nth_imp_all_set
% 0.30/0.51  thf(fact_1309_all__nth__imp__all__set,axiom,
% 0.30/0.51      ! [Xs2: list_o,P: $o > $o,X3: $o] :
% 0.30/0.51        ( ! [I3: nat] :
% 0.30/0.51            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.51           => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
% 0.30/0.51       => ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.30/0.51         => ( P @ X3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_nth_imp_all_set
% 0.30/0.51  thf(fact_1310_all__nth__imp__all__set,axiom,
% 0.30/0.51      ! [Xs2: list_nat,P: nat > $o,X3: nat] :
% 0.30/0.51        ( ! [I3: nat] :
% 0.30/0.51            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.51           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
% 0.30/0.51       => ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.51         => ( P @ X3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % all_nth_imp_all_set
% 0.30/0.51  thf(fact_1311_vebt__member_Osimps_I5_J,axiom,
% 0.30/0.51      ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.30/0.51        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.51        = ( ( X3 != Mi )
% 0.30/0.51         => ( ( X3 != Ma )
% 0.30/0.51           => ( ~ ( ord_less_nat @ X3 @ Mi )
% 0.30/0.51              & ( ~ ( ord_less_nat @ X3 @ Mi )
% 0.30/0.51               => ( ~ ( ord_less_nat @ Ma @ X3 )
% 0.30/0.51                  & ( ~ ( ord_less_nat @ Ma @ X3 )
% 0.30/0.51                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51                       => ( 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 ) ) ) ) ) )
% 0.30/0.51                      & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % vebt_member.simps(5)
% 0.30/0.51  thf(fact_1312_vebt__insert_Osimps_I4_J,axiom,
% 0.30/0.51      ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.30/0.51        ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.51        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 0.30/0.51  
% 0.30/0.51  % vebt_insert.simps(4)
% 0.30/0.51  thf(fact_1313_buildup__gives__empty,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 0.30/0.51        = bot_bot_set_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % buildup_gives_empty
% 0.30/0.51  thf(fact_1314_double__not__eq__Suc__double,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.51       != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % double_not_eq_Suc_double
% 0.30/0.51  thf(fact_1315_Suc__double__not__eq__double,axiom,
% 0.30/0.51      ! [M: nat,N: nat] :
% 0.30/0.51        ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51       != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % Suc_double_not_eq_double
% 0.30/0.51  thf(fact_1316_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.30/0.51      ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 0.30/0.51        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X3 )
% 0.30/0.51        = ( ( X3 = Mi )
% 0.30/0.51          | ( X3 = Ma )
% 0.30/0.51          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51             => ( 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 ) ) ) ) ) )
% 0.30/0.51            & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.membermima.simps(4)
% 0.30/0.51  thf(fact_1317_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.30/0.51      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 0.30/0.51        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X3 )
% 0.30/0.51        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51           => ( 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 ) ) ) ) ) )
% 0.30/0.51          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.naive_member.simps(3)
% 0.30/0.51  thf(fact_1318_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.30/0.51      ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
% 0.30/0.51        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X3 )
% 0.30/0.51        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51           => ( 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 ) ) ) ) ) )
% 0.30/0.51          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.membermima.simps(5)
% 0.30/0.51  thf(fact_1319_vebt__pred_Oelims,axiom,
% 0.30/0.51      ! [X3: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.30/0.51        ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
% 0.30/0.51          = Y3 )
% 0.30/0.51       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.51                ( X3
% 0.30/0.51                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.51           => ( ( Xa2 = zero_zero_nat )
% 0.30/0.51             => ( Y3 != none_nat ) ) )
% 0.30/0.51         => ( ! [A3: $o] :
% 0.30/0.51                ( ? [Uw2: $o] :
% 0.30/0.51                    ( X3
% 0.30/0.51                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 0.30/0.51               => ( ( Xa2
% 0.30/0.51                    = ( suc @ zero_zero_nat ) )
% 0.30/0.51                 => ~ ( ( A3
% 0.30/0.51                       => ( Y3
% 0.30/0.51                          = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.51                      & ( ~ A3
% 0.30/0.51                       => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.51           => ( ! [A3: $o,B3: $o] :
% 0.30/0.51                  ( ( X3
% 0.30/0.51                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.51                 => ( ? [Va2: nat] :
% 0.30/0.51                        ( Xa2
% 0.30/0.51                        = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.51                   => ~ ( ( B3
% 0.30/0.51                         => ( Y3
% 0.30/0.51                            = ( some_nat @ one_one_nat ) ) )
% 0.30/0.51                        & ( ~ B3
% 0.30/0.51                         => ( ( A3
% 0.30/0.51                             => ( Y3
% 0.30/0.51                                = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.51                            & ( ~ A3
% 0.30/0.51                             => ( Y3 = none_nat ) ) ) ) ) ) )
% 0.30/0.51             => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.30/0.51                      ( X3
% 0.30/0.51                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.30/0.51                 => ( Y3 != none_nat ) )
% 0.30/0.51               => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 0.30/0.51                        ( X3
% 0.30/0.51                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 0.30/0.51                   => ( Y3 != none_nat ) )
% 0.30/0.51                 => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 0.30/0.51                          ( X3
% 0.30/0.51                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 0.30/0.51                     => ( Y3 != none_nat ) )
% 0.30/0.51                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.51                          ( ( X3
% 0.30/0.51                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.51                         => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.51                               => ( Y3
% 0.30/0.51                                  = ( some_nat @ Ma2 ) ) )
% 0.30/0.51                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.51                               => ( Y3
% 0.30/0.51                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.51                                    @ ( if_option_nat
% 0.30/0.51                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51                                         != none_nat )
% 0.30/0.51                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51                                      @ ( if_option_nat
% 0.30/0.51                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51                                          = none_nat )
% 0.30/0.51                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.30/0.51                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51                                    @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % vebt_pred.elims
% 0.30/0.51  thf(fact_1320_low__def,axiom,
% 0.30/0.51      ( vEBT_VEBT_low
% 0.30/0.51      = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % low_def
% 0.30/0.51  thf(fact_1321_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 0.30/0.51      ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 0.30/0.51  
% 0.30/0.51  % VEBT_internal.minNull.simps(4)
% 0.30/0.51  thf(fact_1322_valid__tree__deg__neq__0,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT] :
% 0.30/0.51        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % valid_tree_deg_neq_0
% 0.30/0.51  thf(fact_1323_valid__0__not,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT] :
% 0.30/0.51        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % valid_0_not
% 0.30/0.51  thf(fact_1324_buildup__nothing__in__min__max,axiom,
% 0.30/0.51      ! [N: nat,X3: nat] :
% 0.30/0.51        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% 0.30/0.51  
% 0.30/0.51  % buildup_nothing_in_min_max
% 0.30/0.51  thf(fact_1325_buildup__nothing__in__leaf,axiom,
% 0.30/0.51      ! [N: nat,X3: nat] :
% 0.30/0.51        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% 0.30/0.51  
% 0.30/0.51  % buildup_nothing_in_leaf
% 0.30/0.51  thf(fact_1326_deg__not__0,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % deg_not_0
% 0.30/0.51  thf(fact_1327_Leaf__0__not,axiom,
% 0.30/0.51      ! [A: $o,B2: $o] :
% 0.30/0.51        ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % Leaf_0_not
% 0.30/0.51  thf(fact_1328_deg__1__Leafy,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.51       => ( ( N = one_one_nat )
% 0.30/0.51         => ? [A3: $o,B3: $o] :
% 0.30/0.51              ( T
% 0.30/0.51              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % deg_1_Leafy
% 0.30/0.51  thf(fact_1329_deg__1__Leaf,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.30/0.51       => ? [A3: $o,B3: $o] :
% 0.30/0.51            ( T
% 0.30/0.51            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % deg_1_Leaf
% 0.30/0.51  thf(fact_1330_deg1Leaf,axiom,
% 0.30/0.51      ! [T: vEBT_VEBT] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.30/0.51        = ( ? [A5: $o,B4: $o] :
% 0.30/0.51              ( T
% 0.30/0.51              = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % deg1Leaf
% 0.30/0.51  thf(fact_1331_both__member__options__def,axiom,
% 0.30/0.51      ( vEBT_V8194947554948674370ptions
% 0.30/0.51      = ( ^ [T2: vEBT_VEBT,X: nat] :
% 0.30/0.51            ( ( vEBT_V5719532721284313246member @ T2 @ X )
% 0.30/0.51            | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % both_member_options_def
% 0.30/0.51  thf(fact_1332_buildup__gives__valid,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.51       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.30/0.51  
% 0.30/0.51  % buildup_gives_valid
% 0.30/0.51  thf(fact_1333_mod__mod__trivial,axiom,
% 0.30/0.51      ! [A: nat,B2: nat] :
% 0.30/0.51        ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 0.30/0.51        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_mod_trivial
% 0.30/0.51  thf(fact_1334_mod__mod__trivial,axiom,
% 0.30/0.51      ! [A: int,B2: int] :
% 0.30/0.51        ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 0.30/0.51        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_mod_trivial
% 0.30/0.51  thf(fact_1335_mod__mod__trivial,axiom,
% 0.30/0.51      ! [A: code_integer,B2: code_integer] :
% 0.30/0.51        ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 0.30/0.51        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_mod_trivial
% 0.30/0.51  thf(fact_1336_member__valid__both__member__options,axiom,
% 0.30/0.51      ! [Tree: vEBT_VEBT,N: nat,X3: nat] :
% 0.30/0.51        ( ( vEBT_invar_vebt @ Tree @ N )
% 0.30/0.51       => ( ( vEBT_vebt_member @ Tree @ X3 )
% 0.30/0.51         => ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
% 0.30/0.51            | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % member_valid_both_member_options
% 0.30/0.51  thf(fact_1337_le__zero__eq,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.51        = ( N = zero_zero_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % le_zero_eq
% 0.30/0.51  thf(fact_1338_not__gr__zero,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.51        = ( N = zero_zero_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % not_gr_zero
% 0.30/0.51  thf(fact_1339_mult__zero__left,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( times_times_complex @ zero_zero_complex @ A )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_left
% 0.30/0.51  thf(fact_1340_mult__zero__left,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( times_times_real @ zero_zero_real @ A )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_left
% 0.30/0.51  thf(fact_1341_mult__zero__left,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( times_times_rat @ zero_zero_rat @ A )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_left
% 0.30/0.51  thf(fact_1342_mult__zero__left,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( times_times_nat @ zero_zero_nat @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_left
% 0.30/0.51  thf(fact_1343_mult__zero__left,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( times_times_int @ zero_zero_int @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_left
% 0.30/0.51  thf(fact_1344_mult__zero__right,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( times_times_complex @ A @ zero_zero_complex )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_right
% 0.30/0.51  thf(fact_1345_mult__zero__right,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( times_times_real @ A @ zero_zero_real )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_right
% 0.30/0.51  thf(fact_1346_mult__zero__right,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( times_times_rat @ A @ zero_zero_rat )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_right
% 0.30/0.51  thf(fact_1347_mult__zero__right,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( times_times_nat @ A @ zero_zero_nat )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_right
% 0.30/0.51  thf(fact_1348_mult__zero__right,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( times_times_int @ A @ zero_zero_int )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_zero_right
% 0.30/0.51  thf(fact_1349_mult__eq__0__iff,axiom,
% 0.30/0.51      ! [A: complex,B2: complex] :
% 0.30/0.51        ( ( ( times_times_complex @ A @ B2 )
% 0.30/0.51          = zero_zero_complex )
% 0.30/0.51        = ( ( A = zero_zero_complex )
% 0.30/0.51          | ( B2 = zero_zero_complex ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_eq_0_iff
% 0.30/0.51  thf(fact_1350_mult__eq__0__iff,axiom,
% 0.30/0.51      ! [A: real,B2: real] :
% 0.30/0.51        ( ( ( times_times_real @ A @ B2 )
% 0.30/0.51          = zero_zero_real )
% 0.30/0.51        = ( ( A = zero_zero_real )
% 0.30/0.51          | ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_eq_0_iff
% 0.30/0.51  thf(fact_1351_mult__eq__0__iff,axiom,
% 0.30/0.51      ! [A: rat,B2: rat] :
% 0.30/0.51        ( ( ( times_times_rat @ A @ B2 )
% 0.30/0.51          = zero_zero_rat )
% 0.30/0.51        = ( ( A = zero_zero_rat )
% 0.30/0.51          | ( B2 = zero_zero_rat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_eq_0_iff
% 0.30/0.51  thf(fact_1352_mult__eq__0__iff,axiom,
% 0.30/0.51      ! [A: nat,B2: nat] :
% 0.30/0.51        ( ( ( times_times_nat @ A @ B2 )
% 0.30/0.51          = zero_zero_nat )
% 0.30/0.51        = ( ( A = zero_zero_nat )
% 0.30/0.51          | ( B2 = zero_zero_nat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_eq_0_iff
% 0.30/0.51  thf(fact_1353_mult__eq__0__iff,axiom,
% 0.30/0.51      ! [A: int,B2: int] :
% 0.30/0.51        ( ( ( times_times_int @ A @ B2 )
% 0.30/0.51          = zero_zero_int )
% 0.30/0.51        = ( ( A = zero_zero_int )
% 0.30/0.51          | ( B2 = zero_zero_int ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_eq_0_iff
% 0.30/0.51  thf(fact_1354_mult__cancel__left,axiom,
% 0.30/0.51      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.51        ( ( ( times_times_complex @ C @ A )
% 0.30/0.51          = ( times_times_complex @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_complex )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_left
% 0.30/0.51  thf(fact_1355_mult__cancel__left,axiom,
% 0.30/0.51      ! [C: real,A: real,B2: real] :
% 0.30/0.51        ( ( ( times_times_real @ C @ A )
% 0.30/0.51          = ( times_times_real @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_real )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_left
% 0.30/0.51  thf(fact_1356_mult__cancel__left,axiom,
% 0.30/0.51      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.51        ( ( ( times_times_rat @ C @ A )
% 0.30/0.51          = ( times_times_rat @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_rat )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_left
% 0.30/0.51  thf(fact_1357_mult__cancel__left,axiom,
% 0.30/0.51      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.51        ( ( ( times_times_nat @ C @ A )
% 0.30/0.51          = ( times_times_nat @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_nat )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_left
% 0.30/0.51  thf(fact_1358_mult__cancel__left,axiom,
% 0.30/0.51      ! [C: int,A: int,B2: int] :
% 0.30/0.51        ( ( ( times_times_int @ C @ A )
% 0.30/0.51          = ( times_times_int @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_int )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_left
% 0.30/0.51  thf(fact_1359_mult__cancel__right,axiom,
% 0.30/0.51      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.51        ( ( ( times_times_complex @ A @ C )
% 0.30/0.51          = ( times_times_complex @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_complex )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_right
% 0.30/0.51  thf(fact_1360_mult__cancel__right,axiom,
% 0.30/0.51      ! [A: real,C: real,B2: real] :
% 0.30/0.51        ( ( ( times_times_real @ A @ C )
% 0.30/0.51          = ( times_times_real @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_real )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_right
% 0.30/0.51  thf(fact_1361_mult__cancel__right,axiom,
% 0.30/0.51      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.51        ( ( ( times_times_rat @ A @ C )
% 0.30/0.51          = ( times_times_rat @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_rat )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_right
% 0.30/0.51  thf(fact_1362_mult__cancel__right,axiom,
% 0.30/0.51      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.51        ( ( ( times_times_nat @ A @ C )
% 0.30/0.51          = ( times_times_nat @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_nat )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_right
% 0.30/0.51  thf(fact_1363_mult__cancel__right,axiom,
% 0.30/0.51      ! [A: int,C: int,B2: int] :
% 0.30/0.51        ( ( ( times_times_int @ A @ C )
% 0.30/0.51          = ( times_times_int @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_int )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % mult_cancel_right
% 0.30/0.51  thf(fact_1364_add__0,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add_0
% 0.30/0.51  thf(fact_1365_add__0,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add_0
% 0.30/0.51  thf(fact_1366_add__0,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add_0
% 0.30/0.51  thf(fact_1367_add__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add_0
% 0.30/0.51  thf(fact_1368_add__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add_0
% 0.30/0.51  thf(fact_1369_zero__eq__add__iff__both__eq__0,axiom,
% 0.30/0.51      ! [X3: nat,Y3: nat] :
% 0.30/0.51        ( ( zero_zero_nat
% 0.30/0.51          = ( plus_plus_nat @ X3 @ Y3 ) )
% 0.30/0.51        = ( ( X3 = zero_zero_nat )
% 0.30/0.51          & ( Y3 = zero_zero_nat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % zero_eq_add_iff_both_eq_0
% 0.30/0.51  thf(fact_1370_add__eq__0__iff__both__eq__0,axiom,
% 0.30/0.51      ! [X3: nat,Y3: nat] :
% 0.30/0.51        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.30/0.51          = zero_zero_nat )
% 0.30/0.51        = ( ( X3 = zero_zero_nat )
% 0.30/0.51          & ( Y3 = zero_zero_nat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_eq_0_iff_both_eq_0
% 0.30/0.51  thf(fact_1371_add__cancel__right__right,axiom,
% 0.30/0.51      ! [A: complex,B2: complex] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_complex @ A @ B2 ) )
% 0.30/0.51        = ( B2 = zero_zero_complex ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_right
% 0.30/0.51  thf(fact_1372_add__cancel__right__right,axiom,
% 0.30/0.51      ! [A: real,B2: real] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_real @ A @ B2 ) )
% 0.30/0.51        = ( B2 = zero_zero_real ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_right
% 0.30/0.51  thf(fact_1373_add__cancel__right__right,axiom,
% 0.30/0.51      ! [A: rat,B2: rat] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.51        = ( B2 = zero_zero_rat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_right
% 0.30/0.51  thf(fact_1374_add__cancel__right__right,axiom,
% 0.30/0.51      ! [A: nat,B2: nat] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.51        = ( B2 = zero_zero_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_right
% 0.30/0.51  thf(fact_1375_add__cancel__right__right,axiom,
% 0.30/0.51      ! [A: int,B2: int] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_int @ A @ B2 ) )
% 0.30/0.51        = ( B2 = zero_zero_int ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_right
% 0.30/0.51  thf(fact_1376_add__cancel__right__left,axiom,
% 0.30/0.51      ! [A: complex,B2: complex] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_complex @ B2 @ A ) )
% 0.30/0.51        = ( B2 = zero_zero_complex ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_left
% 0.30/0.51  thf(fact_1377_add__cancel__right__left,axiom,
% 0.30/0.51      ! [A: real,B2: real] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_real @ B2 @ A ) )
% 0.30/0.51        = ( B2 = zero_zero_real ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_left
% 0.30/0.51  thf(fact_1378_add__cancel__right__left,axiom,
% 0.30/0.51      ! [A: rat,B2: rat] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_rat @ B2 @ A ) )
% 0.30/0.51        = ( B2 = zero_zero_rat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_left
% 0.30/0.51  thf(fact_1379_add__cancel__right__left,axiom,
% 0.30/0.51      ! [A: nat,B2: nat] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_nat @ B2 @ A ) )
% 0.30/0.51        = ( B2 = zero_zero_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_left
% 0.30/0.51  thf(fact_1380_add__cancel__right__left,axiom,
% 0.30/0.51      ! [A: int,B2: int] :
% 0.30/0.51        ( ( A
% 0.30/0.51          = ( plus_plus_int @ B2 @ A ) )
% 0.30/0.51        = ( B2 = zero_zero_int ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_right_left
% 0.30/0.51  thf(fact_1381_add__cancel__left__right,axiom,
% 0.30/0.51      ! [A: complex,B2: complex] :
% 0.30/0.51        ( ( ( plus_plus_complex @ A @ B2 )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_complex ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_right
% 0.30/0.51  thf(fact_1382_add__cancel__left__right,axiom,
% 0.30/0.51      ! [A: real,B2: real] :
% 0.30/0.51        ( ( ( plus_plus_real @ A @ B2 )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_real ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_right
% 0.30/0.51  thf(fact_1383_add__cancel__left__right,axiom,
% 0.30/0.51      ! [A: rat,B2: rat] :
% 0.30/0.51        ( ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_rat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_right
% 0.30/0.51  thf(fact_1384_add__cancel__left__right,axiom,
% 0.30/0.51      ! [A: nat,B2: nat] :
% 0.30/0.51        ( ( ( plus_plus_nat @ A @ B2 )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_right
% 0.30/0.51  thf(fact_1385_add__cancel__left__right,axiom,
% 0.30/0.51      ! [A: int,B2: int] :
% 0.30/0.51        ( ( ( plus_plus_int @ A @ B2 )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_int ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_right
% 0.30/0.51  thf(fact_1386_add__cancel__left__left,axiom,
% 0.30/0.51      ! [B2: complex,A: complex] :
% 0.30/0.51        ( ( ( plus_plus_complex @ B2 @ A )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_complex ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_left
% 0.30/0.51  thf(fact_1387_add__cancel__left__left,axiom,
% 0.30/0.51      ! [B2: real,A: real] :
% 0.30/0.51        ( ( ( plus_plus_real @ B2 @ A )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_real ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_left
% 0.30/0.51  thf(fact_1388_add__cancel__left__left,axiom,
% 0.30/0.51      ! [B2: rat,A: rat] :
% 0.30/0.51        ( ( ( plus_plus_rat @ B2 @ A )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_rat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_left
% 0.30/0.51  thf(fact_1389_add__cancel__left__left,axiom,
% 0.30/0.51      ! [B2: nat,A: nat] :
% 0.30/0.51        ( ( ( plus_plus_nat @ B2 @ A )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_nat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_left
% 0.30/0.51  thf(fact_1390_add__cancel__left__left,axiom,
% 0.30/0.51      ! [B2: int,A: int] :
% 0.30/0.51        ( ( ( plus_plus_int @ B2 @ A )
% 0.30/0.51          = A )
% 0.30/0.51        = ( B2 = zero_zero_int ) ) ).
% 0.30/0.51  
% 0.30/0.51  % add_cancel_left_left
% 0.30/0.51  thf(fact_1391_double__zero__sym,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( zero_zero_real
% 0.30/0.51          = ( plus_plus_real @ A @ A ) )
% 0.30/0.51        = ( A = zero_zero_real ) ) ).
% 0.30/0.51  
% 0.30/0.51  % double_zero_sym
% 0.30/0.51  thf(fact_1392_double__zero__sym,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( zero_zero_rat
% 0.30/0.51          = ( plus_plus_rat @ A @ A ) )
% 0.30/0.51        = ( A = zero_zero_rat ) ) ).
% 0.30/0.51  
% 0.30/0.51  % double_zero_sym
% 0.30/0.51  thf(fact_1393_double__zero__sym,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( zero_zero_int
% 0.30/0.51          = ( plus_plus_int @ A @ A ) )
% 0.30/0.51        = ( A = zero_zero_int ) ) ).
% 0.30/0.51  
% 0.30/0.51  % double_zero_sym
% 0.30/0.51  thf(fact_1394_add_Oright__neutral,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add.right_neutral
% 0.30/0.51  thf(fact_1395_add_Oright__neutral,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add.right_neutral
% 0.30/0.51  thf(fact_1396_add_Oright__neutral,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add.right_neutral
% 0.30/0.51  thf(fact_1397_add_Oright__neutral,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add.right_neutral
% 0.30/0.51  thf(fact_1398_add_Oright__neutral,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % add.right_neutral
% 0.30/0.51  thf(fact_1399_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( minus_minus_complex @ A @ A )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51  thf(fact_1400_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( minus_minus_real @ A @ A )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51  thf(fact_1401_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( minus_minus_rat @ A @ A )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51  thf(fact_1402_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ A @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51  thf(fact_1403_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( minus_minus_int @ A @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.51  thf(fact_1404_diff__zero,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_zero
% 0.30/0.51  thf(fact_1405_diff__zero,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_zero
% 0.30/0.51  thf(fact_1406_diff__zero,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_zero
% 0.30/0.51  thf(fact_1407_diff__zero,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_zero
% 0.30/0.51  thf(fact_1408_diff__zero,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_zero
% 0.30/0.51  thf(fact_1409_zero__diff,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % zero_diff
% 0.30/0.51  thf(fact_1410_diff__0__right,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_0_right
% 0.30/0.51  thf(fact_1411_diff__0__right,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_0_right
% 0.30/0.51  thf(fact_1412_diff__0__right,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_0_right
% 0.30/0.51  thf(fact_1413_diff__0__right,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_0_right
% 0.30/0.51  thf(fact_1414_diff__self,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( minus_minus_complex @ A @ A )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_self
% 0.30/0.51  thf(fact_1415_diff__self,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( minus_minus_real @ A @ A )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_self
% 0.30/0.51  thf(fact_1416_diff__self,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( minus_minus_rat @ A @ A )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_self
% 0.30/0.51  thf(fact_1417_diff__self,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( minus_minus_int @ A @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % diff_self
% 0.30/0.51  thf(fact_1418_divide__eq__0__iff,axiom,
% 0.30/0.51      ! [A: complex,B2: complex] :
% 0.30/0.51        ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.30/0.51          = zero_zero_complex )
% 0.30/0.51        = ( ( A = zero_zero_complex )
% 0.30/0.51          | ( B2 = zero_zero_complex ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_eq_0_iff
% 0.30/0.51  thf(fact_1419_divide__eq__0__iff,axiom,
% 0.30/0.51      ! [A: real,B2: real] :
% 0.30/0.51        ( ( ( divide_divide_real @ A @ B2 )
% 0.30/0.51          = zero_zero_real )
% 0.30/0.51        = ( ( A = zero_zero_real )
% 0.30/0.51          | ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_eq_0_iff
% 0.30/0.51  thf(fact_1420_divide__eq__0__iff,axiom,
% 0.30/0.51      ! [A: rat,B2: rat] :
% 0.30/0.51        ( ( ( divide_divide_rat @ A @ B2 )
% 0.30/0.51          = zero_zero_rat )
% 0.30/0.51        = ( ( A = zero_zero_rat )
% 0.30/0.51          | ( B2 = zero_zero_rat ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_eq_0_iff
% 0.30/0.51  thf(fact_1421_divide__cancel__left,axiom,
% 0.30/0.51      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.51        ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.30/0.51          = ( divide1717551699836669952omplex @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_complex )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_cancel_left
% 0.30/0.51  thf(fact_1422_divide__cancel__left,axiom,
% 0.30/0.51      ! [C: real,A: real,B2: real] :
% 0.30/0.51        ( ( ( divide_divide_real @ C @ A )
% 0.30/0.51          = ( divide_divide_real @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_real )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_cancel_left
% 0.30/0.51  thf(fact_1423_divide__cancel__left,axiom,
% 0.30/0.51      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.51        ( ( ( divide_divide_rat @ C @ A )
% 0.30/0.51          = ( divide_divide_rat @ C @ B2 ) )
% 0.30/0.51        = ( ( C = zero_zero_rat )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_cancel_left
% 0.30/0.51  thf(fact_1424_divide__cancel__right,axiom,
% 0.30/0.51      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.51        ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.51          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_complex )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_cancel_right
% 0.30/0.51  thf(fact_1425_divide__cancel__right,axiom,
% 0.30/0.51      ! [A: real,C: real,B2: real] :
% 0.30/0.51        ( ( ( divide_divide_real @ A @ C )
% 0.30/0.51          = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_real )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_cancel_right
% 0.30/0.51  thf(fact_1426_divide__cancel__right,axiom,
% 0.30/0.51      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.51        ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.51          = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.51        = ( ( C = zero_zero_rat )
% 0.30/0.51          | ( A = B2 ) ) ) ).
% 0.30/0.51  
% 0.30/0.51  % divide_cancel_right
% 0.30/0.51  thf(fact_1427_division__ring__divide__zero,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % division_ring_divide_zero
% 0.30/0.51  thf(fact_1428_division__ring__divide__zero,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % division_ring_divide_zero
% 0.30/0.51  thf(fact_1429_division__ring__divide__zero,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % division_ring_divide_zero
% 0.30/0.51  thf(fact_1430_div__0,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % div_0
% 0.30/0.51  thf(fact_1431_div__0,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( divide_divide_real @ zero_zero_real @ A )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % div_0
% 0.30/0.51  thf(fact_1432_div__0,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % div_0
% 0.30/0.51  thf(fact_1433_div__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % div_0
% 0.30/0.51  thf(fact_1434_div__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % div_0
% 0.30/0.51  thf(fact_1435_div__by__0,axiom,
% 0.30/0.51      ! [A: complex] :
% 0.30/0.51        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.30/0.51        = zero_zero_complex ) ).
% 0.30/0.51  
% 0.30/0.51  % div_by_0
% 0.30/0.51  thf(fact_1436_div__by__0,axiom,
% 0.30/0.51      ! [A: real] :
% 0.30/0.51        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.51        = zero_zero_real ) ).
% 0.30/0.51  
% 0.30/0.51  % div_by_0
% 0.30/0.51  thf(fact_1437_div__by__0,axiom,
% 0.30/0.51      ! [A: rat] :
% 0.30/0.51        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.30/0.51        = zero_zero_rat ) ).
% 0.30/0.51  
% 0.30/0.51  % div_by_0
% 0.30/0.51  thf(fact_1438_div__by__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % div_by_0
% 0.30/0.51  thf(fact_1439_div__by__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % div_by_0
% 0.30/0.51  thf(fact_1440_bits__div__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_div_0
% 0.30/0.51  thf(fact_1441_bits__div__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_div_0
% 0.30/0.51  thf(fact_1442_bits__div__by__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_div_by_0
% 0.30/0.51  thf(fact_1443_bits__div__by__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_div_by_0
% 0.30/0.51  thf(fact_1444_mod__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_0
% 0.30/0.51  thf(fact_1445_mod__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_0
% 0.30/0.51  thf(fact_1446_mod__0,axiom,
% 0.30/0.51      ! [A: code_integer] :
% 0.30/0.51        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.51        = zero_z3403309356797280102nteger ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_0
% 0.30/0.51  thf(fact_1447_mod__by__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_by_0
% 0.30/0.51  thf(fact_1448_mod__by__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_by_0
% 0.30/0.51  thf(fact_1449_mod__by__0,axiom,
% 0.30/0.51      ! [A: code_integer] :
% 0.30/0.51        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.51        = A ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_by_0
% 0.30/0.51  thf(fact_1450_mod__self,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( modulo_modulo_nat @ A @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_self
% 0.30/0.51  thf(fact_1451_mod__self,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( modulo_modulo_int @ A @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_self
% 0.30/0.51  thf(fact_1452_mod__self,axiom,
% 0.30/0.51      ! [A: code_integer] :
% 0.30/0.51        ( ( modulo364778990260209775nteger @ A @ A )
% 0.30/0.51        = zero_z3403309356797280102nteger ) ).
% 0.30/0.51  
% 0.30/0.51  % mod_self
% 0.30/0.51  thf(fact_1453_bits__mod__0,axiom,
% 0.30/0.51      ! [A: nat] :
% 0.30/0.51        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.51        = zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_mod_0
% 0.30/0.51  thf(fact_1454_bits__mod__0,axiom,
% 0.30/0.51      ! [A: int] :
% 0.30/0.51        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.51        = zero_zero_int ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_mod_0
% 0.30/0.51  thf(fact_1455_bits__mod__0,axiom,
% 0.30/0.51      ! [A: code_integer] :
% 0.30/0.51        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.51        = zero_z3403309356797280102nteger ) ).
% 0.30/0.51  
% 0.30/0.51  % bits_mod_0
% 0.30/0.51  thf(fact_1456_less__nat__zero__code,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.51  
% 0.30/0.51  % less_nat_zero_code
% 0.30/0.51  thf(fact_1457_neq0__conv,axiom,
% 0.30/0.51      ! [N: nat] :
% 0.30/0.51        ( ( N != zero_zero_nat )
% 0.30/0.51        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % neq0_conv
% 0.30/0.52  thf(fact_1458_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( A != zero_zero_nat )
% 0.30/0.52        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % bot_nat_0.not_eq_extremum
% 0.30/0.52  thf(fact_1459_le0,axiom,
% 0.30/0.52      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.52  
% 0.30/0.52  % le0
% 0.30/0.52  thf(fact_1460_bot__nat__0_Oextremum,axiom,
% 0.30/0.52      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.30/0.52  
% 0.30/0.52  % bot_nat_0.extremum
% 0.30/0.52  thf(fact_1461_mod__add__self1,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_self1
% 0.30/0.52  thf(fact_1462_mod__add__self1,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_self1
% 0.30/0.52  thf(fact_1463_mod__add__self1,axiom,
% 0.30/0.52      ! [B2: code_integer,A: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_self1
% 0.30/0.52  thf(fact_1464_mod__add__self2,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_self2
% 0.30/0.52  thf(fact_1465_mod__add__self2,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_self2
% 0.30/0.52  thf(fact_1466_mod__add__self2,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_self2
% 0.30/0.52  thf(fact_1467_minus__mod__self2,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % minus_mod_self2
% 0.30/0.52  thf(fact_1468_minus__mod__self2,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % minus_mod_self2
% 0.30/0.52  thf(fact_1469_Nat_Oadd__0__right,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.30/0.52        = M ) ).
% 0.30/0.52  
% 0.30/0.52  % Nat.add_0_right
% 0.30/0.52  thf(fact_1470_add__is__0,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ( plus_plus_nat @ M @ N )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52        = ( ( M = zero_zero_nat )
% 0.30/0.52          & ( N = zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_is_0
% 0.30/0.52  thf(fact_1471_diff__self__eq__0,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( minus_minus_nat @ M @ M )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_self_eq_0
% 0.30/0.52  thf(fact_1472_diff__0__eq__0,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_0_eq_0
% 0.30/0.52  thf(fact_1473_mult__is__0,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ( times_times_nat @ M @ N )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52        = ( ( M = zero_zero_nat )
% 0.30/0.52          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_is_0
% 0.30/0.52  thf(fact_1474_mult__0__right,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( times_times_nat @ M @ zero_zero_nat )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_0_right
% 0.30/0.52  thf(fact_1475_mult__cancel1,axiom,
% 0.30/0.52      ! [K: nat,M: nat,N: nat] :
% 0.30/0.52        ( ( ( times_times_nat @ K @ M )
% 0.30/0.52          = ( times_times_nat @ K @ N ) )
% 0.30/0.52        = ( ( M = N )
% 0.30/0.52          | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel1
% 0.30/0.52  thf(fact_1476_mult__cancel2,axiom,
% 0.30/0.52      ! [M: nat,K: nat,N: nat] :
% 0.30/0.52        ( ( ( times_times_nat @ M @ K )
% 0.30/0.52          = ( times_times_nat @ N @ K ) )
% 0.30/0.52        = ( ( M = N )
% 0.30/0.52          | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel2
% 0.30/0.52  thf(fact_1477_mod__less,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ M @ N )
% 0.30/0.52       => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.52          = M ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_less
% 0.30/0.52  thf(fact_1478_add__le__same__cancel1,axiom,
% 0.30/0.52      ! [B2: real,A: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel1
% 0.30/0.52  thf(fact_1479_add__le__same__cancel1,axiom,
% 0.30/0.52      ! [B2: rat,A: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel1
% 0.30/0.52  thf(fact_1480_add__le__same__cancel1,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel1
% 0.30/0.52  thf(fact_1481_add__le__same__cancel1,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel1
% 0.30/0.52  thf(fact_1482_add__le__same__cancel2,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel2
% 0.30/0.52  thf(fact_1483_add__le__same__cancel2,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel2
% 0.30/0.52  thf(fact_1484_add__le__same__cancel2,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel2
% 0.30/0.52  thf(fact_1485_add__le__same__cancel2,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_le_same_cancel2
% 0.30/0.52  thf(fact_1486_le__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel1
% 0.30/0.52  thf(fact_1487_le__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel1
% 0.30/0.52  thf(fact_1488_le__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel1
% 0.30/0.52  thf(fact_1489_le__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel1
% 0.30/0.52  thf(fact_1490_le__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel2
% 0.30/0.52  thf(fact_1491_le__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel2
% 0.30/0.52  thf(fact_1492_le__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel2
% 0.30/0.52  thf(fact_1493_le__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_add_same_cancel2
% 0.30/0.52  thf(fact_1494_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.52        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.52  thf(fact_1495_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.30/0.52        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.52  thf(fact_1496_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.30/0.52        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.52  thf(fact_1497_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.52        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.52  thf(fact_1498_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.30/0.52        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.52  thf(fact_1499_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.30/0.52        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.52  thf(fact_1500_diff__ge__0__iff__ge,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_real @ B2 @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_ge_0_iff_ge
% 0.30/0.52  thf(fact_1501_diff__ge__0__iff__ge,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_ge_0_iff_ge
% 0.30/0.52  thf(fact_1502_diff__ge__0__iff__ge,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_ge_0_iff_ge
% 0.30/0.52  thf(fact_1503_add__less__same__cancel1,axiom,
% 0.30/0.52      ! [B2: real,A: real] :
% 0.30/0.52        ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel1
% 0.30/0.52  thf(fact_1504_add__less__same__cancel1,axiom,
% 0.30/0.52      ! [B2: rat,A: rat] :
% 0.30/0.52        ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel1
% 0.30/0.52  thf(fact_1505_add__less__same__cancel1,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel1
% 0.30/0.52  thf(fact_1506_add__less__same__cancel1,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.30/0.52        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel1
% 0.30/0.52  thf(fact_1507_add__less__same__cancel2,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel2
% 0.30/0.52  thf(fact_1508_add__less__same__cancel2,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel2
% 0.30/0.52  thf(fact_1509_add__less__same__cancel2,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel2
% 0.30/0.52  thf(fact_1510_add__less__same__cancel2,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_less_same_cancel2
% 0.30/0.52  thf(fact_1511_less__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel1
% 0.30/0.52  thf(fact_1512_less__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel1
% 0.30/0.52  thf(fact_1513_less__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel1
% 0.30/0.52  thf(fact_1514_less__add__same__cancel1,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel1
% 0.30/0.52  thf(fact_1515_less__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel2
% 0.30/0.52  thf(fact_1516_less__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel2
% 0.30/0.52  thf(fact_1517_less__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel2
% 0.30/0.52  thf(fact_1518_less__add__same__cancel2,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 0.30/0.52        = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_add_same_cancel2
% 0.30/0.52  thf(fact_1519_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.52        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.52  thf(fact_1520_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.30/0.52        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.52  thf(fact_1521_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.30/0.52        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.52  thf(fact_1522_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.52        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.52  thf(fact_1523_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.30/0.52        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.52  thf(fact_1524_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.30/0.52        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.52  thf(fact_1525_diff__gt__0__iff__gt,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_real @ B2 @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_gt_0_iff_gt
% 0.30/0.52  thf(fact_1526_diff__gt__0__iff__gt,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_rat @ B2 @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_gt_0_iff_gt
% 0.30/0.52  thf(fact_1527_diff__gt__0__iff__gt,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.52        = ( ord_less_int @ B2 @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_gt_0_iff_gt
% 0.30/0.52  thf(fact_1528_mult__cancel__left1,axiom,
% 0.30/0.52      ! [C: complex,B2: complex] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_complex @ C @ B2 ) )
% 0.30/0.52        = ( ( C = zero_zero_complex )
% 0.30/0.52          | ( B2 = one_one_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left1
% 0.30/0.52  thf(fact_1529_mult__cancel__left1,axiom,
% 0.30/0.52      ! [C: real,B2: real] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_real @ C @ B2 ) )
% 0.30/0.52        = ( ( C = zero_zero_real )
% 0.30/0.52          | ( B2 = one_one_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left1
% 0.30/0.52  thf(fact_1530_mult__cancel__left1,axiom,
% 0.30/0.52      ! [C: rat,B2: rat] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_rat @ C @ B2 ) )
% 0.30/0.52        = ( ( C = zero_zero_rat )
% 0.30/0.52          | ( B2 = one_one_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left1
% 0.30/0.52  thf(fact_1531_mult__cancel__left1,axiom,
% 0.30/0.52      ! [C: int,B2: int] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_int @ C @ B2 ) )
% 0.30/0.52        = ( ( C = zero_zero_int )
% 0.30/0.52          | ( B2 = one_one_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left1
% 0.30/0.52  thf(fact_1532_mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: complex,A: complex] :
% 0.30/0.52        ( ( ( times_times_complex @ C @ A )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_complex )
% 0.30/0.52          | ( A = one_one_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left2
% 0.30/0.52  thf(fact_1533_mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: real,A: real] :
% 0.30/0.52        ( ( ( times_times_real @ C @ A )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_real )
% 0.30/0.52          | ( A = one_one_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left2
% 0.30/0.52  thf(fact_1534_mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: rat,A: rat] :
% 0.30/0.52        ( ( ( times_times_rat @ C @ A )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_rat )
% 0.30/0.52          | ( A = one_one_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left2
% 0.30/0.52  thf(fact_1535_mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: int,A: int] :
% 0.30/0.52        ( ( ( times_times_int @ C @ A )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_int )
% 0.30/0.52          | ( A = one_one_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_left2
% 0.30/0.52  thf(fact_1536_mult__cancel__right1,axiom,
% 0.30/0.52      ! [C: complex,B2: complex] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_complex @ B2 @ C ) )
% 0.30/0.52        = ( ( C = zero_zero_complex )
% 0.30/0.52          | ( B2 = one_one_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right1
% 0.30/0.52  thf(fact_1537_mult__cancel__right1,axiom,
% 0.30/0.52      ! [C: real,B2: real] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_real @ B2 @ C ) )
% 0.30/0.52        = ( ( C = zero_zero_real )
% 0.30/0.52          | ( B2 = one_one_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right1
% 0.30/0.52  thf(fact_1538_mult__cancel__right1,axiom,
% 0.30/0.52      ! [C: rat,B2: rat] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_rat @ B2 @ C ) )
% 0.30/0.52        = ( ( C = zero_zero_rat )
% 0.30/0.52          | ( B2 = one_one_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right1
% 0.30/0.52  thf(fact_1539_mult__cancel__right1,axiom,
% 0.30/0.52      ! [C: int,B2: int] :
% 0.30/0.52        ( ( C
% 0.30/0.52          = ( times_times_int @ B2 @ C ) )
% 0.30/0.52        = ( ( C = zero_zero_int )
% 0.30/0.52          | ( B2 = one_one_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right1
% 0.30/0.52  thf(fact_1540_mult__cancel__right2,axiom,
% 0.30/0.52      ! [A: complex,C: complex] :
% 0.30/0.52        ( ( ( times_times_complex @ A @ C )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_complex )
% 0.30/0.52          | ( A = one_one_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right2
% 0.30/0.52  thf(fact_1541_mult__cancel__right2,axiom,
% 0.30/0.52      ! [A: real,C: real] :
% 0.30/0.52        ( ( ( times_times_real @ A @ C )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_real )
% 0.30/0.52          | ( A = one_one_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right2
% 0.30/0.52  thf(fact_1542_mult__cancel__right2,axiom,
% 0.30/0.52      ! [A: rat,C: rat] :
% 0.30/0.52        ( ( ( times_times_rat @ A @ C )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_rat )
% 0.30/0.52          | ( A = one_one_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right2
% 0.30/0.52  thf(fact_1543_mult__cancel__right2,axiom,
% 0.30/0.52      ! [A: int,C: int] :
% 0.30/0.52        ( ( ( times_times_int @ A @ C )
% 0.30/0.52          = C )
% 0.30/0.52        = ( ( C = zero_zero_int )
% 0.30/0.52          | ( A = one_one_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_cancel_right2
% 0.30/0.52  thf(fact_1544_sum__squares__eq__zero__iff,axiom,
% 0.30/0.52      ! [X3: real,Y3: real] :
% 0.30/0.52        ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.30/0.52          = zero_zero_real )
% 0.30/0.52        = ( ( X3 = zero_zero_real )
% 0.30/0.52          & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % sum_squares_eq_zero_iff
% 0.30/0.52  thf(fact_1545_sum__squares__eq__zero__iff,axiom,
% 0.30/0.52      ! [X3: rat,Y3: rat] :
% 0.30/0.52        ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.30/0.52          = zero_zero_rat )
% 0.30/0.52        = ( ( X3 = zero_zero_rat )
% 0.30/0.52          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % sum_squares_eq_zero_iff
% 0.30/0.52  thf(fact_1546_sum__squares__eq__zero__iff,axiom,
% 0.30/0.52      ! [X3: int,Y3: int] :
% 0.30/0.52        ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.30/0.52          = zero_zero_int )
% 0.30/0.52        = ( ( X3 = zero_zero_int )
% 0.30/0.52          & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % sum_squares_eq_zero_iff
% 0.30/0.52  thf(fact_1547_diff__numeral__special_I9_J,axiom,
% 0.30/0.52      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.30/0.52      = zero_zero_complex ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_numeral_special(9)
% 0.30/0.52  thf(fact_1548_diff__numeral__special_I9_J,axiom,
% 0.30/0.52      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.30/0.52      = zero_zero_real ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_numeral_special(9)
% 0.30/0.52  thf(fact_1549_diff__numeral__special_I9_J,axiom,
% 0.30/0.52      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.30/0.52      = zero_zero_rat ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_numeral_special(9)
% 0.30/0.52  thf(fact_1550_diff__numeral__special_I9_J,axiom,
% 0.30/0.52      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.30/0.52      = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_numeral_special(9)
% 0.30/0.52  thf(fact_1551_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.52      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.52        ( ( ( C = zero_zero_complex )
% 0.30/0.52         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.30/0.52            = zero_zero_complex ) )
% 0.30/0.52        & ( ( C != zero_zero_complex )
% 0.30/0.52         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.30/0.52            = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_divide_mult_cancel_left_if
% 0.30/0.52  thf(fact_1552_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.52      ! [C: real,A: real,B2: real] :
% 0.30/0.52        ( ( ( C = zero_zero_real )
% 0.30/0.52         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.52            = zero_zero_real ) )
% 0.30/0.52        & ( ( C != zero_zero_real )
% 0.30/0.52         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.52            = ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_divide_mult_cancel_left_if
% 0.30/0.52  thf(fact_1553_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.52      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.52        ( ( ( C = zero_zero_rat )
% 0.30/0.52         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.52            = zero_zero_rat ) )
% 0.30/0.52        & ( ( C != zero_zero_rat )
% 0.30/0.52         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.52            = ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_divide_mult_cancel_left_if
% 0.30/0.52  thf(fact_1554_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.52      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.52        ( ( C != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.30/0.52          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_left
% 0.30/0.52  thf(fact_1555_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.52      ! [C: real,A: real,B2: real] :
% 0.30/0.52        ( ( C != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.52          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_left
% 0.30/0.52  thf(fact_1556_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.52      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.52        ( ( C != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.52          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_left
% 0.30/0.52  thf(fact_1557_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.52        ( ( C != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
% 0.30/0.52          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.52  thf(fact_1558_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: real,A: real,B2: real] :
% 0.30/0.52        ( ( C != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.52          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.52  thf(fact_1559_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.52      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.52        ( ( C != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.52          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.52  thf(fact_1560_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.52      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.52        ( ( C != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
% 0.30/0.52          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_right
% 0.30/0.52  thf(fact_1561_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.52      ! [C: real,A: real,B2: real] :
% 0.30/0.52        ( ( C != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.52          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_right
% 0.30/0.52  thf(fact_1562_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.52      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.52        ( ( C != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.52          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_right
% 0.30/0.52  thf(fact_1563_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.52      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.52        ( ( C != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
% 0.30/0.52          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.52  thf(fact_1564_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.52      ! [C: real,A: real,B2: real] :
% 0.30/0.52        ( ( C != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.52          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.52  thf(fact_1565_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.52      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.52        ( ( C != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.52          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.52  thf(fact_1566_div__mult__mult1__if,axiom,
% 0.30/0.52      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.52        ( ( ( C = zero_zero_nat )
% 0.30/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.30/0.52            = zero_zero_nat ) )
% 0.30/0.52        & ( ( C != zero_zero_nat )
% 0.30/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.30/0.52            = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_mult1_if
% 0.30/0.52  thf(fact_1567_div__mult__mult1__if,axiom,
% 0.30/0.52      ! [C: int,A: int,B2: int] :
% 0.30/0.52        ( ( ( C = zero_zero_int )
% 0.30/0.52         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.52            = zero_zero_int ) )
% 0.30/0.52        & ( ( C != zero_zero_int )
% 0.30/0.52         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.52            = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_mult1_if
% 0.30/0.52  thf(fact_1568_div__mult__mult2,axiom,
% 0.30/0.52      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.52        ( ( C != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.52          = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_mult2
% 0.30/0.52  thf(fact_1569_div__mult__mult2,axiom,
% 0.30/0.52      ! [C: int,A: int,B2: int] :
% 0.30/0.52        ( ( C != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.52          = ( divide_divide_int @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_mult2
% 0.30/0.52  thf(fact_1570_div__mult__mult1,axiom,
% 0.30/0.52      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.52        ( ( C != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.30/0.52          = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_mult1
% 0.30/0.52  thf(fact_1571_div__mult__mult1,axiom,
% 0.30/0.52      ! [C: int,A: int,B2: int] :
% 0.30/0.52        ( ( C != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.52          = ( divide_divide_int @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_mult1
% 0.30/0.52  thf(fact_1572_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52      ! [A: complex,B2: complex] :
% 0.30/0.52        ( ( A != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
% 0.30/0.52          = B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_left
% 0.30/0.52  thf(fact_1573_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( A != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
% 0.30/0.52          = B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_left
% 0.30/0.52  thf(fact_1574_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( A != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ A )
% 0.30/0.52          = B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_left
% 0.30/0.52  thf(fact_1575_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( A != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
% 0.30/0.52          = B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_left
% 0.30/0.52  thf(fact_1576_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( A != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
% 0.30/0.52          = B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_left
% 0.30/0.52  thf(fact_1577_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52      ! [B2: complex,A: complex] :
% 0.30/0.52        ( ( B2 != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
% 0.30/0.52          = A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_right
% 0.30/0.52  thf(fact_1578_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52      ! [B2: real,A: real] :
% 0.30/0.52        ( ( B2 != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
% 0.30/0.52          = A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_right
% 0.30/0.52  thf(fact_1579_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52      ! [B2: rat,A: rat] :
% 0.30/0.52        ( ( B2 != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ B2 )
% 0.30/0.52          = A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_right
% 0.30/0.52  thf(fact_1580_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( B2 != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 0.30/0.52          = A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_right
% 0.30/0.52  thf(fact_1581_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( B2 != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 0.30/0.52          = A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_mult_div_cancel_right
% 0.30/0.52  thf(fact_1582_diff__add__zero,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_add_zero
% 0.30/0.52  thf(fact_1583_divide__eq__1__iff,axiom,
% 0.30/0.52      ! [A: complex,B2: complex] :
% 0.30/0.52        ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.30/0.52          = one_one_complex )
% 0.30/0.52        = ( ( B2 != zero_zero_complex )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_1_iff
% 0.30/0.52  thf(fact_1584_divide__eq__1__iff,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ( divide_divide_real @ A @ B2 )
% 0.30/0.52          = one_one_real )
% 0.30/0.52        = ( ( B2 != zero_zero_real )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_1_iff
% 0.30/0.52  thf(fact_1585_divide__eq__1__iff,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ( divide_divide_rat @ A @ B2 )
% 0.30/0.52          = one_one_rat )
% 0.30/0.52        = ( ( B2 != zero_zero_rat )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_1_iff
% 0.30/0.52  thf(fact_1586_one__eq__divide__iff,axiom,
% 0.30/0.52      ! [A: complex,B2: complex] :
% 0.30/0.52        ( ( one_one_complex
% 0.30/0.52          = ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.52        = ( ( B2 != zero_zero_complex )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_eq_divide_iff
% 0.30/0.52  thf(fact_1587_one__eq__divide__iff,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( one_one_real
% 0.30/0.52          = ( divide_divide_real @ A @ B2 ) )
% 0.30/0.52        = ( ( B2 != zero_zero_real )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_eq_divide_iff
% 0.30/0.52  thf(fact_1588_one__eq__divide__iff,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( one_one_rat
% 0.30/0.52          = ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.52        = ( ( B2 != zero_zero_rat )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_eq_divide_iff
% 0.30/0.52  thf(fact_1589_divide__self,axiom,
% 0.30/0.52      ! [A: complex] :
% 0.30/0.52        ( ( A != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52          = one_one_complex ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_self
% 0.30/0.52  thf(fact_1590_divide__self,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( A != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ A @ A )
% 0.30/0.52          = one_one_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_self
% 0.30/0.52  thf(fact_1591_divide__self,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( A != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52          = one_one_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_self
% 0.30/0.52  thf(fact_1592_divide__self__if,axiom,
% 0.30/0.52      ! [A: complex] :
% 0.30/0.52        ( ( ( A = zero_zero_complex )
% 0.30/0.52         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52            = zero_zero_complex ) )
% 0.30/0.52        & ( ( A != zero_zero_complex )
% 0.30/0.52         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52            = one_one_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_self_if
% 0.30/0.52  thf(fact_1593_divide__self__if,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ( A = zero_zero_real )
% 0.30/0.52         => ( ( divide_divide_real @ A @ A )
% 0.30/0.52            = zero_zero_real ) )
% 0.30/0.52        & ( ( A != zero_zero_real )
% 0.30/0.52         => ( ( divide_divide_real @ A @ A )
% 0.30/0.52            = one_one_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_self_if
% 0.30/0.52  thf(fact_1594_divide__self__if,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ( A = zero_zero_rat )
% 0.30/0.52         => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52            = zero_zero_rat ) )
% 0.30/0.52        & ( ( A != zero_zero_rat )
% 0.30/0.52         => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52            = one_one_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_self_if
% 0.30/0.52  thf(fact_1595_divide__eq__eq__1,axiom,
% 0.30/0.52      ! [B2: real,A: real] :
% 0.30/0.52        ( ( ( divide_divide_real @ B2 @ A )
% 0.30/0.52          = one_one_real )
% 0.30/0.52        = ( ( A != zero_zero_real )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_eq_1
% 0.30/0.52  thf(fact_1596_divide__eq__eq__1,axiom,
% 0.30/0.52      ! [B2: rat,A: rat] :
% 0.30/0.52        ( ( ( divide_divide_rat @ B2 @ A )
% 0.30/0.52          = one_one_rat )
% 0.30/0.52        = ( ( A != zero_zero_rat )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_eq_1
% 0.30/0.52  thf(fact_1597_eq__divide__eq__1,axiom,
% 0.30/0.52      ! [B2: real,A: real] :
% 0.30/0.52        ( ( one_one_real
% 0.30/0.52          = ( divide_divide_real @ B2 @ A ) )
% 0.30/0.52        = ( ( A != zero_zero_real )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % eq_divide_eq_1
% 0.30/0.52  thf(fact_1598_eq__divide__eq__1,axiom,
% 0.30/0.52      ! [B2: rat,A: rat] :
% 0.30/0.52        ( ( one_one_rat
% 0.30/0.52          = ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.52        = ( ( A != zero_zero_rat )
% 0.30/0.52          & ( A = B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % eq_divide_eq_1
% 0.30/0.52  thf(fact_1599_one__divide__eq__0__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ( divide_divide_real @ one_one_real @ A )
% 0.30/0.52          = zero_zero_real )
% 0.30/0.52        = ( A = zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_divide_eq_0_iff
% 0.30/0.52  thf(fact_1600_one__divide__eq__0__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.30/0.52          = zero_zero_rat )
% 0.30/0.52        = ( A = zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_divide_eq_0_iff
% 0.30/0.52  thf(fact_1601_zero__eq__1__divide__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( zero_zero_real
% 0.30/0.52          = ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.52        = ( A = zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_1_divide_iff
% 0.30/0.52  thf(fact_1602_zero__eq__1__divide__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( zero_zero_rat
% 0.30/0.52          = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.52        = ( A = zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_1_divide_iff
% 0.30/0.52  thf(fact_1603_div__self,axiom,
% 0.30/0.52      ! [A: complex] :
% 0.30/0.52        ( ( A != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.52          = one_one_complex ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_self
% 0.30/0.52  thf(fact_1604_div__self,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( A != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ A @ A )
% 0.30/0.52          = one_one_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_self
% 0.30/0.52  thf(fact_1605_div__self,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( A != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ A @ A )
% 0.30/0.52          = one_one_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_self
% 0.30/0.52  thf(fact_1606_div__self,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( A != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ A @ A )
% 0.30/0.52          = one_one_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_self
% 0.30/0.52  thf(fact_1607_div__self,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( A != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ A @ A )
% 0.30/0.52          = one_one_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_self
% 0.30/0.52  thf(fact_1608_power__0__Suc,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.30/0.52        = zero_zero_rat ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_Suc
% 0.30/0.52  thf(fact_1609_power__0__Suc,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_Suc
% 0.30/0.52  thf(fact_1610_power__0__Suc,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.30/0.52        = zero_zero_real ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_Suc
% 0.30/0.52  thf(fact_1611_power__0__Suc,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_Suc
% 0.30/0.52  thf(fact_1612_power__0__Suc,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.30/0.52        = zero_zero_complex ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_Suc
% 0.30/0.52  thf(fact_1613_power__zero__numeral,axiom,
% 0.30/0.52      ! [K: num] :
% 0.30/0.52        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52        = zero_zero_rat ) ).
% 0.30/0.52  
% 0.30/0.52  % power_zero_numeral
% 0.30/0.52  thf(fact_1614_power__zero__numeral,axiom,
% 0.30/0.52      ! [K: num] :
% 0.30/0.52        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % power_zero_numeral
% 0.30/0.52  thf(fact_1615_power__zero__numeral,axiom,
% 0.30/0.52      ! [K: num] :
% 0.30/0.52        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52        = zero_zero_real ) ).
% 0.30/0.52  
% 0.30/0.52  % power_zero_numeral
% 0.30/0.52  thf(fact_1616_power__zero__numeral,axiom,
% 0.30/0.52      ! [K: num] :
% 0.30/0.52        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % power_zero_numeral
% 0.30/0.52  thf(fact_1617_power__zero__numeral,axiom,
% 0.30/0.52      ! [K: num] :
% 0.30/0.52        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52        = zero_zero_complex ) ).
% 0.30/0.52  
% 0.30/0.52  % power_zero_numeral
% 0.30/0.52  thf(fact_1618_mod__mult__self1__is__0,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self1_is_0
% 0.30/0.52  thf(fact_1619_mod__mult__self1__is__0,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self1_is_0
% 0.30/0.52  thf(fact_1620_mod__mult__self1__is__0,axiom,
% 0.30/0.52      ! [B2: code_integer,A: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ B2 )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self1_is_0
% 0.30/0.52  thf(fact_1621_mod__mult__self2__is__0,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self2_is_0
% 0.30/0.52  thf(fact_1622_mod__mult__self2__is__0,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self2_is_0
% 0.30/0.52  thf(fact_1623_mod__mult__self2__is__0,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self2_is_0
% 0.30/0.52  thf(fact_1624_mod__by__1,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_by_1
% 0.30/0.52  thf(fact_1625_mod__by__1,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_by_1
% 0.30/0.52  thf(fact_1626_mod__by__1,axiom,
% 0.30/0.52      ! [A: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_by_1
% 0.30/0.52  thf(fact_1627_bits__mod__by__1,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_mod_by_1
% 0.30/0.52  thf(fact_1628_bits__mod__by__1,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_mod_by_1
% 0.30/0.52  thf(fact_1629_bits__mod__by__1,axiom,
% 0.30/0.52      ! [A: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_mod_by_1
% 0.30/0.52  thf(fact_1630_mod__div__trivial,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_div_trivial
% 0.30/0.52  thf(fact_1631_mod__div__trivial,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_div_trivial
% 0.30/0.52  thf(fact_1632_mod__div__trivial,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_div_trivial
% 0.30/0.52  thf(fact_1633_bits__mod__div__trivial,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_mod_div_trivial
% 0.30/0.52  thf(fact_1634_bits__mod__div__trivial,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_mod_div_trivial
% 0.30/0.52  thf(fact_1635_bits__mod__div__trivial,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_mod_div_trivial
% 0.30/0.52  thf(fact_1636_power__Suc0__right,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = A ) ).
% 0.30/0.52  
% 0.30/0.52  % power_Suc0_right
% 0.30/0.52  thf(fact_1637_power__Suc0__right,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = A ) ).
% 0.30/0.52  
% 0.30/0.52  % power_Suc0_right
% 0.30/0.52  thf(fact_1638_power__Suc0__right,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = A ) ).
% 0.30/0.52  
% 0.30/0.52  % power_Suc0_right
% 0.30/0.52  thf(fact_1639_power__Suc0__right,axiom,
% 0.30/0.52      ! [A: complex] :
% 0.30/0.52        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = A ) ).
% 0.30/0.52  
% 0.30/0.52  % power_Suc0_right
% 0.30/0.52  thf(fact_1640_mod__mult__self1,axiom,
% 0.30/0.52      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self1
% 0.30/0.52  thf(fact_1641_mod__mult__self1,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self1
% 0.30/0.52  thf(fact_1642_mod__mult__self1,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self1
% 0.30/0.52  thf(fact_1643_mod__mult__self2,axiom,
% 0.30/0.52      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self2
% 0.30/0.52  thf(fact_1644_mod__mult__self2,axiom,
% 0.30/0.52      ! [A: int,B2: int,C: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self2
% 0.30/0.52  thf(fact_1645_mod__mult__self2,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self2
% 0.30/0.52  thf(fact_1646_mod__mult__self3,axiom,
% 0.30/0.52      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self3
% 0.30/0.52  thf(fact_1647_mod__mult__self3,axiom,
% 0.30/0.52      ! [C: int,B2: int,A: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self3
% 0.30/0.52  thf(fact_1648_mod__mult__self3,axiom,
% 0.30/0.52      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B2 ) @ A ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self3
% 0.30/0.52  thf(fact_1649_mod__mult__self4,axiom,
% 0.30/0.52      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self4
% 0.30/0.52  thf(fact_1650_mod__mult__self4,axiom,
% 0.30/0.52      ! [B2: int,C: int,A: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self4
% 0.30/0.52  thf(fact_1651_mod__mult__self4,axiom,
% 0.30/0.52      ! [B2: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C ) @ A ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_self4
% 0.30/0.52  thf(fact_1652_zero__less__Suc,axiom,
% 0.30/0.52      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_Suc
% 0.30/0.52  thf(fact_1653_less__Suc0,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = ( N = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_Suc0
% 0.30/0.52  thf(fact_1654_add__gr__0,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.52          | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % add_gr_0
% 0.30/0.52  thf(fact_1655_mult__eq__1__iff,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ( times_times_nat @ M @ N )
% 0.30/0.52          = ( suc @ zero_zero_nat ) )
% 0.30/0.52        = ( ( M
% 0.30/0.52            = ( suc @ zero_zero_nat ) )
% 0.30/0.52          & ( N
% 0.30/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_eq_1_iff
% 0.30/0.52  thf(fact_1656_one__eq__mult__iff,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ( suc @ zero_zero_nat )
% 0.30/0.52          = ( times_times_nat @ M @ N ) )
% 0.30/0.52        = ( ( M
% 0.30/0.52            = ( suc @ zero_zero_nat ) )
% 0.30/0.52          & ( N
% 0.30/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_eq_mult_iff
% 0.30/0.52  thf(fact_1657_zero__less__diff,axiom,
% 0.30/0.52      ! [N: nat,M: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.52        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_diff
% 0.30/0.52  thf(fact_1658_div__by__Suc__0,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = M ) ).
% 0.30/0.52  
% 0.30/0.52  % div_by_Suc_0
% 0.30/0.52  thf(fact_1659_nat__mult__less__cancel__disj,axiom,
% 0.30/0.52      ! [K: nat,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52          & ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nat_mult_less_cancel_disj
% 0.30/0.52  thf(fact_1660_nat__0__less__mult__iff,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nat_0_less_mult_iff
% 0.30/0.52  thf(fact_1661_mult__less__cancel2,axiom,
% 0.30/0.52      ! [M: nat,K: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52          & ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_less_cancel2
% 0.30/0.52  thf(fact_1662_diff__is__0__eq,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ( minus_minus_nat @ M @ N )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_is_0_eq
% 0.30/0.52  thf(fact_1663_diff__is__0__eq_H,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.52       => ( ( minus_minus_nat @ M @ N )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % diff_is_0_eq'
% 0.30/0.52  thf(fact_1664_div__less,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ M @ N )
% 0.30/0.52       => ( ( divide_divide_nat @ M @ N )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_less
% 0.30/0.52  thf(fact_1665_less__one,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ N @ one_one_nat )
% 0.30/0.52        = ( N = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_one
% 0.30/0.52  thf(fact_1666_power__Suc__0,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.52        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_Suc_0
% 0.30/0.52  thf(fact_1667_nat__power__eq__Suc__0__iff,axiom,
% 0.30/0.52      ! [X3: nat,M: nat] :
% 0.30/0.52        ( ( ( power_power_nat @ X3 @ M )
% 0.30/0.52          = ( suc @ zero_zero_nat ) )
% 0.30/0.52        = ( ( M = zero_zero_nat )
% 0.30/0.52          | ( X3
% 0.30/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nat_power_eq_Suc_0_iff
% 0.30/0.52  thf(fact_1668_nat__zero__less__power__iff,axiom,
% 0.30/0.52      ! [X3: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.52          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nat_zero_less_power_iff
% 0.30/0.52  thf(fact_1669_nat__mult__div__cancel__disj,axiom,
% 0.30/0.52      ! [K: nat,M: nat,N: nat] :
% 0.30/0.52        ( ( ( K = zero_zero_nat )
% 0.30/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52            = zero_zero_nat ) )
% 0.30/0.52        & ( ( K != zero_zero_nat )
% 0.30/0.52         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52            = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nat_mult_div_cancel_disj
% 0.30/0.52  thf(fact_1670_mod__by__Suc__0,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_by_Suc_0
% 0.30/0.52  thf(fact_1671_zero__le__divide__1__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.52        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_le_divide_1_iff
% 0.30/0.52  thf(fact_1672_zero__le__divide__1__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.52        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_le_divide_1_iff
% 0.30/0.52  thf(fact_1673_divide__le__0__1__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.52        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_le_0_1_iff
% 0.30/0.52  thf(fact_1674_divide__le__0__1__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.30/0.52        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_le_0_1_iff
% 0.30/0.52  thf(fact_1675_zero__less__divide__1__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.52        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_divide_1_iff
% 0.30/0.52  thf(fact_1676_zero__less__divide__1__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.52        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_divide_1_iff
% 0.30/0.52  thf(fact_1677_less__divide__eq__1__pos,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_divide_eq_1_pos
% 0.30/0.52  thf(fact_1678_less__divide__eq__1__pos,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_divide_eq_1_pos
% 0.30/0.52  thf(fact_1679_less__divide__eq__1__neg,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_real @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_divide_eq_1_neg
% 0.30/0.52  thf(fact_1680_less__divide__eq__1__neg,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % less_divide_eq_1_neg
% 0.30/0.52  thf(fact_1681_divide__less__eq__1__pos,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.30/0.52          = ( ord_less_real @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_less_eq_1_pos
% 0.30/0.52  thf(fact_1682_divide__less__eq__1__pos,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.30/0.52          = ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_less_eq_1_pos
% 0.30/0.52  thf(fact_1683_divide__less__eq__1__neg,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.30/0.52          = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_less_eq_1_neg
% 0.30/0.52  thf(fact_1684_divide__less__eq__1__neg,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.30/0.52          = ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_less_eq_1_neg
% 0.30/0.52  thf(fact_1685_divide__less__0__1__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.52        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_less_0_1_iff
% 0.30/0.52  thf(fact_1686_divide__less__0__1__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.30/0.52        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_less_0_1_iff
% 0.30/0.52  thf(fact_1687_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52      ! [B2: complex,W: num,A: complex] :
% 0.30/0.52        ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.52          = A )
% 0.30/0.52        = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52             != zero_zero_complex )
% 0.30/0.52           => ( B2
% 0.30/0.52              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.30/0.52          & ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52              = zero_zero_complex )
% 0.30/0.52           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_eq_numeral1(1)
% 0.30/0.52  thf(fact_1688_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52      ! [B2: real,W: num,A: real] :
% 0.30/0.52        ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
% 0.30/0.52          = A )
% 0.30/0.52        = ( ( ( ( numeral_numeral_real @ W )
% 0.30/0.52             != zero_zero_real )
% 0.30/0.52           => ( B2
% 0.30/0.52              = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.52          & ( ( ( numeral_numeral_real @ W )
% 0.30/0.52              = zero_zero_real )
% 0.30/0.52           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_eq_numeral1(1)
% 0.30/0.52  thf(fact_1689_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.52      ! [B2: rat,W: num,A: rat] :
% 0.30/0.52        ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
% 0.30/0.52          = A )
% 0.30/0.52        = ( ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52             != zero_zero_rat )
% 0.30/0.52           => ( B2
% 0.30/0.52              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.52          & ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52              = zero_zero_rat )
% 0.30/0.52           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_eq_eq_numeral1(1)
% 0.30/0.52  thf(fact_1690_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52      ! [A: complex,B2: complex,W: num] :
% 0.30/0.52        ( ( A
% 0.30/0.52          = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.52        = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52             != zero_zero_complex )
% 0.30/0.52           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.52              = B2 ) )
% 0.30/0.52          & ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.52              = zero_zero_complex )
% 0.30/0.52           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % eq_divide_eq_numeral1(1)
% 0.30/0.52  thf(fact_1691_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52      ! [A: real,B2: real,W: num] :
% 0.30/0.52        ( ( A
% 0.30/0.52          = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.52        = ( ( ( ( numeral_numeral_real @ W )
% 0.30/0.52             != zero_zero_real )
% 0.30/0.52           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.30/0.52              = B2 ) )
% 0.30/0.52          & ( ( ( numeral_numeral_real @ W )
% 0.30/0.52              = zero_zero_real )
% 0.30/0.52           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % eq_divide_eq_numeral1(1)
% 0.30/0.52  thf(fact_1692_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.52      ! [A: rat,B2: rat,W: num] :
% 0.30/0.52        ( ( A
% 0.30/0.52          = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.52        = ( ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52             != zero_zero_rat )
% 0.30/0.52           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.30/0.52              = B2 ) )
% 0.30/0.52          & ( ( ( numeral_numeral_rat @ W )
% 0.30/0.52              = zero_zero_rat )
% 0.30/0.52           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % eq_divide_eq_numeral1(1)
% 0.30/0.52  thf(fact_1693_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.52      ! [A: complex,B2: complex] :
% 0.30/0.52        ( ( A != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
% 0.30/0.52          = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_divide_mult_cancel_left
% 0.30/0.52  thf(fact_1694_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( A != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
% 0.30/0.52          = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_divide_mult_cancel_left
% 0.30/0.52  thf(fact_1695_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( A != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.52          = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_divide_mult_cancel_left
% 0.30/0.52  thf(fact_1696_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.52      ! [B2: complex,A: complex] :
% 0.30/0.52        ( ( B2 != zero_zero_complex )
% 0.30/0.52       => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
% 0.30/0.52          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_divide_mult_cancel_right
% 0.30/0.52  thf(fact_1697_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.52      ! [B2: real,A: real] :
% 0.30/0.52        ( ( B2 != zero_zero_real )
% 0.30/0.52       => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
% 0.30/0.52          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_divide_mult_cancel_right
% 0.30/0.52  thf(fact_1698_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.52      ! [B2: rat,A: rat] :
% 0.30/0.52        ( ( B2 != zero_zero_rat )
% 0.30/0.52       => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.52          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nonzero_divide_mult_cancel_right
% 0.30/0.52  thf(fact_1699_div__mult__self1,axiom,
% 0.30/0.52      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.52        ( ( B2 != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 0.30/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self1
% 0.30/0.52  thf(fact_1700_div__mult__self1,axiom,
% 0.30/0.52      ! [B2: int,A: int,C: int] :
% 0.30/0.52        ( ( B2 != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 0.30/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self1
% 0.30/0.52  thf(fact_1701_div__mult__self2,axiom,
% 0.30/0.52      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.52        ( ( B2 != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 0.30/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self2
% 0.30/0.52  thf(fact_1702_div__mult__self2,axiom,
% 0.30/0.52      ! [B2: int,A: int,C: int] :
% 0.30/0.52        ( ( B2 != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 0.30/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self2
% 0.30/0.52  thf(fact_1703_div__mult__self3,axiom,
% 0.30/0.52      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.52        ( ( B2 != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 0.30/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self3
% 0.30/0.52  thf(fact_1704_div__mult__self3,axiom,
% 0.30/0.52      ! [B2: int,C: int,A: int] :
% 0.30/0.52        ( ( B2 != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 0.30/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self3
% 0.30/0.52  thf(fact_1705_div__mult__self4,axiom,
% 0.30/0.52      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.52        ( ( B2 != zero_zero_nat )
% 0.30/0.52       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 0.30/0.52          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self4
% 0.30/0.52  thf(fact_1706_div__mult__self4,axiom,
% 0.30/0.52      ! [B2: int,C: int,A: int] :
% 0.30/0.52        ( ( B2 != zero_zero_int )
% 0.30/0.52       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 0.30/0.52          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self4
% 0.30/0.52  thf(fact_1707_power__eq__0__iff,axiom,
% 0.30/0.52      ! [A: rat,N: nat] :
% 0.30/0.52        ( ( ( power_power_rat @ A @ N )
% 0.30/0.52          = zero_zero_rat )
% 0.30/0.52        = ( ( A = zero_zero_rat )
% 0.30/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_eq_0_iff
% 0.30/0.52  thf(fact_1708_power__eq__0__iff,axiom,
% 0.30/0.52      ! [A: nat,N: nat] :
% 0.30/0.52        ( ( ( power_power_nat @ A @ N )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52        = ( ( A = zero_zero_nat )
% 0.30/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_eq_0_iff
% 0.30/0.52  thf(fact_1709_power__eq__0__iff,axiom,
% 0.30/0.52      ! [A: real,N: nat] :
% 0.30/0.52        ( ( ( power_power_real @ A @ N )
% 0.30/0.52          = zero_zero_real )
% 0.30/0.52        = ( ( A = zero_zero_real )
% 0.30/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_eq_0_iff
% 0.30/0.52  thf(fact_1710_power__eq__0__iff,axiom,
% 0.30/0.52      ! [A: int,N: nat] :
% 0.30/0.52        ( ( ( power_power_int @ A @ N )
% 0.30/0.52          = zero_zero_int )
% 0.30/0.52        = ( ( A = zero_zero_int )
% 0.30/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_eq_0_iff
% 0.30/0.52  thf(fact_1711_power__eq__0__iff,axiom,
% 0.30/0.52      ! [A: complex,N: nat] :
% 0.30/0.52        ( ( ( power_power_complex @ A @ N )
% 0.30/0.52          = zero_zero_complex )
% 0.30/0.52        = ( ( A = zero_zero_complex )
% 0.30/0.52          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_eq_0_iff
% 0.30/0.52  thf(fact_1712_Suc__pred,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.52          = N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_pred
% 0.30/0.52  thf(fact_1713_one__le__mult__iff,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.30/0.52        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.52          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % one_le_mult_iff
% 0.30/0.52  thf(fact_1714_nat__mult__le__cancel__disj,axiom,
% 0.30/0.52      ! [K: nat,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % nat_mult_le_cancel_disj
% 0.30/0.52  thf(fact_1715_mult__le__cancel2,axiom,
% 0.30/0.52      ! [M: nat,K: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.30/0.52        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.52         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_le_cancel2
% 0.30/0.52  thf(fact_1716_div__mult__self1__is__m,axiom,
% 0.30/0.52      ! [N: nat,M: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.30/0.52          = M ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self1_is_m
% 0.30/0.52  thf(fact_1717_div__mult__self__is__m,axiom,
% 0.30/0.52      ! [N: nat,M: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.30/0.52          = M ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_mult_self_is_m
% 0.30/0.52  thf(fact_1718_Suc__mod__mult__self1,axiom,
% 0.30/0.52      ! [M: nat,K: nat,N: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.30/0.52        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_mod_mult_self1
% 0.30/0.52  thf(fact_1719_Suc__mod__mult__self2,axiom,
% 0.30/0.52      ! [M: nat,N: nat,K: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.30/0.52        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_mod_mult_self2
% 0.30/0.52  thf(fact_1720_Suc__mod__mult__self3,axiom,
% 0.30/0.52      ! [K: nat,N: nat,M: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.30/0.52        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_mod_mult_self3
% 0.30/0.52  thf(fact_1721_Suc__mod__mult__self4,axiom,
% 0.30/0.52      ! [N: nat,K: nat,M: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.30/0.52        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_mod_mult_self4
% 0.30/0.52  thf(fact_1722_le__divide__eq__1__pos,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_divide_eq_1_pos
% 0.30/0.52  thf(fact_1723_le__divide__eq__1__pos,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_divide_eq_1_pos
% 0.30/0.52  thf(fact_1724_le__divide__eq__1__neg,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_divide_eq_1_neg
% 0.30/0.52  thf(fact_1725_le__divide__eq__1__neg,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.52          = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % le_divide_eq_1_neg
% 0.30/0.52  thf(fact_1726_divide__le__eq__1__pos,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.52       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.30/0.52          = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_le_eq_1_pos
% 0.30/0.52  thf(fact_1727_divide__le__eq__1__pos,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.52       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.30/0.52          = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_le_eq_1_pos
% 0.30/0.52  thf(fact_1728_divide__le__eq__1__neg,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.52       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.30/0.52          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_le_eq_1_neg
% 0.30/0.52  thf(fact_1729_divide__le__eq__1__neg,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.52       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.30/0.52          = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % divide_le_eq_1_neg
% 0.30/0.52  thf(fact_1730_power__strict__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: real,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.52       => ( ( ord_less_real @ B2 @ one_one_real )
% 0.30/0.52         => ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_strict_decreasing_iff
% 0.30/0.52  thf(fact_1731_power__strict__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: rat,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.52       => ( ( ord_less_rat @ B2 @ one_one_rat )
% 0.30/0.52         => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_strict_decreasing_iff
% 0.30/0.52  thf(fact_1732_power__strict__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: nat,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.52       => ( ( ord_less_nat @ B2 @ one_one_nat )
% 0.30/0.52         => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_strict_decreasing_iff
% 0.30/0.52  thf(fact_1733_power__strict__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: int,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.52       => ( ( ord_less_int @ B2 @ one_one_int )
% 0.30/0.52         => ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_strict_decreasing_iff
% 0.30/0.52  thf(fact_1734_power__mono__iff,axiom,
% 0.30/0.52      ! [A: real,B2: real,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 0.30/0.52              = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mono_iff
% 0.30/0.52  thf(fact_1735_power__mono__iff,axiom,
% 0.30/0.52      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 0.30/0.52              = ( ord_less_eq_rat @ A @ B2 ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mono_iff
% 0.30/0.52  thf(fact_1736_power__mono__iff,axiom,
% 0.30/0.52      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 0.30/0.52              = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mono_iff
% 0.30/0.52  thf(fact_1737_power__mono__iff,axiom,
% 0.30/0.52      ! [A: int,B2: int,N: nat] :
% 0.30/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 0.30/0.52              = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mono_iff
% 0.30/0.52  thf(fact_1738_zero__eq__power2,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_rat )
% 0.30/0.52        = ( A = zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_power2
% 0.30/0.52  thf(fact_1739_zero__eq__power2,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52        = ( A = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_power2
% 0.30/0.52  thf(fact_1740_zero__eq__power2,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_real )
% 0.30/0.52        = ( A = zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_power2
% 0.30/0.52  thf(fact_1741_zero__eq__power2,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_int )
% 0.30/0.52        = ( A = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_power2
% 0.30/0.52  thf(fact_1742_zero__eq__power2,axiom,
% 0.30/0.52      ! [A: complex] :
% 0.30/0.52        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_complex )
% 0.30/0.52        = ( A = zero_zero_complex ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_eq_power2
% 0.30/0.52  thf(fact_1743_one__mod__two__eq__one,axiom,
% 0.30/0.52      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52      = one_one_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % one_mod_two_eq_one
% 0.30/0.52  thf(fact_1744_one__mod__two__eq__one,axiom,
% 0.30/0.52      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52      = one_one_int ) ).
% 0.30/0.52  
% 0.30/0.52  % one_mod_two_eq_one
% 0.30/0.52  thf(fact_1745_one__mod__two__eq__one,axiom,
% 0.30/0.52      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.52      = one_one_Code_integer ) ).
% 0.30/0.52  
% 0.30/0.52  % one_mod_two_eq_one
% 0.30/0.52  thf(fact_1746_bits__one__mod__two__eq__one,axiom,
% 0.30/0.52      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52      = one_one_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_one_mod_two_eq_one
% 0.30/0.52  thf(fact_1747_bits__one__mod__two__eq__one,axiom,
% 0.30/0.52      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52      = one_one_int ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_one_mod_two_eq_one
% 0.30/0.52  thf(fact_1748_bits__one__mod__two__eq__one,axiom,
% 0.30/0.52      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.52      = one_one_Code_integer ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_one_mod_two_eq_one
% 0.30/0.52  thf(fact_1749_mod2__Suc__Suc,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52        = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod2_Suc_Suc
% 0.30/0.52  thf(fact_1750_Suc__diff__1,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.30/0.52          = N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_diff_1
% 0.30/0.52  thf(fact_1751_Suc__times__numeral__mod__eq,axiom,
% 0.30/0.52      ! [K: num,N: nat] :
% 0.30/0.52        ( ( ( numeral_numeral_nat @ K )
% 0.30/0.52         != one_one_nat )
% 0.30/0.52       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.52          = one_one_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % Suc_times_numeral_mod_eq
% 0.30/0.52  thf(fact_1752_one__div__two__eq__zero,axiom,
% 0.30/0.52      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52      = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % one_div_two_eq_zero
% 0.30/0.52  thf(fact_1753_one__div__two__eq__zero,axiom,
% 0.30/0.52      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52      = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % one_div_two_eq_zero
% 0.30/0.52  thf(fact_1754_bits__1__div__2,axiom,
% 0.30/0.52      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52      = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_1_div_2
% 0.30/0.52  thf(fact_1755_bits__1__div__2,axiom,
% 0.30/0.52      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52      = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % bits_1_div_2
% 0.30/0.52  thf(fact_1756_power__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: real,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.52       => ( ( ord_less_real @ B2 @ one_one_real )
% 0.30/0.52         => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_decreasing_iff
% 0.30/0.52  thf(fact_1757_power__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: rat,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.52       => ( ( ord_less_rat @ B2 @ one_one_rat )
% 0.30/0.52         => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_decreasing_iff
% 0.30/0.52  thf(fact_1758_power__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: nat,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.52       => ( ( ord_less_nat @ B2 @ one_one_nat )
% 0.30/0.52         => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_decreasing_iff
% 0.30/0.52  thf(fact_1759_power__decreasing__iff,axiom,
% 0.30/0.52      ! [B2: int,M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.52       => ( ( ord_less_int @ B2 @ one_one_int )
% 0.30/0.52         => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 0.30/0.52            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_decreasing_iff
% 0.30/0.52  thf(fact_1760_power2__less__eq__zero__iff,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.30/0.52        = ( A = zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_less_eq_zero_iff
% 0.30/0.52  thf(fact_1761_power2__less__eq__zero__iff,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.30/0.52        = ( A = zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_less_eq_zero_iff
% 0.30/0.52  thf(fact_1762_power2__less__eq__zero__iff,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.30/0.52        = ( A = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_less_eq_zero_iff
% 0.30/0.52  thf(fact_1763_power2__eq__iff__nonneg,axiom,
% 0.30/0.52      ! [X3: real,Y3: real] :
% 0.30/0.52        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.52         => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52              = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_eq_iff_nonneg
% 0.30/0.52  thf(fact_1764_power2__eq__iff__nonneg,axiom,
% 0.30/0.52      ! [X3: rat,Y3: rat] :
% 0.30/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.52         => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52              = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_eq_iff_nonneg
% 0.30/0.52  thf(fact_1765_power2__eq__iff__nonneg,axiom,
% 0.30/0.52      ! [X3: nat,Y3: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.30/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.52         => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52              = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_eq_iff_nonneg
% 0.30/0.52  thf(fact_1766_power2__eq__iff__nonneg,axiom,
% 0.30/0.52      ! [X3: int,Y3: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.52         => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52              = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power2_eq_iff_nonneg
% 0.30/0.52  thf(fact_1767_zero__less__power2,axiom,
% 0.30/0.52      ! [A: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52        = ( A != zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_power2
% 0.30/0.52  thf(fact_1768_zero__less__power2,axiom,
% 0.30/0.52      ! [A: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52        = ( A != zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_power2
% 0.30/0.52  thf(fact_1769_zero__less__power2,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52        = ( A != zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_power2
% 0.30/0.52  thf(fact_1770_sum__power2__eq__zero__iff,axiom,
% 0.30/0.52      ! [X3: rat,Y3: rat] :
% 0.30/0.52        ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52          = zero_zero_rat )
% 0.30/0.52        = ( ( X3 = zero_zero_rat )
% 0.30/0.52          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % sum_power2_eq_zero_iff
% 0.30/0.52  thf(fact_1771_sum__power2__eq__zero__iff,axiom,
% 0.30/0.52      ! [X3: real,Y3: real] :
% 0.30/0.52        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52          = zero_zero_real )
% 0.30/0.52        = ( ( X3 = zero_zero_real )
% 0.30/0.52          & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % sum_power2_eq_zero_iff
% 0.30/0.52  thf(fact_1772_sum__power2__eq__zero__iff,axiom,
% 0.30/0.52      ! [X3: int,Y3: int] :
% 0.30/0.52        ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52          = zero_zero_int )
% 0.30/0.52        = ( ( X3 = zero_zero_int )
% 0.30/0.52          & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % sum_power2_eq_zero_iff
% 0.30/0.52  thf(fact_1773_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52         != zero_zero_nat )
% 0.30/0.52        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = one_one_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod_2_eq_0_eq_1
% 0.30/0.52  thf(fact_1774_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52         != zero_zero_int )
% 0.30/0.52        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52          = one_one_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod_2_eq_0_eq_1
% 0.30/0.52  thf(fact_1775_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.52      ! [A: code_integer] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.52         != zero_z3403309356797280102nteger )
% 0.30/0.52        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.52          = one_one_Code_integer ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod_2_eq_0_eq_1
% 0.30/0.52  thf(fact_1776_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.52      ! [A: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52         != one_one_nat )
% 0.30/0.52        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod_2_eq_1_eq_0
% 0.30/0.52  thf(fact_1777_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.52      ! [A: int] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52         != one_one_int )
% 0.30/0.52        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod_2_eq_1_eq_0
% 0.30/0.52  thf(fact_1778_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.52      ! [A: code_integer] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.52         != one_one_Code_integer )
% 0.30/0.52        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod_2_eq_1_eq_0
% 0.30/0.52  thf(fact_1779_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52         != ( suc @ zero_zero_nat ) )
% 0.30/0.52        = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % not_mod2_eq_Suc_0_eq_0
% 0.30/0.52  thf(fact_1780_add__self__mod__2,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % add_self_mod_2
% 0.30/0.52  thf(fact_1781_mod2__gr__0,axiom,
% 0.30/0.52      ! [M: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.52        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.52          = one_one_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod2_gr_0
% 0.30/0.52  thf(fact_1782_zero__reorient,axiom,
% 0.30/0.52      ! [X3: complex] :
% 0.30/0.52        ( ( zero_zero_complex = X3 )
% 0.30/0.52        = ( X3 = zero_zero_complex ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_reorient
% 0.30/0.52  thf(fact_1783_zero__reorient,axiom,
% 0.30/0.52      ! [X3: real] :
% 0.30/0.52        ( ( zero_zero_real = X3 )
% 0.30/0.52        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_reorient
% 0.30/0.52  thf(fact_1784_zero__reorient,axiom,
% 0.30/0.52      ! [X3: rat] :
% 0.30/0.52        ( ( zero_zero_rat = X3 )
% 0.30/0.52        = ( X3 = zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_reorient
% 0.30/0.52  thf(fact_1785_zero__reorient,axiom,
% 0.30/0.52      ! [X3: nat] :
% 0.30/0.52        ( ( zero_zero_nat = X3 )
% 0.30/0.52        = ( X3 = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_reorient
% 0.30/0.52  thf(fact_1786_zero__reorient,axiom,
% 0.30/0.52      ! [X3: int] :
% 0.30/0.52        ( ( zero_zero_int = X3 )
% 0.30/0.52        = ( X3 = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_reorient
% 0.30/0.52  thf(fact_1787_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.52       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.52  thf(fact_1788_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.52       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.52  thf(fact_1789_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.52       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.52  thf(fact_1790_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.52       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.52  thf(fact_1791_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.52       => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.52  thf(fact_1792_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.52      ! [B2: code_integer,A: code_integer] :
% 0.30/0.52        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.52       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.52  thf(fact_1793_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ A @ B2 )
% 0.30/0.52          = A )
% 0.30/0.52        = ( ( divide_divide_nat @ A @ B2 )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_eq_self_iff_div_eq_0
% 0.30/0.52  thf(fact_1794_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.52          = A )
% 0.30/0.52        = ( ( divide_divide_int @ A @ B2 )
% 0.30/0.52          = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_eq_self_iff_div_eq_0
% 0.30/0.52  thf(fact_1795_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.30/0.52          = A )
% 0.30/0.52        = ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.30/0.52          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_eq_self_iff_div_eq_0
% 0.30/0.52  thf(fact_1796_mod__Suc,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.52            = N )
% 0.30/0.52         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.30/0.52            = zero_zero_nat ) )
% 0.30/0.52        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.52           != N )
% 0.30/0.52         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.30/0.52            = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_Suc
% 0.30/0.52  thf(fact_1797_mod__less__divisor,axiom,
% 0.30/0.52      ! [N: nat,M: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_less_divisor
% 0.30/0.52  thf(fact_1798_mod__eq__0D,axiom,
% 0.30/0.52      ! [M: nat,D: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ M @ D )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52       => ? [Q3: nat] :
% 0.30/0.52            ( M
% 0.30/0.52            = ( times_times_nat @ D @ Q3 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_eq_0D
% 0.30/0.52  thf(fact_1799_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.30/0.52      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.30/0.52        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT_internal.naive_member.simps(2)
% 0.30/0.52  thf(fact_1800_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.30/0.52      vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT_internal.minNull.simps(1)
% 0.30/0.52  thf(fact_1801_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.30/0.52      ! [Uv: $o] :
% 0.30/0.52        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT_internal.minNull.simps(2)
% 0.30/0.52  thf(fact_1802_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.30/0.52      ! [Uu: $o] :
% 0.30/0.52        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT_internal.minNull.simps(3)
% 0.30/0.52  thf(fact_1803_vebt__member_Osimps_I1_J,axiom,
% 0.30/0.52      ! [A: $o,B2: $o,X3: nat] :
% 0.30/0.52        ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 0.30/0.52        = ( ( ( X3 = zero_zero_nat )
% 0.30/0.52           => A )
% 0.30/0.52          & ( ( X3 != zero_zero_nat )
% 0.30/0.52           => ( ( ( X3 = one_one_nat )
% 0.30/0.52               => B2 )
% 0.30/0.52              & ( X3 = one_one_nat ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % vebt_member.simps(1)
% 0.30/0.52  thf(fact_1804_vebt__insert_Osimps_I1_J,axiom,
% 0.30/0.52      ! [X3: nat,A: $o,B2: $o] :
% 0.30/0.52        ( ( ( X3 = zero_zero_nat )
% 0.30/0.52         => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 0.30/0.52            = ( vEBT_Leaf @ $true @ B2 ) ) )
% 0.30/0.52        & ( ( X3 != zero_zero_nat )
% 0.30/0.52         => ( ( ( X3 = one_one_nat )
% 0.30/0.52             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 0.30/0.52                = ( vEBT_Leaf @ A @ $true ) ) )
% 0.30/0.52            & ( ( X3 != one_one_nat )
% 0.30/0.52             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 0.30/0.52                = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % vebt_insert.simps(1)
% 0.30/0.52  thf(fact_1805_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.52      ! [B2: code_integer,A: code_integer] :
% 0.30/0.52        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.52       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.52  thf(fact_1806_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.52      ! [B2: nat,A: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.52       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.52  thf(fact_1807_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.52      ! [B2: int,A: int] :
% 0.30/0.52        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.52       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.52  thf(fact_1808_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.52       => ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 0.30/0.52         => ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.30/0.52            = A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.52  thf(fact_1809_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.52       => ( ( ord_less_nat @ A @ B2 )
% 0.30/0.52         => ( ( modulo_modulo_nat @ A @ B2 )
% 0.30/0.52            = A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.52  thf(fact_1810_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.52       => ( ( ord_less_int @ A @ B2 )
% 0.30/0.52         => ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.52            = A ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.52  thf(fact_1811_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.52      ! [N: num,Q2: num] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.52          = zero_zero_nat )
% 0.30/0.52        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % cong_exp_iff_simps(2)
% 0.30/0.52  thf(fact_1812_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.52      ! [N: num,Q2: num] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.52          = zero_zero_int )
% 0.30/0.52        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.52          = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % cong_exp_iff_simps(2)
% 0.30/0.52  thf(fact_1813_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.52      ! [N: num,Q2: num] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.52          = zero_z3403309356797280102nteger )
% 0.30/0.52        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.52          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.52  
% 0.30/0.52  % cong_exp_iff_simps(2)
% 0.30/0.52  thf(fact_1814_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.52        = zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % cong_exp_iff_simps(1)
% 0.30/0.52  thf(fact_1815_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.52        = zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % cong_exp_iff_simps(1)
% 0.30/0.52  thf(fact_1816_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.52        = zero_z3403309356797280102nteger ) ).
% 0.30/0.52  
% 0.30/0.52  % cong_exp_iff_simps(1)
% 0.30/0.52  thf(fact_1817_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.30/0.52      ! [X3: produc9072475918466114483BT_nat] :
% 0.30/0.52        ( ! [A3: $o,B3: $o,X5: nat] :
% 0.30/0.52            ( X3
% 0.30/0.52           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
% 0.30/0.52       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.30/0.52              ( X3
% 0.30/0.52             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.30/0.52         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 0.30/0.52                ( X3
% 0.30/0.52               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X5 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT_internal.naive_member.cases
% 0.30/0.52  thf(fact_1818_mod__le__divisor,axiom,
% 0.30/0.52      ! [N: nat,M: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_le_divisor
% 0.30/0.52  thf(fact_1819_div__less__mono,axiom,
% 0.30/0.52      ! [A4: nat,B5: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ A4 @ B5 )
% 0.30/0.52       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52         => ( ( ( modulo_modulo_nat @ A4 @ N )
% 0.30/0.52              = zero_zero_nat )
% 0.30/0.52           => ( ( ( modulo_modulo_nat @ B5 @ N )
% 0.30/0.52                = zero_zero_nat )
% 0.30/0.52             => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % div_less_mono
% 0.30/0.52  thf(fact_1820_invar__vebt_Ointros_I1_J,axiom,
% 0.30/0.52      ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % invar_vebt.intros(1)
% 0.30/0.52  thf(fact_1821_power__0__left,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ( N = zero_zero_nat )
% 0.30/0.52         => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.30/0.52            = one_one_rat ) )
% 0.30/0.52        & ( ( N != zero_zero_nat )
% 0.30/0.52         => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.30/0.52            = zero_zero_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_left
% 0.30/0.52  thf(fact_1822_power__0__left,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ( N = zero_zero_nat )
% 0.30/0.52         => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.52            = one_one_nat ) )
% 0.30/0.52        & ( ( N != zero_zero_nat )
% 0.30/0.52         => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.52            = zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_left
% 0.30/0.52  thf(fact_1823_power__0__left,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ( N = zero_zero_nat )
% 0.30/0.52         => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.52            = one_one_real ) )
% 0.30/0.52        & ( ( N != zero_zero_nat )
% 0.30/0.52         => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.52            = zero_zero_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_left
% 0.30/0.52  thf(fact_1824_power__0__left,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ( N = zero_zero_nat )
% 0.30/0.52         => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.52            = one_one_int ) )
% 0.30/0.52        & ( ( N != zero_zero_nat )
% 0.30/0.52         => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.52            = zero_zero_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_left
% 0.30/0.52  thf(fact_1825_power__0__left,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ( N = zero_zero_nat )
% 0.30/0.52         => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.30/0.52            = one_one_complex ) )
% 0.30/0.52        & ( ( N != zero_zero_nat )
% 0.30/0.52         => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.30/0.52            = zero_zero_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_0_left
% 0.30/0.52  thf(fact_1826_zero__power,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.30/0.52          = zero_zero_rat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_power
% 0.30/0.52  thf(fact_1827_zero__power,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.30/0.52          = zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_power
% 0.30/0.52  thf(fact_1828_zero__power,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( power_power_real @ zero_zero_real @ N )
% 0.30/0.52          = zero_zero_real ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_power
% 0.30/0.52  thf(fact_1829_zero__power,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( power_power_int @ zero_zero_int @ N )
% 0.30/0.52          = zero_zero_int ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_power
% 0.30/0.52  thf(fact_1830_zero__power,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52       => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.30/0.52          = zero_zero_complex ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_power
% 0.30/0.52  thf(fact_1831_mod__mult__eq,axiom,
% 0.30/0.52      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_eq
% 0.30/0.52  thf(fact_1832_mod__mult__eq,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_eq
% 0.30/0.52  thf(fact_1833_mod__mult__eq,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_eq
% 0.30/0.52  thf(fact_1834_mod__mult__cong,axiom,
% 0.30/0.52      ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.52          = ( modulo_modulo_nat @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo_modulo_nat @ B2 @ C )
% 0.30/0.52            = ( modulo_modulo_nat @ B @ C ) )
% 0.30/0.52         => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_cong
% 0.30/0.52  thf(fact_1835_mod__mult__cong,axiom,
% 0.30/0.52      ! [A: int,C: int,A2: int,B2: int,B: int] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.52          = ( modulo_modulo_int @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo_modulo_int @ B2 @ C )
% 0.30/0.52            = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.52         => ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_cong
% 0.30/0.52  thf(fact_1836_mod__mult__cong,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.52          = ( modulo364778990260209775nteger @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 0.30/0.52            = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.52         => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_cong
% 0.30/0.52  thf(fact_1837_mod__mult__mult2,axiom,
% 0.30/0.52      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.52        = ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_mult2
% 0.30/0.52  thf(fact_1838_mod__mult__mult2,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.52        = ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_mult2
% 0.30/0.52  thf(fact_1839_mod__mult__mult2,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.52        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_mult2
% 0.30/0.52  thf(fact_1840_mult__mod__right,axiom,
% 0.30/0.52      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.52        ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.52        = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_mod_right
% 0.30/0.52  thf(fact_1841_mult__mod__right,axiom,
% 0.30/0.52      ! [C: int,A: int,B2: int] :
% 0.30/0.52        ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.52        = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_mod_right
% 0.30/0.52  thf(fact_1842_mult__mod__right,axiom,
% 0.30/0.52      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_mod_right
% 0.30/0.52  thf(fact_1843_mod__mult__left__eq,axiom,
% 0.30/0.52      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_left_eq
% 0.30/0.52  thf(fact_1844_mod__mult__left__eq,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_left_eq
% 0.30/0.52  thf(fact_1845_mod__mult__left__eq,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_left_eq
% 0.30/0.52  thf(fact_1846_mod__mult__right__eq,axiom,
% 0.30/0.52      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_right_eq
% 0.30/0.52  thf(fact_1847_mod__mult__right__eq,axiom,
% 0.30/0.52      ! [A: int,B2: int,C: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_right_eq
% 0.30/0.52  thf(fact_1848_mod__mult__right__eq,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_mult_right_eq
% 0.30/0.52  thf(fact_1849_mod__add__eq,axiom,
% 0.30/0.52      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_eq
% 0.30/0.52  thf(fact_1850_mod__add__eq,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_eq
% 0.30/0.52  thf(fact_1851_mod__add__eq,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_eq
% 0.30/0.52  thf(fact_1852_mod__add__cong,axiom,
% 0.30/0.52      ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 0.30/0.52        ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.52          = ( modulo_modulo_nat @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo_modulo_nat @ B2 @ C )
% 0.30/0.52            = ( modulo_modulo_nat @ B @ C ) )
% 0.30/0.52         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_cong
% 0.30/0.52  thf(fact_1853_mod__add__cong,axiom,
% 0.30/0.52      ! [A: int,C: int,A2: int,B2: int,B: int] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.52          = ( modulo_modulo_int @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo_modulo_int @ B2 @ C )
% 0.30/0.52            = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.52         => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_cong
% 0.30/0.52  thf(fact_1854_mod__add__cong,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.52          = ( modulo364778990260209775nteger @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 0.30/0.52            = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.52         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_cong
% 0.30/0.52  thf(fact_1855_mod__add__left__eq,axiom,
% 0.30/0.52      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_left_eq
% 0.30/0.52  thf(fact_1856_mod__add__left__eq,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_left_eq
% 0.30/0.52  thf(fact_1857_mod__add__left__eq,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_left_eq
% 0.30/0.52  thf(fact_1858_mod__add__right__eq,axiom,
% 0.30/0.52      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_right_eq
% 0.30/0.52  thf(fact_1859_mod__add__right__eq,axiom,
% 0.30/0.52      ! [A: int,B2: int,C: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_right_eq
% 0.30/0.52  thf(fact_1860_mod__add__right__eq,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_add_right_eq
% 0.30/0.52  thf(fact_1861_mod__diff__eq,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_eq
% 0.30/0.52  thf(fact_1862_mod__diff__eq,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_eq
% 0.30/0.52  thf(fact_1863_mod__diff__cong,axiom,
% 0.30/0.52      ! [A: int,C: int,A2: int,B2: int,B: int] :
% 0.30/0.52        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.52          = ( modulo_modulo_int @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo_modulo_int @ B2 @ C )
% 0.30/0.52            = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.52         => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_cong
% 0.30/0.52  thf(fact_1864_mod__diff__cong,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 0.30/0.52        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.52          = ( modulo364778990260209775nteger @ A2 @ C ) )
% 0.30/0.52       => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 0.30/0.52            = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.52         => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 0.30/0.52            = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_cong
% 0.30/0.52  thf(fact_1865_mod__diff__left__eq,axiom,
% 0.30/0.52      ! [A: int,C: int,B2: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_left_eq
% 0.30/0.52  thf(fact_1866_mod__diff__left__eq,axiom,
% 0.30/0.52      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_left_eq
% 0.30/0.52  thf(fact_1867_mod__diff__right__eq,axiom,
% 0.30/0.52      ! [A: int,B2: int,C: int] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_right_eq
% 0.30/0.52  thf(fact_1868_mod__diff__right__eq,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_diff_right_eq
% 0.30/0.52  thf(fact_1869_power__mod,axiom,
% 0.30/0.52      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mod
% 0.30/0.52  thf(fact_1870_power__mod,axiom,
% 0.30/0.52      ! [A: int,B2: int,N: nat] :
% 0.30/0.52        ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
% 0.30/0.52        = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mod
% 0.30/0.52  thf(fact_1871_power__mod,axiom,
% 0.30/0.52      ! [A: code_integer,B2: code_integer,N: nat] :
% 0.30/0.52        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ N ) @ B2 )
% 0.30/0.52        = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B2 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % power_mod
% 0.30/0.52  thf(fact_1872_VEBT_Oexhaust,axiom,
% 0.30/0.52      ! [Y3: vEBT_VEBT] :
% 0.30/0.52        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.30/0.52            ( Y3
% 0.30/0.52           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.30/0.52       => ~ ! [X21: $o,X222: $o] :
% 0.30/0.52              ( Y3
% 0.30/0.52             != ( vEBT_Leaf @ X21 @ X222 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT.exhaust
% 0.30/0.52  thf(fact_1873_VEBT_Odistinct_I1_J,axiom,
% 0.30/0.52      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X212: $o,X223: $o] :
% 0.30/0.52        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.30/0.52       != ( vEBT_Leaf @ X212 @ X223 ) ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT.distinct(1)
% 0.30/0.52  thf(fact_1874_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.30/0.52      ! [X3: produc9072475918466114483BT_nat] :
% 0.30/0.52        ( ! [Uu2: $o,Uv2: $o,D3: nat] :
% 0.30/0.52            ( X3
% 0.30/0.52           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
% 0.30/0.52       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 0.30/0.52              ( X3
% 0.30/0.52             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % VEBT_internal.valid'.cases
% 0.30/0.52  thf(fact_1875_mod__Suc__eq,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.30/0.52        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_Suc_eq
% 0.30/0.52  thf(fact_1876_mod__Suc__Suc__eq,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.30/0.52        = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_Suc_Suc_eq
% 0.30/0.52  thf(fact_1877_mod__less__eq__dividend,axiom,
% 0.30/0.52      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.30/0.52  
% 0.30/0.52  % mod_less_eq_dividend
% 0.30/0.52  thf(fact_1878_vebt__pred_Osimps_I1_J,axiom,
% 0.30/0.52      ! [Uu: $o,Uv: $o] :
% 0.30/0.52        ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 0.30/0.52        = none_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % vebt_pred.simps(1)
% 0.30/0.52  thf(fact_1879_le__numeral__extra_I3_J,axiom,
% 0.30/0.52      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/0.52  
% 0.30/0.52  % le_numeral_extra(3)
% 0.30/0.52  thf(fact_1880_le__numeral__extra_I3_J,axiom,
% 0.30/0.52      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.30/0.52  
% 0.30/0.52  % le_numeral_extra(3)
% 0.30/0.52  thf(fact_1881_le__numeral__extra_I3_J,axiom,
% 0.30/0.52      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.30/0.52  
% 0.30/0.52  % le_numeral_extra(3)
% 0.30/0.52  thf(fact_1882_le__numeral__extra_I3_J,axiom,
% 0.30/0.52      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.52  
% 0.30/0.52  % le_numeral_extra(3)
% 0.30/0.52  thf(fact_1883_zero__le,axiom,
% 0.30/0.52      ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_le
% 0.30/0.52  thf(fact_1884_less__numeral__extra_I3_J,axiom,
% 0.30/0.52      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.30/0.52  
% 0.30/0.52  % less_numeral_extra(3)
% 0.30/0.52  thf(fact_1885_less__numeral__extra_I3_J,axiom,
% 0.30/0.52      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.30/0.52  
% 0.30/0.52  % less_numeral_extra(3)
% 0.30/0.52  thf(fact_1886_less__numeral__extra_I3_J,axiom,
% 0.30/0.52      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % less_numeral_extra(3)
% 0.30/0.52  thf(fact_1887_less__numeral__extra_I3_J,axiom,
% 0.30/0.52      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.30/0.52  
% 0.30/0.52  % less_numeral_extra(3)
% 0.30/0.52  thf(fact_1888_field__lbound__gt__zero,axiom,
% 0.30/0.52      ! [D1: real,D22: real] :
% 0.30/0.52        ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.30/0.52       => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.30/0.52         => ? [E2: real] :
% 0.30/0.52              ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.52              & ( ord_less_real @ E2 @ D1 )
% 0.30/0.52              & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % field_lbound_gt_zero
% 0.30/0.52  thf(fact_1889_field__lbound__gt__zero,axiom,
% 0.30/0.52      ! [D1: rat,D22: rat] :
% 0.30/0.52        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.30/0.52       => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.30/0.52         => ? [E2: rat] :
% 0.30/0.52              ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.52              & ( ord_less_rat @ E2 @ D1 )
% 0.30/0.52              & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % field_lbound_gt_zero
% 0.30/0.52  thf(fact_1890_zero__less__iff__neq__zero,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.52        = ( N != zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_less_iff_neq_zero
% 0.30/0.52  thf(fact_1891_gr__implies__not__zero,axiom,
% 0.30/0.52      ! [M: nat,N: nat] :
% 0.30/0.52        ( ( ord_less_nat @ M @ N )
% 0.30/0.52       => ( N != zero_zero_nat ) ) ).
% 0.30/0.52  
% 0.30/0.52  % gr_implies_not_zero
% 0.30/0.52  thf(fact_1892_not__less__zero,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.52  
% 0.30/0.52  % not_less_zero
% 0.30/0.52  thf(fact_1893_gr__zeroI,axiom,
% 0.30/0.52      ! [N: nat] :
% 0.30/0.52        ( ( N != zero_zero_nat )
% 0.30/0.52       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % gr_zeroI
% 0.30/0.52  thf(fact_1894_zero__neq__numeral,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( zero_zero_complex
% 0.30/0.52       != ( numera6690914467698888265omplex @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_neq_numeral
% 0.30/0.52  thf(fact_1895_zero__neq__numeral,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( zero_zero_real
% 0.30/0.52       != ( numeral_numeral_real @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_neq_numeral
% 0.30/0.52  thf(fact_1896_zero__neq__numeral,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( zero_zero_rat
% 0.30/0.52       != ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_neq_numeral
% 0.30/0.52  thf(fact_1897_zero__neq__numeral,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( zero_zero_nat
% 0.30/0.52       != ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_neq_numeral
% 0.30/0.52  thf(fact_1898_zero__neq__numeral,axiom,
% 0.30/0.52      ! [N: num] :
% 0.30/0.52        ( zero_zero_int
% 0.30/0.52       != ( numeral_numeral_int @ N ) ) ).
% 0.30/0.52  
% 0.30/0.52  % zero_neq_numeral
% 0.30/0.52  thf(fact_1899_mult__not__zero,axiom,
% 0.30/0.52      ! [A: complex,B2: complex] :
% 0.30/0.52        ( ( ( times_times_complex @ A @ B2 )
% 0.30/0.52         != zero_zero_complex )
% 0.30/0.52       => ( ( A != zero_zero_complex )
% 0.30/0.52          & ( B2 != zero_zero_complex ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_not_zero
% 0.30/0.52  thf(fact_1900_mult__not__zero,axiom,
% 0.30/0.52      ! [A: real,B2: real] :
% 0.30/0.52        ( ( ( times_times_real @ A @ B2 )
% 0.30/0.52         != zero_zero_real )
% 0.30/0.52       => ( ( A != zero_zero_real )
% 0.30/0.52          & ( B2 != zero_zero_real ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_not_zero
% 0.30/0.52  thf(fact_1901_mult__not__zero,axiom,
% 0.30/0.52      ! [A: rat,B2: rat] :
% 0.30/0.52        ( ( ( times_times_rat @ A @ B2 )
% 0.30/0.52         != zero_zero_rat )
% 0.30/0.52       => ( ( A != zero_zero_rat )
% 0.30/0.52          & ( B2 != zero_zero_rat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_not_zero
% 0.30/0.52  thf(fact_1902_mult__not__zero,axiom,
% 0.30/0.52      ! [A: nat,B2: nat] :
% 0.30/0.52        ( ( ( times_times_nat @ A @ B2 )
% 0.30/0.52         != zero_zero_nat )
% 0.30/0.52       => ( ( A != zero_zero_nat )
% 0.30/0.52          & ( B2 != zero_zero_nat ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_not_zero
% 0.30/0.52  thf(fact_1903_mult__not__zero,axiom,
% 0.30/0.52      ! [A: int,B2: int] :
% 0.30/0.52        ( ( ( times_times_int @ A @ B2 )
% 0.30/0.52         != zero_zero_int )
% 0.30/0.52       => ( ( A != zero_zero_int )
% 0.30/0.52          & ( B2 != zero_zero_int ) ) ) ).
% 0.30/0.52  
% 0.30/0.52  % mult_not_zero
% 0.30/0.52  thf(fact_1904_divisors__zero,axiom,
% 0.30/0.52      ! [A: complex,B2: complex] :
% 0.30/0.53        ( ( ( times_times_complex @ A @ B2 )
% 0.30/0.53          = zero_zero_complex )
% 0.30/0.53       => ( ( A = zero_zero_complex )
% 0.30/0.53          | ( B2 = zero_zero_complex ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divisors_zero
% 0.30/0.53  thf(fact_1905_divisors__zero,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ( times_times_real @ A @ B2 )
% 0.30/0.53          = zero_zero_real )
% 0.30/0.53       => ( ( A = zero_zero_real )
% 0.30/0.53          | ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divisors_zero
% 0.30/0.53  thf(fact_1906_divisors__zero,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ( times_times_rat @ A @ B2 )
% 0.30/0.53          = zero_zero_rat )
% 0.30/0.53       => ( ( A = zero_zero_rat )
% 0.30/0.53          | ( B2 = zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divisors_zero
% 0.30/0.53  thf(fact_1907_divisors__zero,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ( times_times_nat @ A @ B2 )
% 0.30/0.53          = zero_zero_nat )
% 0.30/0.53       => ( ( A = zero_zero_nat )
% 0.30/0.53          | ( B2 = zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divisors_zero
% 0.30/0.53  thf(fact_1908_divisors__zero,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ( times_times_int @ A @ B2 )
% 0.30/0.53          = zero_zero_int )
% 0.30/0.53       => ( ( A = zero_zero_int )
% 0.30/0.53          | ( B2 = zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divisors_zero
% 0.30/0.53  thf(fact_1909_no__zero__divisors,axiom,
% 0.30/0.53      ! [A: complex,B2: complex] :
% 0.30/0.53        ( ( A != zero_zero_complex )
% 0.30/0.53       => ( ( B2 != zero_zero_complex )
% 0.30/0.53         => ( ( times_times_complex @ A @ B2 )
% 0.30/0.53           != zero_zero_complex ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % no_zero_divisors
% 0.30/0.53  thf(fact_1910_no__zero__divisors,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( A != zero_zero_real )
% 0.30/0.53       => ( ( B2 != zero_zero_real )
% 0.30/0.53         => ( ( times_times_real @ A @ B2 )
% 0.30/0.53           != zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % no_zero_divisors
% 0.30/0.53  thf(fact_1911_no__zero__divisors,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( A != zero_zero_rat )
% 0.30/0.53       => ( ( B2 != zero_zero_rat )
% 0.30/0.53         => ( ( times_times_rat @ A @ B2 )
% 0.30/0.53           != zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % no_zero_divisors
% 0.30/0.53  thf(fact_1912_no__zero__divisors,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( A != zero_zero_nat )
% 0.30/0.53       => ( ( B2 != zero_zero_nat )
% 0.30/0.53         => ( ( times_times_nat @ A @ B2 )
% 0.30/0.53           != zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % no_zero_divisors
% 0.30/0.53  thf(fact_1913_no__zero__divisors,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( A != zero_zero_int )
% 0.30/0.53       => ( ( B2 != zero_zero_int )
% 0.30/0.53         => ( ( times_times_int @ A @ B2 )
% 0.30/0.53           != zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % no_zero_divisors
% 0.30/0.53  thf(fact_1914_mult__left__cancel,axiom,
% 0.30/0.53      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.53        ( ( C != zero_zero_complex )
% 0.30/0.53       => ( ( ( times_times_complex @ C @ A )
% 0.30/0.53            = ( times_times_complex @ C @ B2 ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_cancel
% 0.30/0.53  thf(fact_1915_mult__left__cancel,axiom,
% 0.30/0.53      ! [C: real,A: real,B2: real] :
% 0.30/0.53        ( ( C != zero_zero_real )
% 0.30/0.53       => ( ( ( times_times_real @ C @ A )
% 0.30/0.53            = ( times_times_real @ C @ B2 ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_cancel
% 0.30/0.53  thf(fact_1916_mult__left__cancel,axiom,
% 0.30/0.53      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.53        ( ( C != zero_zero_rat )
% 0.30/0.53       => ( ( ( times_times_rat @ C @ A )
% 0.30/0.53            = ( times_times_rat @ C @ B2 ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_cancel
% 0.30/0.53  thf(fact_1917_mult__left__cancel,axiom,
% 0.30/0.53      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.53        ( ( C != zero_zero_nat )
% 0.30/0.53       => ( ( ( times_times_nat @ C @ A )
% 0.30/0.53            = ( times_times_nat @ C @ B2 ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_cancel
% 0.30/0.53  thf(fact_1918_mult__left__cancel,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( C != zero_zero_int )
% 0.30/0.53       => ( ( ( times_times_int @ C @ A )
% 0.30/0.53            = ( times_times_int @ C @ B2 ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_cancel
% 0.30/0.53  thf(fact_1919_mult__right__cancel,axiom,
% 0.30/0.53      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.53        ( ( C != zero_zero_complex )
% 0.30/0.53       => ( ( ( times_times_complex @ A @ C )
% 0.30/0.53            = ( times_times_complex @ B2 @ C ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_cancel
% 0.30/0.53  thf(fact_1920_mult__right__cancel,axiom,
% 0.30/0.53      ! [C: real,A: real,B2: real] :
% 0.30/0.53        ( ( C != zero_zero_real )
% 0.30/0.53       => ( ( ( times_times_real @ A @ C )
% 0.30/0.53            = ( times_times_real @ B2 @ C ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_cancel
% 0.30/0.53  thf(fact_1921_mult__right__cancel,axiom,
% 0.30/0.53      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.53        ( ( C != zero_zero_rat )
% 0.30/0.53       => ( ( ( times_times_rat @ A @ C )
% 0.30/0.53            = ( times_times_rat @ B2 @ C ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_cancel
% 0.30/0.53  thf(fact_1922_mult__right__cancel,axiom,
% 0.30/0.53      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.53        ( ( C != zero_zero_nat )
% 0.30/0.53       => ( ( ( times_times_nat @ A @ C )
% 0.30/0.53            = ( times_times_nat @ B2 @ C ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_cancel
% 0.30/0.53  thf(fact_1923_mult__right__cancel,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( C != zero_zero_int )
% 0.30/0.53       => ( ( ( times_times_int @ A @ C )
% 0.30/0.53            = ( times_times_int @ B2 @ C ) )
% 0.30/0.53          = ( A = B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_cancel
% 0.30/0.53  thf(fact_1924_zero__neq__one,axiom,
% 0.30/0.53      zero_zero_complex != one_one_complex ).
% 0.30/0.53  
% 0.30/0.53  % zero_neq_one
% 0.30/0.53  thf(fact_1925_zero__neq__one,axiom,
% 0.30/0.53      zero_zero_real != one_one_real ).
% 0.30/0.53  
% 0.30/0.53  % zero_neq_one
% 0.30/0.53  thf(fact_1926_zero__neq__one,axiom,
% 0.30/0.53      zero_zero_rat != one_one_rat ).
% 0.30/0.53  
% 0.30/0.53  % zero_neq_one
% 0.30/0.53  thf(fact_1927_zero__neq__one,axiom,
% 0.30/0.53      zero_zero_nat != one_one_nat ).
% 0.30/0.53  
% 0.30/0.53  % zero_neq_one
% 0.30/0.53  thf(fact_1928_zero__neq__one,axiom,
% 0.30/0.53      zero_zero_int != one_one_int ).
% 0.30/0.53  
% 0.30/0.53  % zero_neq_one
% 0.30/0.53  thf(fact_1929_verit__sum__simplify,axiom,
% 0.30/0.53      ! [A: complex] :
% 0.30/0.53        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % verit_sum_simplify
% 0.30/0.53  thf(fact_1930_verit__sum__simplify,axiom,
% 0.30/0.53      ! [A: real] :
% 0.30/0.53        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % verit_sum_simplify
% 0.30/0.53  thf(fact_1931_verit__sum__simplify,axiom,
% 0.30/0.53      ! [A: rat] :
% 0.30/0.53        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % verit_sum_simplify
% 0.30/0.53  thf(fact_1932_verit__sum__simplify,axiom,
% 0.30/0.53      ! [A: nat] :
% 0.30/0.53        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % verit_sum_simplify
% 0.30/0.53  thf(fact_1933_verit__sum__simplify,axiom,
% 0.30/0.53      ! [A: int] :
% 0.30/0.53        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % verit_sum_simplify
% 0.30/0.53  thf(fact_1934_add_Ogroup__left__neutral,axiom,
% 0.30/0.53      ! [A: complex] :
% 0.30/0.53        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.group_left_neutral
% 0.30/0.53  thf(fact_1935_add_Ogroup__left__neutral,axiom,
% 0.30/0.53      ! [A: real] :
% 0.30/0.53        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.group_left_neutral
% 0.30/0.53  thf(fact_1936_add_Ogroup__left__neutral,axiom,
% 0.30/0.53      ! [A: rat] :
% 0.30/0.53        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.group_left_neutral
% 0.30/0.53  thf(fact_1937_add_Ogroup__left__neutral,axiom,
% 0.30/0.53      ! [A: int] :
% 0.30/0.53        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.group_left_neutral
% 0.30/0.53  thf(fact_1938_add_Ocomm__neutral,axiom,
% 0.30/0.53      ! [A: complex] :
% 0.30/0.53        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.comm_neutral
% 0.30/0.53  thf(fact_1939_add_Ocomm__neutral,axiom,
% 0.30/0.53      ! [A: real] :
% 0.30/0.53        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.comm_neutral
% 0.30/0.53  thf(fact_1940_add_Ocomm__neutral,axiom,
% 0.30/0.53      ! [A: rat] :
% 0.30/0.53        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.comm_neutral
% 0.30/0.53  thf(fact_1941_add_Ocomm__neutral,axiom,
% 0.30/0.53      ! [A: nat] :
% 0.30/0.53        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.comm_neutral
% 0.30/0.53  thf(fact_1942_add_Ocomm__neutral,axiom,
% 0.30/0.53      ! [A: int] :
% 0.30/0.53        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % add.comm_neutral
% 0.30/0.53  thf(fact_1943_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.53      ! [A: complex] :
% 0.30/0.53        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % comm_monoid_add_class.add_0
% 0.30/0.53  thf(fact_1944_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.53      ! [A: real] :
% 0.30/0.53        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % comm_monoid_add_class.add_0
% 0.30/0.53  thf(fact_1945_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.53      ! [A: rat] :
% 0.30/0.53        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % comm_monoid_add_class.add_0
% 0.30/0.53  thf(fact_1946_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.53      ! [A: nat] :
% 0.30/0.53        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % comm_monoid_add_class.add_0
% 0.30/0.53  thf(fact_1947_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.53      ! [A: int] :
% 0.30/0.53        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.53        = A ) ).
% 0.30/0.53  
% 0.30/0.53  % comm_monoid_add_class.add_0
% 0.30/0.53  thf(fact_1948_eq__iff__diff__eq__0,axiom,
% 0.30/0.53      ( ( ^ [Y6: complex,Z5: complex] : ( Y6 = Z5 ) )
% 0.30/0.53      = ( ^ [A5: complex,B4: complex] :
% 0.30/0.53            ( ( minus_minus_complex @ A5 @ B4 )
% 0.30/0.53            = zero_zero_complex ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_iff_diff_eq_0
% 0.30/0.53  thf(fact_1949_eq__iff__diff__eq__0,axiom,
% 0.30/0.53      ( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
% 0.30/0.53      = ( ^ [A5: real,B4: real] :
% 0.30/0.53            ( ( minus_minus_real @ A5 @ B4 )
% 0.30/0.53            = zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_iff_diff_eq_0
% 0.30/0.53  thf(fact_1950_eq__iff__diff__eq__0,axiom,
% 0.30/0.53      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.53      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.53            ( ( minus_minus_rat @ A5 @ B4 )
% 0.30/0.53            = zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_iff_diff_eq_0
% 0.30/0.53  thf(fact_1951_eq__iff__diff__eq__0,axiom,
% 0.30/0.53      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.53      = ( ^ [A5: int,B4: int] :
% 0.30/0.53            ( ( minus_minus_int @ A5 @ B4 )
% 0.30/0.53            = zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_iff_diff_eq_0
% 0.30/0.53  thf(fact_1952_power__not__zero,axiom,
% 0.30/0.53      ! [A: rat,N: nat] :
% 0.30/0.53        ( ( A != zero_zero_rat )
% 0.30/0.53       => ( ( power_power_rat @ A @ N )
% 0.30/0.53         != zero_zero_rat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_not_zero
% 0.30/0.53  thf(fact_1953_power__not__zero,axiom,
% 0.30/0.53      ! [A: nat,N: nat] :
% 0.30/0.53        ( ( A != zero_zero_nat )
% 0.30/0.53       => ( ( power_power_nat @ A @ N )
% 0.30/0.53         != zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_not_zero
% 0.30/0.53  thf(fact_1954_power__not__zero,axiom,
% 0.30/0.53      ! [A: real,N: nat] :
% 0.30/0.53        ( ( A != zero_zero_real )
% 0.30/0.53       => ( ( power_power_real @ A @ N )
% 0.30/0.53         != zero_zero_real ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_not_zero
% 0.30/0.53  thf(fact_1955_power__not__zero,axiom,
% 0.30/0.53      ! [A: int,N: nat] :
% 0.30/0.53        ( ( A != zero_zero_int )
% 0.30/0.53       => ( ( power_power_int @ A @ N )
% 0.30/0.53         != zero_zero_int ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_not_zero
% 0.30/0.53  thf(fact_1956_power__not__zero,axiom,
% 0.30/0.53      ! [A: complex,N: nat] :
% 0.30/0.53        ( ( A != zero_zero_complex )
% 0.30/0.53       => ( ( power_power_complex @ A @ N )
% 0.30/0.53         != zero_zero_complex ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_not_zero
% 0.30/0.53  thf(fact_1957_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.30/0.53      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.30/0.53        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.30/0.53  
% 0.30/0.53  % VEBT_internal.membermima.simps(2)
% 0.30/0.53  thf(fact_1958_num_Osize_I4_J,axiom,
% 0.30/0.53      ( ( size_size_num @ one )
% 0.30/0.53      = zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % num.size(4)
% 0.30/0.53  thf(fact_1959_nat_Odistinct_I1_J,axiom,
% 0.30/0.53      ! [X2: nat] :
% 0.30/0.53        ( zero_zero_nat
% 0.30/0.53       != ( suc @ X2 ) ) ).
% 0.30/0.53  
% 0.30/0.53  % nat.distinct(1)
% 0.30/0.53  thf(fact_1960_old_Onat_Odistinct_I2_J,axiom,
% 0.30/0.53      ! [Nat2: nat] :
% 0.30/0.53        ( ( suc @ Nat2 )
% 0.30/0.53       != zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % old.nat.distinct(2)
% 0.30/0.53  thf(fact_1961_old_Onat_Odistinct_I1_J,axiom,
% 0.30/0.53      ! [Nat2: nat] :
% 0.30/0.53        ( zero_zero_nat
% 0.30/0.53       != ( suc @ Nat2 ) ) ).
% 0.30/0.53  
% 0.30/0.53  % old.nat.distinct(1)
% 0.30/0.53  thf(fact_1962_nat_OdiscI,axiom,
% 0.30/0.53      ! [Nat: nat,X2: nat] :
% 0.30/0.53        ( ( Nat
% 0.30/0.53          = ( suc @ X2 ) )
% 0.30/0.53       => ( Nat != zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % nat.discI
% 0.30/0.53  thf(fact_1963_old_Onat_Oexhaust,axiom,
% 0.30/0.53      ! [Y3: nat] :
% 0.30/0.53        ( ( Y3 != zero_zero_nat )
% 0.30/0.53       => ~ ! [Nat3: nat] :
% 0.30/0.53              ( Y3
% 0.30/0.53             != ( suc @ Nat3 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % old.nat.exhaust
% 0.30/0.53  thf(fact_1964_nat__induct,axiom,
% 0.30/0.53      ! [P: nat > $o,N: nat] :
% 0.30/0.53        ( ( P @ zero_zero_nat )
% 0.30/0.53       => ( ! [N3: nat] :
% 0.30/0.53              ( ( P @ N3 )
% 0.30/0.53             => ( P @ ( suc @ N3 ) ) )
% 0.30/0.53         => ( P @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % nat_induct
% 0.30/0.53  thf(fact_1965_diff__induct,axiom,
% 0.30/0.53      ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.53        ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 0.30/0.53       => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.30/0.53         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.53                ( ( P @ X5 @ Y4 )
% 0.30/0.53               => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
% 0.30/0.53           => ( P @ M @ N ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % diff_induct
% 0.30/0.53  thf(fact_1966_zero__induct,axiom,
% 0.30/0.53      ! [P: nat > $o,K: nat] :
% 0.30/0.53        ( ( P @ K )
% 0.30/0.53       => ( ! [N3: nat] :
% 0.30/0.53              ( ( P @ ( suc @ N3 ) )
% 0.30/0.53             => ( P @ N3 ) )
% 0.30/0.53         => ( P @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_induct
% 0.30/0.53  thf(fact_1967_Suc__neq__Zero,axiom,
% 0.30/0.53      ! [M: nat] :
% 0.30/0.53        ( ( suc @ M )
% 0.30/0.53       != zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % Suc_neq_Zero
% 0.30/0.53  thf(fact_1968_Zero__neq__Suc,axiom,
% 0.30/0.53      ! [M: nat] :
% 0.30/0.53        ( zero_zero_nat
% 0.30/0.53       != ( suc @ M ) ) ).
% 0.30/0.53  
% 0.30/0.53  % Zero_neq_Suc
% 0.30/0.53  thf(fact_1969_Zero__not__Suc,axiom,
% 0.30/0.53      ! [M: nat] :
% 0.30/0.53        ( zero_zero_nat
% 0.30/0.53       != ( suc @ M ) ) ).
% 0.30/0.53  
% 0.30/0.53  % Zero_not_Suc
% 0.30/0.53  thf(fact_1970_not0__implies__Suc,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ( ( N != zero_zero_nat )
% 0.30/0.53       => ? [M4: nat] :
% 0.30/0.53            ( N
% 0.30/0.53            = ( suc @ M4 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % not0_implies_Suc
% 0.30/0.53  thf(fact_1971_infinite__descent0,axiom,
% 0.30/0.53      ! [P: nat > $o,N: nat] :
% 0.30/0.53        ( ( P @ zero_zero_nat )
% 0.30/0.53       => ( ! [N3: nat] :
% 0.30/0.53              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.53             => ( ~ ( P @ N3 )
% 0.30/0.53               => ? [M2: nat] :
% 0.30/0.53                    ( ( ord_less_nat @ M2 @ N3 )
% 0.30/0.53                    & ~ ( P @ M2 ) ) ) )
% 0.30/0.53         => ( P @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % infinite_descent0
% 0.30/0.53  thf(fact_1972_gr__implies__not0,axiom,
% 0.30/0.53      ! [M: nat,N: nat] :
% 0.30/0.53        ( ( ord_less_nat @ M @ N )
% 0.30/0.53       => ( N != zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % gr_implies_not0
% 0.30/0.53  thf(fact_1973_less__zeroE,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % less_zeroE
% 0.30/0.53  thf(fact_1974_not__less0,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_less0
% 0.30/0.53  thf(fact_1975_not__gr0,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.30/0.53        = ( N = zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % not_gr0
% 0.30/0.53  thf(fact_1976_gr0I,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ( ( N != zero_zero_nat )
% 0.30/0.53       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % gr0I
% 0.30/0.53  thf(fact_1977_bot__nat__0_Oextremum__strict,axiom,
% 0.30/0.53      ! [A: nat] :
% 0.30/0.53        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % bot_nat_0.extremum_strict
% 0.30/0.53  thf(fact_1978_le__0__eq,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.30/0.53        = ( N = zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % le_0_eq
% 0.30/0.53  thf(fact_1979_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.53      ! [A: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53       => ( A = zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % bot_nat_0.extremum_uniqueI
% 0.30/0.53  thf(fact_1980_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.53      ! [A: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53        = ( A = zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % bot_nat_0.extremum_unique
% 0.30/0.53  thf(fact_1981_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.53      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.30/0.53  
% 0.30/0.53  % less_eq_nat.simps(1)
% 0.30/0.53  thf(fact_1982_add__eq__self__zero,axiom,
% 0.30/0.53      ! [M: nat,N: nat] :
% 0.30/0.53        ( ( ( plus_plus_nat @ M @ N )
% 0.30/0.53          = M )
% 0.30/0.53       => ( N = zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_eq_self_zero
% 0.30/0.53  thf(fact_1983_plus__nat_Oadd__0,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.30/0.53        = N ) ).
% 0.30/0.53  
% 0.30/0.53  % plus_nat.add_0
% 0.30/0.53  thf(fact_1984_diffs0__imp__equal,axiom,
% 0.30/0.53      ! [M: nat,N: nat] :
% 0.30/0.53        ( ( ( minus_minus_nat @ M @ N )
% 0.30/0.53          = zero_zero_nat )
% 0.30/0.53       => ( ( ( minus_minus_nat @ N @ M )
% 0.30/0.53            = zero_zero_nat )
% 0.30/0.53         => ( M = N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % diffs0_imp_equal
% 0.30/0.53  thf(fact_1985_minus__nat_Odiff__0,axiom,
% 0.30/0.53      ! [M: nat] :
% 0.30/0.53        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.30/0.53        = M ) ).
% 0.30/0.53  
% 0.30/0.53  % minus_nat.diff_0
% 0.30/0.53  thf(fact_1986_mult__0,axiom,
% 0.30/0.53      ! [N: nat] :
% 0.30/0.53        ( ( times_times_nat @ zero_zero_nat @ N )
% 0.30/0.53        = zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_0
% 0.30/0.53  thf(fact_1987_nat__mult__eq__cancel__disj,axiom,
% 0.30/0.53      ! [K: nat,M: nat,N: nat] :
% 0.30/0.53        ( ( ( times_times_nat @ K @ M )
% 0.30/0.53          = ( times_times_nat @ K @ N ) )
% 0.30/0.53        = ( ( K = zero_zero_nat )
% 0.30/0.53          | ( M = N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % nat_mult_eq_cancel_disj
% 0.30/0.53  thf(fact_1988_split__mod,axiom,
% 0.30/0.53      ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.53        ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.53        = ( ( ( N = zero_zero_nat )
% 0.30/0.53           => ( P @ M ) )
% 0.30/0.53          & ( ( N != zero_zero_nat )
% 0.30/0.53           => ! [I4: nat,J3: nat] :
% 0.30/0.53                ( ( ord_less_nat @ J3 @ N )
% 0.30/0.53               => ( ( M
% 0.30/0.53                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.30/0.53                 => ( P @ J3 ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mod
% 0.30/0.53  thf(fact_1989_power__eq__imp__eq__base,axiom,
% 0.30/0.53      ! [A: real,N: nat,B2: real] :
% 0.30/0.53        ( ( ( power_power_real @ A @ N )
% 0.30/0.53          = ( power_power_real @ B2 @ N ) )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53             => ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_imp_eq_base
% 0.30/0.53  thf(fact_1990_power__eq__imp__eq__base,axiom,
% 0.30/0.53      ! [A: rat,N: nat,B2: rat] :
% 0.30/0.53        ( ( ( power_power_rat @ A @ N )
% 0.30/0.53          = ( power_power_rat @ B2 @ N ) )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53             => ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_imp_eq_base
% 0.30/0.53  thf(fact_1991_power__eq__imp__eq__base,axiom,
% 0.30/0.53      ! [A: nat,N: nat,B2: nat] :
% 0.30/0.53        ( ( ( power_power_nat @ A @ N )
% 0.30/0.53          = ( power_power_nat @ B2 @ N ) )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53             => ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_imp_eq_base
% 0.30/0.53  thf(fact_1992_power__eq__imp__eq__base,axiom,
% 0.30/0.53      ! [A: int,N: nat,B2: int] :
% 0.30/0.53        ( ( ( power_power_int @ A @ N )
% 0.30/0.53          = ( power_power_int @ B2 @ N ) )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53             => ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_imp_eq_base
% 0.30/0.53  thf(fact_1993_power__eq__iff__eq__base,axiom,
% 0.30/0.53      ! [N: nat,A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.53           => ( ( ( power_power_real @ A @ N )
% 0.30/0.53                = ( power_power_real @ B2 @ N ) )
% 0.30/0.53              = ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_iff_eq_base
% 0.30/0.53  thf(fact_1994_power__eq__iff__eq__base,axiom,
% 0.30/0.53      ! [N: nat,A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.53           => ( ( ( power_power_rat @ A @ N )
% 0.30/0.53                = ( power_power_rat @ B2 @ N ) )
% 0.30/0.53              = ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_iff_eq_base
% 0.30/0.53  thf(fact_1995_power__eq__iff__eq__base,axiom,
% 0.30/0.53      ! [N: nat,A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.53           => ( ( ( power_power_nat @ A @ N )
% 0.30/0.53                = ( power_power_nat @ B2 @ N ) )
% 0.30/0.53              = ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_iff_eq_base
% 0.30/0.53  thf(fact_1996_power__eq__iff__eq__base,axiom,
% 0.30/0.53      ! [N: nat,A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.53           => ( ( ( power_power_int @ A @ N )
% 0.30/0.53                = ( power_power_int @ B2 @ N ) )
% 0.30/0.53              = ( A = B2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_eq_iff_eq_base
% 0.30/0.53  thf(fact_1997_lambda__zero,axiom,
% 0.30/0.53      ( ( ^ [H: complex] : zero_zero_complex )
% 0.30/0.53      = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.30/0.53  
% 0.30/0.53  % lambda_zero
% 0.30/0.53  thf(fact_1998_lambda__zero,axiom,
% 0.30/0.53      ( ( ^ [H: real] : zero_zero_real )
% 0.30/0.53      = ( times_times_real @ zero_zero_real ) ) ).
% 0.30/0.53  
% 0.30/0.53  % lambda_zero
% 0.30/0.53  thf(fact_1999_lambda__zero,axiom,
% 0.30/0.53      ( ( ^ [H: rat] : zero_zero_rat )
% 0.30/0.53      = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % lambda_zero
% 0.30/0.53  thf(fact_2000_lambda__zero,axiom,
% 0.30/0.53      ( ( ^ [H: nat] : zero_zero_nat )
% 0.30/0.53      = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % lambda_zero
% 0.30/0.53  thf(fact_2001_lambda__zero,axiom,
% 0.30/0.53      ( ( ^ [H: int] : zero_zero_int )
% 0.30/0.53      = ( times_times_int @ zero_zero_int ) ) ).
% 0.30/0.53  
% 0.30/0.53  % lambda_zero
% 0.30/0.53  thf(fact_2002_vebt__insert_Osimps_I2_J,axiom,
% 0.30/0.53      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 0.30/0.53        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X3 )
% 0.30/0.53        = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% 0.30/0.53  
% 0.30/0.53  % vebt_insert.simps(2)
% 0.30/0.53  thf(fact_2003_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.30/0.53      ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
% 0.30/0.53        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
% 0.30/0.53        = ( ( X3 = Mi )
% 0.30/0.53          | ( X3 = Ma ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % VEBT_internal.membermima.simps(3)
% 0.30/0.53  thf(fact_2004_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.53      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.53        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.30/0.53       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.53          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.53  thf(fact_2005_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.53      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.53          = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.53  thf(fact_2006_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.53          = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.53  thf(fact_2007_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.53      ! [M: num,Q2: num,N: num] :
% 0.30/0.53        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.53          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.53        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.53          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % cong_exp_iff_simps(9)
% 0.30/0.53  thf(fact_2008_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.53      ! [M: num,Q2: num,N: num] :
% 0.30/0.53        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.53          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.53        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.53          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % cong_exp_iff_simps(9)
% 0.30/0.53  thf(fact_2009_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.53      ! [M: num,Q2: num,N: num] :
% 0.30/0.53        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.53          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.53        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.53          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % cong_exp_iff_simps(9)
% 0.30/0.53  thf(fact_2010_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.53      ! [M: num,N: num] :
% 0.30/0.53        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.53        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % cong_exp_iff_simps(4)
% 0.30/0.53  thf(fact_2011_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.53      ! [M: num,N: num] :
% 0.30/0.53        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.53        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % cong_exp_iff_simps(4)
% 0.30/0.53  thf(fact_2012_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.53      ! [M: num,N: num] :
% 0.30/0.53        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.53        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % cong_exp_iff_simps(4)
% 0.30/0.53  thf(fact_2013_mod__eqE,axiom,
% 0.30/0.53      ! [A: int,C: int,B2: int] :
% 0.30/0.53        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.53          = ( modulo_modulo_int @ B2 @ C ) )
% 0.30/0.53       => ~ ! [D3: int] :
% 0.30/0.53              ( B2
% 0.30/0.53             != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mod_eqE
% 0.30/0.53  thf(fact_2014_mod__eqE,axiom,
% 0.30/0.53      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.53        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.53          = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.30/0.53       => ~ ! [D3: code_integer] :
% 0.30/0.53              ( B2
% 0.30/0.53             != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mod_eqE
% 0.30/0.53  thf(fact_2015_div__add1__eq,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.53        = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % div_add1_eq
% 0.30/0.53  thf(fact_2016_div__add1__eq,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.53        = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % div_add1_eq
% 0.30/0.53  thf(fact_2017_div__add1__eq,axiom,
% 0.30/0.53      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.53        ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.30/0.53        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % div_add1_eq
% 0.30/0.53  thf(fact_2018_Suc__times__mod__eq,axiom,
% 0.30/0.53      ! [M: nat,N: nat] :
% 0.30/0.53        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.53       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.30/0.53          = one_one_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % Suc_times_mod_eq
% 0.30/0.53  thf(fact_2019_mod__induct,axiom,
% 0.30/0.53      ! [P: nat > $o,N: nat,P4: nat,M: nat] :
% 0.30/0.53        ( ( P @ N )
% 0.30/0.53       => ( ( ord_less_nat @ N @ P4 )
% 0.30/0.53         => ( ( ord_less_nat @ M @ P4 )
% 0.30/0.53           => ( ! [N3: nat] :
% 0.30/0.53                  ( ( ord_less_nat @ N3 @ P4 )
% 0.30/0.53                 => ( ( P @ N3 )
% 0.30/0.53                   => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
% 0.30/0.53             => ( P @ M ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mod_induct
% 0.30/0.53  thf(fact_2020_mod__Suc__le__divisor,axiom,
% 0.30/0.53      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.30/0.53  
% 0.30/0.53  % mod_Suc_le_divisor
% 0.30/0.53  thf(fact_2021_power__strict__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,N: nat] :
% 0.30/0.53        ( ( ord_less_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53           => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_strict_mono
% 0.30/0.53  thf(fact_2022_power__strict__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53           => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_strict_mono
% 0.30/0.53  thf(fact_2023_power__strict__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53           => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_strict_mono
% 0.30/0.53  thf(fact_2024_power__strict__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,N: nat] :
% 0.30/0.53        ( ( ord_less_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.53           => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_strict_mono
% 0.30/0.53  thf(fact_2025_mod__if,axiom,
% 0.30/0.53      ( modulo_modulo_nat
% 0.30/0.53      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mod_if
% 0.30/0.53  thf(fact_2026_mod__geq,axiom,
% 0.30/0.53      ! [M: nat,N: nat] :
% 0.30/0.53        ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.53       => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.53          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mod_geq
% 0.30/0.53  thf(fact_2027_vebt__insert_Ocases,axiom,
% 0.30/0.53      ! [X3: produc9072475918466114483BT_nat] :
% 0.30/0.53        ( ! [A3: $o,B3: $o,X5: nat] :
% 0.30/0.53            ( X3
% 0.30/0.53           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
% 0.30/0.53       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 0.30/0.53              ( X3
% 0.30/0.53             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X5 ) )
% 0.30/0.53         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 0.30/0.53                ( X3
% 0.30/0.53               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X5 ) )
% 0.30/0.53           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53                  ( X3
% 0.30/0.53                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) )
% 0.30/0.53             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53                    ( X3
% 0.30/0.53                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % vebt_insert.cases
% 0.30/0.53  thf(fact_2028_vebt__member_Ocases,axiom,
% 0.30/0.53      ! [X3: produc9072475918466114483BT_nat] :
% 0.30/0.53        ( ! [A3: $o,B3: $o,X5: nat] :
% 0.30/0.53            ( X3
% 0.30/0.53           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
% 0.30/0.53       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
% 0.30/0.53              ( X3
% 0.30/0.53             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
% 0.30/0.53         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
% 0.30/0.53                ( X3
% 0.30/0.53               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
% 0.30/0.53           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 0.30/0.53                  ( X3
% 0.30/0.53                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
% 0.30/0.53             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.53                    ( X3
% 0.30/0.53                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % vebt_member.cases
% 0.30/0.53  thf(fact_2029_le__mod__geq,axiom,
% 0.30/0.53      ! [N: nat,M: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.53       => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.53          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % le_mod_geq
% 0.30/0.53  thf(fact_2030_nat__mod__eq__iff,axiom,
% 0.30/0.53      ! [X3: nat,N: nat,Y3: nat] :
% 0.30/0.53        ( ( ( modulo_modulo_nat @ X3 @ N )
% 0.30/0.53          = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.30/0.53        = ( ? [Q1: nat,Q22: nat] :
% 0.30/0.53              ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N @ Q1 ) )
% 0.30/0.53              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % nat_mod_eq_iff
% 0.30/0.53  thf(fact_2031_VEBT__internal_OminNull_Ocases,axiom,
% 0.30/0.53      ! [X3: vEBT_VEBT] :
% 0.30/0.53        ( ( X3
% 0.30/0.53         != ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.53       => ( ! [Uv2: $o] :
% 0.30/0.53              ( X3
% 0.30/0.53             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.53         => ( ! [Uu2: $o] :
% 0.30/0.53                ( X3
% 0.30/0.53               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.53           => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.53                  ( X3
% 0.30/0.53                 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.53             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.53                    ( X3
% 0.30/0.53                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % VEBT_internal.minNull.cases
% 0.30/0.53  thf(fact_2032_vebt__pred_Osimps_I2_J,axiom,
% 0.30/0.53      ! [A: $o,Uw: $o] :
% 0.30/0.53        ( ( A
% 0.30/0.53         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.53            = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53        & ( ~ A
% 0.30/0.53         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.53            = none_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % vebt_pred.simps(2)
% 0.30/0.53  thf(fact_2033_vebt__mint_Osimps_I1_J,axiom,
% 0.30/0.53      ! [A: $o,B2: $o] :
% 0.30/0.53        ( ( A
% 0.30/0.53         => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 0.30/0.53            = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53        & ( ~ A
% 0.30/0.53         => ( ( B2
% 0.30/0.53             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 0.30/0.53                = ( some_nat @ one_one_nat ) ) )
% 0.30/0.53            & ( ~ B2
% 0.30/0.53             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 0.30/0.53                = none_nat ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % vebt_mint.simps(1)
% 0.30/0.53  thf(fact_2034_not__numeral__le__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_le_zero
% 0.30/0.53  thf(fact_2035_not__numeral__le__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_le_zero
% 0.30/0.53  thf(fact_2036_not__numeral__le__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_le_zero
% 0.30/0.53  thf(fact_2037_not__numeral__le__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_le_zero
% 0.30/0.53  thf(fact_2038_zero__le__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_numeral
% 0.30/0.53  thf(fact_2039_zero__le__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_numeral
% 0.30/0.53  thf(fact_2040_zero__le__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_numeral
% 0.30/0.53  thf(fact_2041_zero__le__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_numeral
% 0.30/0.53  thf(fact_2042_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53  thf(fact_2043_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53  thf(fact_2044_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53  thf(fact_2045_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.53  thf(fact_2046_zero__le__mult__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_mult_iff
% 0.30/0.53  thf(fact_2047_zero__le__mult__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_mult_iff
% 0.30/0.53  thf(fact_2048_zero__le__mult__iff,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53            & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53            & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_mult_iff
% 0.30/0.53  thf(fact_2049_mult__nonneg__nonpos2,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos2
% 0.30/0.53  thf(fact_2050_mult__nonneg__nonpos2,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos2
% 0.30/0.53  thf(fact_2051_mult__nonneg__nonpos2,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.30/0.53         => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos2
% 0.30/0.53  thf(fact_2052_mult__nonneg__nonpos2,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos2
% 0.30/0.53  thf(fact_2053_mult__nonpos__nonneg,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonneg
% 0.30/0.53  thf(fact_2054_mult__nonpos__nonneg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonneg
% 0.30/0.53  thf(fact_2055_mult__nonpos__nonneg,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.53         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonneg
% 0.30/0.53  thf(fact_2056_mult__nonpos__nonneg,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonneg
% 0.30/0.53  thf(fact_2057_mult__nonneg__nonpos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos
% 0.30/0.53  thf(fact_2058_mult__nonneg__nonpos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos
% 0.30/0.53  thf(fact_2059_mult__nonneg__nonpos,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.30/0.53         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos
% 0.30/0.53  thf(fact_2060_mult__nonneg__nonpos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonpos
% 0.30/0.53  thf(fact_2061_mult__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonneg
% 0.30/0.53  thf(fact_2062_mult__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonneg
% 0.30/0.53  thf(fact_2063_mult__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.53         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonneg
% 0.30/0.53  thf(fact_2064_mult__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.53         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonneg_nonneg
% 0.30/0.53  thf(fact_2065_split__mult__neg__le,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 0.30/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
% 0.30/0.53       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_neg_le
% 0.30/0.53  thf(fact_2066_split__mult__neg__le,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 0.30/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
% 0.30/0.53       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_neg_le
% 0.30/0.53  thf(fact_2067_split__mult__neg__le,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53            & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
% 0.30/0.53          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53            & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
% 0.30/0.53       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_neg_le
% 0.30/0.53  thf(fact_2068_split__mult__neg__le,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53            & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 0.30/0.53          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53            & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
% 0.30/0.53       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_neg_le
% 0.30/0.53  thf(fact_2069_mult__le__0__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 0.30/0.53        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 0.30/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_le_0_iff
% 0.30/0.53  thf(fact_2070_mult__le__0__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 0.30/0.53        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 0.30/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_le_0_iff
% 0.30/0.53  thf(fact_2071_mult__le__0__iff,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 0.30/0.53        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53            & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 0.30/0.53          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53            & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_le_0_iff
% 0.30/0.53  thf(fact_2072_mult__right__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono
% 0.30/0.53  thf(fact_2073_mult__right__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono
% 0.30/0.53  thf(fact_2074_mult__right__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono
% 0.30/0.53  thf(fact_2075_mult__right__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono
% 0.30/0.53  thf(fact_2076_mult__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: real,A: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono_neg
% 0.30/0.53  thf(fact_2077_mult__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono_neg
% 0.30/0.53  thf(fact_2078_mult__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: int,A: int,C: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_right_mono_neg
% 0.30/0.53  thf(fact_2079_mult__left__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono
% 0.30/0.53  thf(fact_2080_mult__left__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono
% 0.30/0.53  thf(fact_2081_mult__left__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono
% 0.30/0.53  thf(fact_2082_mult__left__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono
% 0.30/0.53  thf(fact_2083_mult__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonpos
% 0.30/0.53  thf(fact_2084_mult__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonpos
% 0.30/0.53  thf(fact_2085_mult__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_nonpos_nonpos
% 0.30/0.53  thf(fact_2086_mult__left__mono__neg,axiom,
% 0.30/0.53      ! [B2: real,A: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono_neg
% 0.30/0.53  thf(fact_2087_mult__left__mono__neg,axiom,
% 0.30/0.53      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono_neg
% 0.30/0.53  thf(fact_2088_mult__left__mono__neg,axiom,
% 0.30/0.53      ! [B2: int,A: int,C: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.53         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_left_mono_neg
% 0.30/0.53  thf(fact_2089_split__mult__pos__le,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 0.30/0.53       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_pos_le
% 0.30/0.53  thf(fact_2090_split__mult__pos__le,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 0.30/0.53       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_pos_le
% 0.30/0.53  thf(fact_2091_split__mult__pos__le,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53            & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53            & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 0.30/0.53       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % split_mult_pos_le
% 0.30/0.53  thf(fact_2092_zero__le__square,axiom,
% 0.30/0.53      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_square
% 0.30/0.53  thf(fact_2093_zero__le__square,axiom,
% 0.30/0.53      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_square
% 0.30/0.53  thf(fact_2094_zero__le__square,axiom,
% 0.30/0.53      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_square
% 0.30/0.53  thf(fact_2095_mult__mono_H,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono'
% 0.30/0.53  thf(fact_2096_mult__mono_H,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono'
% 0.30/0.53  thf(fact_2097_mult__mono_H,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono'
% 0.30/0.53  thf(fact_2098_mult__mono_H,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono'
% 0.30/0.53  thf(fact_2099_mult__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.53           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono
% 0.30/0.53  thf(fact_2100_mult__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.53           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono
% 0.30/0.53  thf(fact_2101_mult__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.53           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono
% 0.30/0.53  thf(fact_2102_mult__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.53           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_mono
% 0.30/0.53  thf(fact_2103_not__numeral__less__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_less_zero
% 0.30/0.53  thf(fact_2104_not__numeral__less__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_less_zero
% 0.30/0.53  thf(fact_2105_not__numeral__less__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_less_zero
% 0.30/0.53  thf(fact_2106_not__numeral__less__zero,axiom,
% 0.30/0.53      ! [N: num] :
% 0.30/0.53        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.30/0.53  
% 0.30/0.53  % not_numeral_less_zero
% 0.30/0.53  thf(fact_2107_zero__less__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_numeral
% 0.30/0.53  thf(fact_2108_zero__less__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_numeral
% 0.30/0.53  thf(fact_2109_zero__less__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_numeral
% 0.30/0.53  thf(fact_2110_zero__less__numeral,axiom,
% 0.30/0.53      ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_numeral
% 0.30/0.53  thf(fact_2111_not__one__le__zero,axiom,
% 0.30/0.53      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_le_zero
% 0.30/0.53  thf(fact_2112_not__one__le__zero,axiom,
% 0.30/0.53      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_le_zero
% 0.30/0.53  thf(fact_2113_not__one__le__zero,axiom,
% 0.30/0.53      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_le_zero
% 0.30/0.53  thf(fact_2114_not__one__le__zero,axiom,
% 0.30/0.53      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_le_zero
% 0.30/0.53  thf(fact_2115_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53  
% 0.30/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53  thf(fact_2116_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53  
% 0.30/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53  thf(fact_2117_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53  
% 0.30/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53  thf(fact_2118_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53  
% 0.30/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.53  thf(fact_2119_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one_class.zero_le_one
% 0.30/0.53  thf(fact_2120_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one_class.zero_le_one
% 0.30/0.53  thf(fact_2121_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one_class.zero_le_one
% 0.30/0.53  thf(fact_2122_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.53      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one_class.zero_le_one
% 0.30/0.53  thf(fact_2123_add__decreasing,axiom,
% 0.30/0.53      ! [A: real,C: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ C @ B2 )
% 0.30/0.53         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing
% 0.30/0.53  thf(fact_2124_add__decreasing,axiom,
% 0.30/0.53      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ C @ B2 )
% 0.30/0.53         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing
% 0.30/0.53  thf(fact_2125_add__decreasing,axiom,
% 0.30/0.53      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_eq_nat @ C @ B2 )
% 0.30/0.53         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing
% 0.30/0.53  thf(fact_2126_add__decreasing,axiom,
% 0.30/0.53      ! [A: int,C: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_eq_int @ C @ B2 )
% 0.30/0.53         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing
% 0.30/0.53  thf(fact_2127_add__increasing,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ B2 @ C )
% 0.30/0.53         => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing
% 0.30/0.53  thf(fact_2128_add__increasing,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.53         => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing
% 0.30/0.53  thf(fact_2129_add__increasing,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.53         => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing
% 0.30/0.53  thf(fact_2130_add__increasing,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.53         => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing
% 0.30/0.53  thf(fact_2131_add__decreasing2,axiom,
% 0.30/0.53      ! [C: real,A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing2
% 0.30/0.53  thf(fact_2132_add__decreasing2,axiom,
% 0.30/0.53      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing2
% 0.30/0.53  thf(fact_2133_add__decreasing2,axiom,
% 0.30/0.53      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing2
% 0.30/0.53  thf(fact_2134_add__decreasing2,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_decreasing2
% 0.30/0.53  thf(fact_2135_add__increasing2,axiom,
% 0.30/0.53      ! [C: real,B2: real,A: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53       => ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.53         => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing2
% 0.30/0.53  thf(fact_2136_add__increasing2,axiom,
% 0.30/0.53      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53       => ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.53         => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing2
% 0.30/0.53  thf(fact_2137_add__increasing2,axiom,
% 0.30/0.53      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.53       => ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.53         => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing2
% 0.30/0.53  thf(fact_2138_add__increasing2,axiom,
% 0.30/0.53      ! [C: int,B2: int,A: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.53       => ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.53         => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_increasing2
% 0.30/0.53  thf(fact_2139_add__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_nonneg
% 0.30/0.53  thf(fact_2140_add__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_nonneg
% 0.30/0.53  thf(fact_2141_add__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.53         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_nonneg
% 0.30/0.53  thf(fact_2142_add__nonneg__nonneg,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.53         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_nonneg
% 0.30/0.53  thf(fact_2143_add__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_nonpos
% 0.30/0.53  thf(fact_2144_add__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_nonpos
% 0.30/0.53  thf(fact_2145_add__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.30/0.53         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_nonpos
% 0.30/0.53  thf(fact_2146_add__nonpos__nonpos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_nonpos
% 0.30/0.53  thf(fact_2147_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.53         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_real )
% 0.30/0.53            = ( ( X3 = zero_zero_real )
% 0.30/0.53              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_eq_0_iff
% 0.30/0.53  thf(fact_2148_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_rat )
% 0.30/0.53            = ( ( X3 = zero_zero_rat )
% 0.30/0.53              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_eq_0_iff
% 0.30/0.53  thf(fact_2149_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: nat,Y3: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.53         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_nat )
% 0.30/0.53            = ( ( X3 = zero_zero_nat )
% 0.30/0.53              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_eq_0_iff
% 0.30/0.53  thf(fact_2150_add__nonneg__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: int,Y3: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.53         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_int )
% 0.30/0.53            = ( ( X3 = zero_zero_int )
% 0.30/0.53              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonneg_eq_0_iff
% 0.30/0.53  thf(fact_2151_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.53         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_real )
% 0.30/0.53            = ( ( X3 = zero_zero_real )
% 0.30/0.53              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_eq_0_iff
% 0.30/0.53  thf(fact_2152_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_rat )
% 0.30/0.53            = ( ( X3 = zero_zero_rat )
% 0.30/0.53              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_eq_0_iff
% 0.30/0.53  thf(fact_2153_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: nat,Y3: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 0.30/0.53         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_nat )
% 0.30/0.53            = ( ( X3 = zero_zero_nat )
% 0.30/0.53              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_eq_0_iff
% 0.30/0.53  thf(fact_2154_add__nonpos__eq__0__iff,axiom,
% 0.30/0.53      ! [X3: int,Y3: int] :
% 0.30/0.53        ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.53         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 0.30/0.53              = zero_zero_int )
% 0.30/0.53            = ( ( X3 = zero_zero_int )
% 0.30/0.53              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_nonpos_eq_0_iff
% 0.30/0.53  thf(fact_2155_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53  thf(fact_2156_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53  thf(fact_2157_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53  thf(fact_2158_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.53  thf(fact_2159_mult__less__cancel__right__disj,axiom,
% 0.30/0.53      ! [A: real,C: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53            & ( ord_less_real @ A @ B2 ) )
% 0.30/0.53          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53            & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_right_disj
% 0.30/0.53  thf(fact_2160_mult__less__cancel__right__disj,axiom,
% 0.30/0.53      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53            & ( ord_less_rat @ A @ B2 ) )
% 0.30/0.53          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_right_disj
% 0.30/0.53  thf(fact_2161_mult__less__cancel__right__disj,axiom,
% 0.30/0.53      ! [A: int,C: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.53        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53            & ( ord_less_int @ A @ B2 ) )
% 0.30/0.53          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53            & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_right_disj
% 0.30/0.53  thf(fact_2162_mult__strict__right__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono
% 0.30/0.53  thf(fact_2163_mult__strict__right__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono
% 0.30/0.53  thf(fact_2164_mult__strict__right__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono
% 0.30/0.53  thf(fact_2165_mult__strict__right__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono
% 0.30/0.53  thf(fact_2166_mult__strict__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: real,A: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono_neg
% 0.30/0.53  thf(fact_2167_mult__strict__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono_neg
% 0.30/0.53  thf(fact_2168_mult__strict__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: int,A: int,C: int] :
% 0.30/0.53        ( ( ord_less_int @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_right_mono_neg
% 0.30/0.53  thf(fact_2169_mult__less__cancel__left__disj,axiom,
% 0.30/0.53      ! [C: real,A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53            & ( ord_less_real @ A @ B2 ) )
% 0.30/0.53          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53            & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_disj
% 0.30/0.53  thf(fact_2170_mult__less__cancel__left__disj,axiom,
% 0.30/0.53      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53            & ( ord_less_rat @ A @ B2 ) )
% 0.30/0.53          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_disj
% 0.30/0.53  thf(fact_2171_mult__less__cancel__left__disj,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53            & ( ord_less_int @ A @ B2 ) )
% 0.30/0.53          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53            & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_disj
% 0.30/0.53  thf(fact_2172_mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono
% 0.30/0.53  thf(fact_2173_mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono
% 0.30/0.53  thf(fact_2174_mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.53         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono
% 0.30/0.53  thf(fact_2175_mult__strict__left__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono
% 0.30/0.53  thf(fact_2176_mult__strict__left__mono__neg,axiom,
% 0.30/0.53      ! [B2: real,A: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono_neg
% 0.30/0.53  thf(fact_2177_mult__strict__left__mono__neg,axiom,
% 0.30/0.53      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono_neg
% 0.30/0.53  thf(fact_2178_mult__strict__left__mono__neg,axiom,
% 0.30/0.53      ! [B2: int,A: int,C: int] :
% 0.30/0.53        ( ( ord_less_int @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_strict_left_mono_neg
% 0.30/0.53  thf(fact_2179_mult__less__cancel__left__pos,axiom,
% 0.30/0.53      ! [C: real,A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.53          = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_pos
% 0.30/0.53  thf(fact_2180_mult__less__cancel__left__pos,axiom,
% 0.30/0.53      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.53          = ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_pos
% 0.30/0.53  thf(fact_2181_mult__less__cancel__left__pos,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.53       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.53          = ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_pos
% 0.30/0.53  thf(fact_2182_mult__less__cancel__left__neg,axiom,
% 0.30/0.53      ! [C: real,A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.53          = ( ord_less_real @ B2 @ A ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_neg
% 0.30/0.53  thf(fact_2183_mult__less__cancel__left__neg,axiom,
% 0.30/0.53      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.53          = ( ord_less_rat @ B2 @ A ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_neg
% 0.30/0.53  thf(fact_2184_mult__less__cancel__left__neg,axiom,
% 0.30/0.53      ! [C: int,A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.53          = ( ord_less_int @ B2 @ A ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_cancel_left_neg
% 0.30/0.53  thf(fact_2185_zero__less__mult__pos2,axiom,
% 0.30/0.53      ! [B2: real,A: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos2
% 0.30/0.53  thf(fact_2186_zero__less__mult__pos2,axiom,
% 0.30/0.53      ! [B2: rat,A: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A ) )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos2
% 0.30/0.53  thf(fact_2187_zero__less__mult__pos2,axiom,
% 0.30/0.53      ! [B2: nat,A: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53         => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos2
% 0.30/0.53  thf(fact_2188_zero__less__mult__pos2,axiom,
% 0.30/0.53      ! [B2: int,A: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53         => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos2
% 0.30/0.53  thf(fact_2189_zero__less__mult__pos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos
% 0.30/0.53  thf(fact_2190_zero__less__mult__pos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos
% 0.30/0.53  thf(fact_2191_zero__less__mult__pos,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53         => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos
% 0.30/0.53  thf(fact_2192_zero__less__mult__pos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53         => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_pos
% 0.30/0.53  thf(fact_2193_zero__less__mult__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_real @ zero_zero_real @ B2 ) )
% 0.30/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_iff
% 0.30/0.53  thf(fact_2194_zero__less__mult__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 0.30/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_iff
% 0.30/0.53  thf(fact_2195_zero__less__mult__iff,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53            & ( ord_less_int @ zero_zero_int @ B2 ) )
% 0.30/0.53          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53            & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_mult_iff
% 0.30/0.53  thf(fact_2196_mult__pos__neg2,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg2
% 0.30/0.53  thf(fact_2197_mult__pos__neg2,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg2
% 0.30/0.53  thf(fact_2198_mult__pos__neg2,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.30/0.53         => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg2
% 0.30/0.53  thf(fact_2199_mult__pos__neg2,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg2
% 0.30/0.53  thf(fact_2200_mult__pos__pos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_pos
% 0.30/0.53  thf(fact_2201_mult__pos__pos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_pos
% 0.30/0.53  thf(fact_2202_mult__pos__pos,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.53         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_pos
% 0.30/0.53  thf(fact_2203_mult__pos__pos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.53         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_pos
% 0.30/0.53  thf(fact_2204_mult__pos__neg,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg
% 0.30/0.53  thf(fact_2205_mult__pos__neg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg
% 0.30/0.53  thf(fact_2206_mult__pos__neg,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.30/0.53         => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg
% 0.30/0.53  thf(fact_2207_mult__pos__neg,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_pos_neg
% 0.30/0.53  thf(fact_2208_mult__neg__pos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.53         => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_pos
% 0.30/0.53  thf(fact_2209_mult__neg__pos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.53         => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_pos
% 0.30/0.53  thf(fact_2210_mult__neg__pos,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.53         => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_pos
% 0.30/0.53  thf(fact_2211_mult__neg__pos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.53         => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_pos
% 0.30/0.53  thf(fact_2212_mult__less__0__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_real @ B2 @ zero_zero_real ) )
% 0.30/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_0_iff
% 0.30/0.53  thf(fact_2213_mult__less__0__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 0.30/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_0_iff
% 0.30/0.53  thf(fact_2214_mult__less__0__iff,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 0.30/0.53        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53            & ( ord_less_int @ B2 @ zero_zero_int ) )
% 0.30/0.53          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53            & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_less_0_iff
% 0.30/0.53  thf(fact_2215_not__square__less__zero,axiom,
% 0.30/0.53      ! [A: real] :
% 0.30/0.53        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.30/0.53  
% 0.30/0.53  % not_square_less_zero
% 0.30/0.53  thf(fact_2216_not__square__less__zero,axiom,
% 0.30/0.53      ! [A: rat] :
% 0.30/0.53        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_square_less_zero
% 0.30/0.53  thf(fact_2217_not__square__less__zero,axiom,
% 0.30/0.53      ! [A: int] :
% 0.30/0.53        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.30/0.53  
% 0.30/0.53  % not_square_less_zero
% 0.30/0.53  thf(fact_2218_mult__neg__neg,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_neg
% 0.30/0.53  thf(fact_2219_mult__neg__neg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_neg
% 0.30/0.53  thf(fact_2220_mult__neg__neg,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % mult_neg_neg
% 0.30/0.53  thf(fact_2221_vebt__maxt_Osimps_I1_J,axiom,
% 0.30/0.53      ! [B2: $o,A: $o] :
% 0.30/0.53        ( ( B2
% 0.30/0.53         => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 0.30/0.53            = ( some_nat @ one_one_nat ) ) )
% 0.30/0.53        & ( ~ B2
% 0.30/0.53         => ( ( A
% 0.30/0.53             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 0.30/0.53                = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.53            & ( ~ A
% 0.30/0.53             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 0.30/0.53                = none_nat ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % vebt_maxt.simps(1)
% 0.30/0.53  thf(fact_2222_le__iff__diff__le__0,axiom,
% 0.30/0.53      ( ord_less_eq_real
% 0.30/0.53      = ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % le_iff_diff_le_0
% 0.30/0.53  thf(fact_2223_le__iff__diff__le__0,axiom,
% 0.30/0.53      ( ord_less_eq_rat
% 0.30/0.53      = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % le_iff_diff_le_0
% 0.30/0.53  thf(fact_2224_le__iff__diff__le__0,axiom,
% 0.30/0.53      ( ord_less_eq_int
% 0.30/0.53      = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % le_iff_diff_le_0
% 0.30/0.53  thf(fact_2225_less__numeral__extra_I1_J,axiom,
% 0.30/0.53      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53  
% 0.30/0.53  % less_numeral_extra(1)
% 0.30/0.53  thf(fact_2226_less__numeral__extra_I1_J,axiom,
% 0.30/0.53      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53  
% 0.30/0.53  % less_numeral_extra(1)
% 0.30/0.53  thf(fact_2227_less__numeral__extra_I1_J,axiom,
% 0.30/0.53      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53  
% 0.30/0.53  % less_numeral_extra(1)
% 0.30/0.53  thf(fact_2228_less__numeral__extra_I1_J,axiom,
% 0.30/0.53      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53  
% 0.30/0.53  % less_numeral_extra(1)
% 0.30/0.53  thf(fact_2229_not__one__less__zero,axiom,
% 0.30/0.53      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_less_zero
% 0.30/0.53  thf(fact_2230_not__one__less__zero,axiom,
% 0.30/0.53      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_less_zero
% 0.30/0.53  thf(fact_2231_not__one__less__zero,axiom,
% 0.30/0.53      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_less_zero
% 0.30/0.53  thf(fact_2232_not__one__less__zero,axiom,
% 0.30/0.53      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.30/0.53  
% 0.30/0.53  % not_one_less_zero
% 0.30/0.53  thf(fact_2233_zero__less__one,axiom,
% 0.30/0.53      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one
% 0.30/0.53  thf(fact_2234_zero__less__one,axiom,
% 0.30/0.53      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one
% 0.30/0.53  thf(fact_2235_zero__less__one,axiom,
% 0.30/0.53      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one
% 0.30/0.53  thf(fact_2236_zero__less__one,axiom,
% 0.30/0.53      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_one
% 0.30/0.53  thf(fact_2237_add__less__zeroD,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.53          | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_less_zeroD
% 0.30/0.53  thf(fact_2238_add__less__zeroD,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 0.30/0.53          | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_less_zeroD
% 0.30/0.53  thf(fact_2239_add__less__zeroD,axiom,
% 0.30/0.53      ! [X3: int,Y3: int] :
% 0.30/0.53        ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y3 ) @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_int @ X3 @ zero_zero_int )
% 0.30/0.53          | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_less_zeroD
% 0.30/0.53  thf(fact_2240_add__neg__neg,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_neg_neg
% 0.30/0.53  thf(fact_2241_add__neg__neg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_neg_neg
% 0.30/0.53  thf(fact_2242_add__neg__neg,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.53       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.30/0.53         => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_neg_neg
% 0.30/0.53  thf(fact_2243_add__neg__neg,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.53       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.53         => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_neg_neg
% 0.30/0.53  thf(fact_2244_add__pos__pos,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_pos_pos
% 0.30/0.53  thf(fact_2245_add__pos__pos,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_pos_pos
% 0.30/0.53  thf(fact_2246_add__pos__pos,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.53         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_pos_pos
% 0.30/0.53  thf(fact_2247_add__pos__pos,axiom,
% 0.30/0.53      ! [A: int,B2: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.53         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % add_pos_pos
% 0.30/0.53  thf(fact_2248_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.30/0.53      ! [A: nat,B2: nat] :
% 0.30/0.53        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.53       => ~ ! [C2: nat] :
% 0.30/0.53              ( ( B2
% 0.30/0.53                = ( plus_plus_nat @ A @ C2 ) )
% 0.30/0.53             => ( C2 = zero_zero_nat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % canonically_ordered_monoid_add_class.lessE
% 0.30/0.53  thf(fact_2249_pos__add__strict,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.53         => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % pos_add_strict
% 0.30/0.53  thf(fact_2250_pos__add__strict,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.53         => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % pos_add_strict
% 0.30/0.53  thf(fact_2251_pos__add__strict,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.53         => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % pos_add_strict
% 0.30/0.53  thf(fact_2252_pos__add__strict,axiom,
% 0.30/0.53      ! [A: int,B2: int,C: int] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.53         => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % pos_add_strict
% 0.30/0.53  thf(fact_2253_divide__right__mono__neg,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_right_mono_neg
% 0.30/0.53  thf(fact_2254_divide__right__mono__neg,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_right_mono_neg
% 0.30/0.53  thf(fact_2255_divide__nonpos__nonpos,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonpos_nonpos
% 0.30/0.53  thf(fact_2256_divide__nonpos__nonpos,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonpos_nonpos
% 0.30/0.53  thf(fact_2257_divide__nonpos__nonneg,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonpos_nonneg
% 0.30/0.53  thf(fact_2258_divide__nonpos__nonneg,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonpos_nonneg
% 0.30/0.53  thf(fact_2259_divide__nonneg__nonpos,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonneg_nonpos
% 0.30/0.53  thf(fact_2260_divide__nonneg__nonpos,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonneg_nonpos
% 0.30/0.53  thf(fact_2261_divide__nonneg__nonneg,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonneg_nonneg
% 0.30/0.53  thf(fact_2262_divide__nonneg__nonneg,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_nonneg_nonneg
% 0.30/0.53  thf(fact_2263_zero__le__divide__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_divide_iff
% 0.30/0.53  thf(fact_2264_zero__le__divide__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 0.30/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_divide_iff
% 0.30/0.53  thf(fact_2265_divide__right__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_right_mono
% 0.30/0.53  thf(fact_2266_divide__right__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_right_mono
% 0.30/0.53  thf(fact_2267_divide__le__0__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 0.30/0.53        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 0.30/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_le_0_iff
% 0.30/0.53  thf(fact_2268_divide__le__0__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 0.30/0.53        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 0.30/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_le_0_iff
% 0.30/0.53  thf(fact_2269_less__iff__diff__less__0,axiom,
% 0.30/0.53      ( ord_less_real
% 0.30/0.53      = ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % less_iff_diff_less_0
% 0.30/0.53  thf(fact_2270_less__iff__diff__less__0,axiom,
% 0.30/0.53      ( ord_less_rat
% 0.30/0.53      = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % less_iff_diff_less_0
% 0.30/0.53  thf(fact_2271_less__iff__diff__less__0,axiom,
% 0.30/0.53      ( ord_less_int
% 0.30/0.53      = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % less_iff_diff_less_0
% 0.30/0.53  thf(fact_2272_divide__strict__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: real,A: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_strict_right_mono_neg
% 0.30/0.53  thf(fact_2273_divide__strict__right__mono__neg,axiom,
% 0.30/0.53      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.53       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_strict_right_mono_neg
% 0.30/0.53  thf(fact_2274_divide__strict__right__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( ord_less_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_strict_right_mono
% 0.30/0.53  thf(fact_2275_divide__strict__right__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_strict_right_mono
% 0.30/0.53  thf(fact_2276_zero__less__divide__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_real @ zero_zero_real @ B2 ) )
% 0.30/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_divide_iff
% 0.30/0.53  thf(fact_2277_zero__less__divide__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 0.30/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_divide_iff
% 0.30/0.53  thf(fact_2278_divide__less__cancel,axiom,
% 0.30/0.53      ! [A: real,C: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.53           => ( ord_less_real @ A @ B2 ) )
% 0.30/0.53          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.53           => ( ord_less_real @ B2 @ A ) )
% 0.30/0.53          & ( C != zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_less_cancel
% 0.30/0.53  thf(fact_2279_divide__less__cancel,axiom,
% 0.30/0.53      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.53           => ( ord_less_rat @ A @ B2 ) )
% 0.30/0.53          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.53           => ( ord_less_rat @ B2 @ A ) )
% 0.30/0.53          & ( C != zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_less_cancel
% 0.30/0.53  thf(fact_2280_divide__less__0__iff,axiom,
% 0.30/0.53      ! [A: real,B2: real] :
% 0.30/0.53        ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 0.30/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53            & ( ord_less_real @ B2 @ zero_zero_real ) )
% 0.30/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53            & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_less_0_iff
% 0.30/0.53  thf(fact_2281_divide__less__0__iff,axiom,
% 0.30/0.53      ! [A: rat,B2: rat] :
% 0.30/0.53        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 0.30/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53            & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 0.30/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53            & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_less_0_iff
% 0.30/0.53  thf(fact_2282_divide__pos__pos,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_pos_pos
% 0.30/0.53  thf(fact_2283_divide__pos__pos,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_pos_pos
% 0.30/0.53  thf(fact_2284_divide__pos__neg,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.53       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_pos_neg
% 0.30/0.53  thf(fact_2285_divide__pos__neg,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.53       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_pos_neg
% 0.30/0.53  thf(fact_2286_divide__neg__pos,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.53         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_neg_pos
% 0.30/0.53  thf(fact_2287_divide__neg__pos,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.53         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_neg_pos
% 0.30/0.53  thf(fact_2288_divide__neg__neg,axiom,
% 0.30/0.53      ! [X3: real,Y3: real] :
% 0.30/0.53        ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.53       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.53         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_neg_neg
% 0.30/0.53  thf(fact_2289_divide__neg__neg,axiom,
% 0.30/0.53      ! [X3: rat,Y3: rat] :
% 0.30/0.53        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 0.30/0.53       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.53         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_neg_neg
% 0.30/0.53  thf(fact_2290_zero__le__power,axiom,
% 0.30/0.53      ! [A: real,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_power
% 0.30/0.53  thf(fact_2291_zero__le__power,axiom,
% 0.30/0.53      ! [A: rat,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_power
% 0.30/0.53  thf(fact_2292_zero__le__power,axiom,
% 0.30/0.53      ! [A: nat,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_power
% 0.30/0.53  thf(fact_2293_zero__le__power,axiom,
% 0.30/0.53      ! [A: int,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_le_power
% 0.30/0.53  thf(fact_2294_power__mono,axiom,
% 0.30/0.53      ! [A: real,B2: real,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_mono
% 0.30/0.53  thf(fact_2295_power__mono,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_mono
% 0.30/0.53  thf(fact_2296_power__mono,axiom,
% 0.30/0.53      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_mono
% 0.30/0.53  thf(fact_2297_power__mono,axiom,
% 0.30/0.53      ! [A: int,B2: int,N: nat] :
% 0.30/0.53        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % power_mono
% 0.30/0.53  thf(fact_2298_zero__less__power,axiom,
% 0.30/0.53      ! [A: real,N: nat] :
% 0.30/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_power
% 0.30/0.53  thf(fact_2299_zero__less__power,axiom,
% 0.30/0.53      ! [A: rat,N: nat] :
% 0.30/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_power
% 0.30/0.53  thf(fact_2300_zero__less__power,axiom,
% 0.30/0.53      ! [A: nat,N: nat] :
% 0.30/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.53       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_power
% 0.30/0.53  thf(fact_2301_zero__less__power,axiom,
% 0.30/0.53      ! [A: int,N: nat] :
% 0.30/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.53       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % zero_less_power
% 0.30/0.53  thf(fact_2302_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.30/0.53      ! [X3: vEBT_VEBT] :
% 0.30/0.53        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.53       => ( ! [Uv2: $o] :
% 0.30/0.53              ( X3
% 0.30/0.53             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.53         => ( ! [Uu2: $o] :
% 0.30/0.53                ( X3
% 0.30/0.53               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.53           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.53                  ( X3
% 0.30/0.53                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % VEBT_internal.minNull.elims(3)
% 0.30/0.53  thf(fact_2303_frac__eq__eq,axiom,
% 0.30/0.53      ! [Y3: complex,Z: complex,X3: complex,W: complex] :
% 0.30/0.53        ( ( Y3 != zero_zero_complex )
% 0.30/0.53       => ( ( Z != zero_zero_complex )
% 0.30/0.53         => ( ( ( divide1717551699836669952omplex @ X3 @ Y3 )
% 0.30/0.53              = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.53            = ( ( times_times_complex @ X3 @ Z )
% 0.30/0.53              = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % frac_eq_eq
% 0.30/0.53  thf(fact_2304_frac__eq__eq,axiom,
% 0.30/0.53      ! [Y3: real,Z: real,X3: real,W: real] :
% 0.30/0.53        ( ( Y3 != zero_zero_real )
% 0.30/0.53       => ( ( Z != zero_zero_real )
% 0.30/0.53         => ( ( ( divide_divide_real @ X3 @ Y3 )
% 0.30/0.53              = ( divide_divide_real @ W @ Z ) )
% 0.30/0.53            = ( ( times_times_real @ X3 @ Z )
% 0.30/0.53              = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % frac_eq_eq
% 0.30/0.53  thf(fact_2305_frac__eq__eq,axiom,
% 0.30/0.53      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 0.30/0.53        ( ( Y3 != zero_zero_rat )
% 0.30/0.53       => ( ( Z != zero_zero_rat )
% 0.30/0.53         => ( ( ( divide_divide_rat @ X3 @ Y3 )
% 0.30/0.53              = ( divide_divide_rat @ W @ Z ) )
% 0.30/0.53            = ( ( times_times_rat @ X3 @ Z )
% 0.30/0.53              = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % frac_eq_eq
% 0.30/0.53  thf(fact_2306_divide__eq__eq,axiom,
% 0.30/0.53      ! [B2: complex,C: complex,A: complex] :
% 0.30/0.53        ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.30/0.53          = A )
% 0.30/0.53        = ( ( ( C != zero_zero_complex )
% 0.30/0.53           => ( B2
% 0.30/0.53              = ( times_times_complex @ A @ C ) ) )
% 0.30/0.53          & ( ( C = zero_zero_complex )
% 0.30/0.53           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_eq_eq
% 0.30/0.53  thf(fact_2307_divide__eq__eq,axiom,
% 0.30/0.53      ! [B2: real,C: real,A: real] :
% 0.30/0.53        ( ( ( divide_divide_real @ B2 @ C )
% 0.30/0.53          = A )
% 0.30/0.53        = ( ( ( C != zero_zero_real )
% 0.30/0.53           => ( B2
% 0.30/0.53              = ( times_times_real @ A @ C ) ) )
% 0.30/0.53          & ( ( C = zero_zero_real )
% 0.30/0.53           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_eq_eq
% 0.30/0.53  thf(fact_2308_divide__eq__eq,axiom,
% 0.30/0.53      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.53        ( ( ( divide_divide_rat @ B2 @ C )
% 0.30/0.53          = A )
% 0.30/0.53        = ( ( ( C != zero_zero_rat )
% 0.30/0.53           => ( B2
% 0.30/0.53              = ( times_times_rat @ A @ C ) ) )
% 0.30/0.53          & ( ( C = zero_zero_rat )
% 0.30/0.53           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_eq_eq
% 0.30/0.53  thf(fact_2309_eq__divide__eq,axiom,
% 0.30/0.53      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.53        ( ( A
% 0.30/0.53          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.53        = ( ( ( C != zero_zero_complex )
% 0.30/0.53           => ( ( times_times_complex @ A @ C )
% 0.30/0.53              = B2 ) )
% 0.30/0.53          & ( ( C = zero_zero_complex )
% 0.30/0.53           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_divide_eq
% 0.30/0.53  thf(fact_2310_eq__divide__eq,axiom,
% 0.30/0.53      ! [A: real,B2: real,C: real] :
% 0.30/0.53        ( ( A
% 0.30/0.53          = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.53        = ( ( ( C != zero_zero_real )
% 0.30/0.53           => ( ( times_times_real @ A @ C )
% 0.30/0.53              = B2 ) )
% 0.30/0.53          & ( ( C = zero_zero_real )
% 0.30/0.53           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_divide_eq
% 0.30/0.53  thf(fact_2311_eq__divide__eq,axiom,
% 0.30/0.53      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.53        ( ( A
% 0.30/0.53          = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.53        = ( ( ( C != zero_zero_rat )
% 0.30/0.53           => ( ( times_times_rat @ A @ C )
% 0.30/0.53              = B2 ) )
% 0.30/0.53          & ( ( C = zero_zero_rat )
% 0.30/0.53           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % eq_divide_eq
% 0.30/0.53  thf(fact_2312_divide__eq__imp,axiom,
% 0.30/0.53      ! [C: complex,B2: complex,A: complex] :
% 0.30/0.53        ( ( C != zero_zero_complex )
% 0.30/0.53       => ( ( B2
% 0.30/0.53            = ( times_times_complex @ A @ C ) )
% 0.30/0.53         => ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.30/0.53            = A ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_eq_imp
% 0.30/0.53  thf(fact_2313_divide__eq__imp,axiom,
% 0.30/0.53      ! [C: real,B2: real,A: real] :
% 0.30/0.53        ( ( C != zero_zero_real )
% 0.30/0.53       => ( ( B2
% 0.30/0.53            = ( times_times_real @ A @ C ) )
% 0.30/0.53         => ( ( divide_divide_real @ B2 @ C )
% 0.30/0.53            = A ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_eq_imp
% 0.30/0.53  thf(fact_2314_divide__eq__imp,axiom,
% 0.30/0.53      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.53        ( ( C != zero_zero_rat )
% 0.30/0.53       => ( ( B2
% 0.30/0.53            = ( times_times_rat @ A @ C ) )
% 0.30/0.53         => ( ( divide_divide_rat @ B2 @ C )
% 0.30/0.53            = A ) ) ) ).
% 0.30/0.53  
% 0.30/0.53  % divide_eq_imp
% 0.30/0.53  thf(fact_2315_eq__divide__imp,axiom,
% 0.30/0.53      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.53        ( ( C != zero_zero_complex )
% 0.30/0.53       => ( ( ( times_times_complex @ A @ C )
% 0.30/0.54            = B2 )
% 0.30/0.54         => ( A
% 0.30/0.54            = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % eq_divide_imp
% 0.30/0.54  thf(fact_2316_eq__divide__imp,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( C != zero_zero_real )
% 0.30/0.54       => ( ( ( times_times_real @ A @ C )
% 0.30/0.54            = B2 )
% 0.30/0.54         => ( A
% 0.30/0.54            = ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % eq_divide_imp
% 0.30/0.54  thf(fact_2317_eq__divide__imp,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( C != zero_zero_rat )
% 0.30/0.54       => ( ( ( times_times_rat @ A @ C )
% 0.30/0.54            = B2 )
% 0.30/0.54         => ( A
% 0.30/0.54            = ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % eq_divide_imp
% 0.30/0.54  thf(fact_2318_nonzero__divide__eq__eq,axiom,
% 0.30/0.54      ! [C: complex,B2: complex,A: complex] :
% 0.30/0.54        ( ( C != zero_zero_complex )
% 0.30/0.54       => ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.30/0.54            = A )
% 0.30/0.54          = ( B2
% 0.30/0.54            = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nonzero_divide_eq_eq
% 0.30/0.54  thf(fact_2319_nonzero__divide__eq__eq,axiom,
% 0.30/0.54      ! [C: real,B2: real,A: real] :
% 0.30/0.54        ( ( C != zero_zero_real )
% 0.30/0.54       => ( ( ( divide_divide_real @ B2 @ C )
% 0.30/0.54            = A )
% 0.30/0.54          = ( B2
% 0.30/0.54            = ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nonzero_divide_eq_eq
% 0.30/0.54  thf(fact_2320_nonzero__divide__eq__eq,axiom,
% 0.30/0.54      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.54        ( ( C != zero_zero_rat )
% 0.30/0.54       => ( ( ( divide_divide_rat @ B2 @ C )
% 0.30/0.54            = A )
% 0.30/0.54          = ( B2
% 0.30/0.54            = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nonzero_divide_eq_eq
% 0.30/0.54  thf(fact_2321_nonzero__eq__divide__eq,axiom,
% 0.30/0.54      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.54        ( ( C != zero_zero_complex )
% 0.30/0.54       => ( ( A
% 0.30/0.54            = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.54          = ( ( times_times_complex @ A @ C )
% 0.30/0.54            = B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nonzero_eq_divide_eq
% 0.30/0.54  thf(fact_2322_nonzero__eq__divide__eq,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( C != zero_zero_real )
% 0.30/0.54       => ( ( A
% 0.30/0.54            = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.54          = ( ( times_times_real @ A @ C )
% 0.30/0.54            = B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nonzero_eq_divide_eq
% 0.30/0.54  thf(fact_2323_nonzero__eq__divide__eq,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( C != zero_zero_rat )
% 0.30/0.54       => ( ( A
% 0.30/0.54            = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.54          = ( ( times_times_rat @ A @ C )
% 0.30/0.54            = B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nonzero_eq_divide_eq
% 0.30/0.54  thf(fact_2324_right__inverse__eq,axiom,
% 0.30/0.54      ! [B2: complex,A: complex] :
% 0.30/0.54        ( ( B2 != zero_zero_complex )
% 0.30/0.54       => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.30/0.54            = one_one_complex )
% 0.30/0.54          = ( A = B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % right_inverse_eq
% 0.30/0.54  thf(fact_2325_right__inverse__eq,axiom,
% 0.30/0.54      ! [B2: real,A: real] :
% 0.30/0.54        ( ( B2 != zero_zero_real )
% 0.30/0.54       => ( ( ( divide_divide_real @ A @ B2 )
% 0.30/0.54            = one_one_real )
% 0.30/0.54          = ( A = B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % right_inverse_eq
% 0.30/0.54  thf(fact_2326_right__inverse__eq,axiom,
% 0.30/0.54      ! [B2: rat,A: rat] :
% 0.30/0.54        ( ( B2 != zero_zero_rat )
% 0.30/0.54       => ( ( ( divide_divide_rat @ A @ B2 )
% 0.30/0.54            = one_one_rat )
% 0.30/0.54          = ( A = B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % right_inverse_eq
% 0.30/0.54  thf(fact_2327_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.30/0.54      ! [X3: vEBT_VEBT] :
% 0.30/0.54        ( ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.54       => ( ( X3
% 0.30/0.54           != ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.54         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.54                ( X3
% 0.30/0.54               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % VEBT_internal.minNull.elims(2)
% 0.30/0.54  thf(fact_2328_power__0,axiom,
% 0.30/0.54      ! [A: rat] :
% 0.30/0.54        ( ( power_power_rat @ A @ zero_zero_nat )
% 0.30/0.54        = one_one_rat ) ).
% 0.30/0.54  
% 0.30/0.54  % power_0
% 0.30/0.54  thf(fact_2329_power__0,axiom,
% 0.30/0.54      ! [A: nat] :
% 0.30/0.54        ( ( power_power_nat @ A @ zero_zero_nat )
% 0.30/0.54        = one_one_nat ) ).
% 0.30/0.54  
% 0.30/0.54  % power_0
% 0.30/0.54  thf(fact_2330_power__0,axiom,
% 0.30/0.54      ! [A: real] :
% 0.30/0.54        ( ( power_power_real @ A @ zero_zero_nat )
% 0.30/0.54        = one_one_real ) ).
% 0.30/0.54  
% 0.30/0.54  % power_0
% 0.30/0.54  thf(fact_2331_power__0,axiom,
% 0.30/0.54      ! [A: int] :
% 0.30/0.54        ( ( power_power_int @ A @ zero_zero_nat )
% 0.30/0.54        = one_one_int ) ).
% 0.30/0.54  
% 0.30/0.54  % power_0
% 0.30/0.54  thf(fact_2332_power__0,axiom,
% 0.30/0.54      ! [A: complex] :
% 0.30/0.54        ( ( power_power_complex @ A @ zero_zero_nat )
% 0.30/0.54        = one_one_complex ) ).
% 0.30/0.54  
% 0.30/0.54  % power_0
% 0.30/0.54  thf(fact_2333_less__Suc__eq__0__disj,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.30/0.54        = ( ( M = zero_zero_nat )
% 0.30/0.54          | ? [J3: nat] :
% 0.30/0.54              ( ( M
% 0.30/0.54                = ( suc @ J3 ) )
% 0.30/0.54              & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % less_Suc_eq_0_disj
% 0.30/0.54  thf(fact_2334_gr0__implies__Suc,axiom,
% 0.30/0.54      ! [N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54       => ? [M4: nat] :
% 0.30/0.54            ( N
% 0.30/0.54            = ( suc @ M4 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % gr0_implies_Suc
% 0.30/0.54  thf(fact_2335_All__less__Suc2,axiom,
% 0.30/0.54      ! [N: nat,P: nat > $o] :
% 0.30/0.54        ( ( ! [I4: nat] :
% 0.30/0.54              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.54             => ( P @ I4 ) ) )
% 0.30/0.54        = ( ( P @ zero_zero_nat )
% 0.30/0.54          & ! [I4: nat] :
% 0.30/0.54              ( ( ord_less_nat @ I4 @ N )
% 0.30/0.54             => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % All_less_Suc2
% 0.30/0.54  thf(fact_2336_gr0__conv__Suc,axiom,
% 0.30/0.54      ! [N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54        = ( ? [M3: nat] :
% 0.30/0.54              ( N
% 0.30/0.54              = ( suc @ M3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % gr0_conv_Suc
% 0.30/0.54  thf(fact_2337_Ex__less__Suc2,axiom,
% 0.30/0.54      ! [N: nat,P: nat > $o] :
% 0.30/0.54        ( ( ? [I4: nat] :
% 0.30/0.54              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.30/0.54              & ( P @ I4 ) ) )
% 0.30/0.54        = ( ( P @ zero_zero_nat )
% 0.30/0.54          | ? [I4: nat] :
% 0.30/0.54              ( ( ord_less_nat @ I4 @ N )
% 0.30/0.54              & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % Ex_less_Suc2
% 0.30/0.54  thf(fact_2338_add__is__1,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( ( plus_plus_nat @ M @ N )
% 0.30/0.54          = ( suc @ zero_zero_nat ) )
% 0.30/0.54        = ( ( ( M
% 0.30/0.54              = ( suc @ zero_zero_nat ) )
% 0.30/0.54            & ( N = zero_zero_nat ) )
% 0.30/0.54          | ( ( M = zero_zero_nat )
% 0.30/0.54            & ( N
% 0.30/0.54              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_is_1
% 0.30/0.54  thf(fact_2339_one__is__add,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( ( suc @ zero_zero_nat )
% 0.30/0.54          = ( plus_plus_nat @ M @ N ) )
% 0.30/0.54        = ( ( ( M
% 0.30/0.54              = ( suc @ zero_zero_nat ) )
% 0.30/0.54            & ( N = zero_zero_nat ) )
% 0.30/0.54          | ( ( M = zero_zero_nat )
% 0.30/0.54            & ( N
% 0.30/0.54              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % one_is_add
% 0.30/0.54  thf(fact_2340_ex__least__nat__le,axiom,
% 0.30/0.54      ! [P: nat > $o,N: nat] :
% 0.30/0.54        ( ( P @ N )
% 0.30/0.54       => ( ~ ( P @ zero_zero_nat )
% 0.30/0.54         => ? [K3: nat] :
% 0.30/0.54              ( ( ord_less_eq_nat @ K3 @ N )
% 0.30/0.54              & ! [I2: nat] :
% 0.30/0.54                  ( ( ord_less_nat @ I2 @ K3 )
% 0.30/0.54                 => ~ ( P @ I2 ) )
% 0.30/0.54              & ( P @ K3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % ex_least_nat_le
% 0.30/0.54  thf(fact_2341_option_Osize_I4_J,axiom,
% 0.30/0.54      ! [X2: nat] :
% 0.30/0.54        ( ( size_size_option_nat @ ( some_nat @ X2 ) )
% 0.30/0.54        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % option.size(4)
% 0.30/0.54  thf(fact_2342_option_Osize_I4_J,axiom,
% 0.30/0.54      ! [X2: product_prod_nat_nat] :
% 0.30/0.54        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
% 0.30/0.54        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % option.size(4)
% 0.30/0.54  thf(fact_2343_option_Osize_I4_J,axiom,
% 0.30/0.54      ! [X2: num] :
% 0.30/0.54        ( ( size_size_option_num @ ( some_num @ X2 ) )
% 0.30/0.54        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % option.size(4)
% 0.30/0.54  thf(fact_2344_less__imp__add__positive,axiom,
% 0.30/0.54      ! [I: nat,J: nat] :
% 0.30/0.54        ( ( ord_less_nat @ I @ J )
% 0.30/0.54       => ? [K3: nat] :
% 0.30/0.54            ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.30/0.54            & ( ( plus_plus_nat @ I @ K3 )
% 0.30/0.54              = J ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % less_imp_add_positive
% 0.30/0.54  thf(fact_2345_option_Osize_I3_J,axiom,
% 0.30/0.54      ( ( size_size_option_nat @ none_nat )
% 0.30/0.54      = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % option.size(3)
% 0.30/0.54  thf(fact_2346_option_Osize_I3_J,axiom,
% 0.30/0.54      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.30/0.54      = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % option.size(3)
% 0.30/0.54  thf(fact_2347_option_Osize_I3_J,axiom,
% 0.30/0.54      ( ( size_size_option_num @ none_num )
% 0.30/0.54      = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % option.size(3)
% 0.30/0.54  thf(fact_2348_diff__less,axiom,
% 0.30/0.54      ! [N: nat,M: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54         => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_less
% 0.30/0.54  thf(fact_2349_nat__mult__less__cancel1,axiom,
% 0.30/0.54      ! [K: nat,M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.54          = ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_mult_less_cancel1
% 0.30/0.54  thf(fact_2350_nat__mult__eq__cancel1,axiom,
% 0.30/0.54      ! [K: nat,M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54       => ( ( ( times_times_nat @ K @ M )
% 0.30/0.54            = ( times_times_nat @ K @ N ) )
% 0.30/0.54          = ( M = N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_mult_eq_cancel1
% 0.30/0.54  thf(fact_2351_mult__less__mono2,axiom,
% 0.30/0.54      ! [I: nat,J: nat,K: nat] :
% 0.30/0.54        ( ( ord_less_nat @ I @ J )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_mono2
% 0.30/0.54  thf(fact_2352_mult__less__mono1,axiom,
% 0.30/0.54      ! [I: nat,J: nat,K: nat] :
% 0.30/0.54        ( ( ord_less_nat @ I @ J )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_mono1
% 0.30/0.54  thf(fact_2353_One__nat__def,axiom,
% 0.30/0.54      ( one_one_nat
% 0.30/0.54      = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % One_nat_def
% 0.30/0.54  thf(fact_2354_divmod__digit__0_I2_J,axiom,
% 0.30/0.54      ! [B2: nat,A: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.54       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 0.30/0.54            = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_0(2)
% 0.30/0.54  thf(fact_2355_divmod__digit__0_I2_J,axiom,
% 0.30/0.54      ! [B2: int,A: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.54       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 0.30/0.54            = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_0(2)
% 0.30/0.54  thf(fact_2356_divmod__digit__0_I2_J,axiom,
% 0.30/0.54      ! [B2: code_integer,A: code_integer] :
% 0.30/0.54        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.54       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 0.30/0.54            = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_0(2)
% 0.30/0.54  thf(fact_2357_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( ( divide_divide_nat @ M @ N )
% 0.30/0.54          = zero_zero_nat )
% 0.30/0.54        = ( ( ord_less_nat @ M @ N )
% 0.30/0.54          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % Euclidean_Division.div_eq_0_iff
% 0.30/0.54  thf(fact_2358_diff__add__0,axiom,
% 0.30/0.54      ! [N: nat,M: nat] :
% 0.30/0.54        ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.54        = zero_zero_nat ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_add_0
% 0.30/0.54  thf(fact_2359_bits__stable__imp__add__self,axiom,
% 0.30/0.54      ! [A: nat] :
% 0.30/0.54        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.54          = A )
% 0.30/0.54       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.54          = zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % bits_stable_imp_add_self
% 0.30/0.54  thf(fact_2360_bits__stable__imp__add__self,axiom,
% 0.30/0.54      ! [A: int] :
% 0.30/0.54        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.54          = A )
% 0.30/0.54       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.54          = zero_zero_int ) ) ).
% 0.30/0.54  
% 0.30/0.54  % bits_stable_imp_add_self
% 0.30/0.54  thf(fact_2361_bits__stable__imp__add__self,axiom,
% 0.30/0.54      ! [A: code_integer] :
% 0.30/0.54        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.54          = A )
% 0.30/0.54       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.54          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.54  
% 0.30/0.54  % bits_stable_imp_add_self
% 0.30/0.54  thf(fact_2362_nat__power__less__imp__less,axiom,
% 0.30/0.54      ! [I: nat,M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.30/0.54       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.30/0.54         => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_power_less_imp_less
% 0.30/0.54  thf(fact_2363_mult__eq__self__implies__10,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( M
% 0.30/0.54          = ( times_times_nat @ M @ N ) )
% 0.30/0.54       => ( ( N = one_one_nat )
% 0.30/0.54          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_eq_self_implies_10
% 0.30/0.54  thf(fact_2364_vebt__insert_Osimps_I3_J,axiom,
% 0.30/0.54      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 0.30/0.54        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X3 )
% 0.30/0.54        = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 0.30/0.54  
% 0.30/0.54  % vebt_insert.simps(3)
% 0.30/0.54  thf(fact_2365_vebt__member_Osimps_I3_J,axiom,
% 0.30/0.54      ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
% 0.30/0.54        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).
% 0.30/0.54  
% 0.30/0.54  % vebt_member.simps(3)
% 0.30/0.54  thf(fact_2366_verit__le__mono__div,axiom,
% 0.30/0.54      ! [A4: nat,B5: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ A4 @ B5 )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54         => ( ord_less_eq_nat
% 0.30/0.54            @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
% 0.30/0.54              @ ( if_nat
% 0.30/0.54                @ ( ( modulo_modulo_nat @ B5 @ N )
% 0.30/0.54                  = zero_zero_nat )
% 0.30/0.54                @ one_one_nat
% 0.30/0.54                @ zero_zero_nat ) )
% 0.30/0.54            @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % verit_le_mono_div
% 0.30/0.54  thf(fact_2367_VEBT__internal_Omembermima_Ocases,axiom,
% 0.30/0.54      ! [X3: produc9072475918466114483BT_nat] :
% 0.30/0.54        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.30/0.54            ( X3
% 0.30/0.54           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.30/0.54       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.30/0.54              ( X3
% 0.30/0.54             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.30/0.54         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
% 0.30/0.54                ( X3
% 0.30/0.54               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
% 0.30/0.54           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 0.30/0.54                  ( X3
% 0.30/0.54                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X5 ) )
% 0.30/0.54             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
% 0.30/0.54                    ( X3
% 0.30/0.54                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % VEBT_internal.membermima.cases
% 0.30/0.54  thf(fact_2368_vebt__pred_Ocases,axiom,
% 0.30/0.54      ! [X3: produc9072475918466114483BT_nat] :
% 0.30/0.54        ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.54            ( X3
% 0.30/0.54           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 0.30/0.54       => ( ! [A3: $o,Uw2: $o] :
% 0.30/0.54              ( X3
% 0.30/0.54             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.54         => ( ! [A3: $o,B3: $o,Va2: nat] :
% 0.30/0.54                ( X3
% 0.30/0.54               != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 0.30/0.54           => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 0.30/0.54                  ( X3
% 0.30/0.54                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 0.30/0.54             => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 0.30/0.54                    ( X3
% 0.30/0.54                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
% 0.30/0.54               => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 0.30/0.54                      ( X3
% 0.30/0.54                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
% 0.30/0.54                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.30/0.54                        ( X3
% 0.30/0.54                       != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % vebt_pred.cases
% 0.30/0.54  thf(fact_2369_vebt__pred_Osimps_I3_J,axiom,
% 0.30/0.54      ! [B2: $o,A: $o,Va: nat] :
% 0.30/0.54        ( ( B2
% 0.30/0.54         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.54            = ( some_nat @ one_one_nat ) ) )
% 0.30/0.54        & ( ~ B2
% 0.30/0.54         => ( ( A
% 0.30/0.54             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.54                = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.54            & ( ~ A
% 0.30/0.54             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.54                = none_nat ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % vebt_pred.simps(3)
% 0.30/0.54  thf(fact_2370_divmod__digit__0_I1_J,axiom,
% 0.30/0.54      ! [B2: nat,A: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.54       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.54            = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_0(1)
% 0.30/0.54  thf(fact_2371_divmod__digit__0_I1_J,axiom,
% 0.30/0.54      ! [B2: int,A: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.54       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.54            = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_0(1)
% 0.30/0.54  thf(fact_2372_divmod__digit__0_I1_J,axiom,
% 0.30/0.54      ! [B2: code_integer,A: code_integer] :
% 0.30/0.54        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.54       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.54            = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_0(1)
% 0.30/0.54  thf(fact_2373_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.54      ! [M: num,Q2: num] :
% 0.30/0.54        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.54       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cong_exp_iff_simps(8)
% 0.30/0.54  thf(fact_2374_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.54      ! [M: num,Q2: num] :
% 0.30/0.54        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.54       != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cong_exp_iff_simps(8)
% 0.30/0.54  thf(fact_2375_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.54      ! [M: num,Q2: num] :
% 0.30/0.54        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.54       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cong_exp_iff_simps(8)
% 0.30/0.54  thf(fact_2376_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.54      ! [Q2: num,N: num] :
% 0.30/0.54        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.54       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cong_exp_iff_simps(6)
% 0.30/0.54  thf(fact_2377_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.54      ! [Q2: num,N: num] :
% 0.30/0.54        ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.54       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cong_exp_iff_simps(6)
% 0.30/0.54  thf(fact_2378_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.54      ! [Q2: num,N: num] :
% 0.30/0.54        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.54       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cong_exp_iff_simps(6)
% 0.30/0.54  thf(fact_2379_div__mult1__eq,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.54        ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.54        = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mult1_eq
% 0.30/0.54  thf(fact_2380_div__mult1__eq,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int] :
% 0.30/0.54        ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.54        = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mult1_eq
% 0.30/0.54  thf(fact_2381_div__mult1__eq,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.54        ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.30/0.54        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mult1_eq
% 0.30/0.54  thf(fact_2382_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.54      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.54        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 0.30/0.54        = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cancel_div_mod_rules(2)
% 0.30/0.54  thf(fact_2383_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.54      ! [B2: int,A: int,C: int] :
% 0.30/0.54        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 0.30/0.54        = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cancel_div_mod_rules(2)
% 0.30/0.54  thf(fact_2384_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.54      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.54        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 0.30/0.54        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cancel_div_mod_rules(2)
% 0.30/0.54  thf(fact_2385_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.54        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 0.30/0.54        = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cancel_div_mod_rules(1)
% 0.30/0.54  thf(fact_2386_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int] :
% 0.30/0.54        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 0.30/0.54        = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cancel_div_mod_rules(1)
% 0.30/0.54  thf(fact_2387_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.54        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 0.30/0.54        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.54  
% 0.30/0.54  % cancel_div_mod_rules(1)
% 0.30/0.54  thf(fact_2388_mod__div__decomp,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( A
% 0.30/0.54        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_div_decomp
% 0.30/0.54  thf(fact_2389_mod__div__decomp,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( A
% 0.30/0.54        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_div_decomp
% 0.30/0.54  thf(fact_2390_mod__div__decomp,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( A
% 0.30/0.54        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_div_decomp
% 0.30/0.54  thf(fact_2391_div__mult__mod__eq,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mult_mod_eq
% 0.30/0.54  thf(fact_2392_div__mult__mod__eq,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mult_mod_eq
% 0.30/0.54  thf(fact_2393_div__mult__mod__eq,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mult_mod_eq
% 0.30/0.54  thf(fact_2394_mod__div__mult__eq,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_div_mult_eq
% 0.30/0.54  thf(fact_2395_mod__div__mult__eq,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_div_mult_eq
% 0.30/0.54  thf(fact_2396_mod__div__mult__eq,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_div_mult_eq
% 0.30/0.54  thf(fact_2397_mod__mult__div__eq,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_mult_div_eq
% 0.30/0.54  thf(fact_2398_mod__mult__div__eq,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_mult_div_eq
% 0.30/0.54  thf(fact_2399_mod__mult__div__eq,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_mult_div_eq
% 0.30/0.54  thf(fact_2400_mult__div__mod__eq,axiom,
% 0.30/0.54      ! [B2: nat,A: nat] :
% 0.30/0.54        ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_div_mod_eq
% 0.30/0.54  thf(fact_2401_mult__div__mod__eq,axiom,
% 0.30/0.54      ! [B2: int,A: int] :
% 0.30/0.54        ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_div_mod_eq
% 0.30/0.54  thf(fact_2402_mult__div__mod__eq,axiom,
% 0.30/0.54      ! [B2: code_integer,A: code_integer] :
% 0.30/0.54        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.54        = A ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_div_mod_eq
% 0.30/0.54  thf(fact_2403_minus__div__mult__eq__mod,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 0.30/0.54        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_div_mult_eq_mod
% 0.30/0.54  thf(fact_2404_minus__div__mult__eq__mod,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 0.30/0.54        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_div_mult_eq_mod
% 0.30/0.54  thf(fact_2405_minus__div__mult__eq__mod,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 0.30/0.54        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_div_mult_eq_mod
% 0.30/0.54  thf(fact_2406_minus__mod__eq__div__mult,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.54        = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mod_eq_div_mult
% 0.30/0.54  thf(fact_2407_minus__mod__eq__div__mult,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.54        = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mod_eq_div_mult
% 0.30/0.54  thf(fact_2408_minus__mod__eq__div__mult,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.54        = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mod_eq_div_mult
% 0.30/0.54  thf(fact_2409_minus__mod__eq__mult__div,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.54        = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mod_eq_mult_div
% 0.30/0.54  thf(fact_2410_minus__mod__eq__mult__div,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.54        = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mod_eq_mult_div
% 0.30/0.54  thf(fact_2411_minus__mod__eq__mult__div,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.54        = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mod_eq_mult_div
% 0.30/0.54  thf(fact_2412_minus__mult__div__eq__mod,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 0.30/0.54        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mult_div_eq_mod
% 0.30/0.54  thf(fact_2413_minus__mult__div__eq__mod,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 0.30/0.54        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mult_div_eq_mod
% 0.30/0.54  thf(fact_2414_minus__mult__div__eq__mod,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 0.30/0.54        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % minus_mult_div_eq_mod
% 0.30/0.54  thf(fact_2415_mod__eq__nat1E,axiom,
% 0.30/0.54      ! [M: nat,Q2: nat,N: nat] :
% 0.30/0.54        ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.30/0.54          = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.54         => ~ ! [S3: nat] :
% 0.30/0.54                ( M
% 0.30/0.54               != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_eq_nat1E
% 0.30/0.54  thf(fact_2416_mod__eq__nat2E,axiom,
% 0.30/0.54      ! [M: nat,Q2: nat,N: nat] :
% 0.30/0.54        ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.30/0.54          = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.54         => ~ ! [S3: nat] :
% 0.30/0.54                ( N
% 0.30/0.54               != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_eq_nat2E
% 0.30/0.54  thf(fact_2417_nat__mod__eq__lemma,axiom,
% 0.30/0.54      ! [X3: nat,N: nat,Y3: nat] :
% 0.30/0.54        ( ( ( modulo_modulo_nat @ X3 @ N )
% 0.30/0.54          = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.54         => ? [Q3: nat] :
% 0.30/0.54              ( X3
% 0.30/0.54              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_mod_eq_lemma
% 0.30/0.54  thf(fact_2418_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.30/0.54      ! [X3: vEBT_VEBT,Y3: $o] :
% 0.30/0.54        ( ( ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.54          = Y3 )
% 0.30/0.54       => ( ( ( X3
% 0.30/0.54              = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.54           => ~ Y3 )
% 0.30/0.54         => ( ( ? [Uv2: $o] :
% 0.30/0.54                  ( X3
% 0.30/0.54                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.54             => Y3 )
% 0.30/0.54           => ( ( ? [Uu2: $o] :
% 0.30/0.54                    ( X3
% 0.30/0.54                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.54               => Y3 )
% 0.30/0.54             => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.54                      ( X3
% 0.30/0.54                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.54                 => ~ Y3 )
% 0.30/0.54               => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.54                        ( X3
% 0.30/0.54                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.30/0.54                   => Y3 ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % VEBT_internal.minNull.elims(1)
% 0.30/0.54  thf(fact_2419_div__mod__decomp,axiom,
% 0.30/0.54      ! [A4: nat,N: nat] :
% 0.30/0.54        ( A4
% 0.30/0.54        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_mod_decomp
% 0.30/0.54  thf(fact_2420_mod__mult2__eq,axiom,
% 0.30/0.54      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.54        ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.30/0.54        = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_mult2_eq
% 0.30/0.54  thf(fact_2421_modulo__nat__def,axiom,
% 0.30/0.54      ( modulo_modulo_nat
% 0.30/0.54      = ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % modulo_nat_def
% 0.30/0.54  thf(fact_2422_mult__less__le__imp__less,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.54        ( ( ord_less_real @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54           => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_le_imp_less
% 0.30/0.54  thf(fact_2423_mult__less__le__imp__less,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.54        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_le_imp_less
% 0.30/0.54  thf(fact_2424_mult__less__le__imp__less,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.54        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_le_imp_less
% 0.30/0.54  thf(fact_2425_mult__less__le__imp__less,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.54        ( ( ord_less_int @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54           => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_le_imp_less
% 0.30/0.54  thf(fact_2426_mult__le__less__imp__less,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_real @ C @ D )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_less_imp_less
% 0.30/0.54  thf(fact_2427_mult__le__less__imp__less,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_rat @ C @ D )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_less_imp_less
% 0.30/0.54  thf(fact_2428_mult__le__less__imp__less,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_nat @ C @ D )
% 0.30/0.54         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_less_imp_less
% 0.30/0.54  thf(fact_2429_mult__le__less__imp__less,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_int @ C @ D )
% 0.30/0.54         => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_less_imp_less
% 0.30/0.54  thf(fact_2430_mult__right__le__imp__le,axiom,
% 0.30/0.54      ! [A: real,C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54         => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_imp_le
% 0.30/0.54  thf(fact_2431_mult__right__le__imp__le,axiom,
% 0.30/0.54      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54         => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_imp_le
% 0.30/0.54  thf(fact_2432_mult__right__le__imp__le,axiom,
% 0.30/0.54      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54         => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_imp_le
% 0.30/0.54  thf(fact_2433_mult__right__le__imp__le,axiom,
% 0.30/0.54      ! [A: int,C: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54         => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_imp_le
% 0.30/0.54  thf(fact_2434_mult__left__le__imp__le,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54         => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_imp_le
% 0.30/0.54  thf(fact_2435_mult__left__le__imp__le,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54         => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_imp_le
% 0.30/0.54  thf(fact_2436_mult__left__le__imp__le,axiom,
% 0.30/0.54      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54         => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_imp_le
% 0.30/0.54  thf(fact_2437_mult__left__le__imp__le,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54         => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_imp_le
% 0.30/0.54  thf(fact_2438_mult__le__cancel__left__pos,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left_pos
% 0.30/0.54  thf(fact_2439_mult__le__cancel__left__pos,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54          = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left_pos
% 0.30/0.54  thf(fact_2440_mult__le__cancel__left__pos,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54          = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left_pos
% 0.30/0.54  thf(fact_2441_mult__le__cancel__left__neg,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54          = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left_neg
% 0.30/0.54  thf(fact_2442_mult__le__cancel__left__neg,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54          = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left_neg
% 0.30/0.54  thf(fact_2443_mult__le__cancel__left__neg,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54          = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left_neg
% 0.30/0.54  thf(fact_2444_mult__less__cancel__right,axiom,
% 0.30/0.54      ! [A: real,C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right
% 0.30/0.54  thf(fact_2445_mult__less__cancel__right,axiom,
% 0.30/0.54      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right
% 0.30/0.54  thf(fact_2446_mult__less__cancel__right,axiom,
% 0.30/0.54      ! [A: int,C: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_int @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right
% 0.30/0.54  thf(fact_2447_mult__strict__mono_H,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.54        ( ( ord_less_real @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_real @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono'
% 0.30/0.54  thf(fact_2448_mult__strict__mono_H,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.54        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_rat @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono'
% 0.30/0.54  thf(fact_2449_mult__strict__mono_H,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.54        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_nat @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono'
% 0.30/0.54  thf(fact_2450_mult__strict__mono_H,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.54        ( ( ord_less_int @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_int @ C @ D )
% 0.30/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono'
% 0.30/0.54  thf(fact_2451_mult__right__less__imp__less,axiom,
% 0.30/0.54      ! [A: real,C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54         => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_less_imp_less
% 0.30/0.54  thf(fact_2452_mult__right__less__imp__less,axiom,
% 0.30/0.54      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_less_imp_less
% 0.30/0.54  thf(fact_2453_mult__right__less__imp__less,axiom,
% 0.30/0.54      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_less_imp_less
% 0.30/0.54  thf(fact_2454_mult__right__less__imp__less,axiom,
% 0.30/0.54      ! [A: int,C: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54         => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_less_imp_less
% 0.30/0.54  thf(fact_2455_mult__less__cancel__left,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left
% 0.30/0.54  thf(fact_2456_mult__less__cancel__left,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left
% 0.30/0.54  thf(fact_2457_mult__less__cancel__left,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_int @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left
% 0.30/0.54  thf(fact_2458_mult__strict__mono,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.54        ( ( ord_less_real @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_real @ C @ D )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono
% 0.30/0.54  thf(fact_2459_mult__strict__mono,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.54        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_rat @ C @ D )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono
% 0.30/0.54  thf(fact_2460_mult__strict__mono,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.54        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_nat @ C @ D )
% 0.30/0.54         => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono
% 0.30/0.54  thf(fact_2461_mult__strict__mono,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.54        ( ( ord_less_int @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_int @ C @ D )
% 0.30/0.54         => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_strict_mono
% 0.30/0.54  thf(fact_2462_mult__left__less__imp__less,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54         => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_less_imp_less
% 0.30/0.54  thf(fact_2463_mult__left__less__imp__less,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_less_imp_less
% 0.30/0.54  thf(fact_2464_mult__left__less__imp__less,axiom,
% 0.30/0.54      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_less_imp_less
% 0.30/0.54  thf(fact_2465_mult__left__less__imp__less,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54         => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_less_imp_less
% 0.30/0.54  thf(fact_2466_mult__le__cancel__right,axiom,
% 0.30/0.54      ! [A: real,C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_eq_real @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right
% 0.30/0.54  thf(fact_2467_mult__le__cancel__right,axiom,
% 0.30/0.54      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_eq_rat @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right
% 0.30/0.54  thf(fact_2468_mult__le__cancel__right,axiom,
% 0.30/0.54      ! [A: int,C: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_eq_int @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right
% 0.30/0.54  thf(fact_2469_mult__le__cancel__left,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_eq_real @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left
% 0.30/0.54  thf(fact_2470_mult__le__cancel__left,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_eq_rat @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left
% 0.30/0.54  thf(fact_2471_mult__le__cancel__left,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_eq_int @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_left
% 0.30/0.54  thf(fact_2472_field__le__epsilon,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ! [E2: real] :
% 0.30/0.54            ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.54           => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 0.30/0.54       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % field_le_epsilon
% 0.30/0.54  thf(fact_2473_field__le__epsilon,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ! [E2: rat] :
% 0.30/0.54            ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.54           => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 0.30/0.54       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % field_le_epsilon
% 0.30/0.54  thf(fact_2474_add__strict__increasing2,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.54         => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing2
% 0.30/0.54  thf(fact_2475_add__strict__increasing2,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.54         => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing2
% 0.30/0.54  thf(fact_2476_add__strict__increasing2,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.54         => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing2
% 0.30/0.54  thf(fact_2477_add__strict__increasing2,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.54         => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing2
% 0.30/0.54  thf(fact_2478_add__strict__increasing,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54       => ( ( ord_less_eq_real @ B2 @ C )
% 0.30/0.54         => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing
% 0.30/0.54  thf(fact_2479_add__strict__increasing,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.54         => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing
% 0.30/0.54  thf(fact_2480_add__strict__increasing,axiom,
% 0.30/0.54      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.54         => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing
% 0.30/0.54  thf(fact_2481_add__strict__increasing,axiom,
% 0.30/0.54      ! [A: int,B2: int,C: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.54         => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_strict_increasing
% 0.30/0.54  thf(fact_2482_add__pos__nonneg,axiom,
% 0.30/0.54      ! [A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.54         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_pos_nonneg
% 0.30/0.54  thf(fact_2483_add__pos__nonneg,axiom,
% 0.30/0.54      ! [A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.54         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_pos_nonneg
% 0.30/0.54  thf(fact_2484_add__pos__nonneg,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.54         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_pos_nonneg
% 0.30/0.54  thf(fact_2485_add__pos__nonneg,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.54         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_pos_nonneg
% 0.30/0.54  thf(fact_2486_add__nonpos__neg,axiom,
% 0.30/0.54      ! [A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.30/0.54         => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonpos_neg
% 0.30/0.54  thf(fact_2487_add__nonpos__neg,axiom,
% 0.30/0.54      ! [A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.30/0.54         => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonpos_neg
% 0.30/0.54  thf(fact_2488_add__nonpos__neg,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 0.30/0.54         => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonpos_neg
% 0.30/0.54  thf(fact_2489_add__nonpos__neg,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.54         => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonpos_neg
% 0.30/0.54  thf(fact_2490_add__nonneg__pos,axiom,
% 0.30/0.54      ! [A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.54         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonneg_pos
% 0.30/0.54  thf(fact_2491_add__nonneg__pos,axiom,
% 0.30/0.54      ! [A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.54         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonneg_pos
% 0.30/0.54  thf(fact_2492_add__nonneg__pos,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.54         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonneg_pos
% 0.30/0.54  thf(fact_2493_add__nonneg__pos,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.54         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_nonneg_pos
% 0.30/0.54  thf(fact_2494_add__neg__nonpos,axiom,
% 0.30/0.54      ! [A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 0.30/0.54         => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_neg_nonpos
% 0.30/0.54  thf(fact_2495_add__neg__nonpos,axiom,
% 0.30/0.54      ! [A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 0.30/0.54         => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_neg_nonpos
% 0.30/0.54  thf(fact_2496_add__neg__nonpos,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.54       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 0.30/0.54         => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_neg_nonpos
% 0.30/0.54  thf(fact_2497_add__neg__nonpos,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.54       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.30/0.54         => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_neg_nonpos
% 0.30/0.54  thf(fact_2498_divide__nonpos__pos,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonpos_pos
% 0.30/0.54  thf(fact_2499_divide__nonpos__pos,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonpos_pos
% 0.30/0.54  thf(fact_2500_divide__nonpos__neg,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.54         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonpos_neg
% 0.30/0.54  thf(fact_2501_divide__nonpos__neg,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.54         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonpos_neg
% 0.30/0.54  thf(fact_2502_divide__nonneg__pos,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.54       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonneg_pos
% 0.30/0.54  thf(fact_2503_divide__nonneg__pos,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.54       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonneg_pos
% 0.30/0.54  thf(fact_2504_divide__nonneg__neg,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.54       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.30/0.54         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonneg_neg
% 0.30/0.54  thf(fact_2505_divide__nonneg__neg,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.54       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.30/0.54         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_nonneg_neg
% 0.30/0.54  thf(fact_2506_divide__le__cancel,axiom,
% 0.30/0.54      ! [A: real,C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_eq_real @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_le_cancel
% 0.30/0.54  thf(fact_2507_divide__le__cancel,axiom,
% 0.30/0.54      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_eq_rat @ A @ B2 ) )
% 0.30/0.54          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_le_cancel
% 0.30/0.54  thf(fact_2508_frac__less2,axiom,
% 0.30/0.54      ! [X3: real,Y3: real,W: real,Z: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.54           => ( ( ord_less_real @ W @ Z )
% 0.30/0.54             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % frac_less2
% 0.30/0.54  thf(fact_2509_frac__less2,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat,W: rat,Z: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.54           => ( ( ord_less_rat @ W @ Z )
% 0.30/0.54             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % frac_less2
% 0.30/0.54  thf(fact_2510_frac__less,axiom,
% 0.30/0.54      ! [X3: real,Y3: real,W: real,Z: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.54       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.54           => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.54             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % frac_less
% 0.30/0.54  thf(fact_2511_frac__less,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat,W: rat,Z: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.54       => ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.54           => ( ( ord_less_eq_rat @ W @ Z )
% 0.30/0.54             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % frac_less
% 0.30/0.54  thf(fact_2512_frac__le,axiom,
% 0.30/0.54      ! [Y3: real,X3: real,W: real,Z: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.54       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.54           => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.54             => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % frac_le
% 0.30/0.54  thf(fact_2513_frac__le,axiom,
% 0.30/0.54      ! [Y3: rat,X3: rat,W: rat,Z: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.54           => ( ( ord_less_eq_rat @ W @ Z )
% 0.30/0.54             => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % frac_le
% 0.30/0.54  thf(fact_2514_div__positive,axiom,
% 0.30/0.54      ! [B2: nat,A: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.54       => ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.54         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_positive
% 0.30/0.54  thf(fact_2515_div__positive,axiom,
% 0.30/0.54      ! [B2: int,A: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.54       => ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.54         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_positive
% 0.30/0.54  thf(fact_2516_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_nat @ A @ B2 )
% 0.30/0.54         => ( ( divide_divide_nat @ A @ B2 )
% 0.30/0.54            = zero_zero_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.54  thf(fact_2517_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_int @ A @ B2 )
% 0.30/0.54         => ( ( divide_divide_int @ A @ B2 )
% 0.30/0.54            = zero_zero_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.54  thf(fact_2518_mult__left__le__one__le,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.54         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.54           => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_one_le
% 0.30/0.54  thf(fact_2519_mult__left__le__one__le,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_one_le
% 0.30/0.54  thf(fact_2520_mult__left__le__one__le,axiom,
% 0.30/0.54      ! [X3: int,Y3: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.54         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.30/0.54           => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le_one_le
% 0.30/0.54  thf(fact_2521_mult__right__le__one__le,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.54         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.54           => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_one_le
% 0.30/0.54  thf(fact_2522_mult__right__le__one__le,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_one_le
% 0.30/0.54  thf(fact_2523_mult__right__le__one__le,axiom,
% 0.30/0.54      ! [X3: int,Y3: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.54         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.30/0.54           => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_right_le_one_le
% 0.30/0.54  thf(fact_2524_mult__le__one,axiom,
% 0.30/0.54      ! [A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.54         => ( ( ord_less_eq_real @ B2 @ one_one_real )
% 0.30/0.54           => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_one
% 0.30/0.54  thf(fact_2525_mult__le__one,axiom,
% 0.30/0.54      ! [A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.54         => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ one_one_rat ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_one
% 0.30/0.54  thf(fact_2526_mult__le__one,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.54         => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
% 0.30/0.54           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_one
% 0.30/0.54  thf(fact_2527_mult__le__one,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.54         => ( ( ord_less_eq_int @ B2 @ one_one_int )
% 0.30/0.54           => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_one
% 0.30/0.54  thf(fact_2528_mult__left__le,axiom,
% 0.30/0.54      ! [C: real,A: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ C @ one_one_real )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le
% 0.30/0.54  thf(fact_2529_mult__left__le,axiom,
% 0.30/0.54      ! [C: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le
% 0.30/0.54  thf(fact_2530_mult__left__le,axiom,
% 0.30/0.54      ! [C: nat,A: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le
% 0.30/0.54  thf(fact_2531_mult__left__le,axiom,
% 0.30/0.54      ! [C: int,A: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ C @ one_one_int )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_left_le
% 0.30/0.54  thf(fact_2532_sum__squares__le__zero__iff,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 0.30/0.54        = ( ( X3 = zero_zero_real )
% 0.30/0.54          & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_le_zero_iff
% 0.30/0.54  thf(fact_2533_sum__squares__le__zero__iff,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 0.30/0.54        = ( ( X3 = zero_zero_rat )
% 0.30/0.54          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_le_zero_iff
% 0.30/0.54  thf(fact_2534_sum__squares__le__zero__iff,axiom,
% 0.30/0.54      ! [X3: int,Y3: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 0.30/0.54        = ( ( X3 = zero_zero_int )
% 0.30/0.54          & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_le_zero_iff
% 0.30/0.54  thf(fact_2535_sum__squares__ge__zero,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_ge_zero
% 0.30/0.54  thf(fact_2536_sum__squares__ge__zero,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_ge_zero
% 0.30/0.54  thf(fact_2537_sum__squares__ge__zero,axiom,
% 0.30/0.54      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_ge_zero
% 0.30/0.54  thf(fact_2538_power__less__imp__less__base,axiom,
% 0.30/0.54      ! [A: real,N: nat,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.54         => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_less_imp_less_base
% 0.30/0.54  thf(fact_2539_power__less__imp__less__base,axiom,
% 0.30/0.54      ! [A: rat,N: nat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.54         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_less_imp_less_base
% 0.30/0.54  thf(fact_2540_power__less__imp__less__base,axiom,
% 0.30/0.54      ! [A: nat,N: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.54         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_less_imp_less_base
% 0.30/0.54  thf(fact_2541_power__less__imp__less__base,axiom,
% 0.30/0.54      ! [A: int,N: nat,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.54         => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_less_imp_less_base
% 0.30/0.54  thf(fact_2542_sum__squares__gt__zero__iff,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 0.30/0.54        = ( ( X3 != zero_zero_real )
% 0.30/0.54          | ( Y3 != zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_gt_zero_iff
% 0.30/0.54  thf(fact_2543_sum__squares__gt__zero__iff,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 0.30/0.54        = ( ( X3 != zero_zero_rat )
% 0.30/0.54          | ( Y3 != zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_gt_zero_iff
% 0.30/0.54  thf(fact_2544_sum__squares__gt__zero__iff,axiom,
% 0.30/0.54      ! [X3: int,Y3: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 0.30/0.54        = ( ( X3 != zero_zero_int )
% 0.30/0.54          | ( Y3 != zero_zero_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % sum_squares_gt_zero_iff
% 0.30/0.54  thf(fact_2545_not__sum__squares__lt__zero,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 0.30/0.54  
% 0.30/0.54  % not_sum_squares_lt_zero
% 0.30/0.54  thf(fact_2546_not__sum__squares__lt__zero,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 0.30/0.54  
% 0.30/0.54  % not_sum_squares_lt_zero
% 0.30/0.54  thf(fact_2547_not__sum__squares__lt__zero,axiom,
% 0.30/0.54      ! [X3: int,Y3: int] :
% 0.30/0.54        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 0.30/0.54  
% 0.30/0.54  % not_sum_squares_lt_zero
% 0.30/0.54  thf(fact_2548_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.54      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.54          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.54  thf(fact_2549_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.54      ! [C: int,A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.54          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.54  thf(fact_2550_zero__less__two,axiom,
% 0.30/0.54      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.30/0.54  
% 0.30/0.54  % zero_less_two
% 0.30/0.54  thf(fact_2551_zero__less__two,axiom,
% 0.30/0.54      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.30/0.54  
% 0.30/0.54  % zero_less_two
% 0.30/0.54  thf(fact_2552_zero__less__two,axiom,
% 0.30/0.54      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/0.54  
% 0.30/0.54  % zero_less_two
% 0.30/0.54  thf(fact_2553_zero__less__two,axiom,
% 0.30/0.54      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.30/0.54  
% 0.30/0.54  % zero_less_two
% 0.30/0.54  thf(fact_2554_divide__strict__left__mono__neg,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real] :
% 0.30/0.54        ( ( ord_less_real @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.54           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_strict_left_mono_neg
% 0.30/0.54  thf(fact_2555_divide__strict__left__mono__neg,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.54       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.54           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_strict_left_mono_neg
% 0.30/0.54  thf(fact_2556_divide__strict__left__mono,axiom,
% 0.30/0.54      ! [B2: real,A: real,C: real] :
% 0.30/0.54        ( ( ord_less_real @ B2 @ A )
% 0.30/0.54       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.54           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_strict_left_mono
% 0.30/0.54  thf(fact_2557_divide__strict__left__mono,axiom,
% 0.30/0.54      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.54       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.54           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_strict_left_mono
% 0.30/0.54  thf(fact_2558_mult__imp__less__div__pos,axiom,
% 0.30/0.54      ! [Y3: real,Z: real,X3: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54       => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X3 )
% 0.30/0.54         => ( ord_less_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_imp_less_div_pos
% 0.30/0.54  thf(fact_2559_mult__imp__less__div__pos,axiom,
% 0.30/0.54      ! [Y3: rat,Z: rat,X3: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54       => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X3 )
% 0.30/0.54         => ( ord_less_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_imp_less_div_pos
% 0.30/0.54  thf(fact_2560_mult__imp__div__pos__less,axiom,
% 0.30/0.54      ! [Y3: real,X3: real,Z: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.54       => ( ( ord_less_real @ X3 @ ( times_times_real @ Z @ Y3 ) )
% 0.30/0.54         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_imp_div_pos_less
% 0.30/0.54  thf(fact_2561_mult__imp__div__pos__less,axiom,
% 0.30/0.54      ! [Y3: rat,X3: rat,Z: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.54       => ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) )
% 0.30/0.54         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_imp_div_pos_less
% 0.30/0.54  thf(fact_2562_pos__less__divide__eq,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54       => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.54          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % pos_less_divide_eq
% 0.30/0.54  thf(fact_2563_pos__less__divide__eq,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.54          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % pos_less_divide_eq
% 0.30/0.54  thf(fact_2564_pos__divide__less__eq,axiom,
% 0.30/0.54      ! [C: real,B2: real,A: real] :
% 0.30/0.54        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.54          = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % pos_divide_less_eq
% 0.30/0.54  thf(fact_2565_pos__divide__less__eq,axiom,
% 0.30/0.54      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.54          = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % pos_divide_less_eq
% 0.30/0.54  thf(fact_2566_neg__less__divide__eq,axiom,
% 0.30/0.54      ! [C: real,A: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.54          = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % neg_less_divide_eq
% 0.30/0.54  thf(fact_2567_neg__less__divide__eq,axiom,
% 0.30/0.54      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.54          = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % neg_less_divide_eq
% 0.30/0.54  thf(fact_2568_neg__divide__less__eq,axiom,
% 0.30/0.54      ! [C: real,B2: real,A: real] :
% 0.30/0.54        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.54          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % neg_divide_less_eq
% 0.30/0.54  thf(fact_2569_neg__divide__less__eq,axiom,
% 0.30/0.54      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.54          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % neg_divide_less_eq
% 0.30/0.54  thf(fact_2570_less__divide__eq,axiom,
% 0.30/0.54      ! [A: real,B2: real,C: real] :
% 0.30/0.54        ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.30/0.54          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54               => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.30/0.54              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % less_divide_eq
% 0.30/0.54  thf(fact_2571_less__divide__eq,axiom,
% 0.30/0.54      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.30/0.54          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54               => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.54              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % less_divide_eq
% 0.30/0.54  thf(fact_2572_divide__less__eq,axiom,
% 0.30/0.54      ! [B2: real,C: real,A: real] :
% 0.30/0.54        ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.30/0.54          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.30/0.54              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_less_eq
% 0.30/0.54  thf(fact_2573_divide__less__eq,axiom,
% 0.30/0.54      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.54          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.30/0.54              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_less_eq
% 0.30/0.54  thf(fact_2574_less__divide__eq__1,axiom,
% 0.30/0.54      ! [B2: real,A: real] :
% 0.30/0.54        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54            & ( ord_less_real @ A @ B2 ) )
% 0.30/0.54          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54            & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % less_divide_eq_1
% 0.30/0.54  thf(fact_2575_less__divide__eq__1,axiom,
% 0.30/0.54      ! [B2: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54            & ( ord_less_rat @ A @ B2 ) )
% 0.30/0.54          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54            & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % less_divide_eq_1
% 0.30/0.54  thf(fact_2576_divide__less__eq__1,axiom,
% 0.30/0.54      ! [B2: real,A: real] :
% 0.30/0.54        ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54            & ( ord_less_real @ B2 @ A ) )
% 0.30/0.54          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54            & ( ord_less_real @ A @ B2 ) )
% 0.30/0.54          | ( A = zero_zero_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_less_eq_1
% 0.30/0.54  thf(fact_2577_divide__less__eq__1,axiom,
% 0.30/0.54      ! [B2: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54            & ( ord_less_rat @ B2 @ A ) )
% 0.30/0.54          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54            & ( ord_less_rat @ A @ B2 ) )
% 0.30/0.54          | ( A = zero_zero_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_less_eq_1
% 0.30/0.54  thf(fact_2578_power__le__one,axiom,
% 0.30/0.54      ! [A: real,N: nat] :
% 0.30/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.54         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_one
% 0.30/0.54  thf(fact_2579_power__le__one,axiom,
% 0.30/0.54      ! [A: rat,N: nat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.54         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_one
% 0.30/0.54  thf(fact_2580_power__le__one,axiom,
% 0.30/0.54      ! [A: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.54         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_one
% 0.30/0.54  thf(fact_2581_power__le__one,axiom,
% 0.30/0.54      ! [A: int,N: nat] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.54         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_one
% 0.30/0.54  thf(fact_2582_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.54      ! [B2: complex,C: complex,W: num] :
% 0.30/0.54        ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.30/0.54          = ( numera6690914467698888265omplex @ W ) )
% 0.30/0.54        = ( ( ( C != zero_zero_complex )
% 0.30/0.54           => ( B2
% 0.30/0.54              = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.30/0.54          & ( ( C = zero_zero_complex )
% 0.30/0.54           => ( ( numera6690914467698888265omplex @ W )
% 0.30/0.54              = zero_zero_complex ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_eq_eq_numeral(1)
% 0.30/0.54  thf(fact_2583_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.54      ! [B2: real,C: real,W: num] :
% 0.30/0.54        ( ( ( divide_divide_real @ B2 @ C )
% 0.30/0.54          = ( numeral_numeral_real @ W ) )
% 0.30/0.54        = ( ( ( C != zero_zero_real )
% 0.30/0.54           => ( B2
% 0.30/0.54              = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.54          & ( ( C = zero_zero_real )
% 0.30/0.54           => ( ( numeral_numeral_real @ W )
% 0.30/0.54              = zero_zero_real ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_eq_eq_numeral(1)
% 0.30/0.54  thf(fact_2584_divide__eq__eq__numeral_I1_J,axiom,
% 0.30/0.54      ! [B2: rat,C: rat,W: num] :
% 0.30/0.54        ( ( ( divide_divide_rat @ B2 @ C )
% 0.30/0.54          = ( numeral_numeral_rat @ W ) )
% 0.30/0.54        = ( ( ( C != zero_zero_rat )
% 0.30/0.54           => ( B2
% 0.30/0.54              = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.54          & ( ( C = zero_zero_rat )
% 0.30/0.54           => ( ( numeral_numeral_rat @ W )
% 0.30/0.54              = zero_zero_rat ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_eq_eq_numeral(1)
% 0.30/0.54  thf(fact_2585_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54      ! [W: num,B2: complex,C: complex] :
% 0.30/0.54        ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.54          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.54        = ( ( ( C != zero_zero_complex )
% 0.30/0.54           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.30/0.54              = B2 ) )
% 0.30/0.54          & ( ( C = zero_zero_complex )
% 0.30/0.54           => ( ( numera6690914467698888265omplex @ W )
% 0.30/0.54              = zero_zero_complex ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % eq_divide_eq_numeral(1)
% 0.30/0.54  thf(fact_2586_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54      ! [W: num,B2: real,C: real] :
% 0.30/0.54        ( ( ( numeral_numeral_real @ W )
% 0.30/0.54          = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( C != zero_zero_real )
% 0.30/0.54           => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.30/0.54              = B2 ) )
% 0.30/0.54          & ( ( C = zero_zero_real )
% 0.30/0.54           => ( ( numeral_numeral_real @ W )
% 0.30/0.54              = zero_zero_real ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % eq_divide_eq_numeral(1)
% 0.30/0.54  thf(fact_2587_eq__divide__eq__numeral_I1_J,axiom,
% 0.30/0.54      ! [W: num,B2: rat,C: rat] :
% 0.30/0.54        ( ( ( numeral_numeral_rat @ W )
% 0.30/0.54          = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( C != zero_zero_rat )
% 0.30/0.54           => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.30/0.54              = B2 ) )
% 0.30/0.54          & ( ( C = zero_zero_rat )
% 0.30/0.54           => ( ( numeral_numeral_rat @ W )
% 0.30/0.54              = zero_zero_rat ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % eq_divide_eq_numeral(1)
% 0.30/0.54  thf(fact_2588_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.54      ! [Z: complex,A: complex,B2: complex] :
% 0.30/0.54        ( ( ( Z = zero_zero_complex )
% 0.30/0.54         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.30/0.54            = B2 ) )
% 0.30/0.54        & ( ( Z != zero_zero_complex )
% 0.30/0.54         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.30/0.54            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(2)
% 0.30/0.54  thf(fact_2589_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.54      ! [Z: real,A: real,B2: real] :
% 0.30/0.54        ( ( ( Z = zero_zero_real )
% 0.30/0.54         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.30/0.54            = B2 ) )
% 0.30/0.54        & ( ( Z != zero_zero_real )
% 0.30/0.54         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.30/0.54            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(2)
% 0.30/0.54  thf(fact_2590_add__divide__eq__if__simps_I2_J,axiom,
% 0.30/0.54      ! [Z: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ( Z = zero_zero_rat )
% 0.30/0.54         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.30/0.54            = B2 ) )
% 0.30/0.54        & ( ( Z != zero_zero_rat )
% 0.30/0.54         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.30/0.54            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(2)
% 0.30/0.54  thf(fact_2591_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.54      ! [Z: complex,A: complex,B2: complex] :
% 0.30/0.54        ( ( ( Z = zero_zero_complex )
% 0.30/0.54         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.30/0.54            = A ) )
% 0.30/0.54        & ( ( Z != zero_zero_complex )
% 0.30/0.54         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.30/0.54            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(1)
% 0.30/0.54  thf(fact_2592_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.54      ! [Z: real,A: real,B2: real] :
% 0.30/0.54        ( ( ( Z = zero_zero_real )
% 0.30/0.54         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.30/0.54            = A ) )
% 0.30/0.54        & ( ( Z != zero_zero_real )
% 0.30/0.54         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.30/0.54            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(1)
% 0.30/0.54  thf(fact_2593_add__divide__eq__if__simps_I1_J,axiom,
% 0.30/0.54      ! [Z: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ( Z = zero_zero_rat )
% 0.30/0.54         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.30/0.54            = A ) )
% 0.30/0.54        & ( ( Z != zero_zero_rat )
% 0.30/0.54         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.30/0.54            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(1)
% 0.30/0.54  thf(fact_2594_add__frac__eq,axiom,
% 0.30/0.54      ! [Y3: complex,Z: complex,X3: complex,W: complex] :
% 0.30/0.54        ( ( Y3 != zero_zero_complex )
% 0.30/0.54       => ( ( Z != zero_zero_complex )
% 0.30/0.54         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.54            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_frac_eq
% 0.30/0.54  thf(fact_2595_add__frac__eq,axiom,
% 0.30/0.54      ! [Y3: real,Z: real,X3: real,W: real] :
% 0.30/0.54        ( ( Y3 != zero_zero_real )
% 0.30/0.54       => ( ( Z != zero_zero_real )
% 0.30/0.54         => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.54            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_frac_eq
% 0.30/0.54  thf(fact_2596_add__frac__eq,axiom,
% 0.30/0.54      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 0.30/0.54        ( ( Y3 != zero_zero_rat )
% 0.30/0.54       => ( ( Z != zero_zero_rat )
% 0.30/0.54         => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.54            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_frac_eq
% 0.30/0.54  thf(fact_2597_add__frac__num,axiom,
% 0.30/0.54      ! [Y3: complex,X3: complex,Z: complex] :
% 0.30/0.54        ( ( Y3 != zero_zero_complex )
% 0.30/0.54       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ Z )
% 0.30/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_frac_num
% 0.30/0.54  thf(fact_2598_add__frac__num,axiom,
% 0.30/0.54      ! [Y3: real,X3: real,Z: real] :
% 0.30/0.54        ( ( Y3 != zero_zero_real )
% 0.30/0.54       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z )
% 0.30/0.54          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_frac_num
% 0.30/0.54  thf(fact_2599_add__frac__num,axiom,
% 0.30/0.54      ! [Y3: rat,X3: rat,Z: rat] :
% 0.30/0.54        ( ( Y3 != zero_zero_rat )
% 0.30/0.54       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z )
% 0.30/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_frac_num
% 0.30/0.54  thf(fact_2600_add__num__frac,axiom,
% 0.30/0.54      ! [Y3: complex,Z: complex,X3: complex] :
% 0.30/0.54        ( ( Y3 != zero_zero_complex )
% 0.30/0.54       => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 0.30/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_num_frac
% 0.30/0.54  thf(fact_2601_add__num__frac,axiom,
% 0.30/0.54      ! [Y3: real,Z: real,X3: real] :
% 0.30/0.54        ( ( Y3 != zero_zero_real )
% 0.30/0.54       => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.54          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_num_frac
% 0.30/0.54  thf(fact_2602_add__num__frac,axiom,
% 0.30/0.54      ! [Y3: rat,Z: rat,X3: rat] :
% 0.30/0.54        ( ( Y3 != zero_zero_rat )
% 0.30/0.54       => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) )
% 0.30/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_num_frac
% 0.30/0.54  thf(fact_2603_add__divide__eq__iff,axiom,
% 0.30/0.54      ! [Z: complex,X3: complex,Y3: complex] :
% 0.30/0.54        ( ( Z != zero_zero_complex )
% 0.30/0.54       => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.30/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_iff
% 0.30/0.54  thf(fact_2604_add__divide__eq__iff,axiom,
% 0.30/0.54      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.54        ( ( Z != zero_zero_real )
% 0.30/0.54       => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.30/0.54          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_iff
% 0.30/0.54  thf(fact_2605_add__divide__eq__iff,axiom,
% 0.30/0.54      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.54        ( ( Z != zero_zero_rat )
% 0.30/0.54       => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.30/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_iff
% 0.30/0.54  thf(fact_2606_divide__add__eq__iff,axiom,
% 0.30/0.54      ! [Z: complex,X3: complex,Y3: complex] :
% 0.30/0.54        ( ( Z != zero_zero_complex )
% 0.30/0.54       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y3 )
% 0.30/0.54          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_add_eq_iff
% 0.30/0.54  thf(fact_2607_divide__add__eq__iff,axiom,
% 0.30/0.54      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.54        ( ( Z != zero_zero_real )
% 0.30/0.54       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z ) @ Y3 )
% 0.30/0.54          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_add_eq_iff
% 0.30/0.54  thf(fact_2608_divide__add__eq__iff,axiom,
% 0.30/0.54      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.54        ( ( Z != zero_zero_rat )
% 0.30/0.54       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z ) @ Y3 )
% 0.30/0.54          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_add_eq_iff
% 0.30/0.54  thf(fact_2609_power__le__imp__le__base,axiom,
% 0.30/0.54      ! [A: real,N: nat,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.54         => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_imp_le_base
% 0.30/0.54  thf(fact_2610_power__le__imp__le__base,axiom,
% 0.30/0.54      ! [A: rat,N: nat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.54         => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_imp_le_base
% 0.30/0.54  thf(fact_2611_power__le__imp__le__base,axiom,
% 0.30/0.54      ! [A: nat,N: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.54         => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_imp_le_base
% 0.30/0.54  thf(fact_2612_power__le__imp__le__base,axiom,
% 0.30/0.54      ! [A: int,N: nat,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.54         => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_le_imp_le_base
% 0.30/0.54  thf(fact_2613_power__inject__base,axiom,
% 0.30/0.54      ! [A: real,N: nat,B2: real] :
% 0.30/0.54        ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.30/0.54          = ( power_power_real @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.54           => ( A = B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_inject_base
% 0.30/0.54  thf(fact_2614_power__inject__base,axiom,
% 0.30/0.54      ! [A: rat,N: nat,B2: rat] :
% 0.30/0.54        ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.30/0.54          = ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.54           => ( A = B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_inject_base
% 0.30/0.54  thf(fact_2615_power__inject__base,axiom,
% 0.30/0.54      ! [A: nat,N: nat,B2: nat] :
% 0.30/0.54        ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.30/0.54          = ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 0.30/0.54           => ( A = B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_inject_base
% 0.30/0.54  thf(fact_2616_power__inject__base,axiom,
% 0.30/0.54      ! [A: int,N: nat,B2: int] :
% 0.30/0.54        ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.30/0.54          = ( power_power_int @ B2 @ ( suc @ N ) ) )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.54           => ( A = B2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_inject_base
% 0.30/0.54  thf(fact_2617_div__add__self1,axiom,
% 0.30/0.54      ! [B2: nat,A: nat] :
% 0.30/0.54        ( ( B2 != zero_zero_nat )
% 0.30/0.54       => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 0.30/0.54          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_add_self1
% 0.30/0.54  thf(fact_2618_div__add__self1,axiom,
% 0.30/0.54      ! [B2: int,A: int] :
% 0.30/0.54        ( ( B2 != zero_zero_int )
% 0.30/0.54       => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 0.30/0.54          = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_add_self1
% 0.30/0.54  thf(fact_2619_div__add__self2,axiom,
% 0.30/0.54      ! [B2: nat,A: nat] :
% 0.30/0.54        ( ( B2 != zero_zero_nat )
% 0.30/0.54       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 0.30/0.54          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_add_self2
% 0.30/0.54  thf(fact_2620_div__add__self2,axiom,
% 0.30/0.54      ! [B2: int,A: int] :
% 0.30/0.54        ( ( B2 != zero_zero_int )
% 0.30/0.54       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 0.30/0.54          = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_add_self2
% 0.30/0.54  thf(fact_2621_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.54      ! [Z: complex,A: complex,B2: complex] :
% 0.30/0.54        ( ( ( Z = zero_zero_complex )
% 0.30/0.54         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.30/0.54            = A ) )
% 0.30/0.54        & ( ( Z != zero_zero_complex )
% 0.30/0.54         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 0.30/0.54            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(4)
% 0.30/0.54  thf(fact_2622_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.54      ! [Z: real,A: real,B2: real] :
% 0.30/0.54        ( ( ( Z = zero_zero_real )
% 0.30/0.54         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.30/0.54            = A ) )
% 0.30/0.54        & ( ( Z != zero_zero_real )
% 0.30/0.54         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 0.30/0.54            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(4)
% 0.30/0.54  thf(fact_2623_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.54      ! [Z: rat,A: rat,B2: rat] :
% 0.30/0.54        ( ( ( Z = zero_zero_rat )
% 0.30/0.54         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.30/0.54            = A ) )
% 0.30/0.54        & ( ( Z != zero_zero_rat )
% 0.30/0.54         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 0.30/0.54            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % add_divide_eq_if_simps(4)
% 0.30/0.54  thf(fact_2624_diff__frac__eq,axiom,
% 0.30/0.54      ! [Y3: complex,Z: complex,X3: complex,W: complex] :
% 0.30/0.54        ( ( Y3 != zero_zero_complex )
% 0.30/0.54       => ( ( Z != zero_zero_complex )
% 0.30/0.54         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.54            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_frac_eq
% 0.30/0.54  thf(fact_2625_diff__frac__eq,axiom,
% 0.30/0.54      ! [Y3: real,Z: real,X3: real,W: real] :
% 0.30/0.54        ( ( Y3 != zero_zero_real )
% 0.30/0.54       => ( ( Z != zero_zero_real )
% 0.30/0.54         => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.54            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_frac_eq
% 0.30/0.54  thf(fact_2626_diff__frac__eq,axiom,
% 0.30/0.54      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 0.30/0.54        ( ( Y3 != zero_zero_rat )
% 0.30/0.54       => ( ( Z != zero_zero_rat )
% 0.30/0.54         => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.54            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_frac_eq
% 0.30/0.54  thf(fact_2627_diff__divide__eq__iff,axiom,
% 0.30/0.54      ! [Z: complex,X3: complex,Y3: complex] :
% 0.30/0.54        ( ( Z != zero_zero_complex )
% 0.30/0.54       => ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.30/0.54          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_divide_eq_iff
% 0.30/0.54  thf(fact_2628_diff__divide__eq__iff,axiom,
% 0.30/0.54      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.54        ( ( Z != zero_zero_real )
% 0.30/0.54       => ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.30/0.54          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_divide_eq_iff
% 0.30/0.54  thf(fact_2629_diff__divide__eq__iff,axiom,
% 0.30/0.54      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.54        ( ( Z != zero_zero_rat )
% 0.30/0.54       => ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.30/0.54          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_divide_eq_iff
% 0.30/0.54  thf(fact_2630_divide__diff__eq__iff,axiom,
% 0.30/0.54      ! [Z: complex,X3: complex,Y3: complex] :
% 0.30/0.54        ( ( Z != zero_zero_complex )
% 0.30/0.54       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y3 )
% 0.30/0.54          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_diff_eq_iff
% 0.30/0.54  thf(fact_2631_divide__diff__eq__iff,axiom,
% 0.30/0.54      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.54        ( ( Z != zero_zero_real )
% 0.30/0.54       => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z ) @ Y3 )
% 0.30/0.54          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_diff_eq_iff
% 0.30/0.54  thf(fact_2632_divide__diff__eq__iff,axiom,
% 0.30/0.54      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.54        ( ( Z != zero_zero_rat )
% 0.30/0.54       => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z ) @ Y3 )
% 0.30/0.54          = ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divide_diff_eq_iff
% 0.30/0.54  thf(fact_2633_mod__double__modulus,axiom,
% 0.30/0.54      ! [M: code_integer,X3: code_integer] :
% 0.30/0.54        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.30/0.54       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.30/0.54         => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.54              = ( modulo364778990260209775nteger @ X3 @ M ) )
% 0.30/0.54            | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.54              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_double_modulus
% 0.30/0.54  thf(fact_2634_mod__double__modulus,axiom,
% 0.30/0.54      ! [M: nat,X3: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.30/0.54         => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.54              = ( modulo_modulo_nat @ X3 @ M ) )
% 0.30/0.54            | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.54              = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_double_modulus
% 0.30/0.54  thf(fact_2635_mod__double__modulus,axiom,
% 0.30/0.54      ! [M: int,X3: int] :
% 0.30/0.54        ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.54         => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.54              = ( modulo_modulo_int @ X3 @ M ) )
% 0.30/0.54            | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.54              = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mod_double_modulus
% 0.30/0.54  thf(fact_2636_divmod__digit__1_I2_J,axiom,
% 0.30/0.54      ! [A: code_integer,B2: code_integer] :
% 0.30/0.54        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.54       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.54         => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.54           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54              = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_1(2)
% 0.30/0.54  thf(fact_2637_divmod__digit__1_I2_J,axiom,
% 0.30/0.54      ! [A: nat,B2: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.54         => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.54           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54              = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_1(2)
% 0.30/0.54  thf(fact_2638_divmod__digit__1_I2_J,axiom,
% 0.30/0.54      ! [A: int,B2: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.54         => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.54           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 0.30/0.54              = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % divmod_digit_1(2)
% 0.30/0.54  thf(fact_2639_vebt__member_Osimps_I4_J,axiom,
% 0.30/0.54      ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 0.30/0.54        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).
% 0.30/0.54  
% 0.30/0.54  % vebt_member.simps(4)
% 0.30/0.54  thf(fact_2640_numeral__1__eq__Suc__0,axiom,
% 0.30/0.54      ( ( numeral_numeral_nat @ one )
% 0.30/0.54      = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.54  
% 0.30/0.54  % numeral_1_eq_Suc_0
% 0.30/0.54  thf(fact_2641_num_Osize_I5_J,axiom,
% 0.30/0.54      ! [X2: num] :
% 0.30/0.54        ( ( size_size_num @ ( bit0 @ X2 ) )
% 0.30/0.54        = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % num.size(5)
% 0.30/0.54  thf(fact_2642_ex__least__nat__less,axiom,
% 0.30/0.54      ! [P: nat > $o,N: nat] :
% 0.30/0.54        ( ( P @ N )
% 0.30/0.54       => ( ~ ( P @ zero_zero_nat )
% 0.30/0.54         => ? [K3: nat] :
% 0.30/0.54              ( ( ord_less_nat @ K3 @ N )
% 0.30/0.54              & ! [I2: nat] :
% 0.30/0.54                  ( ( ord_less_eq_nat @ I2 @ K3 )
% 0.30/0.54                 => ~ ( P @ I2 ) )
% 0.30/0.54              & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % ex_least_nat_less
% 0.30/0.54  thf(fact_2643_diff__Suc__less,axiom,
% 0.30/0.54      ! [N: nat,I: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54       => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.30/0.54  
% 0.30/0.54  % diff_Suc_less
% 0.30/0.54  thf(fact_2644_one__less__mult,axiom,
% 0.30/0.54      ! [N: nat,M: nat] :
% 0.30/0.54        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.54       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % one_less_mult
% 0.30/0.54  thf(fact_2645_n__less__m__mult__n,axiom,
% 0.30/0.54      ! [N: nat,M: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54         => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % n_less_m_mult_n
% 0.30/0.54  thf(fact_2646_n__less__n__mult__m,axiom,
% 0.30/0.54      ! [N: nat,M: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54         => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % n_less_n_mult_m
% 0.30/0.54  thf(fact_2647_length__pos__if__in__set,axiom,
% 0.30/0.54      ! [X3: real,Xs2: list_real] :
% 0.30/0.54        ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 0.30/0.54       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % length_pos_if_in_set
% 0.30/0.54  thf(fact_2648_length__pos__if__in__set,axiom,
% 0.30/0.54      ! [X3: complex,Xs2: list_complex] :
% 0.30/0.54        ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.54       => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % length_pos_if_in_set
% 0.30/0.54  thf(fact_2649_length__pos__if__in__set,axiom,
% 0.30/0.54      ! [X3: int,Xs2: list_int] :
% 0.30/0.54        ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.30/0.54       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % length_pos_if_in_set
% 0.30/0.54  thf(fact_2650_length__pos__if__in__set,axiom,
% 0.30/0.54      ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.30/0.54        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.54       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % length_pos_if_in_set
% 0.30/0.54  thf(fact_2651_length__pos__if__in__set,axiom,
% 0.30/0.54      ! [X3: $o,Xs2: list_o] :
% 0.30/0.54        ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.30/0.54       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % length_pos_if_in_set
% 0.30/0.54  thf(fact_2652_length__pos__if__in__set,axiom,
% 0.30/0.54      ! [X3: nat,Xs2: list_nat] :
% 0.30/0.54        ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.54       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % length_pos_if_in_set
% 0.30/0.54  thf(fact_2653_nat__induct__non__zero,axiom,
% 0.30/0.54      ! [N: nat,P: nat > $o] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.54       => ( ( P @ one_one_nat )
% 0.30/0.54         => ( ! [N3: nat] :
% 0.30/0.54                ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.54               => ( ( P @ N3 )
% 0.30/0.54                 => ( P @ ( suc @ N3 ) ) ) )
% 0.30/0.54           => ( P @ N ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_induct_non_zero
% 0.30/0.54  thf(fact_2654_nat__mult__le__cancel1,axiom,
% 0.30/0.54      ! [K: nat,M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.54          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_mult_le_cancel1
% 0.30/0.54  thf(fact_2655_nat__diff__split,axiom,
% 0.30/0.54      ! [P: nat > $o,A: nat,B2: nat] :
% 0.30/0.54        ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_nat @ A @ B2 )
% 0.30/0.54           => ( P @ zero_zero_nat ) )
% 0.30/0.54          & ! [D2: nat] :
% 0.30/0.54              ( ( A
% 0.30/0.54                = ( plus_plus_nat @ B2 @ D2 ) )
% 0.30/0.54             => ( P @ D2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_diff_split
% 0.30/0.54  thf(fact_2656_nat__diff__split__asm,axiom,
% 0.30/0.54      ! [P: nat > $o,A: nat,B2: nat] :
% 0.30/0.54        ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.54        = ( ~ ( ( ( ord_less_nat @ A @ B2 )
% 0.30/0.54                & ~ ( P @ zero_zero_nat ) )
% 0.30/0.54              | ? [D2: nat] :
% 0.30/0.54                  ( ( A
% 0.30/0.54                    = ( plus_plus_nat @ B2 @ D2 ) )
% 0.30/0.54                  & ~ ( P @ D2 ) ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_diff_split_asm
% 0.30/0.54  thf(fact_2657_power__gt__expt,axiom,
% 0.30/0.54      ! [N: nat,K: nat] :
% 0.30/0.54        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.54       => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % power_gt_expt
% 0.30/0.54  thf(fact_2658_div__greater__zero__iff,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.54        = ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.54          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_greater_zero_iff
% 0.30/0.54  thf(fact_2659_div__le__mono2,axiom,
% 0.30/0.54      ! [M: nat,N: nat,K: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.54         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_le_mono2
% 0.30/0.54  thf(fact_2660_nat__one__le__power,axiom,
% 0.30/0.54      ! [I: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.30/0.54       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_one_le_power
% 0.30/0.54  thf(fact_2661_div__less__iff__less__mult,axiom,
% 0.30/0.54      ! [Q2: nat,M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.30/0.54       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 0.30/0.54          = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_less_iff_less_mult
% 0.30/0.54  thf(fact_2662_nat__mult__div__cancel1,axiom,
% 0.30/0.54      ! [K: nat,M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.54          = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_mult_div_cancel1
% 0.30/0.54  thf(fact_2663_div__eq__dividend__iff,axiom,
% 0.30/0.54      ! [M: nat,N: nat] :
% 0.30/0.54        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54       => ( ( ( divide_divide_nat @ M @ N )
% 0.30/0.54            = M )
% 0.30/0.54          = ( N = one_one_nat ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_eq_dividend_iff
% 0.30/0.54  thf(fact_2664_div__less__dividend,axiom,
% 0.30/0.54      ! [N: nat,M: nat] :
% 0.30/0.54        ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.54       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.54         => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % div_less_dividend
% 0.30/0.54  thf(fact_2665_nat__induct2,axiom,
% 0.30/0.54      ! [P: nat > $o,N: nat] :
% 0.30/0.54        ( ( P @ zero_zero_nat )
% 0.30/0.54       => ( ( P @ one_one_nat )
% 0.30/0.54         => ( ! [N3: nat] :
% 0.30/0.54                ( ( P @ N3 )
% 0.30/0.54               => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.54           => ( P @ N ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % nat_induct2
% 0.30/0.54  thf(fact_2666_vebt__mint_Ocases,axiom,
% 0.30/0.54      ! [X3: vEBT_VEBT] :
% 0.30/0.54        ( ! [A3: $o,B3: $o] :
% 0.30/0.54            ( X3
% 0.30/0.54           != ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.54       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.54              ( X3
% 0.30/0.54             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.54         => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.54                ( X3
% 0.30/0.54               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % vebt_mint.cases
% 0.30/0.54  thf(fact_2667_field__le__mult__one__interval,axiom,
% 0.30/0.54      ! [X3: real,Y3: real] :
% 0.30/0.54        ( ! [Z4: real] :
% 0.30/0.54            ( ( ord_less_real @ zero_zero_real @ Z4 )
% 0.30/0.54           => ( ( ord_less_real @ Z4 @ one_one_real )
% 0.30/0.54             => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X3 ) @ Y3 ) ) )
% 0.30/0.54       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % field_le_mult_one_interval
% 0.30/0.54  thf(fact_2668_field__le__mult__one__interval,axiom,
% 0.30/0.54      ! [X3: rat,Y3: rat] :
% 0.30/0.54        ( ! [Z4: rat] :
% 0.30/0.54            ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 0.30/0.54           => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 0.30/0.54             => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X3 ) @ Y3 ) ) )
% 0.30/0.54       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.54  
% 0.30/0.54  % field_le_mult_one_interval
% 0.30/0.54  thf(fact_2669_mult__less__cancel__right2,axiom,
% 0.30/0.54      ! [A: real,C: real] :
% 0.30/0.54        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.30/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ A @ one_one_real ) )
% 0.30/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right2
% 0.30/0.54  thf(fact_2670_mult__less__cancel__right2,axiom,
% 0.30/0.54      ! [A: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.30/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.30/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right2
% 0.30/0.54  thf(fact_2671_mult__less__cancel__right2,axiom,
% 0.30/0.54      ! [A: int,C: int] :
% 0.30/0.54        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.30/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_int @ A @ one_one_int ) )
% 0.30/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right2
% 0.30/0.54  thf(fact_2672_mult__less__cancel__right1,axiom,
% 0.30/0.54      ! [C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ one_one_real @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right1
% 0.30/0.54  thf(fact_2673_mult__less__cancel__right1,axiom,
% 0.30/0.54      ! [C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ one_one_rat @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right1
% 0.30/0.54  thf(fact_2674_mult__less__cancel__right1,axiom,
% 0.30/0.54      ! [C: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_int @ one_one_int @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_right1
% 0.30/0.54  thf(fact_2675_mult__less__cancel__left2,axiom,
% 0.30/0.54      ! [C: real,A: real] :
% 0.30/0.54        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.30/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ A @ one_one_real ) )
% 0.30/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left2
% 0.30/0.54  thf(fact_2676_mult__less__cancel__left2,axiom,
% 0.30/0.54      ! [C: rat,A: rat] :
% 0.30/0.54        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.30/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.30/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left2
% 0.30/0.54  thf(fact_2677_mult__less__cancel__left2,axiom,
% 0.30/0.54      ! [C: int,A: int] :
% 0.30/0.54        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.30/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_int @ A @ one_one_int ) )
% 0.30/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left2
% 0.30/0.54  thf(fact_2678_mult__less__cancel__left1,axiom,
% 0.30/0.54      ! [C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_real @ one_one_real @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left1
% 0.30/0.54  thf(fact_2679_mult__less__cancel__left1,axiom,
% 0.30/0.54      ! [C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_rat @ one_one_rat @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left1
% 0.30/0.54  thf(fact_2680_mult__less__cancel__left1,axiom,
% 0.30/0.54      ! [C: int,B2: int] :
% 0.30/0.54        ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
% 0.30/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_int @ one_one_int @ B2 ) )
% 0.30/0.54          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_less_cancel_left1
% 0.30/0.54  thf(fact_2681_mult__le__cancel__right2,axiom,
% 0.30/0.54      ! [A: real,C: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.30/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right2
% 0.30/0.54  thf(fact_2682_mult__le__cancel__right2,axiom,
% 0.30/0.54      ! [A: rat,C: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.30/0.54          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right2
% 0.30/0.54  thf(fact_2683_mult__le__cancel__right2,axiom,
% 0.30/0.54      ! [A: int,C: int] :
% 0.30/0.54        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.30/0.54        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.30/0.54          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right2
% 0.30/0.54  thf(fact_2684_mult__le__cancel__right1,axiom,
% 0.30/0.54      ! [C: real,B2: real] :
% 0.30/0.54        ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54           => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 0.30/0.54          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54           => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 0.30/0.54  
% 0.30/0.54  % mult_le_cancel_right1
% 0.30/0.54  thf(fact_2685_mult__le__cancel__right1,axiom,
% 0.30/0.54      ! [C: rat,B2: rat] :
% 0.30/0.54        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54           => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 0.30/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55           => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_right1
% 0.30/0.55  thf(fact_2686_mult__le__cancel__right1,axiom,
% 0.30/0.55      ! [C: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55           => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 0.30/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.55           => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_right1
% 0.30/0.55  thf(fact_2687_mult__le__cancel__left2,axiom,
% 0.30/0.55      ! [C: real,A: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.30/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_left2
% 0.30/0.55  thf(fact_2688_mult__le__cancel__left2,axiom,
% 0.30/0.55      ! [C: rat,A: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.30/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_left2
% 0.30/0.55  thf(fact_2689_mult__le__cancel__left2,axiom,
% 0.30/0.55      ! [C: int,A: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.30/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.30/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.55           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_left2
% 0.30/0.55  thf(fact_2690_mult__le__cancel__left1,axiom,
% 0.30/0.55      ! [C: real,B2: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 0.30/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55           => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_left1
% 0.30/0.55  thf(fact_2691_mult__le__cancel__left1,axiom,
% 0.30/0.55      ! [C: rat,B2: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 0.30/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55           => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_left1
% 0.30/0.55  thf(fact_2692_mult__le__cancel__left1,axiom,
% 0.30/0.55      ! [C: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
% 0.30/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55           => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 0.30/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.55           => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_le_cancel_left1
% 0.30/0.55  thf(fact_2693_divide__left__mono__neg,axiom,
% 0.30/0.55      ! [A: real,B2: real,C: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.55       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.55           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_left_mono_neg
% 0.30/0.55  thf(fact_2694_divide__left__mono__neg,axiom,
% 0.30/0.55      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.55       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.55           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_left_mono_neg
% 0.30/0.55  thf(fact_2695_mult__imp__le__div__pos,axiom,
% 0.30/0.55      ! [Y3: real,Z: real,X3: real] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X3 )
% 0.30/0.55         => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_imp_le_div_pos
% 0.30/0.55  thf(fact_2696_mult__imp__le__div__pos,axiom,
% 0.30/0.55      ! [Y3: rat,Z: rat,X3: rat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X3 )
% 0.30/0.55         => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_imp_le_div_pos
% 0.30/0.55  thf(fact_2697_mult__imp__div__pos__le,axiom,
% 0.30/0.55      ! [Y3: real,X3: real,Z: real] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.55       => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z @ Y3 ) )
% 0.30/0.55         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_imp_div_pos_le
% 0.30/0.55  thf(fact_2698_mult__imp__div__pos__le,axiom,
% 0.30/0.55      ! [Y3: rat,X3: rat,Z: rat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55       => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) )
% 0.30/0.55         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_imp_div_pos_le
% 0.30/0.55  thf(fact_2699_pos__le__divide__eq,axiom,
% 0.30/0.55      ! [C: real,A: real,B2: real] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.55          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_le_divide_eq
% 0.30/0.55  thf(fact_2700_pos__le__divide__eq,axiom,
% 0.30/0.55      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.55          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_le_divide_eq
% 0.30/0.55  thf(fact_2701_pos__divide__le__eq,axiom,
% 0.30/0.55      ! [C: real,B2: real,A: real] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.55          = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_divide_le_eq
% 0.30/0.55  thf(fact_2702_pos__divide__le__eq,axiom,
% 0.30/0.55      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.55          = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_divide_le_eq
% 0.30/0.55  thf(fact_2703_neg__le__divide__eq,axiom,
% 0.30/0.55      ! [C: real,A: real,B2: real] :
% 0.30/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.55          = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_le_divide_eq
% 0.30/0.55  thf(fact_2704_neg__le__divide__eq,axiom,
% 0.30/0.55      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.55          = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_le_divide_eq
% 0.30/0.55  thf(fact_2705_neg__divide__le__eq,axiom,
% 0.30/0.55      ! [C: real,B2: real,A: real] :
% 0.30/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.55          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_divide_le_eq
% 0.30/0.55  thf(fact_2706_neg__divide__le__eq,axiom,
% 0.30/0.55      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.55          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_divide_le_eq
% 0.30/0.55  thf(fact_2707_divide__left__mono,axiom,
% 0.30/0.55      ! [B2: real,A: real,C: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.55           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_left_mono
% 0.30/0.55  thf(fact_2708_divide__left__mono,axiom,
% 0.30/0.55      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.55           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_left_mono
% 0.30/0.55  thf(fact_2709_le__divide__eq,axiom,
% 0.30/0.55      ! [A: real,B2: real,C: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.30/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.30/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_divide_eq
% 0.30/0.55  thf(fact_2710_le__divide__eq,axiom,
% 0.30/0.55      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.30/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_divide_eq
% 0.30/0.55  thf(fact_2711_divide__le__eq,axiom,
% 0.30/0.55      ! [B2: real,C: real,A: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 0.30/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 0.30/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_le_eq
% 0.30/0.55  thf(fact_2712_divide__le__eq,axiom,
% 0.30/0.55      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 0.30/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_le_eq
% 0.30/0.55  thf(fact_2713_le__divide__eq__1,axiom,
% 0.30/0.55      ! [B2: real,A: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55            & ( ord_less_eq_real @ A @ B2 ) )
% 0.30/0.55          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55            & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_divide_eq_1
% 0.30/0.55  thf(fact_2714_le__divide__eq__1,axiom,
% 0.30/0.55      ! [B2: rat,A: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55            & ( ord_less_eq_rat @ A @ B2 ) )
% 0.30/0.55          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55            & ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_divide_eq_1
% 0.30/0.55  thf(fact_2715_divide__le__eq__1,axiom,
% 0.30/0.55      ! [B2: real,A: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55            & ( ord_less_eq_real @ B2 @ A ) )
% 0.30/0.55          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55            & ( ord_less_eq_real @ A @ B2 ) )
% 0.30/0.55          | ( A = zero_zero_real ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_le_eq_1
% 0.30/0.55  thf(fact_2716_divide__le__eq__1,axiom,
% 0.30/0.55      ! [B2: rat,A: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55            & ( ord_less_eq_rat @ B2 @ A ) )
% 0.30/0.55          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55            & ( ord_less_eq_rat @ A @ B2 ) )
% 0.30/0.55          | ( A = zero_zero_rat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_le_eq_1
% 0.30/0.55  thf(fact_2717_convex__bound__le,axiom,
% 0.30/0.55      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ X3 @ A )
% 0.30/0.55       => ( ( ord_less_eq_real @ Y3 @ A )
% 0.30/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.30/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.30/0.55             => ( ( ( plus_plus_real @ U @ V )
% 0.30/0.55                  = one_one_real )
% 0.30/0.55               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % convex_bound_le
% 0.30/0.55  thf(fact_2718_convex__bound__le,axiom,
% 0.30/0.55      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ X3 @ A )
% 0.30/0.55       => ( ( ord_less_eq_rat @ Y3 @ A )
% 0.30/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.30/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.30/0.55             => ( ( ( plus_plus_rat @ U @ V )
% 0.30/0.55                  = one_one_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % convex_bound_le
% 0.30/0.55  thf(fact_2719_convex__bound__le,axiom,
% 0.30/0.55      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ X3 @ A )
% 0.30/0.55       => ( ( ord_less_eq_int @ Y3 @ A )
% 0.30/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.30/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.30/0.55             => ( ( ( plus_plus_int @ U @ V )
% 0.30/0.55                  = one_one_int )
% 0.30/0.55               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % convex_bound_le
% 0.30/0.55  thf(fact_2720_divide__less__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [B2: real,C: real,W: num] :
% 0.30/0.55        ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.30/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_less_eq_numeral(1)
% 0.30/0.55  thf(fact_2721_divide__less__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [B2: rat,C: rat,W: num] :
% 0.30/0.55        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.30/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_less_eq_numeral(1)
% 0.30/0.55  thf(fact_2722_less__divide__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [W: num,B2: real,C: real] :
% 0.30/0.55        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.30/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % less_divide_eq_numeral(1)
% 0.30/0.55  thf(fact_2723_less__divide__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [W: num,B2: rat,C: rat] :
% 0.30/0.55        ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.30/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % less_divide_eq_numeral(1)
% 0.30/0.55  thf(fact_2724_frac__le__eq,axiom,
% 0.30/0.55      ! [Y3: real,Z: real,X3: real,W: real] :
% 0.30/0.55        ( ( Y3 != zero_zero_real )
% 0.30/0.55       => ( ( Z != zero_zero_real )
% 0.30/0.55         => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.55            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % frac_le_eq
% 0.30/0.55  thf(fact_2725_frac__le__eq,axiom,
% 0.30/0.55      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 0.30/0.55        ( ( Y3 != zero_zero_rat )
% 0.30/0.55       => ( ( Z != zero_zero_rat )
% 0.30/0.55         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.55            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % frac_le_eq
% 0.30/0.55  thf(fact_2726_divmod__digit__1_I1_J,axiom,
% 0.30/0.55      ! [A: code_integer,B2: code_integer] :
% 0.30/0.55        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.55       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.55         => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.55           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
% 0.30/0.55              = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divmod_digit_1(1)
% 0.30/0.55  thf(fact_2727_divmod__digit__1_I1_J,axiom,
% 0.30/0.55      ! [A: nat,B2: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 0.30/0.55         => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.55           => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
% 0.30/0.55              = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divmod_digit_1(1)
% 0.30/0.55  thf(fact_2728_divmod__digit__1_I1_J,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.55         => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 0.30/0.55           => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
% 0.30/0.55              = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divmod_digit_1(1)
% 0.30/0.55  thf(fact_2729_frac__less__eq,axiom,
% 0.30/0.55      ! [Y3: real,Z: real,X3: real,W: real] :
% 0.30/0.55        ( ( Y3 != zero_zero_real )
% 0.30/0.55       => ( ( Z != zero_zero_real )
% 0.30/0.55         => ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.30/0.55            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % frac_less_eq
% 0.30/0.55  thf(fact_2730_frac__less__eq,axiom,
% 0.30/0.55      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 0.30/0.55        ( ( Y3 != zero_zero_rat )
% 0.30/0.55       => ( ( Z != zero_zero_rat )
% 0.30/0.55         => ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.30/0.55            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % frac_less_eq
% 0.30/0.55  thf(fact_2731_power__Suc__less,axiom,
% 0.30/0.55      ! [A: real,N: nat] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55       => ( ( ord_less_real @ A @ one_one_real )
% 0.30/0.55         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less
% 0.30/0.55  thf(fact_2732_power__Suc__less,axiom,
% 0.30/0.55      ! [A: rat,N: nat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.30/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less
% 0.30/0.55  thf(fact_2733_power__Suc__less,axiom,
% 0.30/0.55      ! [A: nat,N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.30/0.55         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less
% 0.30/0.55  thf(fact_2734_power__Suc__less,axiom,
% 0.30/0.55      ! [A: int,N: nat] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ A @ one_one_int )
% 0.30/0.55         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less
% 0.30/0.55  thf(fact_2735_power__Suc__le__self,axiom,
% 0.30/0.55      ! [A: real,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.55         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_le_self
% 0.30/0.55  thf(fact_2736_power__Suc__le__self,axiom,
% 0.30/0.55      ! [A: rat,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.55         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_le_self
% 0.30/0.55  thf(fact_2737_power__Suc__le__self,axiom,
% 0.30/0.55      ! [A: nat,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.55         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_le_self
% 0.30/0.55  thf(fact_2738_power__Suc__le__self,axiom,
% 0.30/0.55      ! [A: int,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.55         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_le_self
% 0.30/0.55  thf(fact_2739_power__Suc__less__one,axiom,
% 0.30/0.55      ! [A: real,N: nat] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55       => ( ( ord_less_real @ A @ one_one_real )
% 0.30/0.55         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less_one
% 0.30/0.55  thf(fact_2740_power__Suc__less__one,axiom,
% 0.30/0.55      ! [A: rat,N: nat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.30/0.55         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less_one
% 0.30/0.55  thf(fact_2741_power__Suc__less__one,axiom,
% 0.30/0.55      ! [A: nat,N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.30/0.55         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less_one
% 0.30/0.55  thf(fact_2742_power__Suc__less__one,axiom,
% 0.30/0.55      ! [A: int,N: nat] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ A @ one_one_int )
% 0.30/0.55         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_Suc_less_one
% 0.30/0.55  thf(fact_2743_power__strict__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: real] :
% 0.30/0.55        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55         => ( ( ord_less_real @ A @ one_one_real )
% 0.30/0.55           => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_strict_decreasing
% 0.30/0.55  thf(fact_2744_power__strict__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: rat] :
% 0.30/0.55        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55         => ( ( ord_less_rat @ A @ one_one_rat )
% 0.30/0.55           => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_strict_decreasing
% 0.30/0.55  thf(fact_2745_power__strict__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: nat] :
% 0.30/0.55        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55         => ( ( ord_less_nat @ A @ one_one_nat )
% 0.30/0.55           => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_strict_decreasing
% 0.30/0.55  thf(fact_2746_power__strict__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: int] :
% 0.30/0.55        ( ( ord_less_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55         => ( ( ord_less_int @ A @ one_one_int )
% 0.30/0.55           => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_strict_decreasing
% 0.30/0.55  thf(fact_2747_power__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: real] :
% 0.30/0.55        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55         => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.55           => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_decreasing
% 0.30/0.55  thf(fact_2748_power__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: rat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.55           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_decreasing
% 0.30/0.55  thf(fact_2749_power__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.55           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_decreasing
% 0.30/0.55  thf(fact_2750_power__decreasing,axiom,
% 0.30/0.55      ! [N: nat,N4: nat,A: int] :
% 0.30/0.55        ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55         => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.55           => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_decreasing
% 0.30/0.55  thf(fact_2751_vebt__mint_Oelims,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.55        ( ( ( vEBT_vebt_mint @ X3 )
% 0.30/0.55          = Y3 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ~ ( ( A3
% 0.30/0.55                   => ( Y3
% 0.30/0.55                      = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55                  & ( ~ A3
% 0.30/0.55                   => ( ( B3
% 0.30/0.55                       => ( Y3
% 0.30/0.55                          = ( some_nat @ one_one_nat ) ) )
% 0.30/0.55                      & ( ~ B3
% 0.30/0.55                       => ( Y3 = none_nat ) ) ) ) ) )
% 0.30/0.55         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55             => ( Y3 != none_nat ) )
% 0.30/0.55           => ~ ! [Mi2: nat] :
% 0.30/0.55                  ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.55                 => ( Y3
% 0.30/0.55                   != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_mint.elims
% 0.30/0.55  thf(fact_2752_zero__power2,axiom,
% 0.30/0.55      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55      = zero_zero_rat ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_power2
% 0.30/0.55  thf(fact_2753_zero__power2,axiom,
% 0.30/0.55      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55      = zero_zero_nat ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_power2
% 0.30/0.55  thf(fact_2754_zero__power2,axiom,
% 0.30/0.55      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55      = zero_zero_real ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_power2
% 0.30/0.55  thf(fact_2755_zero__power2,axiom,
% 0.30/0.55      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55      = zero_zero_int ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_power2
% 0.30/0.55  thf(fact_2756_zero__power2,axiom,
% 0.30/0.55      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55      = zero_zero_complex ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_power2
% 0.30/0.55  thf(fact_2757_vebt__maxt_Oelims,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.55        ( ( ( vEBT_vebt_maxt @ X3 )
% 0.30/0.55          = Y3 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ~ ( ( B3
% 0.30/0.55                   => ( Y3
% 0.30/0.55                      = ( some_nat @ one_one_nat ) ) )
% 0.30/0.55                  & ( ~ B3
% 0.30/0.55                   => ( ( A3
% 0.30/0.55                       => ( Y3
% 0.30/0.55                          = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55                      & ( ~ A3
% 0.30/0.55                       => ( Y3 = none_nat ) ) ) ) ) )
% 0.30/0.55         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55             => ( Y3 != none_nat ) )
% 0.30/0.55           => ~ ! [Mi2: nat,Ma2: nat] :
% 0.30/0.55                  ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.55                 => ( Y3
% 0.30/0.55                   != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_maxt.elims
% 0.30/0.55  thf(fact_2758_self__le__power,axiom,
% 0.30/0.55      ! [A: real,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % self_le_power
% 0.30/0.55  thf(fact_2759_self__le__power,axiom,
% 0.30/0.55      ! [A: rat,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % self_le_power
% 0.30/0.55  thf(fact_2760_self__le__power,axiom,
% 0.30/0.55      ! [A: nat,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % self_le_power
% 0.30/0.55  thf(fact_2761_self__le__power,axiom,
% 0.30/0.55      ! [A: int,N: nat] :
% 0.30/0.55        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % self_le_power
% 0.30/0.55  thf(fact_2762_one__less__power,axiom,
% 0.30/0.55      ! [A: real,N: nat] :
% 0.30/0.55        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % one_less_power
% 0.30/0.55  thf(fact_2763_one__less__power,axiom,
% 0.30/0.55      ! [A: rat,N: nat] :
% 0.30/0.55        ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % one_less_power
% 0.30/0.55  thf(fact_2764_one__less__power,axiom,
% 0.30/0.55      ! [A: nat,N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % one_less_power
% 0.30/0.55  thf(fact_2765_one__less__power,axiom,
% 0.30/0.55      ! [A: int,N: nat] :
% 0.30/0.55        ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % one_less_power
% 0.30/0.55  thf(fact_2766_numeral__2__eq__2,axiom,
% 0.30/0.55      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.30/0.55      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % numeral_2_eq_2
% 0.30/0.55  thf(fact_2767_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_2768_power__diff,axiom,
% 0.30/0.55      ! [A: complex,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_complex )
% 0.30/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff
% 0.30/0.55  thf(fact_2769_power__diff,axiom,
% 0.30/0.55      ! [A: real,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_real )
% 0.30/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff
% 0.30/0.55  thf(fact_2770_power__diff,axiom,
% 0.30/0.55      ! [A: rat,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_rat )
% 0.30/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff
% 0.30/0.55  thf(fact_2771_power__diff,axiom,
% 0.30/0.55      ! [A: nat,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_nat )
% 0.30/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff
% 0.30/0.55  thf(fact_2772_power__diff,axiom,
% 0.30/0.55      ! [A: int,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_int )
% 0.30/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.55            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff
% 0.30/0.55  thf(fact_2773_div__if,axiom,
% 0.30/0.55      ( divide_divide_nat
% 0.30/0.55      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.55            ( if_nat
% 0.30/0.55            @ ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.55              | ( N2 = zero_zero_nat ) )
% 0.30/0.55            @ zero_zero_nat
% 0.30/0.55            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_if
% 0.30/0.55  thf(fact_2774_div__geq,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ~ ( ord_less_nat @ M @ N )
% 0.30/0.55         => ( ( divide_divide_nat @ M @ N )
% 0.30/0.55            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_geq
% 0.30/0.55  thf(fact_2775_Suc__pred_H,axiom,
% 0.30/0.55      ! [N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( N
% 0.30/0.55          = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Suc_pred'
% 0.30/0.55  thf(fact_2776_Suc__diff__eq__diff__pred,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.30/0.55          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Suc_diff_eq_diff_pred
% 0.30/0.55  thf(fact_2777_add__eq__if,axiom,
% 0.30/0.55      ( plus_plus_nat
% 0.30/0.55      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % add_eq_if
% 0.30/0.55  thf(fact_2778_less__eq__div__iff__mult__less__eq,axiom,
% 0.30/0.55      ! [Q2: nat,M: nat,N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.30/0.55       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 0.30/0.55          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % less_eq_div_iff_mult_less_eq
% 0.30/0.55  thf(fact_2779_dividend__less__times__div,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % dividend_less_times_div
% 0.30/0.55  thf(fact_2780_dividend__less__div__times,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % dividend_less_div_times
% 0.30/0.55  thf(fact_2781_split__div,axiom,
% 0.30/0.55      ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.55        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.55        = ( ( ( N = zero_zero_nat )
% 0.30/0.55           => ( P @ zero_zero_nat ) )
% 0.30/0.55          & ( ( N != zero_zero_nat )
% 0.30/0.55           => ! [I4: nat,J3: nat] :
% 0.30/0.55                ( ( ord_less_nat @ J3 @ N )
% 0.30/0.55               => ( ( M
% 0.30/0.55                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.30/0.55                 => ( P @ I4 ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % split_div
% 0.30/0.55  thf(fact_2782_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.55          = Y3 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ( Y3
% 0.30/0.55                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                       => A3 )
% 0.30/0.55                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55                           => B3 )
% 0.30/0.55                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.30/0.55         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.55             => Y3 )
% 0.30/0.55           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                  ( ? [S3: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.30/0.55                 => ( Y3
% 0.30/0.55                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                           => ( 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 ) ) ) ) ) )
% 0.30/0.55                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.naive_member.elims(1)
% 0.30/0.55  thf(fact_2783_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                   => A3 )
% 0.30/0.55                  & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55                   => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55                       => B3 )
% 0.30/0.55                      & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                ( ? [S3: vEBT_VEBT] :
% 0.30/0.55                    ( X3
% 0.30/0.55                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.30/0.55               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                     => ( 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 ) ) ) ) ) )
% 0.30/0.55                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.naive_member.elims(2)
% 0.30/0.55  thf(fact_2784_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                 => A3 )
% 0.30/0.55                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55                 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55                     => B3 )
% 0.30/0.55                    & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55                ( X3
% 0.30/0.55               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.55           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                  ( ? [S3: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.30/0.55                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                     => ( 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 ) ) ) ) ) )
% 0.30/0.55                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.naive_member.elims(3)
% 0.30/0.55  thf(fact_2785_mult__eq__if,axiom,
% 0.30/0.55      ( times_times_nat
% 0.30/0.55      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_eq_if
% 0.30/0.55  thf(fact_2786_vebt__pred_Osimps_I5_J,axiom,
% 0.30/0.55      ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 0.30/0.55        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
% 0.30/0.55        = none_nat ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_pred.simps(5)
% 0.30/0.55  thf(fact_2787_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.55          = Y3 )
% 0.30/0.55       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.55                ( X3
% 0.30/0.55                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55           => Y3 )
% 0.30/0.55         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.55             => Y3 )
% 0.30/0.55           => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.55                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.55                 => ( Y3
% 0.30/0.55                    = ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.55                          | ( Xa2 = Ma2 ) ) ) ) )
% 0.30/0.55             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                    ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.55                        ( X3
% 0.30/0.55                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.55                   => ( Y3
% 0.30/0.55                      = ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.55                            | ( Xa2 = Ma2 )
% 0.30/0.55                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                               => ( 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 ) ) ) ) ) )
% 0.30/0.55                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.30/0.55               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                      ( ? [Vd2: vEBT_VEBT] :
% 0.30/0.55                          ( X3
% 0.30/0.55                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.30/0.55                     => ( Y3
% 0.30/0.55                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                               => ( 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 ) ) ) ) ) )
% 0.30/0.55                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.membermima.elims(1)
% 0.30/0.55  thf(fact_2788_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.55       => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.55              ( X3
% 0.30/0.55             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55                ( X3
% 0.30/0.55               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.55           => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.55                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.55                 => ( ( Xa2 = Mi2 )
% 0.30/0.55                    | ( Xa2 = Ma2 ) ) )
% 0.30/0.55             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                    ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.55                        ( X3
% 0.30/0.55                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.55                   => ( ( Xa2 = Mi2 )
% 0.30/0.55                      | ( Xa2 = Ma2 )
% 0.30/0.55                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                         => ( 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 ) ) ) ) ) )
% 0.30/0.55                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.30/0.55               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                      ( ? [Vd2: vEBT_VEBT] :
% 0.30/0.55                          ( X3
% 0.30/0.55                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.30/0.55                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                         => ( 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 ) ) ) ) ) )
% 0.30/0.55                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.membermima.elims(3)
% 0.30/0.55  thf(fact_2789_convex__bound__lt,axiom,
% 0.30/0.55      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 0.30/0.55        ( ( ord_less_real @ X3 @ A )
% 0.30/0.55       => ( ( ord_less_real @ Y3 @ A )
% 0.30/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.30/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.30/0.55             => ( ( ( plus_plus_real @ U @ V )
% 0.30/0.55                  = one_one_real )
% 0.30/0.55               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % convex_bound_lt
% 0.30/0.55  thf(fact_2790_convex__bound__lt,axiom,
% 0.30/0.55      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.30/0.55        ( ( ord_less_rat @ X3 @ A )
% 0.30/0.55       => ( ( ord_less_rat @ Y3 @ A )
% 0.30/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.30/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.30/0.55             => ( ( ( plus_plus_rat @ U @ V )
% 0.30/0.55                  = one_one_rat )
% 0.30/0.55               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % convex_bound_lt
% 0.30/0.55  thf(fact_2791_convex__bound__lt,axiom,
% 0.30/0.55      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 0.30/0.55        ( ( ord_less_int @ X3 @ A )
% 0.30/0.55       => ( ( ord_less_int @ Y3 @ A )
% 0.30/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.30/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.30/0.55             => ( ( ( plus_plus_int @ U @ V )
% 0.30/0.55                  = one_one_int )
% 0.30/0.55               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % convex_bound_lt
% 0.30/0.55  thf(fact_2792_le__divide__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [W: num,B2: real,C: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.30/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_divide_eq_numeral(1)
% 0.30/0.55  thf(fact_2793_le__divide__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [W: num,B2: rat,C: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.30/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_divide_eq_numeral(1)
% 0.30/0.55  thf(fact_2794_divide__le__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [B2: real,C: real,W: num] :
% 0.30/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 0.30/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.30/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 0.30/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_le_eq_numeral(1)
% 0.30/0.55  thf(fact_2795_divide__le__eq__numeral_I1_J,axiom,
% 0.30/0.55      ! [B2: rat,C: rat,W: num] :
% 0.30/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.30/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.30/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 0.30/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % divide_le_eq_numeral(1)
% 0.30/0.55  thf(fact_2796_half__gt__zero,axiom,
% 0.30/0.55      ! [A: real] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % half_gt_zero
% 0.30/0.55  thf(fact_2797_half__gt__zero,axiom,
% 0.30/0.55      ! [A: rat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % half_gt_zero
% 0.30/0.55  thf(fact_2798_half__gt__zero__iff,axiom,
% 0.30/0.55      ! [A: real] :
% 0.30/0.55        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.55        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.55  
% 0.30/0.55  % half_gt_zero_iff
% 0.30/0.55  thf(fact_2799_half__gt__zero__iff,axiom,
% 0.30/0.55      ! [A: rat] :
% 0.30/0.55        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.55        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.55  
% 0.30/0.55  % half_gt_zero_iff
% 0.30/0.55  thf(fact_2800_scaling__mono,axiom,
% 0.30/0.55      ! [U: real,V: real,R: real,S: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ U @ V )
% 0.30/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.30/0.55         => ( ( ord_less_eq_real @ R @ S )
% 0.30/0.55           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % scaling_mono
% 0.30/0.55  thf(fact_2801_scaling__mono,axiom,
% 0.30/0.55      ! [U: rat,V: rat,R: rat,S: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ U @ V )
% 0.30/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 0.30/0.55         => ( ( ord_less_eq_rat @ R @ S )
% 0.30/0.55           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % scaling_mono
% 0.30/0.55  thf(fact_2802_zero__le__power2,axiom,
% 0.30/0.55      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_power2
% 0.30/0.55  thf(fact_2803_zero__le__power2,axiom,
% 0.30/0.55      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_power2
% 0.30/0.55  thf(fact_2804_zero__le__power2,axiom,
% 0.30/0.55      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_power2
% 0.30/0.55  thf(fact_2805_power2__eq__imp__eq,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55          = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 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           => ( X3 = Y3 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_eq_imp_eq
% 0.30/0.55  thf(fact_2806_power2__eq__imp__eq,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55          = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 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           => ( X3 = Y3 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_eq_imp_eq
% 0.30/0.55  thf(fact_2807_power2__eq__imp__eq,axiom,
% 0.30/0.55      ! [X3: nat,Y3: nat] :
% 0.30/0.55        ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55          = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.30/0.55         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.55           => ( X3 = Y3 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_eq_imp_eq
% 0.30/0.55  thf(fact_2808_power2__eq__imp__eq,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55          = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.55           => ( X3 = Y3 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_eq_imp_eq
% 0.30/0.55  thf(fact_2809_power2__le__imp__le,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55         => ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_le_imp_le
% 0.30/0.55  thf(fact_2810_power2__le__imp__le,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55         => ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_le_imp_le
% 0.30/0.55  thf(fact_2811_power2__le__imp__le,axiom,
% 0.30/0.55      ! [X3: nat,Y3: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.55         => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_le_imp_le
% 0.30/0.55  thf(fact_2812_power2__le__imp__le,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.55         => ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_le_imp_le
% 0.30/0.55  thf(fact_2813_power2__less__0,axiom,
% 0.30/0.55      ! [A: real] :
% 0.30/0.55        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_0
% 0.30/0.55  thf(fact_2814_power2__less__0,axiom,
% 0.30/0.55      ! [A: rat] :
% 0.30/0.55        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_0
% 0.30/0.55  thf(fact_2815_power2__less__0,axiom,
% 0.30/0.55      ! [A: int] :
% 0.30/0.55        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_0
% 0.30/0.55  thf(fact_2816_exp__add__not__zero__imp__left,axiom,
% 0.30/0.55      ! [M: nat,N: nat] :
% 0.30/0.55        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55         != zero_zero_nat )
% 0.30/0.55       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.55         != zero_zero_nat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % exp_add_not_zero_imp_left
% 0.30/0.55  thf(fact_2817_exp__add__not__zero__imp__left,axiom,
% 0.30/0.55      ! [M: nat,N: nat] :
% 0.30/0.55        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55         != zero_zero_int )
% 0.30/0.55       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.30/0.55         != zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % exp_add_not_zero_imp_left
% 0.30/0.55  thf(fact_2818_exp__add__not__zero__imp__right,axiom,
% 0.30/0.55      ! [M: nat,N: nat] :
% 0.30/0.55        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55         != zero_zero_nat )
% 0.30/0.55       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.55         != zero_zero_nat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % exp_add_not_zero_imp_right
% 0.30/0.55  thf(fact_2819_exp__add__not__zero__imp__right,axiom,
% 0.30/0.55      ! [M: nat,N: nat] :
% 0.30/0.55        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.55         != zero_zero_int )
% 0.30/0.55       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.55         != zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % exp_add_not_zero_imp_right
% 0.30/0.55  thf(fact_2820_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.55         != zero_zero_nat )
% 0.30/0.55       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.55         != zero_zero_nat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.55  thf(fact_2821_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.55         != zero_zero_int )
% 0.30/0.55       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.55         != zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.55  thf(fact_2822_power__diff__power__eq,axiom,
% 0.30/0.55      ! [A: nat,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_nat )
% 0.30/0.55       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.55              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.30/0.55          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.30/0.55           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.55              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff_power_eq
% 0.30/0.55  thf(fact_2823_power__diff__power__eq,axiom,
% 0.30/0.55      ! [A: int,N: nat,M: nat] :
% 0.30/0.55        ( ( A != zero_zero_int )
% 0.30/0.55       => ( ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.55              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.30/0.55          & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.30/0.55           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.30/0.55              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_diff_power_eq
% 0.30/0.55  thf(fact_2824_less__2__cases,axiom,
% 0.30/0.55      ! [N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55       => ( ( N = zero_zero_nat )
% 0.30/0.55          | ( N
% 0.30/0.55            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % less_2_cases
% 0.30/0.55  thf(fact_2825_less__2__cases__iff,axiom,
% 0.30/0.55      ! [N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( ( N = zero_zero_nat )
% 0.30/0.55          | ( N
% 0.30/0.55            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % less_2_cases_iff
% 0.30/0.55  thf(fact_2826_power__eq__if,axiom,
% 0.30/0.55      ( power_power_complex
% 0.30/0.55      = ( ^ [P5: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_eq_if
% 0.30/0.55  thf(fact_2827_power__eq__if,axiom,
% 0.30/0.55      ( power_power_real
% 0.30/0.55      = ( ^ [P5: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_eq_if
% 0.30/0.55  thf(fact_2828_power__eq__if,axiom,
% 0.30/0.55      ( power_power_rat
% 0.30/0.55      = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_eq_if
% 0.30/0.55  thf(fact_2829_power__eq__if,axiom,
% 0.30/0.55      ( power_power_nat
% 0.30/0.55      = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_eq_if
% 0.30/0.55  thf(fact_2830_power__eq__if,axiom,
% 0.30/0.55      ( power_power_int
% 0.30/0.55      = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_eq_if
% 0.30/0.55  thf(fact_2831_power__minus__mult,axiom,
% 0.30/0.55      ! [N: nat,A: complex] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55          = ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_minus_mult
% 0.30/0.55  thf(fact_2832_power__minus__mult,axiom,
% 0.30/0.55      ! [N: nat,A: real] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55          = ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_minus_mult
% 0.30/0.55  thf(fact_2833_power__minus__mult,axiom,
% 0.30/0.55      ! [N: nat,A: rat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55          = ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_minus_mult
% 0.30/0.55  thf(fact_2834_power__minus__mult,axiom,
% 0.30/0.55      ! [N: nat,A: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55          = ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_minus_mult
% 0.30/0.55  thf(fact_2835_power__minus__mult,axiom,
% 0.30/0.55      ! [N: nat,A: int] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.30/0.55          = ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power_minus_mult
% 0.30/0.55  thf(fact_2836_le__div__geq,axiom,
% 0.30/0.55      ! [N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.55         => ( ( divide_divide_nat @ M @ N )
% 0.30/0.55            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % le_div_geq
% 0.30/0.55  thf(fact_2837_split__div_H,axiom,
% 0.30/0.55      ! [P: nat > $o,M: nat,N: nat] :
% 0.30/0.55        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.55        = ( ( ( N = zero_zero_nat )
% 0.30/0.55            & ( P @ zero_zero_nat ) )
% 0.30/0.55          | ? [Q4: nat] :
% 0.30/0.55              ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 0.30/0.55              & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 0.30/0.55              & ( P @ Q4 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % split_div'
% 0.30/0.55  thf(fact_2838_div__exp__mod__exp__eq,axiom,
% 0.30/0.55      ! [A: nat,N: nat,M: nat] :
% 0.30/0.55        ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55        = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_exp_mod_exp_eq
% 0.30/0.55  thf(fact_2839_div__exp__mod__exp__eq,axiom,
% 0.30/0.55      ! [A: int,N: nat,M: nat] :
% 0.30/0.55        ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55        = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_exp_mod_exp_eq
% 0.30/0.55  thf(fact_2840_div__exp__mod__exp__eq,axiom,
% 0.30/0.55      ! [A: code_integer,N: nat,M: nat] :
% 0.30/0.55        ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55        = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_exp_mod_exp_eq
% 0.30/0.55  thf(fact_2841_vebt__pred_Osimps_I6_J,axiom,
% 0.30/0.55      ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 0.30/0.55        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
% 0.30/0.55        = none_nat ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_pred.simps(6)
% 0.30/0.55  thf(fact_2842_power2__less__imp__less,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55         => ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_imp_less
% 0.30/0.55  thf(fact_2843_power2__less__imp__less,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55         => ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_imp_less
% 0.30/0.55  thf(fact_2844_power2__less__imp__less,axiom,
% 0.30/0.55      ! [X3: nat,Y3: nat] :
% 0.30/0.55        ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.30/0.55         => ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_imp_less
% 0.30/0.55  thf(fact_2845_power2__less__imp__less,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.55         => ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_less_imp_less
% 0.30/0.55  thf(fact_2846_sum__power2__ge__zero,axiom,
% 0.30/0.55      ! [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 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_ge_zero
% 0.30/0.55  thf(fact_2847_sum__power2__ge__zero,axiom,
% 0.30/0.55      ! [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 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_ge_zero
% 0.30/0.55  thf(fact_2848_sum__power2__ge__zero,axiom,
% 0.30/0.55      ! [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 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_ge_zero
% 0.30/0.55  thf(fact_2849_sum__power2__le__zero__iff,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ( ( 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 )
% 0.30/0.55        = ( ( X3 = zero_zero_real )
% 0.30/0.55          & ( Y3 = zero_zero_real ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_le_zero_iff
% 0.30/0.55  thf(fact_2850_sum__power2__le__zero__iff,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ( ( 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 )
% 0.30/0.55        = ( ( X3 = zero_zero_rat )
% 0.30/0.55          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_le_zero_iff
% 0.30/0.55  thf(fact_2851_sum__power2__le__zero__iff,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ( ( 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 )
% 0.30/0.55        = ( ( X3 = zero_zero_int )
% 0.30/0.55          & ( Y3 = zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_le_zero_iff
% 0.30/0.55  thf(fact_2852_not__sum__power2__lt__zero,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ~ ( 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 ) ).
% 0.30/0.55  
% 0.30/0.55  % not_sum_power2_lt_zero
% 0.30/0.55  thf(fact_2853_not__sum__power2__lt__zero,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ~ ( 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 ) ).
% 0.30/0.55  
% 0.30/0.55  % not_sum_power2_lt_zero
% 0.30/0.55  thf(fact_2854_not__sum__power2__lt__zero,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ~ ( 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 ) ).
% 0.30/0.55  
% 0.30/0.55  % not_sum_power2_lt_zero
% 0.30/0.55  thf(fact_2855_sum__power2__gt__zero__iff,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ( ( 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 ) ) ) ) )
% 0.30/0.55        = ( ( X3 != zero_zero_real )
% 0.30/0.55          | ( Y3 != zero_zero_real ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_gt_zero_iff
% 0.30/0.55  thf(fact_2856_sum__power2__gt__zero__iff,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ( ( 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 ) ) ) ) )
% 0.30/0.55        = ( ( X3 != zero_zero_rat )
% 0.30/0.55          | ( Y3 != zero_zero_rat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_gt_zero_iff
% 0.30/0.55  thf(fact_2857_sum__power2__gt__zero__iff,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ( ( 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 ) ) ) ) )
% 0.30/0.55        = ( ( X3 != zero_zero_int )
% 0.30/0.55          | ( Y3 != zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_power2_gt_zero_iff
% 0.30/0.55  thf(fact_2858_zero__le__even__power_H,axiom,
% 0.30/0.55      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_even_power'
% 0.30/0.55  thf(fact_2859_zero__le__even__power_H,axiom,
% 0.30/0.55      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_even_power'
% 0.30/0.55  thf(fact_2860_zero__le__even__power_H,axiom,
% 0.30/0.55      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_even_power'
% 0.30/0.55  thf(fact_2861_nat__bit__induct,axiom,
% 0.30/0.55      ! [P: nat > $o,N: nat] :
% 0.30/0.55        ( ( P @ zero_zero_nat )
% 0.30/0.55       => ( ! [N3: nat] :
% 0.30/0.55              ( ( P @ N3 )
% 0.30/0.55             => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.55               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.30/0.55         => ( ! [N3: nat] :
% 0.30/0.55                ( ( P @ N3 )
% 0.30/0.55               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.30/0.55           => ( P @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % nat_bit_induct
% 0.30/0.55  thf(fact_2862_div__2__gt__zero,axiom,
% 0.30/0.55      ! [N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.55       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_2_gt_zero
% 0.30/0.55  thf(fact_2863_Suc__n__div__2__gt__zero,axiom,
% 0.30/0.55      ! [N: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Suc_n_div_2_gt_zero
% 0.30/0.55  thf(fact_2864_mult__exp__mod__exp__eq,axiom,
% 0.30/0.55      ! [M: nat,N: nat,A: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.55       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.55          = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_exp_mod_exp_eq
% 0.30/0.55  thf(fact_2865_mult__exp__mod__exp__eq,axiom,
% 0.30/0.55      ! [M: nat,N: nat,A: int] :
% 0.30/0.55        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.55       => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.55          = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_exp_mod_exp_eq
% 0.30/0.55  thf(fact_2866_mult__exp__mod__exp__eq,axiom,
% 0.30/0.55      ! [M: nat,N: nat,A: code_integer] :
% 0.30/0.55        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.55       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.55          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % mult_exp_mod_exp_eq
% 0.30/0.55  thf(fact_2867_vebt__member_Oelims_I2_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                   => A3 )
% 0.30/0.55                  & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55                   => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55                       => B3 )
% 0.30/0.55                      & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55                    ( X3
% 0.30/0.55                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.55               => ~ ( ( Xa2 != Mi2 )
% 0.30/0.55                   => ( ( Xa2 != Ma2 )
% 0.30/0.55                     => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55                        & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55                         => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                             => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                                 => ( 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 ) ) ) ) ) )
% 0.30/0.55                                & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_member.elims(2)
% 0.30/0.55  thf(fact_2868_vebt__member_Oelims_I1_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.55        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.55          = Y3 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ( Y3
% 0.30/0.55                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                       => A3 )
% 0.30/0.55                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55                           => B3 )
% 0.30/0.55                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.30/0.55         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55             => Y3 )
% 0.30/0.55           => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55                    ( X3
% 0.30/0.55                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55               => Y3 )
% 0.30/0.55             => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55                 => Y3 )
% 0.30/0.55               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                      ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55                          ( X3
% 0.30/0.55                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.55                     => ( Y3
% 0.30/0.55                        = ( ~ ( ( Xa2 != Mi2 )
% 0.30/0.55                             => ( ( Xa2 != Ma2 )
% 0.30/0.55                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                                           => ( 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 ) ) ) ) ) )
% 0.30/0.55                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_member.elims(1)
% 0.30/0.55  thf(fact_2869_vebt__member_Oelims_I3_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.55       => ( ! [A3: $o,B3: $o] :
% 0.30/0.55              ( ( X3
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55             => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                 => A3 )
% 0.30/0.55                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.55                 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.55                     => B3 )
% 0.30/0.55                    & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.55         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.55                ( X3
% 0.30/0.55               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.55           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.55             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55                    ( X3
% 0.30/0.55                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.55               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                      ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.55                          ( X3
% 0.30/0.55                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.55                     => ( ( Xa2 != Mi2 )
% 0.30/0.55                       => ( ( Xa2 != Ma2 )
% 0.30/0.55                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.55                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                                     => ( 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 ) ) ) ) ) )
% 0.30/0.55                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_member.elims(3)
% 0.30/0.55  thf(fact_2870_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55      ! [A: real,N: 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 ) ) @ N ) ) ) )
% 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_2871_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55      ! [A: rat,N: 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 ) ) @ N ) ) ) )
% 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_2872_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55      ! [A: int,N: 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 ) ) @ N ) ) ) )
% 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_2873_odd__power__less__zero,axiom,
% 0.30/0.55      ! [A: real,N: 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 ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_power_less_zero
% 0.30/0.55  thf(fact_2874_odd__power__less__zero,axiom,
% 0.30/0.55      ! [A: rat,N: 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 ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_power_less_zero
% 0.30/0.55  thf(fact_2875_odd__power__less__zero,axiom,
% 0.30/0.55      ! [A: int,N: 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 ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_power_less_zero
% 0.30/0.55  thf(fact_2876_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.30/0.55      ! [X3: nat,N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55           => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.exp_split_high_low(1)
% 0.30/0.55  thf(fact_2877_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.30/0.55      ! [X3: nat,N: nat,M: nat] :
% 0.30/0.55        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.30/0.55       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.55         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55           => ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.exp_split_high_low(2)
% 0.30/0.55  thf(fact_2878_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.55        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.55       => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.55              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.55                  ( X3
% 0.30/0.55                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.55             => ~ ( ( Xa2 = Mi2 )
% 0.30/0.55                  | ( Xa2 = Ma2 ) ) )
% 0.30/0.55         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.55                    ( X3
% 0.30/0.55                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.55               => ~ ( ( Xa2 = Mi2 )
% 0.30/0.55                    | ( Xa2 = Ma2 )
% 0.30/0.55                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                       => ( 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 ) ) ) ) ) )
% 0.30/0.55                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.30/0.55           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.55                  ( ? [Vd2: vEBT_VEBT] :
% 0.30/0.55                      ( X3
% 0.30/0.55                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.30/0.55                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                       => ( 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 ) ) ) ) ) )
% 0.30/0.55                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.membermima.elims(2)
% 0.30/0.55  thf(fact_2879_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_2880_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_2881_invar__vebt_Osimps,axiom,
% 0.30/0.55      ( vEBT_invar_vebt
% 0.30/0.55      = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 0.30/0.55            ( ( ? [A5: $o,B4: $o] :
% 0.30/0.55                  ( A1
% 0.30/0.55                  = ( vEBT_Leaf @ A5 @ B4 ) )
% 0.30/0.55              & ( A22
% 0.30/0.55                = ( suc @ zero_zero_nat ) ) )
% 0.30/0.55            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 0.30/0.55                ( ( A1
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.30/0.55                & ! [X: vEBT_VEBT] :
% 0.30/0.55                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.30/0.55                & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 0.30/0.55                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.55                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.55                & ( A22
% 0.30/0.55                  = ( plus_plus_nat @ N2 @ N2 ) )
% 0.30/0.55                & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
% 0.30/0.55                & ! [X: vEBT_VEBT] :
% 0.30/0.55                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.55            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 0.30/0.55                ( ( A1
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.30/0.55                & ! [X: vEBT_VEBT] :
% 0.30/0.55                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.30/0.55                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 0.30/0.55                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.55                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.30/0.55                & ( A22
% 0.30/0.55                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.30/0.55                & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
% 0.30/0.55                & ! [X: vEBT_VEBT] :
% 0.30/0.55                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.55            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.30/0.55                ( ( A1
% 0.30/0.55                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.30/0.55                & ! [X: vEBT_VEBT] :
% 0.30/0.55                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.30/0.55                & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 0.30/0.55                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.55                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.55                & ( A22
% 0.30/0.55                  = ( plus_plus_nat @ N2 @ N2 ) )
% 0.30/0.55                & ! [I4: nat] :
% 0.30/0.55                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.55                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
% 0.30/0.55                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.30/0.55                & ( ( Mi3 = Ma3 )
% 0.30/0.55                 => ! [X: vEBT_VEBT] :
% 0.30/0.55                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.55                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.55                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.30/0.55                & ( ( Mi3 != Ma3 )
% 0.30/0.55                 => ! [I4: nat] :
% 0.30/0.55                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.55                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.30/0.55                            = I4 )
% 0.30/0.55                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.30/0.55                        & ! [X: nat] :
% 0.30/0.55                            ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 0.30/0.55                                = I4 )
% 0.30/0.55                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 0.30/0.55                           => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.55                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 0.30/0.55            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.30/0.55                ( ( A1
% 0.30/0.55                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.30/0.55                & ! [X: vEBT_VEBT] :
% 0.30/0.55                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.30/0.55                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 0.30/0.55                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.30/0.55                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.30/0.55                & ( A22
% 0.30/0.55                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.30/0.55                & ! [I4: nat] :
% 0.30/0.55                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.30/0.55                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
% 0.30/0.55                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.30/0.55                & ( ( Mi3 = Ma3 )
% 0.30/0.55                 => ! [X: vEBT_VEBT] :
% 0.30/0.55                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.30/0.55                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.55                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.55                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.30/0.55                & ( ( Mi3 != Ma3 )
% 0.30/0.55                 => ! [I4: nat] :
% 0.30/0.55                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.30/0.55                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.30/0.55                            = I4 )
% 0.30/0.55                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.30/0.55                        & ! [X: nat] :
% 0.30/0.55                            ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 0.30/0.55                                = I4 )
% 0.30/0.55                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 0.30/0.55                           => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.55                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % invar_vebt.simps
% 0.30/0.55  thf(fact_2882_invar__vebt_Ocases,axiom,
% 0.30/0.55      ! [A12: vEBT_VEBT,A23: nat] :
% 0.30/0.55        ( ( vEBT_invar_vebt @ A12 @ A23 )
% 0.30/0.55       => ( ( ? [A3: $o,B3: $o] :
% 0.30/0.55                ( A12
% 0.30/0.55                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55           => ( A23
% 0.30/0.55             != ( suc @ zero_zero_nat ) ) )
% 0.30/0.55         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.30/0.55                ( ( A12
% 0.30/0.55                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.55               => ( ( A23 = Deg2 )
% 0.30/0.55                 => ( ! [X6: vEBT_VEBT] :
% 0.30/0.55                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                       => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 0.30/0.55                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                       => ( ( M4 = N3 )
% 0.30/0.55                         => ( ( Deg2
% 0.30/0.55                              = ( plus_plus_nat @ N3 @ M4 ) )
% 0.30/0.55                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.30/0.55                             => ~ ! [X6: vEBT_VEBT] :
% 0.30/0.55                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.30/0.55           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.30/0.55                  ( ( A12
% 0.30/0.55                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.55                 => ( ( A23 = Deg2 )
% 0.30/0.55                   => ( ! [X6: vEBT_VEBT] :
% 0.30/0.55                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                         => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 0.30/0.55                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                         => ( ( M4
% 0.30/0.55                              = ( suc @ N3 ) )
% 0.30/0.55                           => ( ( Deg2
% 0.30/0.55                                = ( plus_plus_nat @ N3 @ M4 ) )
% 0.30/0.55                             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.30/0.55                               => ~ ! [X6: vEBT_VEBT] :
% 0.30/0.55                                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.30/0.55             => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.30/0.55                    ( ( A12
% 0.30/0.55                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.55                   => ( ( A23 = Deg2 )
% 0.30/0.55                     => ( ! [X6: vEBT_VEBT] :
% 0.30/0.55                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                           => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 0.30/0.55                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                           => ( ( M4 = N3 )
% 0.30/0.55                             => ( ( Deg2
% 0.30/0.55                                  = ( plus_plus_nat @ N3 @ M4 ) )
% 0.30/0.55                               => ( ! [I2: nat] :
% 0.30/0.55                                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
% 0.30/0.55                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.55                                 => ( ( ( Mi2 = Ma2 )
% 0.30/0.55                                     => ! [X6: vEBT_VEBT] :
% 0.30/0.55                                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
% 0.30/0.55                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.30/0.55                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.55                                       => ~ ( ( Mi2 != Ma2 )
% 0.30/0.55                                           => ! [I2: nat] :
% 0.30/0.55                                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 0.30/0.55                                                      = I2 )
% 0.30/0.55                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 0.30/0.55                                                  & ! [X6: nat] :
% 0.30/0.55                                                      ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
% 0.30/0.55                                                          = I2 )
% 0.30/0.55                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
% 0.30/0.55                                                     => ( ( ord_less_nat @ Mi2 @ X6 )
% 0.30/0.55                                                        & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.55               => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.30/0.55                      ( ( A12
% 0.30/0.55                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.55                     => ( ( A23 = Deg2 )
% 0.30/0.55                       => ( ! [X6: vEBT_VEBT] :
% 0.30/0.55                              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                             => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 0.30/0.55                         => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.30/0.55                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.55                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                             => ( ( M4
% 0.30/0.55                                  = ( suc @ N3 ) )
% 0.30/0.55                               => ( ( Deg2
% 0.30/0.55                                    = ( plus_plus_nat @ N3 @ M4 ) )
% 0.30/0.55                                 => ( ! [I2: nat] :
% 0.30/0.55                                        ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                                       => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
% 0.30/0.55                                          = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.55                                   => ( ( ( Mi2 = Ma2 )
% 0.30/0.55                                       => ! [X6: vEBT_VEBT] :
% 0.30/0.55                                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.55                                           => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
% 0.30/0.55                                     => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.30/0.55                                       => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.55                                         => ~ ( ( Mi2 != Ma2 )
% 0.30/0.55                                             => ! [I2: nat] :
% 0.30/0.55                                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.30/0.55                                                 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 0.30/0.55                                                        = I2 )
% 0.30/0.55                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 0.30/0.55                                                    & ! [X6: nat] :
% 0.30/0.55                                                        ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
% 0.30/0.55                                                            = I2 )
% 0.30/0.55                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
% 0.30/0.55                                                       => ( ( ord_less_nat @ Mi2 @ X6 )
% 0.30/0.55                                                          & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % invar_vebt.cases
% 0.30/0.55  thf(fact_2883_vebt__member_Osimps_I2_J,axiom,
% 0.30/0.55      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
% 0.30/0.55        ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_member.simps(2)
% 0.30/0.55  thf(fact_2884_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 0.30/0.55      ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.30/0.55        ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 0.30/0.55  
% 0.30/0.55  % VEBT_internal.minNull.simps(5)
% 0.30/0.55  thf(fact_2885_inrange,axiom,
% 0.30/0.55      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.55        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.55       => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % inrange
% 0.30/0.55  thf(fact_2886_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_2887_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_2888_vebt__pred_Opelims,axiom,
% 0.30/0.55      ! [X3: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.30/0.55        ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
% 0.30/0.55          = Y3 )
% 0.30/0.55       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.55         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.55                ( ( X3
% 0.30/0.55                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.55               => ( ( Xa2 = zero_zero_nat )
% 0.30/0.55                 => ( ( Y3 = none_nat )
% 0.30/0.55                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 0.30/0.55           => ( ! [A3: $o,Uw2: $o] :
% 0.30/0.55                  ( ( X3
% 0.30/0.55                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 0.30/0.55                 => ( ( Xa2
% 0.30/0.55                      = ( suc @ zero_zero_nat ) )
% 0.30/0.55                   => ( ( ( A3
% 0.30/0.55                         => ( Y3
% 0.30/0.55                            = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55                        & ( ~ A3
% 0.30/0.55                         => ( Y3 = none_nat ) ) )
% 0.30/0.55                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.30/0.55             => ( ! [A3: $o,B3: $o] :
% 0.30/0.55                    ( ( X3
% 0.30/0.55                      = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.55                   => ! [Va2: nat] :
% 0.30/0.55                        ( ( Xa2
% 0.30/0.55                          = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.55                       => ( ( ( B3
% 0.30/0.55                             => ( Y3
% 0.30/0.55                                = ( some_nat @ one_one_nat ) ) )
% 0.30/0.55                            & ( ~ B3
% 0.30/0.55                             => ( ( A3
% 0.30/0.55                                 => ( Y3
% 0.30/0.55                                    = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55                                & ( ~ A3
% 0.30/0.55                                 => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.55                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 0.30/0.55               => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.30/0.55                      ( ( X3
% 0.30/0.55                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.30/0.55                     => ( ( Y3 = none_nat )
% 0.30/0.55                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 0.30/0.55                 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 0.30/0.55                        ( ( X3
% 0.30/0.55                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 0.30/0.55                       => ( ( Y3 = none_nat )
% 0.30/0.55                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
% 0.30/0.55                   => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 0.30/0.55                          ( ( X3
% 0.30/0.55                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 0.30/0.55                         => ( ( Y3 = none_nat )
% 0.30/0.55                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
% 0.30/0.55                     => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.55                            ( ( X3
% 0.30/0.55                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.55                           => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                                 => ( Y3
% 0.30/0.55                                    = ( some_nat @ Ma2 ) ) )
% 0.30/0.55                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.55                                 => ( Y3
% 0.30/0.55                                    = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.55                                      @ ( if_option_nat
% 0.30/0.55                                        @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55                                           != none_nat )
% 0.30/0.55                                          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.55                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.55                                        @ ( if_option_nat
% 0.30/0.55                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.55                                            = none_nat )
% 0.30/0.55                                          @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.30/0.55                                          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.55                                      @ none_nat ) ) ) )
% 0.30/0.55                             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % vebt_pred.pelims
% 0.30/0.55  thf(fact_2889_Diff__eq__empty__iff,axiom,
% 0.30/0.55      ! [A4: set_nat,B5: set_nat] :
% 0.30/0.55        ( ( ( minus_minus_set_nat @ A4 @ B5 )
% 0.30/0.55          = bot_bot_set_nat )
% 0.30/0.55        = ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Diff_eq_empty_iff
% 0.30/0.55  thf(fact_2890_Diff__eq__empty__iff,axiom,
% 0.30/0.55      ! [A4: set_real,B5: set_real] :
% 0.30/0.55        ( ( ( minus_minus_set_real @ A4 @ B5 )
% 0.30/0.55          = bot_bot_set_real )
% 0.30/0.55        = ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Diff_eq_empty_iff
% 0.30/0.55  thf(fact_2891_Diff__eq__empty__iff,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( ( minus_minus_set_int @ A4 @ B5 )
% 0.30/0.55          = bot_bot_set_int )
% 0.30/0.55        = ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Diff_eq_empty_iff
% 0.30/0.55  thf(fact_2892_empty__subsetI,axiom,
% 0.30/0.55      ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% 0.30/0.55  
% 0.30/0.55  % empty_subsetI
% 0.30/0.55  thf(fact_2893_empty__subsetI,axiom,
% 0.30/0.55      ! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).
% 0.30/0.55  
% 0.30/0.55  % empty_subsetI
% 0.30/0.55  thf(fact_2894_empty__subsetI,axiom,
% 0.30/0.55      ! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).
% 0.30/0.55  
% 0.30/0.55  % empty_subsetI
% 0.30/0.55  thf(fact_2895_subset__empty,axiom,
% 0.30/0.55      ! [A4: set_nat] :
% 0.30/0.55        ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
% 0.30/0.55        = ( A4 = bot_bot_set_nat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_empty
% 0.30/0.55  thf(fact_2896_subset__empty,axiom,
% 0.30/0.55      ! [A4: set_real] :
% 0.30/0.55        ( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
% 0.30/0.55        = ( A4 = bot_bot_set_real ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_empty
% 0.30/0.55  thf(fact_2897_subset__empty,axiom,
% 0.30/0.55      ! [A4: set_int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
% 0.30/0.55        = ( A4 = bot_bot_set_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_empty
% 0.30/0.55  thf(fact_2898_double__eq__0__iff,axiom,
% 0.30/0.55      ! [A: real] :
% 0.30/0.55        ( ( ( plus_plus_real @ A @ A )
% 0.30/0.55          = zero_zero_real )
% 0.30/0.55        = ( A = zero_zero_real ) ) ).
% 0.30/0.55  
% 0.30/0.55  % double_eq_0_iff
% 0.30/0.55  thf(fact_2899_double__eq__0__iff,axiom,
% 0.30/0.55      ! [A: rat] :
% 0.30/0.55        ( ( ( plus_plus_rat @ A @ A )
% 0.30/0.55          = zero_zero_rat )
% 0.30/0.55        = ( A = zero_zero_rat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % double_eq_0_iff
% 0.30/0.55  thf(fact_2900_double__eq__0__iff,axiom,
% 0.30/0.55      ! [A: int] :
% 0.30/0.55        ( ( ( plus_plus_int @ A @ A )
% 0.30/0.55          = zero_zero_int )
% 0.30/0.55        = ( A = zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % double_eq_0_iff
% 0.30/0.55  thf(fact_2901_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_2902_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_2903_set__bit__Suc,axiom,
% 0.30/0.55      ! [N: nat,A: code_integer] :
% 0.30/0.55        ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.30/0.55        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_Suc
% 0.30/0.55  thf(fact_2904_set__bit__Suc,axiom,
% 0.30/0.55      ! [N: nat,A: int] :
% 0.30/0.55        ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.30/0.55        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_Suc
% 0.30/0.55  thf(fact_2905_set__bit__Suc,axiom,
% 0.30/0.55      ! [N: nat,A: nat] :
% 0.30/0.55        ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.30/0.55        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_Suc
% 0.30/0.55  thf(fact_2906_flip__bit__Suc,axiom,
% 0.30/0.55      ! [N: nat,A: code_integer] :
% 0.30/0.55        ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.30/0.55        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % flip_bit_Suc
% 0.30/0.55  thf(fact_2907_flip__bit__Suc,axiom,
% 0.30/0.55      ! [N: nat,A: int] :
% 0.30/0.55        ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.30/0.55        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % flip_bit_Suc
% 0.30/0.55  thf(fact_2908_flip__bit__Suc,axiom,
% 0.30/0.55      ! [N: nat,A: nat] :
% 0.30/0.55        ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.30/0.55        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % flip_bit_Suc
% 0.30/0.55  thf(fact_2909_subsetI,axiom,
% 0.30/0.55      ! [A4: set_nat,B5: set_nat] :
% 0.30/0.55        ( ! [X5: nat] :
% 0.30/0.55            ( ( member_nat @ X5 @ A4 )
% 0.30/0.55           => ( member_nat @ X5 @ B5 ) )
% 0.30/0.55       => ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetI
% 0.30/0.55  thf(fact_2910_subsetI,axiom,
% 0.30/0.55      ! [A4: set_real,B5: set_real] :
% 0.30/0.55        ( ! [X5: real] :
% 0.30/0.55            ( ( member_real @ X5 @ A4 )
% 0.30/0.55           => ( member_real @ X5 @ B5 ) )
% 0.30/0.55       => ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetI
% 0.30/0.55  thf(fact_2911_subsetI,axiom,
% 0.30/0.55      ! [A4: set_complex,B5: set_complex] :
% 0.30/0.55        ( ! [X5: complex] :
% 0.30/0.55            ( ( member_complex @ X5 @ A4 )
% 0.30/0.55           => ( member_complex @ X5 @ B5 ) )
% 0.30/0.55       => ( ord_le211207098394363844omplex @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetI
% 0.30/0.55  thf(fact_2912_subsetI,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ! [X5: int] :
% 0.30/0.55            ( ( member_int @ X5 @ A4 )
% 0.30/0.55           => ( member_int @ X5 @ B5 ) )
% 0.30/0.55       => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetI
% 0.30/0.55  thf(fact_2913_subset__antisym,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.55         => ( A4 = B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_antisym
% 0.30/0.55  thf(fact_2914_psubsetI,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55       => ( ( A4 != B5 )
% 0.30/0.55         => ( ord_less_set_int @ A4 @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % psubsetI
% 0.30/0.55  thf(fact_2915_div__pos__pos__trivial,axiom,
% 0.30/0.55      ! [K: int,L: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.55       => ( ( ord_less_int @ K @ L )
% 0.30/0.55         => ( ( divide_divide_int @ K @ L )
% 0.30/0.55            = zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_pos_pos_trivial
% 0.30/0.55  thf(fact_2916_div__neg__neg__trivial,axiom,
% 0.30/0.55      ! [K: int,L: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.55       => ( ( ord_less_int @ L @ K )
% 0.30/0.55         => ( ( divide_divide_int @ K @ L )
% 0.30/0.55            = zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_neg_neg_trivial
% 0.30/0.55  thf(fact_2917_i0__less,axiom,
% 0.30/0.55      ! [N: extended_enat] :
% 0.30/0.55        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.30/0.55        = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % i0_less
% 0.30/0.55  thf(fact_2918_idiff__0__right,axiom,
% 0.30/0.55      ! [N: extended_enat] :
% 0.30/0.55        ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.30/0.55        = N ) ).
% 0.30/0.55  
% 0.30/0.55  % idiff_0_right
% 0.30/0.55  thf(fact_2919_idiff__0,axiom,
% 0.30/0.55      ! [N: extended_enat] :
% 0.30/0.55        ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.30/0.55        = zero_z5237406670263579293d_enat ) ).
% 0.30/0.55  
% 0.30/0.55  % idiff_0
% 0.30/0.55  thf(fact_2920_not__real__square__gt__zero,axiom,
% 0.30/0.55      ! [X3: real] :
% 0.30/0.55        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
% 0.30/0.55        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.55  
% 0.30/0.55  % not_real_square_gt_zero
% 0.30/0.55  thf(fact_2921_zmod__numeral__Bit0,axiom,
% 0.30/0.55      ! [V: num,W: num] :
% 0.30/0.55        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.55        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zmod_numeral_Bit0
% 0.30/0.55  thf(fact_2922_half__nonnegative__int__iff,axiom,
% 0.30/0.55      ! [K: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.55        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.55  
% 0.30/0.55  % half_nonnegative_int_iff
% 0.30/0.55  thf(fact_2923_half__negative__int__iff,axiom,
% 0.30/0.55      ! [K: int] :
% 0.30/0.55        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.30/0.55        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % half_negative_int_iff
% 0.30/0.55  thf(fact_2924_psubset__imp__ex__mem,axiom,
% 0.30/0.55      ! [A4: set_nat,B5: set_nat] :
% 0.30/0.55        ( ( ord_less_set_nat @ A4 @ B5 )
% 0.30/0.55       => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % psubset_imp_ex_mem
% 0.30/0.55  thf(fact_2925_psubset__imp__ex__mem,axiom,
% 0.30/0.55      ! [A4: set_real,B5: set_real] :
% 0.30/0.55        ( ( ord_less_set_real @ A4 @ B5 )
% 0.30/0.55       => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % psubset_imp_ex_mem
% 0.30/0.55  thf(fact_2926_psubset__imp__ex__mem,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( ord_less_set_int @ A4 @ B5 )
% 0.30/0.55       => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % psubset_imp_ex_mem
% 0.30/0.55  thf(fact_2927_psubset__imp__ex__mem,axiom,
% 0.30/0.55      ! [A4: set_complex,B5: set_complex] :
% 0.30/0.55        ( ( ord_less_set_complex @ A4 @ B5 )
% 0.30/0.55       => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % psubset_imp_ex_mem
% 0.30/0.55  thf(fact_2928_zdiv__mono__strict,axiom,
% 0.30/0.55      ! [A4: int,B5: int,N: int] :
% 0.30/0.55        ( ( ord_less_int @ A4 @ B5 )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.30/0.55         => ( ( ( modulo_modulo_int @ A4 @ N )
% 0.30/0.55              = zero_zero_int )
% 0.30/0.55           => ( ( ( modulo_modulo_int @ B5 @ N )
% 0.30/0.55                = zero_zero_int )
% 0.30/0.55             => ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_mono_strict
% 0.30/0.55  thf(fact_2929_verit__le__mono__div__int,axiom,
% 0.30/0.55      ! [A4: int,B5: int,N: int] :
% 0.30/0.55        ( ( ord_less_int @ A4 @ B5 )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.30/0.55         => ( ord_less_eq_int
% 0.30/0.55            @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
% 0.30/0.55              @ ( if_int
% 0.30/0.55                @ ( ( modulo_modulo_int @ B5 @ N )
% 0.30/0.55                  = zero_zero_int )
% 0.30/0.55                @ one_one_int
% 0.30/0.55                @ zero_zero_int ) )
% 0.30/0.55            @ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % verit_le_mono_div_int
% 0.30/0.55  thf(fact_2930_zmod__zmult2__eq,axiom,
% 0.30/0.55      ! [C: int,A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.55          = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zmod_zmult2_eq
% 0.30/0.55  thf(fact_2931_split__pos__lemma,axiom,
% 0.30/0.55      ! [K: int,P: int > int > $o,N: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.30/0.55          = ( ! [I4: int,J3: int] :
% 0.30/0.55                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.55                  & ( ord_less_int @ J3 @ K )
% 0.30/0.55                  & ( N
% 0.30/0.55                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.55               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % split_pos_lemma
% 0.30/0.55  thf(fact_2932_split__neg__lemma,axiom,
% 0.30/0.55      ! [K: int,P: int > int > $o,N: int] :
% 0.30/0.55        ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.55       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.30/0.55          = ( ! [I4: int,J3: int] :
% 0.30/0.55                ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.55                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.55                  & ( N
% 0.30/0.55                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.55               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % split_neg_lemma
% 0.30/0.55  thf(fact_2933_div__mod__decomp__int,axiom,
% 0.30/0.55      ! [A4: int,N: int] :
% 0.30/0.55        ( A4
% 0.30/0.55        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_mod_decomp_int
% 0.30/0.55  thf(fact_2934_div__neg__pos__less0,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.55         => ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_neg_pos_less0
% 0.30/0.55  thf(fact_2935_neg__imp__zdiv__neg__iff,axiom,
% 0.30/0.55      ! [B2: int,A: int] :
% 0.30/0.55        ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.55       => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 0.30/0.55          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_imp_zdiv_neg_iff
% 0.30/0.55  thf(fact_2936_pos__imp__zdiv__neg__iff,axiom,
% 0.30/0.55      ! [B2: int,A: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.55       => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 0.30/0.55          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_imp_zdiv_neg_iff
% 0.30/0.55  thf(fact_2937_div__pos__geq,axiom,
% 0.30/0.55      ! [L: int,K: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.55       => ( ( ord_less_eq_int @ L @ K )
% 0.30/0.55         => ( ( divide_divide_int @ K @ L )
% 0.30/0.55            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_pos_geq
% 0.30/0.55  thf(fact_2938_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 0.30/0.55          = ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.55            & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % nonneg1_imp_zdiv_pos_iff
% 0.30/0.55  thf(fact_2939_pos__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.55      ! [B2: int,A: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 0.30/0.55          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_imp_zdiv_nonneg_iff
% 0.30/0.55  thf(fact_2940_neg__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.55      ! [B2: int,A: int] :
% 0.30/0.55        ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 0.30/0.55          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_imp_zdiv_nonneg_iff
% 0.30/0.55  thf(fact_2941_pos__imp__zdiv__pos__iff,axiom,
% 0.30/0.55      ! [K: int,I: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.30/0.55          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_imp_zdiv_pos_iff
% 0.30/0.55  thf(fact_2942_div__nonpos__pos__le0,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.55         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_nonpos_pos_le0
% 0.30/0.55  thf(fact_2943_div__nonneg__neg__le0,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.55         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_nonneg_neg_le0
% 0.30/0.55  thf(fact_2944_int__div__less__self,axiom,
% 0.30/0.55      ! [X3: int,K: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ X3 )
% 0.30/0.55       => ( ( ord_less_int @ one_one_int @ K )
% 0.30/0.55         => ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % int_div_less_self
% 0.30/0.55  thf(fact_2945_div__positive__int,axiom,
% 0.30/0.55      ! [L: int,K: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ L @ K )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.55         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_positive_int
% 0.30/0.55  thf(fact_2946_div__int__pos__iff,axiom,
% 0.30/0.55      ! [K: int,L: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.30/0.55        = ( ( K = zero_zero_int )
% 0.30/0.55          | ( L = zero_zero_int )
% 0.30/0.55          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.55            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.30/0.55          | ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.55            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_int_pos_iff
% 0.30/0.55  thf(fact_2947_zdiv__mono2__neg,axiom,
% 0.30/0.55      ! [A: int,B: int,B2: int] :
% 0.30/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55         => ( ( ord_less_eq_int @ B @ B2 )
% 0.30/0.55           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_mono2_neg
% 0.30/0.55  thf(fact_2948_zdiv__mono1__neg,axiom,
% 0.30/0.55      ! [A: int,A2: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ A @ A2 )
% 0.30/0.55       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.55         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_mono1_neg
% 0.30/0.55  thf(fact_2949_zdiv__eq__0__iff,axiom,
% 0.30/0.55      ! [I: int,K: int] :
% 0.30/0.55        ( ( ( divide_divide_int @ I @ K )
% 0.30/0.55          = zero_zero_int )
% 0.30/0.55        = ( ( K = zero_zero_int )
% 0.30/0.55          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.55            & ( ord_less_int @ I @ K ) )
% 0.30/0.55          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.30/0.55            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_eq_0_iff
% 0.30/0.55  thf(fact_2950_zdiv__mono2,axiom,
% 0.30/0.55      ! [A: int,B: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55         => ( ( ord_less_eq_int @ B @ B2 )
% 0.30/0.55           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_mono2
% 0.30/0.55  thf(fact_2951_zdiv__mono1,axiom,
% 0.30/0.55      ! [A: int,A2: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ A @ A2 )
% 0.30/0.55       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.55         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_mono1
% 0.30/0.55  thf(fact_2952_zdiv__zmult2__eq,axiom,
% 0.30/0.55      ! [C: int,A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.55          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zdiv_zmult2_eq
% 0.30/0.55  thf(fact_2953_int__div__pos__eq,axiom,
% 0.30/0.55      ! [A: int,B2: int,Q2: int,R: int] :
% 0.30/0.55        ( ( A
% 0.30/0.55          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 0.30/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.30/0.55         => ( ( ord_less_int @ R @ B2 )
% 0.30/0.55           => ( ( divide_divide_int @ A @ B2 )
% 0.30/0.55              = Q2 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % int_div_pos_eq
% 0.30/0.55  thf(fact_2954_int__div__neg__eq,axiom,
% 0.30/0.55      ! [A: int,B2: int,Q2: int,R: int] :
% 0.30/0.55        ( ( A
% 0.30/0.55          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 0.30/0.55       => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.30/0.55         => ( ( ord_less_int @ B2 @ R )
% 0.30/0.55           => ( ( divide_divide_int @ A @ B2 )
% 0.30/0.55              = Q2 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % int_div_neg_eq
% 0.30/0.55  thf(fact_2955_split__zdiv,axiom,
% 0.30/0.55      ! [P: int > $o,N: int,K: int] :
% 0.30/0.55        ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.30/0.55        = ( ( ( K = zero_zero_int )
% 0.30/0.55           => ( P @ zero_zero_int ) )
% 0.30/0.55          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.55           => ! [I4: int,J3: int] :
% 0.30/0.55                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.55                  & ( ord_less_int @ J3 @ K )
% 0.30/0.55                  & ( N
% 0.30/0.55                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.55               => ( P @ I4 ) ) )
% 0.30/0.55          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.55           => ! [I4: int,J3: int] :
% 0.30/0.55                ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.55                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.55                  & ( N
% 0.30/0.55                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.55               => ( P @ I4 ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % split_zdiv
% 0.30/0.55  thf(fact_2956_neg__zmod__mult__2,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.55       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.55          = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % neg_zmod_mult_2
% 0.30/0.55  thf(fact_2957_pos__zmod__mult__2,axiom,
% 0.30/0.55      ! [A: int,B2: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.55          = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos_zmod_mult_2
% 0.30/0.55  thf(fact_2958_enat__0__less__mult__iff,axiom,
% 0.30/0.55      ! [M: extended_enat,N: extended_enat] :
% 0.30/0.55        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.30/0.55        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.30/0.55          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % enat_0_less_mult_iff
% 0.30/0.55  thf(fact_2959_not__iless0,axiom,
% 0.30/0.55      ! [N: extended_enat] :
% 0.30/0.55        ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.30/0.55  
% 0.30/0.55  % not_iless0
% 0.30/0.55  thf(fact_2960_iadd__is__0,axiom,
% 0.30/0.55      ! [M: extended_enat,N: extended_enat] :
% 0.30/0.55        ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.30/0.55          = zero_z5237406670263579293d_enat )
% 0.30/0.55        = ( ( M = zero_z5237406670263579293d_enat )
% 0.30/0.55          & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % iadd_is_0
% 0.30/0.55  thf(fact_2961_i0__lb,axiom,
% 0.30/0.55      ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.30/0.55  
% 0.30/0.55  % i0_lb
% 0.30/0.55  thf(fact_2962_ile0__eq,axiom,
% 0.30/0.55      ! [N: extended_enat] :
% 0.30/0.55        ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.30/0.55        = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % ile0_eq
% 0.30/0.55  thf(fact_2963_ex__nat__less,axiom,
% 0.30/0.55      ! [N: nat,P: nat > $o] :
% 0.30/0.55        ( ( ? [M3: nat] :
% 0.30/0.55              ( ( ord_less_eq_nat @ M3 @ N )
% 0.30/0.55              & ( P @ M3 ) ) )
% 0.30/0.55        = ( ? [X: nat] :
% 0.30/0.55              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.55              & ( P @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % ex_nat_less
% 0.30/0.55  thf(fact_2964_all__nat__less,axiom,
% 0.30/0.55      ! [N: nat,P: nat > $o] :
% 0.30/0.55        ( ( ! [M3: nat] :
% 0.30/0.55              ( ( ord_less_eq_nat @ M3 @ N )
% 0.30/0.55             => ( P @ M3 ) ) )
% 0.30/0.55        = ( ! [X: nat] :
% 0.30/0.55              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.55             => ( P @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % all_nat_less
% 0.30/0.55  thf(fact_2965_not__psubset__empty,axiom,
% 0.30/0.55      ! [A4: set_nat] :
% 0.30/0.55        ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% 0.30/0.55  
% 0.30/0.55  % not_psubset_empty
% 0.30/0.55  thf(fact_2966_not__psubset__empty,axiom,
% 0.30/0.55      ! [A4: set_int] :
% 0.30/0.55        ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).
% 0.30/0.55  
% 0.30/0.55  % not_psubset_empty
% 0.30/0.55  thf(fact_2967_not__psubset__empty,axiom,
% 0.30/0.55      ! [A4: set_real] :
% 0.30/0.55        ~ ( ord_less_set_real @ A4 @ bot_bot_set_real ) ).
% 0.30/0.55  
% 0.30/0.55  % not_psubset_empty
% 0.30/0.55  thf(fact_2968_in__mono,axiom,
% 0.30/0.55      ! [A4: set_nat,B5: set_nat,X3: nat] :
% 0.30/0.55        ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.55       => ( ( member_nat @ X3 @ A4 )
% 0.30/0.55         => ( member_nat @ X3 @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % in_mono
% 0.30/0.55  thf(fact_2969_in__mono,axiom,
% 0.30/0.55      ! [A4: set_real,B5: set_real,X3: real] :
% 0.30/0.55        ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.55       => ( ( member_real @ X3 @ A4 )
% 0.30/0.55         => ( member_real @ X3 @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % in_mono
% 0.30/0.55  thf(fact_2970_in__mono,axiom,
% 0.30/0.55      ! [A4: set_complex,B5: set_complex,X3: complex] :
% 0.30/0.55        ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.55       => ( ( member_complex @ X3 @ A4 )
% 0.30/0.55         => ( member_complex @ X3 @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % in_mono
% 0.30/0.55  thf(fact_2971_in__mono,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int,X3: int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55       => ( ( member_int @ X3 @ A4 )
% 0.30/0.55         => ( member_int @ X3 @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % in_mono
% 0.30/0.55  thf(fact_2972_subsetD,axiom,
% 0.30/0.55      ! [A4: set_nat,B5: set_nat,C: nat] :
% 0.30/0.55        ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.55       => ( ( member_nat @ C @ A4 )
% 0.30/0.55         => ( member_nat @ C @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetD
% 0.30/0.55  thf(fact_2973_subsetD,axiom,
% 0.30/0.55      ! [A4: set_real,B5: set_real,C: real] :
% 0.30/0.55        ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.55       => ( ( member_real @ C @ A4 )
% 0.30/0.55         => ( member_real @ C @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetD
% 0.30/0.55  thf(fact_2974_subsetD,axiom,
% 0.30/0.55      ! [A4: set_complex,B5: set_complex,C: complex] :
% 0.30/0.55        ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.55       => ( ( member_complex @ C @ A4 )
% 0.30/0.55         => ( member_complex @ C @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetD
% 0.30/0.55  thf(fact_2975_subsetD,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int,C: int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55       => ( ( member_int @ C @ A4 )
% 0.30/0.55         => ( member_int @ C @ B5 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subsetD
% 0.30/0.55  thf(fact_2976_equalityE,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( A4 = B5 )
% 0.30/0.55       => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55           => ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % equalityE
% 0.30/0.55  thf(fact_2977_subset__eq,axiom,
% 0.30/0.55      ( ord_less_eq_set_nat
% 0.30/0.55      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.55          ! [X: nat] :
% 0.30/0.55            ( ( member_nat @ X @ A6 )
% 0.30/0.55           => ( member_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_eq
% 0.30/0.55  thf(fact_2978_subset__eq,axiom,
% 0.30/0.55      ( ord_less_eq_set_real
% 0.30/0.55      = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.55          ! [X: real] :
% 0.30/0.55            ( ( member_real @ X @ A6 )
% 0.30/0.55           => ( member_real @ X @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_eq
% 0.30/0.55  thf(fact_2979_subset__eq,axiom,
% 0.30/0.55      ( ord_le211207098394363844omplex
% 0.30/0.55      = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.55          ! [X: complex] :
% 0.30/0.55            ( ( member_complex @ X @ A6 )
% 0.30/0.55           => ( member_complex @ X @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_eq
% 0.30/0.55  thf(fact_2980_subset__eq,axiom,
% 0.30/0.55      ( ord_less_eq_set_int
% 0.30/0.55      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.55          ! [X: int] :
% 0.30/0.55            ( ( member_int @ X @ A6 )
% 0.30/0.55           => ( member_int @ X @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_eq
% 0.30/0.55  thf(fact_2981_equalityD1,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( A4 = B5 )
% 0.30/0.55       => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % equalityD1
% 0.30/0.55  thf(fact_2982_equalityD2,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] :
% 0.30/0.55        ( ( A4 = B5 )
% 0.30/0.55       => ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % equalityD2
% 0.30/0.55  thf(fact_2983_subset__iff,axiom,
% 0.30/0.55      ( ord_less_eq_set_nat
% 0.30/0.55      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.55          ! [T2: nat] :
% 0.30/0.55            ( ( member_nat @ T2 @ A6 )
% 0.30/0.55           => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_iff
% 0.30/0.55  thf(fact_2984_subset__iff,axiom,
% 0.30/0.55      ( ord_less_eq_set_real
% 0.30/0.55      = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.55          ! [T2: real] :
% 0.30/0.55            ( ( member_real @ T2 @ A6 )
% 0.30/0.55           => ( member_real @ T2 @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_iff
% 0.30/0.55  thf(fact_2985_subset__iff,axiom,
% 0.30/0.55      ( ord_le211207098394363844omplex
% 0.30/0.55      = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.55          ! [T2: complex] :
% 0.30/0.55            ( ( member_complex @ T2 @ A6 )
% 0.30/0.55           => ( member_complex @ T2 @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_iff
% 0.30/0.55  thf(fact_2986_subset__iff,axiom,
% 0.30/0.55      ( ord_less_eq_set_int
% 0.30/0.55      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.55          ! [T2: int] :
% 0.30/0.55            ( ( member_int @ T2 @ A6 )
% 0.30/0.55           => ( member_int @ T2 @ B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_iff
% 0.30/0.55  thf(fact_2987_subset__refl,axiom,
% 0.30/0.55      ! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_refl
% 0.30/0.55  thf(fact_2988_Collect__mono,axiom,
% 0.30/0.55      ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.55        ( ! [X5: complex] :
% 0.30/0.55            ( ( P @ X5 )
% 0.30/0.55           => ( Q @ X5 ) )
% 0.30/0.55       => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono
% 0.30/0.55  thf(fact_2989_Collect__mono,axiom,
% 0.30/0.55      ! [P: real > $o,Q: real > $o] :
% 0.30/0.55        ( ! [X5: real] :
% 0.30/0.55            ( ( P @ X5 )
% 0.30/0.55           => ( Q @ X5 ) )
% 0.30/0.55       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono
% 0.30/0.55  thf(fact_2990_Collect__mono,axiom,
% 0.30/0.55      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.55        ( ! [X5: list_nat] :
% 0.30/0.55            ( ( P @ X5 )
% 0.30/0.55           => ( Q @ X5 ) )
% 0.30/0.55       => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono
% 0.30/0.55  thf(fact_2991_Collect__mono,axiom,
% 0.30/0.55      ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.55        ( ! [X5: nat] :
% 0.30/0.55            ( ( P @ X5 )
% 0.30/0.55           => ( Q @ X5 ) )
% 0.30/0.55       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono
% 0.30/0.55  thf(fact_2992_Collect__mono,axiom,
% 0.30/0.55      ! [P: int > $o,Q: int > $o] :
% 0.30/0.55        ( ! [X5: int] :
% 0.30/0.55            ( ( P @ X5 )
% 0.30/0.55           => ( Q @ X5 ) )
% 0.30/0.55       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono
% 0.30/0.55  thf(fact_2993_subset__trans,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int,C4: set_int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55       => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 0.30/0.55         => ( ord_less_eq_set_int @ A4 @ C4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_trans
% 0.30/0.55  thf(fact_2994_set__eq__subset,axiom,
% 0.30/0.55      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.55      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.55            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.30/0.55            & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_eq_subset
% 0.30/0.55  thf(fact_2995_Collect__mono__iff,axiom,
% 0.30/0.55      ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.55        ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 0.30/0.55        = ( ! [X: complex] :
% 0.30/0.55              ( ( P @ X )
% 0.30/0.55             => ( Q @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono_iff
% 0.30/0.55  thf(fact_2996_Collect__mono__iff,axiom,
% 0.30/0.55      ! [P: real > $o,Q: real > $o] :
% 0.30/0.55        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.30/0.55        = ( ! [X: real] :
% 0.30/0.55              ( ( P @ X )
% 0.30/0.55             => ( Q @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono_iff
% 0.30/0.55  thf(fact_2997_Collect__mono__iff,axiom,
% 0.30/0.55      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.55        ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.30/0.55        = ( ! [X: list_nat] :
% 0.30/0.55              ( ( P @ X )
% 0.30/0.55             => ( Q @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono_iff
% 0.30/0.55  thf(fact_2998_Collect__mono__iff,axiom,
% 0.30/0.55      ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.55        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.30/0.55        = ( ! [X: nat] :
% 0.30/0.55              ( ( P @ X )
% 0.30/0.55             => ( Q @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono_iff
% 0.30/0.55  thf(fact_2999_Collect__mono__iff,axiom,
% 0.30/0.55      ! [P: int > $o,Q: int > $o] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.30/0.55        = ( ! [X: int] :
% 0.30/0.55              ( ( P @ X )
% 0.30/0.55             => ( Q @ X ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Collect_mono_iff
% 0.30/0.55  thf(fact_3000_Diff__mono,axiom,
% 0.30/0.55      ! [A4: set_int,C4: set_int,D4: set_int,B5: set_int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ C4 )
% 0.30/0.55       => ( ( ord_less_eq_set_int @ D4 @ B5 )
% 0.30/0.55         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % Diff_mono
% 0.30/0.55  thf(fact_3001_Diff__subset,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 ) ).
% 0.30/0.55  
% 0.30/0.55  % Diff_subset
% 0.30/0.55  thf(fact_3002_double__diff,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int,C4: set_int] :
% 0.30/0.55        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.55       => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 0.30/0.55         => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A4 ) )
% 0.30/0.55            = A4 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % double_diff
% 0.30/0.55  thf(fact_3003_subset__iff__psubset__eq,axiom,
% 0.30/0.55      ( ord_less_eq_set_int
% 0.30/0.55      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.55            ( ( ord_less_set_int @ A6 @ B6 )
% 0.30/0.55            | ( A6 = B6 ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % subset_iff_psubset_eq
% 0.30/0.55  thf(fact_3004_subset__psubset__trans,axiom,
% 0.30/0.55      ! [A4: set_int,B5: set_int,C4: set_int] :
% 0.30/0.56        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.56       => ( ( ord_less_set_int @ B5 @ C4 )
% 0.30/0.56         => ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % subset_psubset_trans
% 0.30/0.56  thf(fact_3005_subset__not__subset__eq,axiom,
% 0.30/0.56      ( ord_less_set_int
% 0.30/0.56      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.30/0.56            & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % subset_not_subset_eq
% 0.30/0.56  thf(fact_3006_psubset__subset__trans,axiom,
% 0.30/0.56      ! [A4: set_int,B5: set_int,C4: set_int] :
% 0.30/0.56        ( ( ord_less_set_int @ A4 @ B5 )
% 0.30/0.56       => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 0.30/0.56         => ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubset_subset_trans
% 0.30/0.56  thf(fact_3007_psubset__imp__subset,axiom,
% 0.30/0.56      ! [A4: set_int,B5: set_int] :
% 0.30/0.56        ( ( ord_less_set_int @ A4 @ B5 )
% 0.30/0.56       => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubset_imp_subset
% 0.30/0.56  thf(fact_3008_psubset__eq,axiom,
% 0.30/0.56      ( ord_less_set_int
% 0.30/0.56      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.30/0.56            & ( A6 != B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubset_eq
% 0.30/0.56  thf(fact_3009_psubsetE,axiom,
% 0.30/0.56      ! [A4: set_int,B5: set_int] :
% 0.30/0.56        ( ( ord_less_set_int @ A4 @ B5 )
% 0.30/0.56       => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.56           => ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubsetE
% 0.30/0.56  thf(fact_3010_not__exp__less__eq__0__int,axiom,
% 0.30/0.56      ! [N: nat] :
% 0.30/0.56        ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % not_exp_less_eq_0_int
% 0.30/0.56  thf(fact_3011_realpow__pos__nth2,axiom,
% 0.30/0.56      ! [A: real,N: nat] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.56       => ? [R3: real] :
% 0.30/0.56            ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.56            & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.30/0.56              = A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % realpow_pos_nth2
% 0.30/0.56  thf(fact_3012_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_3013_realpow__pos__nth__unique,axiom,
% 0.30/0.56      ! [N: nat,A: real] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.56       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.56         => ? [X5: real] :
% 0.30/0.56              ( ( ord_less_real @ zero_zero_real @ X5 )
% 0.30/0.56              & ( ( power_power_real @ X5 @ N )
% 0.30/0.56                = A )
% 0.30/0.56              & ! [Y5: real] :
% 0.30/0.56                  ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.30/0.56                    & ( ( power_power_real @ Y5 @ N )
% 0.30/0.56                      = A ) )
% 0.30/0.56                 => ( Y5 = X5 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % realpow_pos_nth_unique
% 0.30/0.56  thf(fact_3014_realpow__pos__nth,axiom,
% 0.30/0.56      ! [N: nat,A: real] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.56       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.56         => ? [R3: real] :
% 0.30/0.56              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.56              & ( ( power_power_real @ R3 @ N )
% 0.30/0.56                = A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % realpow_pos_nth
% 0.30/0.56  thf(fact_3015_neg__zdiv__mult__2,axiom,
% 0.30/0.56      ! [A: int,B2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.56       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.56          = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_zdiv_mult_2
% 0.30/0.56  thf(fact_3016_pos__zdiv__mult__2,axiom,
% 0.30/0.56      ! [A: int,B2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.56          = ( divide_divide_int @ B2 @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_zdiv_mult_2
% 0.30/0.56  thf(fact_3017_less__eq__set__def,axiom,
% 0.30/0.56      ( ord_less_eq_set_nat
% 0.30/0.56      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56            ( ord_less_eq_nat_o
% 0.30/0.56            @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.30/0.56            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_eq_set_def
% 0.30/0.56  thf(fact_3018_less__eq__set__def,axiom,
% 0.30/0.56      ( ord_less_eq_set_real
% 0.30/0.56      = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56            ( ord_less_eq_real_o
% 0.30/0.56            @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.30/0.56            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_eq_set_def
% 0.30/0.56  thf(fact_3019_less__eq__set__def,axiom,
% 0.30/0.56      ( ord_le211207098394363844omplex
% 0.30/0.56      = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56            ( ord_le4573692005234683329plex_o
% 0.30/0.56            @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 0.30/0.56            @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_eq_set_def
% 0.30/0.56  thf(fact_3020_less__eq__set__def,axiom,
% 0.30/0.56      ( ord_less_eq_set_int
% 0.30/0.56      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56            ( ord_less_eq_int_o
% 0.30/0.56            @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.30/0.56            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_eq_set_def
% 0.30/0.56  thf(fact_3021_Collect__subset,axiom,
% 0.30/0.56      ! [A4: set_complex,P: complex > $o] :
% 0.30/0.56        ( ord_le211207098394363844omplex
% 0.30/0.56        @ ( collect_complex
% 0.30/0.56          @ ^ [X: complex] :
% 0.30/0.56              ( ( member_complex @ X @ A4 )
% 0.30/0.56              & ( P @ X ) ) )
% 0.30/0.56        @ A4 ) ).
% 0.30/0.56  
% 0.30/0.56  % Collect_subset
% 0.30/0.56  thf(fact_3022_Collect__subset,axiom,
% 0.30/0.56      ! [A4: set_real,P: real > $o] :
% 0.30/0.56        ( ord_less_eq_set_real
% 0.30/0.56        @ ( collect_real
% 0.30/0.56          @ ^ [X: real] :
% 0.30/0.56              ( ( member_real @ X @ A4 )
% 0.30/0.56              & ( P @ X ) ) )
% 0.30/0.56        @ A4 ) ).
% 0.30/0.56  
% 0.30/0.56  % Collect_subset
% 0.30/0.56  thf(fact_3023_Collect__subset,axiom,
% 0.30/0.56      ! [A4: set_list_nat,P: list_nat > $o] :
% 0.30/0.56        ( ord_le6045566169113846134st_nat
% 0.30/0.56        @ ( collect_list_nat
% 0.30/0.56          @ ^ [X: list_nat] :
% 0.30/0.56              ( ( member_list_nat @ X @ A4 )
% 0.30/0.56              & ( P @ X ) ) )
% 0.30/0.56        @ A4 ) ).
% 0.30/0.56  
% 0.30/0.56  % Collect_subset
% 0.30/0.56  thf(fact_3024_Collect__subset,axiom,
% 0.30/0.56      ! [A4: set_nat,P: nat > $o] :
% 0.30/0.56        ( ord_less_eq_set_nat
% 0.30/0.56        @ ( collect_nat
% 0.30/0.56          @ ^ [X: nat] :
% 0.30/0.56              ( ( member_nat @ X @ A4 )
% 0.30/0.56              & ( P @ X ) ) )
% 0.30/0.56        @ A4 ) ).
% 0.30/0.56  
% 0.30/0.56  % Collect_subset
% 0.30/0.56  thf(fact_3025_Collect__subset,axiom,
% 0.30/0.56      ! [A4: set_int,P: int > $o] :
% 0.30/0.56        ( ord_less_eq_set_int
% 0.30/0.56        @ ( collect_int
% 0.30/0.56          @ ^ [X: int] :
% 0.30/0.56              ( ( member_int @ X @ A4 )
% 0.30/0.56              & ( P @ X ) ) )
% 0.30/0.56        @ A4 ) ).
% 0.30/0.56  
% 0.30/0.56  % Collect_subset
% 0.30/0.56  thf(fact_3026_int__power__div__base,axiom,
% 0.30/0.56      ! [M: nat,K: int] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.30/0.56            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_power_div_base
% 0.30/0.56  thf(fact_3027_unset__bit__Suc,axiom,
% 0.30/0.56      ! [N: nat,A: code_integer] :
% 0.30/0.56        ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.30/0.56        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unset_bit_Suc
% 0.30/0.56  thf(fact_3028_unset__bit__Suc,axiom,
% 0.30/0.56      ! [N: nat,A: int] :
% 0.30/0.56        ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.30/0.56        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unset_bit_Suc
% 0.30/0.56  thf(fact_3029_unset__bit__Suc,axiom,
% 0.30/0.56      ! [N: nat,A: nat] :
% 0.30/0.56        ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.30/0.56        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unset_bit_Suc
% 0.30/0.56  thf(fact_3030_vebt__member_Opelims_I3_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.56        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [A3: $o,B3: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.56               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.30/0.56                 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.56                     => A3 )
% 0.30/0.56                    & ( ( Xa2 != zero_zero_nat )
% 0.30/0.56                     => ( ( ( Xa2 = one_one_nat )
% 0.30/0.56                         => B3 )
% 0.30/0.56                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.56           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.56                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.30/0.56             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.56                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 0.30/0.56               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.56                      ( ( X3
% 0.30/0.56                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.56                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 0.30/0.56                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.56                        ( ( X3
% 0.30/0.56                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.56                       => ( ( 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 ) )
% 0.30/0.56                         => ( ( Xa2 != Mi2 )
% 0.30/0.56                           => ( ( Xa2 != Ma2 )
% 0.30/0.56                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.56                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.56                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.56                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.56                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                                         => ( 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 ) ) ) ) ) )
% 0.30/0.56                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % vebt_member.pelims(3)
% 0.30/0.56  thf(fact_3031_vebt__member_Opelims_I1_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.56        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.56          = Y3 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [A3: $o,B3: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.56               => ( ( Y3
% 0.30/0.56                    = ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.56                       => A3 )
% 0.30/0.56                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.56                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.56                           => B3 )
% 0.30/0.56                          & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.56                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 0.30/0.56           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.56                 => ( ~ Y3
% 0.30/0.56                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.56             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.56                   => ( ~ Y3
% 0.30/0.56                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.30/0.56               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.56                      ( ( X3
% 0.30/0.56                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.56                     => ( ~ Y3
% 0.30/0.56                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.56                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.56                        ( ( X3
% 0.30/0.56                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.56                       => ( ( Y3
% 0.30/0.56                            = ( ( Xa2 != Mi2 )
% 0.30/0.56                             => ( ( Xa2 != Ma2 )
% 0.30/0.56                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.56                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.56                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.56                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.56                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                                           => ( 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 ) ) ) ) ) )
% 0.30/0.56                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 0.30/0.56                         => ~ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % vebt_member.pelims(1)
% 0.30/0.56  thf(fact_3032_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.56        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [A3: $o,B3: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.56               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.30/0.56                 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.56                     => A3 )
% 0.30/0.56                    & ( ( Xa2 != zero_zero_nat )
% 0.30/0.56                     => ( ( ( Xa2 = one_one_nat )
% 0.30/0.56                         => B3 )
% 0.30/0.56                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.56           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.56                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.30/0.56             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.30/0.56                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
% 0.30/0.56                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                         => ( 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 ) ) ) ) ) )
% 0.30/0.56                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % VEBT_internal.naive_member.pelims(3)
% 0.30/0.56  thf(fact_3033_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.56        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [A3: $o,B3: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.56               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.30/0.56                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.56                       => A3 )
% 0.30/0.56                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.56                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.56                           => B3 )
% 0.30/0.56                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.56           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.30/0.56                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
% 0.30/0.56                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                         => ( 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 ) ) ) ) ) )
% 0.30/0.56                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % VEBT_internal.naive_member.pelims(2)
% 0.30/0.56  thf(fact_3034_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.56        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.56          = Y3 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [A3: $o,B3: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.56               => ( ( Y3
% 0.30/0.56                    = ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.56                       => A3 )
% 0.30/0.56                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.56                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.56                           => B3 )
% 0.30/0.56                          & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.56                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 0.30/0.56           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.56                 => ( ~ Y3
% 0.30/0.56                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.56             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.30/0.56                   => ( ( Y3
% 0.30/0.56                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                           => ( 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 ) ) ) ) ) )
% 0.30/0.56                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.30/0.56                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % VEBT_internal.naive_member.pelims(1)
% 0.30/0.56  thf(fact_3035_vebt__member_Opelims_I2_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.56        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [A3: $o,B3: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.56               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 0.30/0.56                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.56                       => A3 )
% 0.30/0.56                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.56                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.56                           => B3 )
% 0.30/0.56                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.56           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.56                 => ( ( 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 ) )
% 0.30/0.56                   => ~ ( ( Xa2 != Mi2 )
% 0.30/0.56                       => ( ( Xa2 != Ma2 )
% 0.30/0.56                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.56                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.56                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.56                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.56                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                                     => ( 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 ) ) ) ) ) )
% 0.30/0.56                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % vebt_member.pelims(2)
% 0.30/0.56  thf(fact_3036_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.56        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.56               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 0.30/0.56           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.56                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 0.30/0.56             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.56                   => ( ( 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 ) )
% 0.30/0.56                     => ( ( Xa2 = Mi2 )
% 0.30/0.56                        | ( Xa2 = Ma2 ) ) ) )
% 0.30/0.56               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.56                      ( ( X3
% 0.30/0.56                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.56                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.30/0.56                       => ( ( Xa2 = Mi2 )
% 0.30/0.56                          | ( Xa2 = Ma2 )
% 0.30/0.56                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                             => ( 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 ) ) ) ) ) )
% 0.30/0.56                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.30/0.56                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.56                        ( ( X3
% 0.30/0.56                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.30/0.56                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 0.30/0.56                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                             => ( 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 ) ) ) ) ) )
% 0.30/0.56                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % VEBT_internal.membermima.pelims(3)
% 0.30/0.56  thf(fact_3037_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.56        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.56          = Y3 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.56               => ( ~ Y3
% 0.30/0.56                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.30/0.56           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.56                 => ( ~ Y3
% 0.30/0.56                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 0.30/0.56             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.56                   => ( ( Y3
% 0.30/0.56                        = ( ( Xa2 = Mi2 )
% 0.30/0.56                          | ( Xa2 = Ma2 ) ) )
% 0.30/0.56                     => ~ ( 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 ) ) ) )
% 0.30/0.56               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.56                      ( ( X3
% 0.30/0.56                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.56                     => ( ( Y3
% 0.30/0.56                          = ( ( Xa2 = Mi2 )
% 0.30/0.56                            | ( Xa2 = Ma2 )
% 0.30/0.56                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                               => ( 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 ) ) ) ) ) )
% 0.30/0.56                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.30/0.56                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.56                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.56                        ( ( X3
% 0.30/0.56                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.30/0.56                       => ( ( Y3
% 0.30/0.56                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                               => ( 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 ) ) ) ) ) )
% 0.30/0.56                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.30/0.56                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % VEBT_internal.membermima.pelims(1)
% 0.30/0.56  thf(fact_3038_set__bit__nonnegative__int__iff,axiom,
% 0.30/0.56      ! [N: nat,K: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.30/0.56        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_bit_nonnegative_int_iff
% 0.30/0.56  thf(fact_3039_flip__bit__nonnegative__int__iff,axiom,
% 0.30/0.56      ! [N: nat,K: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.30/0.56        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.56  
% 0.30/0.56  % flip_bit_nonnegative_int_iff
% 0.30/0.56  thf(fact_3040_unset__bit__nonnegative__int__iff,axiom,
% 0.30/0.56      ! [N: nat,K: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.30/0.56        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unset_bit_nonnegative_int_iff
% 0.30/0.56  thf(fact_3041_set__bit__negative__int__iff,axiom,
% 0.30/0.56      ! [N: nat,K: int] :
% 0.30/0.56        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.56        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_bit_negative_int_iff
% 0.30/0.56  thf(fact_3042_flip__bit__negative__int__iff,axiom,
% 0.30/0.56      ! [N: nat,K: int] :
% 0.30/0.56        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.56        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % flip_bit_negative_int_iff
% 0.30/0.56  thf(fact_3043_unset__bit__negative__int__iff,axiom,
% 0.30/0.56      ! [N: nat,K: int] :
% 0.30/0.56        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ 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_3044_zle__add1__eq__le,axiom,
% 0.30/0.56      ! [W: int,Z: int] :
% 0.30/0.56        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.30/0.56        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zle_add1_eq_le
% 0.30/0.56  thf(fact_3045_zle__diff1__eq,axiom,
% 0.30/0.56      ! [W: int,Z: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.30/0.56        = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zle_diff1_eq
% 0.30/0.56  thf(fact_3046_mod__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         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56            = K ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_neg_neg_trivial
% 0.30/0.56  thf(fact_3047_mod__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         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56            = K ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_pos_pos_trivial
% 0.30/0.56  thf(fact_3048_split__zmod,axiom,
% 0.30/0.56      ! [P: int > $o,N: int,K: int] :
% 0.30/0.56        ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.30/0.56        = ( ( ( K = zero_zero_int )
% 0.30/0.56           => ( P @ N ) )
% 0.30/0.56          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56           => ! [I4: 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                  & ( N
% 0.30/0.56                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.56               => ( P @ J3 ) ) )
% 0.30/0.56          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.56           => ! [I4: 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                  & ( N
% 0.30/0.56                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.30/0.56               => ( P @ J3 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % split_zmod
% 0.30/0.56  thf(fact_3049_q__pos__lemma,axiom,
% 0.30/0.56      ! [B: int,Q5: int,R4: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.30/0.56       => ( ( ord_less_int @ R4 @ B )
% 0.30/0.56         => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56           => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % q_pos_lemma
% 0.30/0.56  thf(fact_3050_int__mod__neg__eq,axiom,
% 0.30/0.56      ! [A: int,B2: int,Q2: int,R: int] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.30/0.56         => ( ( ord_less_int @ B2 @ R )
% 0.30/0.56           => ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.56              = R ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_mod_neg_eq
% 0.30/0.56  thf(fact_3051_int__mod__pos__eq,axiom,
% 0.30/0.56      ! [A: int,B2: int,Q2: int,R: int] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.30/0.56         => ( ( ord_less_int @ R @ B2 )
% 0.30/0.56           => ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.56              = R ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_mod_pos_eq
% 0.30/0.56  thf(fact_3052_zdiv__mono2__lemma,axiom,
% 0.30/0.56      ! [B2: int,Q2: int,R: int,B: int,Q5: int,R4: int] :
% 0.30/0.56        ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R )
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.30/0.56         => ( ( ord_less_int @ R4 @ B )
% 0.30/0.56           => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.30/0.56             => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56               => ( ( ord_less_eq_int @ B @ B2 )
% 0.30/0.56                 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono2_lemma
% 0.30/0.56  thf(fact_3053_zdiv__mono2__neg__lemma,axiom,
% 0.30/0.56      ! [B2: int,Q2: int,R: int,B: int,Q5: int,R4: int] :
% 0.30/0.56        ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R )
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 0.30/0.56       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.30/0.56         => ( ( ord_less_int @ R @ B2 )
% 0.30/0.56           => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.30/0.56             => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56               => ( ( ord_less_eq_int @ B @ B2 )
% 0.30/0.56                 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono2_neg_lemma
% 0.30/0.56  thf(fact_3054_unique__quotient__lemma,axiom,
% 0.30/0.56      ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.30/0.56         => ( ( ord_less_int @ R4 @ B2 )
% 0.30/0.56           => ( ( ord_less_int @ R @ B2 )
% 0.30/0.56             => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_quotient_lemma
% 0.30/0.56  thf(fact_3055_unique__quotient__lemma__neg,axiom,
% 0.30/0.56      ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.30/0.56         => ( ( ord_less_int @ B2 @ R )
% 0.30/0.56           => ( ( ord_less_int @ B2 @ R4 )
% 0.30/0.56             => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_quotient_lemma_neg
% 0.30/0.56  thf(fact_3056_zmod__eq__0__iff,axiom,
% 0.30/0.56      ! [M: int,D: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ M @ D )
% 0.30/0.56          = zero_zero_int )
% 0.30/0.56        = ( ? [Q4: int] :
% 0.30/0.56              ( M
% 0.30/0.56              = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zmod_eq_0_iff
% 0.30/0.56  thf(fact_3057_zmod__eq__0D,axiom,
% 0.30/0.56      ! [M: int,D: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ M @ D )
% 0.30/0.56          = zero_zero_int )
% 0.30/0.56       => ? [Q3: int] :
% 0.30/0.56            ( M
% 0.30/0.56            = ( times_times_int @ D @ Q3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zmod_eq_0D
% 0.30/0.56  thf(fact_3058_pos__zmult__eq__1__iff,axiom,
% 0.30/0.56      ! [M: int,N: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.56       => ( ( ( times_times_int @ M @ N )
% 0.30/0.56            = one_one_int )
% 0.30/0.56          = ( ( M = one_one_int )
% 0.30/0.56            & ( N = one_one_int ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_zmult_eq_1_iff
% 0.30/0.56  thf(fact_3059_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_3060_neg__mod__conj,axiom,
% 0.30/0.56      ! [B2: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ B2 @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ zero_zero_int )
% 0.30/0.56          & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_mod_conj
% 0.30/0.56  thf(fact_3061_pos__mod__conj,axiom,
% 0.30/0.56      ! [B2: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.56          & ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_mod_conj
% 0.30/0.56  thf(fact_3062_zmod__trivial__iff,axiom,
% 0.30/0.56      ! [I: int,K: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ I @ K )
% 0.30/0.56          = I )
% 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  % zmod_trivial_iff
% 0.30/0.56  thf(fact_3063_int__one__le__iff__zero__less,axiom,
% 0.30/0.56      ! [Z: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.30/0.56        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_one_le_iff_zero_less
% 0.30/0.56  thf(fact_3064_zmod__le__nonneg__dividend,axiom,
% 0.30/0.56      ! [M: int,K: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.30/0.56       => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zmod_le_nonneg_dividend
% 0.30/0.56  thf(fact_3065_neg__mod__sign,axiom,
% 0.30/0.56      ! [L: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_mod_sign
% 0.30/0.56  thf(fact_3066_Euclidean__Division_Opos__mod__sign,axiom,
% 0.30/0.56      ! [L: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Euclidean_Division.pos_mod_sign
% 0.30/0.56  thf(fact_3067_zero__one__enat__neq_I1_J,axiom,
% 0.30/0.56      zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.30/0.56  
% 0.30/0.56  % zero_one_enat_neq(1)
% 0.30/0.56  thf(fact_3068_mod__pos__neg__trivial,axiom,
% 0.30/0.56      ! [K: int,L: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.30/0.56         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_pos_neg_trivial
% 0.30/0.56  thf(fact_3069_zless__imp__add1__zle,axiom,
% 0.30/0.56      ! [W: int,Z: int] :
% 0.30/0.56        ( ( ord_less_int @ W @ Z )
% 0.30/0.56       => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zless_imp_add1_zle
% 0.30/0.56  thf(fact_3070_int__less__induct,axiom,
% 0.30/0.56      ! [I: int,K: int,P: int > $o] :
% 0.30/0.56        ( ( ord_less_int @ I @ K )
% 0.30/0.56       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.30/0.56         => ( ! [I3: int] :
% 0.30/0.56                ( ( ord_less_int @ I3 @ K )
% 0.30/0.56               => ( ( P @ I3 )
% 0.30/0.56                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.56           => ( P @ I ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_less_induct
% 0.30/0.56  thf(fact_3071_mod__pos__geq,axiom,
% 0.30/0.56      ! [L: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56       => ( ( ord_less_eq_int @ L @ K )
% 0.30/0.56         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_pos_geq
% 0.30/0.56  thf(fact_3072_odd__less__0__iff,axiom,
% 0.30/0.56      ! [Z: int] :
% 0.30/0.56        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.30/0.56        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_less_0_iff
% 0.30/0.56  thf(fact_3073_zless__add1__eq,axiom,
% 0.30/0.56      ! [W: int,Z: int] :
% 0.30/0.56        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.30/0.56        = ( ( ord_less_int @ W @ Z )
% 0.30/0.56          | ( W = Z ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zless_add1_eq
% 0.30/0.56  thf(fact_3074_le__imp__0__less,axiom,
% 0.30/0.56      ! [Z: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.56       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % le_imp_0_less
% 0.30/0.56  thf(fact_3075_int__le__induct,axiom,
% 0.30/0.56      ! [I: int,K: int,P: int > $o] :
% 0.30/0.56        ( ( ord_less_eq_int @ I @ K )
% 0.30/0.56       => ( ( P @ K )
% 0.30/0.56         => ( ! [I3: int] :
% 0.30/0.56                ( ( ord_less_eq_int @ I3 @ K )
% 0.30/0.56               => ( ( P @ I3 )
% 0.30/0.56                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.56           => ( P @ I ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_le_induct
% 0.30/0.56  thf(fact_3076_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_3077_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_3078_add1__zle__eq,axiom,
% 0.30/0.56      ! [W: int,Z: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.30/0.56        = ( ord_less_int @ W @ Z ) ) ).
% 0.30/0.56  
% 0.30/0.56  % add1_zle_eq
% 0.30/0.56  thf(fact_3079_int__induct,axiom,
% 0.30/0.56      ! [P: int > $o,K: int,I: int] :
% 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         => ( ! [I3: int] :
% 0.30/0.56                ( ( ord_less_eq_int @ I3 @ K )
% 0.30/0.56               => ( ( P @ I3 )
% 0.30/0.56                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.56           => ( P @ I ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_induct
% 0.30/0.56  thf(fact_3080_Euclidean__Division_Opos__mod__bound,axiom,
% 0.30/0.56      ! [L: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Euclidean_Division.pos_mod_bound
% 0.30/0.56  thf(fact_3081_neg__mod__bound,axiom,
% 0.30/0.56      ! [L: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_mod_bound
% 0.30/0.56  thf(fact_3082_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_3083_unset__bit__less__eq,axiom,
% 0.30/0.56      ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.30/0.56  
% 0.30/0.56  % unset_bit_less_eq
% 0.30/0.56  thf(fact_3084_set__bit__greater__eq,axiom,
% 0.30/0.56      ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_bit_greater_eq
% 0.30/0.56  thf(fact_3085_verit__la__generic,axiom,
% 0.30/0.56      ! [A: int,X3: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ A @ X3 )
% 0.30/0.56        | ( A = X3 )
% 0.30/0.56        | ( ord_less_eq_int @ X3 @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % verit_la_generic
% 0.30/0.56  thf(fact_3086_less__eq__int__code_I1_J,axiom,
% 0.30/0.56      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.56  
% 0.30/0.56  % less_eq_int_code(1)
% 0.30/0.56  thf(fact_3087_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_3088_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_3089_imult__is__0,axiom,
% 0.30/0.56      ! [M: extended_enat,N: extended_enat] :
% 0.30/0.56        ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.30/0.56          = zero_z5237406670263579293d_enat )
% 0.30/0.56        = ( ( M = zero_z5237406670263579293d_enat )
% 0.30/0.56          | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % imult_is_0
% 0.30/0.56  thf(fact_3090_int__distrib_I1_J,axiom,
% 0.30/0.56      ! [Z1: int,Z22: int,W: int] :
% 0.30/0.56        ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.30/0.56        = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_distrib(1)
% 0.30/0.56  thf(fact_3091_int__distrib_I2_J,axiom,
% 0.30/0.56      ! [W: int,Z1: int,Z22: int] :
% 0.30/0.56        ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.30/0.56        = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_distrib(2)
% 0.30/0.56  thf(fact_3092_psubsetD,axiom,
% 0.30/0.56      ! [A4: set_nat,B5: set_nat,C: nat] :
% 0.30/0.56        ( ( ord_less_set_nat @ A4 @ B5 )
% 0.30/0.56       => ( ( member_nat @ C @ A4 )
% 0.30/0.56         => ( member_nat @ C @ B5 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubsetD
% 0.30/0.56  thf(fact_3093_psubsetD,axiom,
% 0.30/0.56      ! [A4: set_real,B5: set_real,C: real] :
% 0.30/0.56        ( ( ord_less_set_real @ A4 @ B5 )
% 0.30/0.56       => ( ( member_real @ C @ A4 )
% 0.30/0.56         => ( member_real @ C @ B5 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubsetD
% 0.30/0.56  thf(fact_3094_psubsetD,axiom,
% 0.30/0.56      ! [A4: set_int,B5: set_int,C: int] :
% 0.30/0.56        ( ( ord_less_set_int @ A4 @ B5 )
% 0.30/0.56       => ( ( member_int @ C @ A4 )
% 0.30/0.56         => ( member_int @ C @ B5 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubsetD
% 0.30/0.56  thf(fact_3095_psubsetD,axiom,
% 0.30/0.56      ! [A4: set_complex,B5: set_complex,C: complex] :
% 0.30/0.56        ( ( ord_less_set_complex @ A4 @ B5 )
% 0.30/0.56       => ( ( member_complex @ C @ A4 )
% 0.30/0.56         => ( member_complex @ C @ B5 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % psubsetD
% 0.30/0.56  thf(fact_3096_less__set__def,axiom,
% 0.30/0.56      ( ord_less_set_nat
% 0.30/0.56      = ( ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.56            ( ord_less_nat_o
% 0.30/0.56            @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.30/0.56            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_set_def
% 0.30/0.56  thf(fact_3097_less__set__def,axiom,
% 0.30/0.56      ( ord_less_set_real
% 0.30/0.56      = ( ^ [A6: set_real,B6: set_real] :
% 0.30/0.56            ( ord_less_real_o
% 0.30/0.56            @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.30/0.56            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_set_def
% 0.30/0.56  thf(fact_3098_less__set__def,axiom,
% 0.30/0.56      ( ord_less_set_int
% 0.30/0.56      = ( ^ [A6: set_int,B6: set_int] :
% 0.30/0.56            ( ord_less_int_o
% 0.30/0.56            @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.30/0.56            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_set_def
% 0.30/0.56  thf(fact_3099_less__set__def,axiom,
% 0.30/0.56      ( ord_less_set_complex
% 0.30/0.56      = ( ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.56            ( ord_less_complex_o
% 0.30/0.56            @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 0.30/0.56            @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_set_def
% 0.30/0.56  thf(fact_3100_int__distrib_I4_J,axiom,
% 0.30/0.56      ! [W: int,Z1: int,Z22: int] :
% 0.30/0.56        ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.30/0.56        = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_distrib(4)
% 0.30/0.56  thf(fact_3101_int__distrib_I3_J,axiom,
% 0.30/0.56      ! [Z1: int,Z22: int,W: int] :
% 0.30/0.56        ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.30/0.56        = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_distrib(3)
% 0.30/0.56  thf(fact_3102_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.30/0.56      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.56        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.56       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.56         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.56                ( ( X3
% 0.30/0.56                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.56               => ( ( 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 ) )
% 0.30/0.56                 => ~ ( ( Xa2 = Mi2 )
% 0.30/0.56                      | ( Xa2 = Ma2 ) ) ) )
% 0.30/0.56           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.56                  ( ( X3
% 0.30/0.56                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.56                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.30/0.56                   => ~ ( ( Xa2 = Mi2 )
% 0.30/0.56                        | ( Xa2 = Ma2 )
% 0.30/0.56                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                           => ( 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 ) ) ) ) ) )
% 0.30/0.56                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.30/0.56             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.56                    ( ( X3
% 0.30/0.56                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.30/0.56                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 0.30/0.56                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.56                           => ( 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 ) ) ) ) ) )
% 0.30/0.56                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % VEBT_internal.membermima.pelims(2)
% 0.30/0.56  thf(fact_3103_atLeastatMost__empty,axiom,
% 0.30/0.56      ! [B2: rat,A: rat] :
% 0.30/0.56        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.56       => ( ( set_or633870826150836451st_rat @ A @ B2 )
% 0.30/0.56          = bot_bot_set_rat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty
% 0.30/0.56  thf(fact_3104_atLeastatMost__empty,axiom,
% 0.30/0.56      ! [B2: num,A: num] :
% 0.30/0.56        ( ( ord_less_num @ B2 @ A )
% 0.30/0.56       => ( ( set_or7049704709247886629st_num @ A @ B2 )
% 0.30/0.56          = bot_bot_set_num ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty
% 0.30/0.56  thf(fact_3105_atLeastatMost__empty,axiom,
% 0.30/0.56      ! [B2: nat,A: nat] :
% 0.30/0.56        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.56       => ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 0.30/0.56          = bot_bot_set_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty
% 0.30/0.56  thf(fact_3106_atLeastatMost__empty,axiom,
% 0.30/0.56      ! [B2: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ B2 @ A )
% 0.30/0.56       => ( ( set_or1266510415728281911st_int @ A @ B2 )
% 0.30/0.56          = bot_bot_set_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty
% 0.30/0.56  thf(fact_3107_atLeastatMost__empty,axiom,
% 0.30/0.56      ! [B2: real,A: real] :
% 0.30/0.56        ( ( ord_less_real @ B2 @ A )
% 0.30/0.56       => ( ( set_or1222579329274155063t_real @ A @ B2 )
% 0.30/0.56          = bot_bot_set_real ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty
% 0.30/0.56  thf(fact_3108_atLeastatMost__subset__iff,axiom,
% 0.30/0.56      ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
% 0.30/0.56        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.56          | ( ( ord_less_eq_set_int @ C @ A )
% 0.30/0.56            & ( ord_less_eq_set_int @ B2 @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_subset_iff
% 0.30/0.56  thf(fact_3109_atLeastatMost__subset__iff,axiom,
% 0.30/0.56      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.56        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.56          | ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.56            & ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_subset_iff
% 0.30/0.56  thf(fact_3110_atLeastatMost__subset__iff,axiom,
% 0.30/0.56      ! [A: num,B2: num,C: num,D: num] :
% 0.30/0.56        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_num @ A @ B2 )
% 0.30/0.56          | ( ( ord_less_eq_num @ C @ A )
% 0.30/0.56            & ( ord_less_eq_num @ B2 @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_subset_iff
% 0.30/0.56  thf(fact_3111_atLeastatMost__subset__iff,axiom,
% 0.30/0.56      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.56        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.56          | ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.56            & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_subset_iff
% 0.30/0.56  thf(fact_3112_atLeastatMost__subset__iff,axiom,
% 0.30/0.56      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.56        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_int @ A @ B2 )
% 0.30/0.56          | ( ( ord_less_eq_int @ C @ A )
% 0.30/0.56            & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_subset_iff
% 0.30/0.56  thf(fact_3113_atLeastatMost__subset__iff,axiom,
% 0.30/0.56      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.56        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_real @ A @ B2 )
% 0.30/0.56          | ( ( ord_less_eq_real @ C @ A )
% 0.30/0.56            & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_subset_iff
% 0.30/0.56  thf(fact_3114_atLeastatMost__empty__iff2,axiom,
% 0.30/0.56      ! [A: set_int,B2: set_int] :
% 0.30/0.56        ( ( bot_bot_set_set_int
% 0.30/0.56          = ( set_or370866239135849197et_int @ A @ B2 ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff2
% 0.30/0.56  thf(fact_3115_atLeastatMost__empty__iff2,axiom,
% 0.30/0.56      ! [A: rat,B2: rat] :
% 0.30/0.56        ( ( bot_bot_set_rat
% 0.30/0.56          = ( set_or633870826150836451st_rat @ A @ B2 ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff2
% 0.30/0.56  thf(fact_3116_atLeastatMost__empty__iff2,axiom,
% 0.30/0.56      ! [A: num,B2: num] :
% 0.30/0.56        ( ( bot_bot_set_num
% 0.30/0.56          = ( set_or7049704709247886629st_num @ A @ B2 ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff2
% 0.30/0.56  thf(fact_3117_atLeastatMost__empty__iff2,axiom,
% 0.30/0.56      ! [A: nat,B2: nat] :
% 0.30/0.56        ( ( bot_bot_set_nat
% 0.30/0.56          = ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff2
% 0.30/0.56  thf(fact_3118_atLeastatMost__empty__iff2,axiom,
% 0.30/0.56      ! [A: int,B2: int] :
% 0.30/0.56        ( ( bot_bot_set_int
% 0.30/0.56          = ( set_or1266510415728281911st_int @ A @ B2 ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff2
% 0.30/0.56  thf(fact_3119_atLeastatMost__empty__iff2,axiom,
% 0.30/0.56      ! [A: real,B2: real] :
% 0.30/0.56        ( ( bot_bot_set_real
% 0.30/0.56          = ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.30/0.56        = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff2
% 0.30/0.56  thf(fact_3120_atLeastatMost__empty__iff,axiom,
% 0.30/0.56      ! [A: set_int,B2: set_int] :
% 0.30/0.56        ( ( ( set_or370866239135849197et_int @ A @ B2 )
% 0.30/0.56          = bot_bot_set_set_int )
% 0.30/0.56        = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff
% 0.30/0.56  thf(fact_3121_atLeastatMost__empty__iff,axiom,
% 0.30/0.56      ! [A: rat,B2: rat] :
% 0.30/0.56        ( ( ( set_or633870826150836451st_rat @ A @ B2 )
% 0.30/0.56          = bot_bot_set_rat )
% 0.30/0.56        = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff
% 0.30/0.56  thf(fact_3122_atLeastatMost__empty__iff,axiom,
% 0.30/0.56      ! [A: num,B2: num] :
% 0.30/0.56        ( ( ( set_or7049704709247886629st_num @ A @ B2 )
% 0.30/0.56          = bot_bot_set_num )
% 0.30/0.56        = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff
% 0.30/0.56  thf(fact_3123_atLeastatMost__empty__iff,axiom,
% 0.30/0.56      ! [A: nat,B2: nat] :
% 0.30/0.56        ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 0.30/0.56          = bot_bot_set_nat )
% 0.30/0.56        = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff
% 0.30/0.56  thf(fact_3124_atLeastatMost__empty__iff,axiom,
% 0.30/0.56      ! [A: int,B2: int] :
% 0.30/0.56        ( ( ( set_or1266510415728281911st_int @ A @ B2 )
% 0.30/0.56          = bot_bot_set_int )
% 0.30/0.56        = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff
% 0.30/0.56  thf(fact_3125_atLeastatMost__empty__iff,axiom,
% 0.30/0.56      ! [A: real,B2: real] :
% 0.30/0.56        ( ( ( set_or1222579329274155063t_real @ A @ B2 )
% 0.30/0.56          = bot_bot_set_real )
% 0.30/0.56        = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_empty_iff
% 0.30/0.56  thf(fact_3126_decr__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       => ( ! [X5: int] :
% 0.30/0.56              ( ( P @ X5 )
% 0.30/0.56             => ( P @ ( minus_minus_int @ X5 @ D ) ) )
% 0.30/0.56         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.56           => ! [X6: int] :
% 0.30/0.56                ( ( P @ X6 )
% 0.30/0.56               => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % decr_mult_lemma
% 0.30/0.56  thf(fact_3127_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       => ( ! [X5: int] :
% 0.30/0.56              ( ( P @ X5 )
% 0.30/0.56             => ( P @ ( plus_plus_int @ X5 @ D ) ) )
% 0.30/0.56         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.56           => ! [X6: int] :
% 0.30/0.56                ( ( P @ X6 )
% 0.30/0.56               => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % incr_mult_lemma
% 0.30/0.56  thf(fact_3128_atLeastAtMost__iff,axiom,
% 0.30/0.56      ! [I: set_int,L: set_int,U: set_int] :
% 0.30/0.56        ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 0.30/0.56        = ( ( ord_less_eq_set_int @ L @ I )
% 0.30/0.56          & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastAtMost_iff
% 0.30/0.56  thf(fact_3129_atLeastAtMost__iff,axiom,
% 0.30/0.56      ! [I: rat,L: rat,U: rat] :
% 0.30/0.56        ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 0.30/0.56        = ( ( ord_less_eq_rat @ L @ I )
% 0.30/0.56          & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastAtMost_iff
% 0.30/0.56  thf(fact_3130_atLeastAtMost__iff,axiom,
% 0.30/0.56      ! [I: num,L: num,U: num] :
% 0.30/0.56        ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 0.30/0.56        = ( ( ord_less_eq_num @ L @ I )
% 0.30/0.56          & ( ord_less_eq_num @ I @ U ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastAtMost_iff
% 0.30/0.56  thf(fact_3131_atLeastAtMost__iff,axiom,
% 0.30/0.56      ! [I: nat,L: nat,U: nat] :
% 0.30/0.56        ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.30/0.56        = ( ( ord_less_eq_nat @ L @ I )
% 0.30/0.56          & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastAtMost_iff
% 0.30/0.56  thf(fact_3132_atLeastAtMost__iff,axiom,
% 0.30/0.56      ! [I: int,L: int,U: int] :
% 0.30/0.56        ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.30/0.56        = ( ( ord_less_eq_int @ L @ I )
% 0.30/0.56          & ( ord_less_eq_int @ I @ U ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastAtMost_iff
% 0.30/0.56  thf(fact_3133_atLeastAtMost__iff,axiom,
% 0.30/0.56      ! [I: real,L: real,U: real] :
% 0.30/0.56        ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 0.30/0.56        = ( ( ord_less_eq_real @ L @ I )
% 0.30/0.56          & ( ord_less_eq_real @ I @ U ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastAtMost_iff
% 0.30/0.56  thf(fact_3134_Icc__eq__Icc,axiom,
% 0.30/0.56      ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 0.30/0.56        ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 0.30/0.56          = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 0.30/0.56        = ( ( ( L = L3 )
% 0.30/0.56            & ( H2 = H3 ) )
% 0.30/0.56          | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.30/0.56            & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Icc_eq_Icc
% 0.30/0.56  thf(fact_3135_Icc__eq__Icc,axiom,
% 0.30/0.56      ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 0.30/0.56        ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 0.30/0.56          = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.30/0.56        = ( ( ( L = L3 )
% 0.30/0.56            & ( H2 = H3 ) )
% 0.30/0.56          | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.30/0.56            & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Icc_eq_Icc
% 0.30/0.56  thf(fact_3136_Icc__eq__Icc,axiom,
% 0.30/0.56      ! [L: num,H2: num,L3: num,H3: num] :
% 0.30/0.56        ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 0.30/0.56          = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.30/0.56        = ( ( ( L = L3 )
% 0.30/0.56            & ( H2 = H3 ) )
% 0.30/0.56          | ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.30/0.56            & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Icc_eq_Icc
% 0.30/0.56  thf(fact_3137_Icc__eq__Icc,axiom,
% 0.30/0.56      ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 0.30/0.56        ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 0.30/0.56          = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.30/0.56        = ( ( ( L = L3 )
% 0.30/0.56            & ( H2 = H3 ) )
% 0.30/0.56          | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.30/0.56            & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Icc_eq_Icc
% 0.30/0.56  thf(fact_3138_Icc__eq__Icc,axiom,
% 0.30/0.56      ! [L: int,H2: int,L3: int,H3: int] :
% 0.30/0.56        ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 0.30/0.56          = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.30/0.56        = ( ( ( L = L3 )
% 0.30/0.56            & ( H2 = H3 ) )
% 0.30/0.56          | ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.30/0.56            & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Icc_eq_Icc
% 0.30/0.56  thf(fact_3139_Icc__eq__Icc,axiom,
% 0.30/0.56      ! [L: real,H2: real,L3: real,H3: real] :
% 0.30/0.56        ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 0.30/0.56          = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.30/0.56        = ( ( ( L = L3 )
% 0.30/0.56            & ( H2 = H3 ) )
% 0.30/0.56          | ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.30/0.56            & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Icc_eq_Icc
% 0.30/0.56  thf(fact_3140_minf_I7_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_real @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(7)
% 0.30/0.56  thf(fact_3141_minf_I7_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_rat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(7)
% 0.30/0.56  thf(fact_3142_minf_I7_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_num @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(7)
% 0.30/0.56  thf(fact_3143_minf_I7_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_nat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(7)
% 0.30/0.56  thf(fact_3144_minf_I7_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_int @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(7)
% 0.30/0.56  thf(fact_3145_minf_I5_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_real @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(5)
% 0.30/0.56  thf(fact_3146_minf_I5_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_rat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(5)
% 0.30/0.56  thf(fact_3147_minf_I5_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_num @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(5)
% 0.30/0.56  thf(fact_3148_minf_I5_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_nat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(5)
% 0.30/0.56  thf(fact_3149_minf_I5_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_int @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(5)
% 0.30/0.56  thf(fact_3150_minf_I4_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(4)
% 0.30/0.56  thf(fact_3151_minf_I4_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(4)
% 0.30/0.56  thf(fact_3152_minf_I4_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(4)
% 0.30/0.56  thf(fact_3153_minf_I4_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(4)
% 0.30/0.56  thf(fact_3154_minf_I4_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(4)
% 0.30/0.56  thf(fact_3155_minf_I3_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(3)
% 0.30/0.56  thf(fact_3156_minf_I3_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(3)
% 0.30/0.56  thf(fact_3157_minf_I3_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(3)
% 0.30/0.56  thf(fact_3158_minf_I3_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(3)
% 0.30/0.56  thf(fact_3159_minf_I3_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(3)
% 0.30/0.56  thf(fact_3160_minf_I2_J,axiom,
% 0.30/0.56      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56        ( ? [Z3: real] :
% 0.30/0.56          ! [X5: real] :
% 0.30/0.56            ( ( ord_less_real @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: real] :
% 0.30/0.56            ! [X5: real] :
% 0.30/0.56              ( ( ord_less_real @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: real] :
% 0.30/0.56            ! [X6: real] :
% 0.30/0.56              ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(2)
% 0.30/0.56  thf(fact_3161_minf_I2_J,axiom,
% 0.30/0.56      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56        ( ? [Z3: rat] :
% 0.30/0.56          ! [X5: rat] :
% 0.30/0.56            ( ( ord_less_rat @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: rat] :
% 0.30/0.56            ! [X5: rat] :
% 0.30/0.56              ( ( ord_less_rat @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: rat] :
% 0.30/0.56            ! [X6: rat] :
% 0.30/0.56              ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(2)
% 0.30/0.56  thf(fact_3162_minf_I2_J,axiom,
% 0.30/0.56      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56        ( ? [Z3: num] :
% 0.30/0.56          ! [X5: num] :
% 0.30/0.56            ( ( ord_less_num @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: num] :
% 0.30/0.56            ! [X5: num] :
% 0.30/0.56              ( ( ord_less_num @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: num] :
% 0.30/0.56            ! [X6: num] :
% 0.30/0.56              ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(2)
% 0.30/0.56  thf(fact_3163_minf_I2_J,axiom,
% 0.30/0.56      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56        ( ? [Z3: nat] :
% 0.30/0.56          ! [X5: nat] :
% 0.30/0.56            ( ( ord_less_nat @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: nat] :
% 0.30/0.56            ! [X5: nat] :
% 0.30/0.56              ( ( ord_less_nat @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: nat] :
% 0.30/0.56            ! [X6: nat] :
% 0.30/0.56              ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(2)
% 0.30/0.56  thf(fact_3164_minf_I2_J,axiom,
% 0.30/0.56      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56        ( ? [Z3: int] :
% 0.30/0.56          ! [X5: int] :
% 0.30/0.56            ( ( ord_less_int @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: int] :
% 0.30/0.56            ! [X5: int] :
% 0.30/0.56              ( ( ord_less_int @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: int] :
% 0.30/0.56            ! [X6: int] :
% 0.30/0.56              ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(2)
% 0.30/0.56  thf(fact_3165_minf_I1_J,axiom,
% 0.30/0.56      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56        ( ? [Z3: real] :
% 0.30/0.56          ! [X5: real] :
% 0.30/0.56            ( ( ord_less_real @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: real] :
% 0.30/0.56            ! [X5: real] :
% 0.30/0.56              ( ( ord_less_real @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: real] :
% 0.30/0.56            ! [X6: real] :
% 0.30/0.56              ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(1)
% 0.30/0.56  thf(fact_3166_minf_I1_J,axiom,
% 0.30/0.56      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56        ( ? [Z3: rat] :
% 0.30/0.56          ! [X5: rat] :
% 0.30/0.56            ( ( ord_less_rat @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: rat] :
% 0.30/0.56            ! [X5: rat] :
% 0.30/0.56              ( ( ord_less_rat @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: rat] :
% 0.30/0.56            ! [X6: rat] :
% 0.30/0.56              ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(1)
% 0.30/0.56  thf(fact_3167_minf_I1_J,axiom,
% 0.30/0.56      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56        ( ? [Z3: num] :
% 0.30/0.56          ! [X5: num] :
% 0.30/0.56            ( ( ord_less_num @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: num] :
% 0.30/0.56            ! [X5: num] :
% 0.30/0.56              ( ( ord_less_num @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: num] :
% 0.30/0.56            ! [X6: num] :
% 0.30/0.56              ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(1)
% 0.30/0.56  thf(fact_3168_minf_I1_J,axiom,
% 0.30/0.56      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56        ( ? [Z3: nat] :
% 0.30/0.56          ! [X5: nat] :
% 0.30/0.56            ( ( ord_less_nat @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: nat] :
% 0.30/0.56            ! [X5: nat] :
% 0.30/0.56              ( ( ord_less_nat @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: nat] :
% 0.30/0.56            ! [X6: nat] :
% 0.30/0.56              ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(1)
% 0.30/0.56  thf(fact_3169_minf_I1_J,axiom,
% 0.30/0.56      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56        ( ? [Z3: int] :
% 0.30/0.56          ! [X5: int] :
% 0.30/0.56            ( ( ord_less_int @ X5 @ Z3 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: int] :
% 0.30/0.56            ! [X5: int] :
% 0.30/0.56              ( ( ord_less_int @ X5 @ Z3 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: int] :
% 0.30/0.56            ! [X6: int] :
% 0.30/0.56              ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(1)
% 0.30/0.56  thf(fact_3170_pinf_I7_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_real @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(7)
% 0.30/0.56  thf(fact_3171_pinf_I7_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_rat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(7)
% 0.30/0.56  thf(fact_3172_pinf_I7_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_num @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(7)
% 0.30/0.56  thf(fact_3173_pinf_I7_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_nat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(7)
% 0.30/0.56  thf(fact_3174_pinf_I7_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_int @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(7)
% 0.30/0.56  thf(fact_3175_pinf_I5_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_real @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(5)
% 0.30/0.56  thf(fact_3176_pinf_I5_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_rat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(5)
% 0.30/0.56  thf(fact_3177_pinf_I5_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_num @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(5)
% 0.30/0.56  thf(fact_3178_pinf_I5_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_nat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(5)
% 0.30/0.56  thf(fact_3179_pinf_I5_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_int @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(5)
% 0.30/0.56  thf(fact_3180_pinf_I4_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(4)
% 0.30/0.56  thf(fact_3181_pinf_I4_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(4)
% 0.30/0.56  thf(fact_3182_pinf_I4_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(4)
% 0.30/0.56  thf(fact_3183_pinf_I4_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(4)
% 0.30/0.56  thf(fact_3184_pinf_I4_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(4)
% 0.30/0.56  thf(fact_3185_pinf_I3_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(3)
% 0.30/0.56  thf(fact_3186_pinf_I3_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(3)
% 0.30/0.56  thf(fact_3187_pinf_I3_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(3)
% 0.30/0.56  thf(fact_3188_pinf_I3_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(3)
% 0.30/0.56  thf(fact_3189_pinf_I3_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56       => ( X6 != T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(3)
% 0.30/0.56  thf(fact_3190_pinf_I2_J,axiom,
% 0.30/0.56      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56        ( ? [Z3: real] :
% 0.30/0.56          ! [X5: real] :
% 0.30/0.56            ( ( ord_less_real @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: real] :
% 0.30/0.56            ! [X5: real] :
% 0.30/0.56              ( ( ord_less_real @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: real] :
% 0.30/0.56            ! [X6: real] :
% 0.30/0.56              ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(2)
% 0.30/0.56  thf(fact_3191_pinf_I2_J,axiom,
% 0.30/0.56      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56        ( ? [Z3: rat] :
% 0.30/0.56          ! [X5: rat] :
% 0.30/0.56            ( ( ord_less_rat @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: rat] :
% 0.30/0.56            ! [X5: rat] :
% 0.30/0.56              ( ( ord_less_rat @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: rat] :
% 0.30/0.56            ! [X6: rat] :
% 0.30/0.56              ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(2)
% 0.30/0.56  thf(fact_3192_pinf_I2_J,axiom,
% 0.30/0.56      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56        ( ? [Z3: num] :
% 0.30/0.56          ! [X5: num] :
% 0.30/0.56            ( ( ord_less_num @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: num] :
% 0.30/0.56            ! [X5: num] :
% 0.30/0.56              ( ( ord_less_num @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: num] :
% 0.30/0.56            ! [X6: num] :
% 0.30/0.56              ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(2)
% 0.30/0.56  thf(fact_3193_pinf_I2_J,axiom,
% 0.30/0.56      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56        ( ? [Z3: nat] :
% 0.30/0.56          ! [X5: nat] :
% 0.30/0.56            ( ( ord_less_nat @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: nat] :
% 0.30/0.56            ! [X5: nat] :
% 0.30/0.56              ( ( ord_less_nat @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: nat] :
% 0.30/0.56            ! [X6: nat] :
% 0.30/0.56              ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(2)
% 0.30/0.56  thf(fact_3194_pinf_I2_J,axiom,
% 0.30/0.56      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56        ( ? [Z3: int] :
% 0.30/0.56          ! [X5: int] :
% 0.30/0.56            ( ( ord_less_int @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: int] :
% 0.30/0.56            ! [X5: int] :
% 0.30/0.56              ( ( ord_less_int @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: int] :
% 0.30/0.56            ! [X6: int] :
% 0.30/0.56              ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  | ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(2)
% 0.30/0.56  thf(fact_3195_pinf_I1_J,axiom,
% 0.30/0.56      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.30/0.56        ( ? [Z3: real] :
% 0.30/0.56          ! [X5: real] :
% 0.30/0.56            ( ( ord_less_real @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: real] :
% 0.30/0.56            ! [X5: real] :
% 0.30/0.56              ( ( ord_less_real @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: real] :
% 0.30/0.56            ! [X6: real] :
% 0.30/0.56              ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(1)
% 0.30/0.56  thf(fact_3196_pinf_I1_J,axiom,
% 0.30/0.56      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.30/0.56        ( ? [Z3: rat] :
% 0.30/0.56          ! [X5: rat] :
% 0.30/0.56            ( ( ord_less_rat @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: rat] :
% 0.30/0.56            ! [X5: rat] :
% 0.30/0.56              ( ( ord_less_rat @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: rat] :
% 0.30/0.56            ! [X6: rat] :
% 0.30/0.56              ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(1)
% 0.30/0.56  thf(fact_3197_pinf_I1_J,axiom,
% 0.30/0.56      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.30/0.56        ( ? [Z3: num] :
% 0.30/0.56          ! [X5: num] :
% 0.30/0.56            ( ( ord_less_num @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: num] :
% 0.30/0.56            ! [X5: num] :
% 0.30/0.56              ( ( ord_less_num @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: num] :
% 0.30/0.56            ! [X6: num] :
% 0.30/0.56              ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(1)
% 0.30/0.56  thf(fact_3198_pinf_I1_J,axiom,
% 0.30/0.56      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.30/0.56        ( ? [Z3: nat] :
% 0.30/0.56          ! [X5: nat] :
% 0.30/0.56            ( ( ord_less_nat @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: nat] :
% 0.30/0.56            ! [X5: nat] :
% 0.30/0.56              ( ( ord_less_nat @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: nat] :
% 0.30/0.56            ! [X6: nat] :
% 0.30/0.56              ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(1)
% 0.30/0.56  thf(fact_3199_pinf_I1_J,axiom,
% 0.30/0.56      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.30/0.56        ( ? [Z3: int] :
% 0.30/0.56          ! [X5: int] :
% 0.30/0.56            ( ( ord_less_int @ Z3 @ X5 )
% 0.30/0.56           => ( ( P @ X5 )
% 0.30/0.56              = ( P6 @ X5 ) ) )
% 0.30/0.56       => ( ? [Z3: int] :
% 0.30/0.56            ! [X5: int] :
% 0.30/0.56              ( ( ord_less_int @ Z3 @ X5 )
% 0.30/0.56             => ( ( Q @ X5 )
% 0.30/0.56                = ( Q6 @ X5 ) ) )
% 0.30/0.56         => ? [Z4: int] :
% 0.30/0.56            ! [X6: int] :
% 0.30/0.56              ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56             => ( ( ( P @ X6 )
% 0.30/0.56                  & ( Q @ X6 ) )
% 0.30/0.56                = ( ( P6 @ X6 )
% 0.30/0.56                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(1)
% 0.30/0.56  thf(fact_3200_bounded__Max__nat,axiom,
% 0.30/0.56      ! [P: nat > $o,X3: nat,M7: nat] :
% 0.30/0.56        ( ( P @ X3 )
% 0.30/0.56       => ( ! [X5: nat] :
% 0.30/0.56              ( ( P @ X5 )
% 0.30/0.56             => ( ord_less_eq_nat @ X5 @ M7 ) )
% 0.30/0.56         => ~ ! [M4: nat] :
% 0.30/0.56                ( ( P @ M4 )
% 0.30/0.56               => ~ ! [X6: nat] :
% 0.30/0.56                      ( ( P @ X6 )
% 0.30/0.56                     => ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bounded_Max_nat
% 0.30/0.56  thf(fact_3201_periodic__finite__ex,axiom,
% 0.30/0.56      ! [D: int,P: int > $o] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56       => ( ! [X5: int,K3: int] :
% 0.30/0.56              ( ( P @ X5 )
% 0.30/0.56              = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.30/0.56         => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.30/0.56            = ( ? [X: int] :
% 0.30/0.56                  ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.30/0.56                  & ( P @ X ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % periodic_finite_ex
% 0.30/0.56  thf(fact_3202_aset_I7_J,axiom,
% 0.30/0.56      ! [D4: int,A4: set_int,T: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ! [X6: int] :
% 0.30/0.56            ( ! [Xa3: int] :
% 0.30/0.56                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56               => ! [Xb3: int] :
% 0.30/0.56                    ( ( member_int @ Xb3 @ A4 )
% 0.30/0.56                   => ( X6
% 0.30/0.56                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56           => ( ( ord_less_int @ T @ X6 )
% 0.30/0.56             => ( ord_less_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % aset(7)
% 0.30/0.56  thf(fact_3203_aset_I5_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,A4: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ T @ A4 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ A4 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( ord_less_int @ X6 @ T )
% 0.30/0.56               => ( ord_less_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % aset(5)
% 0.30/0.56  thf(fact_3204_aset_I4_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,A4: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ T @ A4 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ A4 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( X6 != T )
% 0.30/0.56               => ( ( plus_plus_int @ X6 @ D4 )
% 0.30/0.56                 != T ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % aset(4)
% 0.30/0.56  thf(fact_3205_aset_I3_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,A4: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ A4 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( X6 = T )
% 0.30/0.56               => ( ( plus_plus_int @ X6 @ D4 )
% 0.30/0.56                  = T ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % aset(3)
% 0.30/0.56  thf(fact_3206_bset_I7_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,B5: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ T @ B5 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ B5 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( ord_less_int @ T @ X6 )
% 0.30/0.56               => ( ord_less_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bset(7)
% 0.30/0.56  thf(fact_3207_bset_I5_J,axiom,
% 0.30/0.56      ! [D4: int,B5: set_int,T: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ! [X6: int] :
% 0.30/0.56            ( ! [Xa3: int] :
% 0.30/0.56                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56               => ! [Xb3: int] :
% 0.30/0.56                    ( ( member_int @ Xb3 @ B5 )
% 0.30/0.56                   => ( X6
% 0.30/0.56                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56           => ( ( ord_less_int @ X6 @ T )
% 0.30/0.56             => ( ord_less_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bset(5)
% 0.30/0.56  thf(fact_3208_bset_I4_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,B5: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ T @ B5 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ B5 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( X6 != T )
% 0.30/0.56               => ( ( minus_minus_int @ X6 @ D4 )
% 0.30/0.56                 != T ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bset(4)
% 0.30/0.56  thf(fact_3209_bset_I3_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,B5: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ B5 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( X6 = T )
% 0.30/0.56               => ( ( minus_minus_int @ X6 @ D4 )
% 0.30/0.56                  = T ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bset(3)
% 0.30/0.56  thf(fact_3210_aset_I8_J,axiom,
% 0.30/0.56      ! [D4: int,A4: set_int,T: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ! [X6: int] :
% 0.30/0.56            ( ! [Xa3: int] :
% 0.30/0.56                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56               => ! [Xb3: int] :
% 0.30/0.56                    ( ( member_int @ Xb3 @ A4 )
% 0.30/0.56                   => ( X6
% 0.30/0.56                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56           => ( ( ord_less_eq_int @ T @ X6 )
% 0.30/0.56             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % aset(8)
% 0.30/0.56  thf(fact_3211_aset_I6_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,A4: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ A4 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( ord_less_eq_int @ X6 @ T )
% 0.30/0.56               => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % aset(6)
% 0.30/0.56  thf(fact_3212_bset_I8_J,axiom,
% 0.30/0.56      ! [D4: int,T: int,B5: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 0.30/0.56         => ! [X6: int] :
% 0.30/0.56              ( ! [Xa3: int] :
% 0.30/0.56                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                 => ! [Xb3: int] :
% 0.30/0.56                      ( ( member_int @ Xb3 @ B5 )
% 0.30/0.56                     => ( X6
% 0.30/0.56                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56             => ( ( ord_less_eq_int @ T @ X6 )
% 0.30/0.56               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bset(8)
% 0.30/0.56  thf(fact_3213_bset_I6_J,axiom,
% 0.30/0.56      ! [D4: int,B5: set_int,T: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ! [X6: int] :
% 0.30/0.56            ( ! [Xa3: int] :
% 0.30/0.56                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56               => ! [Xb3: int] :
% 0.30/0.56                    ( ( member_int @ Xb3 @ B5 )
% 0.30/0.56                   => ( X6
% 0.30/0.56                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.30/0.56           => ( ( ord_less_eq_int @ X6 @ T )
% 0.30/0.56             => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % bset(6)
% 0.30/0.56  thf(fact_3214_cppi,axiom,
% 0.30/0.56      ! [D4: int,P: int > $o,P6: int > $o,A4: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ? [Z3: int] :
% 0.30/0.56            ! [X5: int] :
% 0.30/0.56              ( ( ord_less_int @ Z3 @ X5 )
% 0.30/0.56             => ( ( P @ X5 )
% 0.30/0.56                = ( P6 @ X5 ) ) )
% 0.30/0.56         => ( ! [X5: int] :
% 0.30/0.56                ( ! [Xa: int] :
% 0.30/0.56                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                   => ! [Xb: int] :
% 0.30/0.56                        ( ( member_int @ Xb @ A4 )
% 0.30/0.56                       => ( X5
% 0.30/0.56                         != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.30/0.56               => ( ( P @ X5 )
% 0.30/0.56                 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.30/0.56           => ( ! [X5: int,K3: int] :
% 0.30/0.56                  ( ( P6 @ X5 )
% 0.30/0.56                  = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56             => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.30/0.56                = ( ? [X: int] :
% 0.30/0.56                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                      & ( P6 @ X ) )
% 0.30/0.56                  | ? [X: int] :
% 0.30/0.56                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                      & ? [Y: int] :
% 0.30/0.56                          ( ( member_int @ Y @ A4 )
% 0.30/0.56                          & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % cppi
% 0.30/0.56  thf(fact_3215_cpmi,axiom,
% 0.30/0.56      ! [D4: int,P: int > $o,P6: int > $o,B5: set_int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.30/0.56       => ( ? [Z3: int] :
% 0.30/0.56            ! [X5: int] :
% 0.30/0.56              ( ( ord_less_int @ X5 @ Z3 )
% 0.30/0.56             => ( ( P @ X5 )
% 0.30/0.56                = ( P6 @ X5 ) ) )
% 0.30/0.56         => ( ! [X5: int] :
% 0.30/0.56                ( ! [Xa: int] :
% 0.30/0.56                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                   => ! [Xb: int] :
% 0.30/0.56                        ( ( member_int @ Xb @ B5 )
% 0.30/0.56                       => ( X5
% 0.30/0.56                         != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.30/0.56               => ( ( P @ X5 )
% 0.30/0.56                 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.30/0.56           => ( ! [X5: int,K3: int] :
% 0.30/0.56                  ( ( P6 @ X5 )
% 0.30/0.56                  = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56             => ( ( ? [X7: int] : ( P @ X7 ) )
% 0.30/0.56                = ( ? [X: int] :
% 0.30/0.56                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                      & ( P6 @ X ) )
% 0.30/0.56                  | ? [X: int] :
% 0.30/0.56                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.30/0.56                      & ? [Y: int] :
% 0.30/0.56                          ( ( member_int @ Y @ B5 )
% 0.30/0.56                          & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % cpmi
% 0.30/0.56  thf(fact_3216_minf_I8_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(8)
% 0.30/0.56  thf(fact_3217_minf_I8_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_eq_rat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(8)
% 0.30/0.56  thf(fact_3218_minf_I8_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(8)
% 0.30/0.56  thf(fact_3219_minf_I8_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(8)
% 0.30/0.56  thf(fact_3220_minf_I8_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56       => ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(8)
% 0.30/0.56  thf(fact_3221_minf_I6_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_eq_real @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(6)
% 0.30/0.56  thf(fact_3222_minf_I6_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_eq_rat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(6)
% 0.30/0.56  thf(fact_3223_minf_I6_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_eq_num @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(6)
% 0.30/0.56  thf(fact_3224_minf_I6_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_eq_nat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(6)
% 0.30/0.56  thf(fact_3225_minf_I6_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.56       => ( ord_less_eq_int @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(6)
% 0.30/0.56  thf(fact_3226_pinf_I8_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_eq_real @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(8)
% 0.30/0.56  thf(fact_3227_pinf_I8_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_eq_rat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(8)
% 0.30/0.56  thf(fact_3228_pinf_I8_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_eq_num @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(8)
% 0.30/0.56  thf(fact_3229_pinf_I8_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_eq_nat @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(8)
% 0.30/0.56  thf(fact_3230_pinf_I8_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56       => ( ord_less_eq_int @ T @ X6 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(8)
% 0.30/0.56  thf(fact_3231_pinf_I6_J,axiom,
% 0.30/0.56      ! [T: real] :
% 0.30/0.56      ? [Z4: real] :
% 0.30/0.56      ! [X6: real] :
% 0.30/0.56        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(6)
% 0.30/0.56  thf(fact_3232_pinf_I6_J,axiom,
% 0.30/0.56      ! [T: rat] :
% 0.30/0.56      ? [Z4: rat] :
% 0.30/0.56      ! [X6: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_eq_rat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(6)
% 0.30/0.56  thf(fact_3233_pinf_I6_J,axiom,
% 0.30/0.56      ! [T: num] :
% 0.30/0.56      ? [Z4: num] :
% 0.30/0.56      ! [X6: num] :
% 0.30/0.56        ( ( ord_less_num @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(6)
% 0.30/0.56  thf(fact_3234_pinf_I6_J,axiom,
% 0.30/0.56      ! [T: nat] :
% 0.30/0.56      ? [Z4: nat] :
% 0.30/0.56      ! [X6: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(6)
% 0.30/0.56  thf(fact_3235_pinf_I6_J,axiom,
% 0.30/0.56      ! [T: int] :
% 0.30/0.56      ? [Z4: int] :
% 0.30/0.56      ! [X6: int] :
% 0.30/0.56        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.56       => ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(6)
% 0.30/0.56  thf(fact_3236_inf__period_I2_J,axiom,
% 0.30/0.56      ! [P: complex > $o,D4: complex,Q: complex > $o] :
% 0.30/0.56        ( ! [X5: complex,K3: complex] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: complex,K3: complex] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: complex,K4: complex] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                | ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
% 0.30/0.56                | ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(2)
% 0.30/0.56  thf(fact_3237_inf__period_I2_J,axiom,
% 0.30/0.56      ! [P: real > $o,D4: real,Q: real > $o] :
% 0.30/0.56        ( ! [X5: real,K3: real] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: real,K3: real] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: real,K4: real] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                | ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.30/0.56                | ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(2)
% 0.30/0.56  thf(fact_3238_inf__period_I2_J,axiom,
% 0.30/0.56      ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.30/0.56        ( ! [X5: rat,K3: rat] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: rat,K3: rat] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: rat,K4: rat] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                | ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.30/0.56                | ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(2)
% 0.30/0.56  thf(fact_3239_inf__period_I2_J,axiom,
% 0.30/0.56      ! [P: int > $o,D4: int,Q: int > $o] :
% 0.30/0.56        ( ! [X5: int,K3: int] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: int,K3: int] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: int,K4: int] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                | ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.30/0.56                | ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(2)
% 0.30/0.56  thf(fact_3240_inf__period_I1_J,axiom,
% 0.30/0.56      ! [P: complex > $o,D4: complex,Q: complex > $o] :
% 0.30/0.56        ( ! [X5: complex,K3: complex] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: complex,K3: complex] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: complex,K4: complex] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                & ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
% 0.30/0.56                & ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(1)
% 0.30/0.56  thf(fact_3241_inf__period_I1_J,axiom,
% 0.30/0.56      ! [P: real > $o,D4: real,Q: real > $o] :
% 0.30/0.56        ( ! [X5: real,K3: real] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: real,K3: real] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: real,K4: real] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                & ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.30/0.56                & ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(1)
% 0.30/0.56  thf(fact_3242_inf__period_I1_J,axiom,
% 0.30/0.56      ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.30/0.56        ( ! [X5: rat,K3: rat] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: rat,K3: rat] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: rat,K4: rat] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                & ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.30/0.56                & ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(1)
% 0.30/0.56  thf(fact_3243_inf__period_I1_J,axiom,
% 0.30/0.56      ! [P: int > $o,D4: int,Q: int > $o] :
% 0.30/0.56        ( ! [X5: int,K3: int] :
% 0.30/0.56            ( ( P @ X5 )
% 0.30/0.56            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56       => ( ! [X5: int,K3: int] :
% 0.30/0.56              ( ( Q @ X5 )
% 0.30/0.56              = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.30/0.56         => ! [X6: int,K4: int] :
% 0.30/0.56              ( ( ( P @ X6 )
% 0.30/0.56                & ( Q @ X6 ) )
% 0.30/0.56              = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.30/0.56                & ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % inf_period(1)
% 0.30/0.56  thf(fact_3244_imp__le__cong,axiom,
% 0.30/0.56      ! [X3: int,X8: int,P: $o,P6: $o] :
% 0.30/0.56        ( ( X3 = X8 )
% 0.30/0.56       => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.30/0.56           => ( P = P6 ) )
% 0.30/0.56         => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.56             => P )
% 0.30/0.56            = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.30/0.56             => P6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % imp_le_cong
% 0.30/0.56  thf(fact_3245_conj__le__cong,axiom,
% 0.30/0.56      ! [X3: int,X8: int,P: $o,P6: $o] :
% 0.30/0.56        ( ( X3 = X8 )
% 0.30/0.56       => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.30/0.56           => ( P = P6 ) )
% 0.30/0.56         => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.56              & P )
% 0.30/0.56            = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.30/0.56              & P6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % conj_le_cong
% 0.30/0.56  thf(fact_3246_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56      ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
% 0.30/0.56        ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.30/0.56        = ( ( ~ ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.56            | ( ( ord_less_eq_set_int @ C @ A )
% 0.30/0.56              & ( ord_less_eq_set_int @ B2 @ D )
% 0.30/0.56              & ( ( ord_less_set_int @ C @ A )
% 0.30/0.56                | ( ord_less_set_int @ B2 @ D ) ) ) )
% 0.30/0.56          & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_psubset_iff
% 0.30/0.56  thf(fact_3247_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.56        ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.30/0.56        = ( ( ~ ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.56            | ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.56              & ( ord_less_eq_rat @ B2 @ D )
% 0.30/0.56              & ( ( ord_less_rat @ C @ A )
% 0.30/0.56                | ( ord_less_rat @ B2 @ D ) ) ) )
% 0.30/0.56          & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_psubset_iff
% 0.30/0.56  thf(fact_3248_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56      ! [A: num,B2: num,C: num,D: num] :
% 0.30/0.56        ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.30/0.56        = ( ( ~ ( ord_less_eq_num @ A @ B2 )
% 0.30/0.56            | ( ( ord_less_eq_num @ C @ A )
% 0.30/0.56              & ( ord_less_eq_num @ B2 @ D )
% 0.30/0.56              & ( ( ord_less_num @ C @ A )
% 0.30/0.56                | ( ord_less_num @ B2 @ D ) ) ) )
% 0.30/0.56          & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_psubset_iff
% 0.30/0.56  thf(fact_3249_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.56        ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.30/0.56        = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.56            | ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.56              & ( ord_less_eq_nat @ B2 @ D )
% 0.30/0.56              & ( ( ord_less_nat @ C @ A )
% 0.30/0.56                | ( ord_less_nat @ B2 @ D ) ) ) )
% 0.30/0.56          & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_psubset_iff
% 0.30/0.56  thf(fact_3250_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.56        ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.30/0.56        = ( ( ~ ( ord_less_eq_int @ A @ B2 )
% 0.30/0.56            | ( ( ord_less_eq_int @ C @ A )
% 0.30/0.56              & ( ord_less_eq_int @ B2 @ D )
% 0.30/0.56              & ( ( ord_less_int @ C @ A )
% 0.30/0.56                | ( ord_less_int @ B2 @ D ) ) ) )
% 0.30/0.56          & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_psubset_iff
% 0.30/0.56  thf(fact_3251_atLeastatMost__psubset__iff,axiom,
% 0.30/0.56      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.56        ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.30/0.56        = ( ( ~ ( ord_less_eq_real @ A @ B2 )
% 0.30/0.56            | ( ( ord_less_eq_real @ C @ A )
% 0.30/0.56              & ( ord_less_eq_real @ B2 @ D )
% 0.30/0.56              & ( ( ord_less_real @ C @ A )
% 0.30/0.56                | ( ord_less_real @ B2 @ D ) ) ) )
% 0.30/0.56          & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % atLeastatMost_psubset_iff
% 0.30/0.56  thf(fact_3252_minusinfinity,axiom,
% 0.30/0.56      ! [D: int,P1: int > $o,P: int > $o] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56       => ( ! [X5: int,K3: int] :
% 0.30/0.56              ( ( P1 @ X5 )
% 0.30/0.56              = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.30/0.56         => ( ? [Z3: int] :
% 0.30/0.56              ! [X5: int] :
% 0.30/0.56                ( ( ord_less_int @ X5 @ Z3 )
% 0.30/0.56               => ( ( P @ X5 )
% 0.30/0.56                  = ( P1 @ X5 ) ) )
% 0.30/0.56           => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.30/0.56             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minusinfinity
% 0.30/0.56  thf(fact_3253_plusinfinity,axiom,
% 0.30/0.56      ! [D: int,P6: int > $o,P: int > $o] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56       => ( ! [X5: int,K3: int] :
% 0.30/0.56              ( ( P6 @ X5 )
% 0.30/0.56              = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.30/0.56         => ( ? [Z3: int] :
% 0.30/0.56              ! [X5: int] :
% 0.30/0.56                ( ( ord_less_int @ Z3 @ X5 )
% 0.30/0.56               => ( ( P @ X5 )
% 0.30/0.56                  = ( P6 @ X5 ) ) )
% 0.30/0.56           => ( ? [X_12: int] : ( P6 @ X_12 )
% 0.30/0.56             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % plusinfinity
% 0.30/0.56  thf(fact_3254_Bolzano,axiom,
% 0.30/0.56      ! [A: real,B2: real,P: real > real > $o] :
% 0.30/0.56        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.56       => ( ! [A3: real,B3: real,C2: real] :
% 0.30/0.56              ( ( P @ A3 @ B3 )
% 0.30/0.56             => ( ( P @ B3 @ C2 )
% 0.30/0.56               => ( ( ord_less_eq_real @ A3 @ B3 )
% 0.30/0.56                 => ( ( ord_less_eq_real @ B3 @ C2 )
% 0.30/0.56                   => ( P @ A3 @ C2 ) ) ) ) )
% 0.30/0.56         => ( ! [X5: real] :
% 0.30/0.56                ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.56               => ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.56                 => ? [D5: real] :
% 0.30/0.56                      ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.30/0.56                      & ! [A3: real,B3: real] :
% 0.30/0.56                          ( ( ( ord_less_eq_real @ A3 @ X5 )
% 0.30/0.56                            & ( ord_less_eq_real @ X5 @ B3 )
% 0.30/0.56                            & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D5 ) )
% 0.30/0.56                         => ( P @ A3 @ B3 ) ) ) ) )
% 0.30/0.56           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Bolzano
% 0.30/0.56  thf(fact_3255_mult__le__cancel__iff1,axiom,
% 0.30/0.56      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.56       => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 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_3256_mult__le__cancel__iff1,axiom,
% 0.30/0.56      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.30/0.56       => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 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_3257_mult__le__cancel__iff1,axiom,
% 0.30/0.56      ! [Z: int,X3: int,Y3: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.56       => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 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_3258_mult__le__cancel__iff2,axiom,
% 0.30/0.56      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.56       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X3 ) @ ( times_times_real @ Z @ 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_3259_mult__le__cancel__iff2,axiom,
% 0.30/0.56      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.30/0.56       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X3 ) @ ( times_times_rat @ Z @ 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_3260_mult__le__cancel__iff2,axiom,
% 0.30/0.56      ! [Z: int,X3: int,Y3: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.56       => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X3 ) @ ( times_times_int @ Z @ 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_3261_divides__aux__eq,axiom,
% 0.30/0.56      ! [Q2: nat,R: nat] :
% 0.30/0.56        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.30/0.56        = ( R = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % divides_aux_eq
% 0.30/0.56  thf(fact_3262_divides__aux__eq,axiom,
% 0.30/0.56      ! [Q2: int,R: int] :
% 0.30/0.56        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56        = ( R = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % divides_aux_eq
% 0.30/0.56  thf(fact_3263_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_int,Ys: list_int] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( product_Pair_int_int @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3264_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_int,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3265_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_int,Ys: list_o] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( product_Pair_int_o @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3266_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_int,Ys: list_nat] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( product_Pair_int_nat @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3267_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3268_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3269_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3270_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3271_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_o,Ys: list_int] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3272_product__nth,axiom,
% 0.30/0.56      ! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.56       => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
% 0.30/0.56          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % product_nth
% 0.30/0.56  thf(fact_3273_neg__eucl__rel__int__mult__2,axiom,
% 0.30/0.56      ! [B2: int,A: int,Q2: int,R: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 0.30/0.56       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56         => ( 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 ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_eucl_rel_int_mult_2
% 0.30/0.56  thf(fact_3274_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3275_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
% 0.30/0.56        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3276_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 0.30/0.56        ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3277_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3278_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_o,Ys: list_o] :
% 0.30/0.56        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3279_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_o,Ys: list_nat] :
% 0.30/0.56        ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3280_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3281_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_nat,Ys: list_o] :
% 0.30/0.56        ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3282_length__product,axiom,
% 0.30/0.56      ! [Xs2: list_nat,Ys: list_nat] :
% 0.30/0.56        ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs2 @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3283_unique__remainder,axiom,
% 0.30/0.56      ! [A: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
% 0.30/0.56        ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56       => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 0.30/0.56         => ( R = R4 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_remainder
% 0.30/0.56  thf(fact_3284_unique__quotient,axiom,
% 0.30/0.56      ! [A: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
% 0.30/0.56        ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56       => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 0.30/0.56         => ( Q2 = Q5 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_quotient
% 0.30/0.56  thf(fact_3285_eucl__rel__int__by0,axiom,
% 0.30/0.56      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.30/0.56  
% 0.30/0.56  % eucl_rel_int_by0
% 0.30/0.56  thf(fact_3286_div__int__unique,axiom,
% 0.30/0.56      ! [K: int,L: int,Q2: int,R: int] :
% 0.30/0.56        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56       => ( ( divide_divide_int @ K @ L )
% 0.30/0.56          = Q2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_int_unique
% 0.30/0.56  thf(fact_3287_mod__int__unique,axiom,
% 0.30/0.56      ! [K: int,L: int,Q2: int,R: int] :
% 0.30/0.56        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56       => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.56          = R ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_int_unique
% 0.30/0.56  thf(fact_3288_eucl__rel__int__dividesI,axiom,
% 0.30/0.56      ! [L: int,K: int,Q2: int] :
% 0.30/0.56        ( ( L != zero_zero_int )
% 0.30/0.56       => ( ( K
% 0.30/0.56            = ( times_times_int @ Q2 @ L ) )
% 0.30/0.56         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % eucl_rel_int_dividesI
% 0.30/0.56  thf(fact_3289_eucl__rel__int,axiom,
% 0.30/0.56      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % eucl_rel_int
% 0.30/0.56  thf(fact_3290_eucl__rel__int__iff,axiom,
% 0.30/0.56      ! [K: int,L: int,Q2: int,R: int] :
% 0.30/0.56        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56        = ( ( K
% 0.30/0.56            = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R ) )
% 0.30/0.56          & ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56           => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.30/0.56              & ( ord_less_int @ R @ L ) ) )
% 0.30/0.56          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56               => ( ( ord_less_int @ L @ R )
% 0.30/0.56                  & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 0.30/0.56              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.56               => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % eucl_rel_int_iff
% 0.30/0.56  thf(fact_3291_mult__less__iff1,axiom,
% 0.30/0.56      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.56       => ( ( ord_less_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 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_3292_mult__less__iff1,axiom,
% 0.30/0.56      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.30/0.56       => ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 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_3293_mult__less__iff1,axiom,
% 0.30/0.56      ! [Z: int,X3: int,Y3: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.56       => ( ( ord_less_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 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_3294_pos__eucl__rel__int__mult__2,axiom,
% 0.30/0.56      ! [B2: int,A: int,Q2: int,R: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.56       => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.56         => ( 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 ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_eucl_rel_int_mult_2
% 0.30/0.56  thf(fact_3295_obtain__set__pred,axiom,
% 0.30/0.56      ! [Z: nat,X3: nat,A4: set_nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z @ X3 )
% 0.30/0.56       => ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
% 0.30/0.56         => ( ( finite_finite_nat @ A4 )
% 0.30/0.56           => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X3 @ X_1 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % obtain_set_pred
% 0.30/0.56  thf(fact_3296_set__vebt__finite,axiom,
% 0.30/0.56      ! [T: vEBT_VEBT,N: nat] :
% 0.30/0.56        ( ( vEBT_invar_vebt @ T @ N )
% 0.30/0.56       => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_vebt_finite
% 0.30/0.56  thf(fact_3297_pred__none__empty,axiom,
% 0.30/0.56      ! [Xs2: set_nat,A: nat] :
% 0.30/0.56        ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
% 0.30/0.56       => ( ( finite_finite_nat @ Xs2 )
% 0.30/0.56         => ~ ? [X6: nat] :
% 0.30/0.56                ( ( member_nat @ X6 @ Xs2 )
% 0.30/0.56                & ( ord_less_nat @ X6 @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pred_none_empty
% 0.30/0.56  thf(fact_3298_prod_Oinject,axiom,
% 0.30/0.56      ! [X1: int,X2: int,Y1: int,Y2: int] :
% 0.30/0.56        ( ( ( product_Pair_int_int @ X1 @ X2 )
% 0.30/0.56          = ( product_Pair_int_int @ Y1 @ Y2 ) )
% 0.30/0.56        = ( ( X1 = Y1 )
% 0.30/0.56          & ( X2 = Y2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % prod.inject
% 0.30/0.56  thf(fact_3299_prod_Oinject,axiom,
% 0.30/0.56      ! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 0.30/0.56        ( ( ( produc6137756002093451184nteger @ X1 @ X2 )
% 0.30/0.56          = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
% 0.30/0.56        = ( ( X1 = Y1 )
% 0.30/0.56          & ( X2 = Y2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % prod.inject
% 0.30/0.56  thf(fact_3300_prod_Oinject,axiom,
% 0.30/0.56      ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 0.30/0.56        ( ( ( produc8603105652947943368nteger @ X1 @ X2 )
% 0.30/0.56          = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
% 0.30/0.56        = ( ( X1 = Y1 )
% 0.30/0.56          & ( X2 = Y2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % prod.inject
% 0.30/0.56  thf(fact_3301_prod_Oinject,axiom,
% 0.30/0.56      ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 0.30/0.56        ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
% 0.30/0.56          = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
% 0.30/0.56        = ( ( X1 = Y1 )
% 0.30/0.56          & ( X2 = Y2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % prod.inject
% 0.30/0.56  thf(fact_3302_prod_Oinject,axiom,
% 0.30/0.56      ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 0.30/0.56        ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
% 0.30/0.56          = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
% 0.30/0.56        = ( ( X1 = Y1 )
% 0.30/0.56          & ( X2 = Y2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % prod.inject
% 0.30/0.56  thf(fact_3303_old_Oprod_Oinject,axiom,
% 0.30/0.57      ! [A: int,B2: int,A2: int,B: int] :
% 0.30/0.57        ( ( ( product_Pair_int_int @ A @ B2 )
% 0.30/0.57          = ( product_Pair_int_int @ A2 @ B ) )
% 0.30/0.57        = ( ( A = A2 )
% 0.30/0.57          & ( B2 = B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.inject
% 0.30/0.57  thf(fact_3304_old_Oprod_Oinject,axiom,
% 0.30/0.57      ! [A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.57        ( ( ( produc6137756002093451184nteger @ A @ B2 )
% 0.30/0.57          = ( produc6137756002093451184nteger @ A2 @ B ) )
% 0.30/0.57        = ( ( A = A2 )
% 0.30/0.57          & ( B2 = B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.inject
% 0.30/0.57  thf(fact_3305_old_Oprod_Oinject,axiom,
% 0.30/0.57      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.57        ( ( ( produc8603105652947943368nteger @ A @ B2 )
% 0.30/0.57          = ( produc8603105652947943368nteger @ A2 @ B ) )
% 0.30/0.57        = ( ( A = A2 )
% 0.30/0.57          & ( B2 = B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.inject
% 0.30/0.57  thf(fact_3306_old_Oprod_Oinject,axiom,
% 0.30/0.57      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.57        ( ( ( produc5700946648718959541nt_int @ A @ B2 )
% 0.30/0.57          = ( produc5700946648718959541nt_int @ A2 @ B ) )
% 0.30/0.57        = ( ( A = A2 )
% 0.30/0.57          & ( B2 = B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.inject
% 0.30/0.57  thf(fact_3307_old_Oprod_Oinject,axiom,
% 0.30/0.57      ! [A: int > option6357759511663192854e_term,B2: product_prod_int_int,A2: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.57        ( ( ( produc4305682042979456191nt_int @ A @ B2 )
% 0.30/0.57          = ( produc4305682042979456191nt_int @ A2 @ B ) )
% 0.30/0.57        = ( ( A = A2 )
% 0.30/0.57          & ( B2 = B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.inject
% 0.30/0.57  thf(fact_3308_List_Ofinite__set,axiom,
% 0.30/0.57      ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % List.finite_set
% 0.30/0.57  thf(fact_3309_List_Ofinite__set,axiom,
% 0.30/0.57      ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % List.finite_set
% 0.30/0.57  thf(fact_3310_List_Ofinite__set,axiom,
% 0.30/0.57      ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % List.finite_set
% 0.30/0.57  thf(fact_3311_List_Ofinite__set,axiom,
% 0.30/0.57      ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % List.finite_set
% 0.30/0.57  thf(fact_3312_infinite__Icc__iff,axiom,
% 0.30/0.57      ! [A: rat,B2: rat] :
% 0.30/0.57        ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) ) )
% 0.30/0.57        = ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_Icc_iff
% 0.30/0.57  thf(fact_3313_infinite__Icc__iff,axiom,
% 0.30/0.57      ! [A: real,B2: real] :
% 0.30/0.57        ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) )
% 0.30/0.57        = ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_Icc_iff
% 0.30/0.57  thf(fact_3314_bounded__nat__set__is__finite,axiom,
% 0.30/0.57      ! [N4: set_nat,N: nat] :
% 0.30/0.57        ( ! [X5: nat] :
% 0.30/0.57            ( ( member_nat @ X5 @ N4 )
% 0.30/0.57           => ( ord_less_nat @ X5 @ N ) )
% 0.30/0.57       => ( finite_finite_nat @ N4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bounded_nat_set_is_finite
% 0.30/0.57  thf(fact_3315_finite__nat__set__iff__bounded,axiom,
% 0.30/0.57      ( finite_finite_nat
% 0.30/0.57      = ( ^ [N6: set_nat] :
% 0.30/0.57          ? [M3: nat] :
% 0.30/0.57          ! [X: nat] :
% 0.30/0.57            ( ( member_nat @ X @ N6 )
% 0.30/0.57           => ( ord_less_nat @ X @ M3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_nat_set_iff_bounded
% 0.30/0.57  thf(fact_3316_finite__nat__set__iff__bounded__le,axiom,
% 0.30/0.57      ( finite_finite_nat
% 0.30/0.57      = ( ^ [N6: set_nat] :
% 0.30/0.57          ? [M3: nat] :
% 0.30/0.57          ! [X: nat] :
% 0.30/0.57            ( ( member_nat @ X @ N6 )
% 0.30/0.57           => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_nat_set_iff_bounded_le
% 0.30/0.57  thf(fact_3317_finite__list,axiom,
% 0.30/0.57      ! [A4: set_VEBT_VEBT] :
% 0.30/0.57        ( ( finite5795047828879050333T_VEBT @ A4 )
% 0.30/0.57       => ? [Xs3: list_VEBT_VEBT] :
% 0.30/0.57            ( ( set_VEBT_VEBT2 @ Xs3 )
% 0.30/0.57            = A4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_list
% 0.30/0.57  thf(fact_3318_finite__list,axiom,
% 0.30/0.57      ! [A4: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ? [Xs3: list_nat] :
% 0.30/0.57            ( ( set_nat2 @ Xs3 )
% 0.30/0.57            = A4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_list
% 0.30/0.57  thf(fact_3319_finite__list,axiom,
% 0.30/0.57      ! [A4: set_int] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ? [Xs3: list_int] :
% 0.30/0.57            ( ( set_int2 @ Xs3 )
% 0.30/0.57            = A4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_list
% 0.30/0.57  thf(fact_3320_finite__list,axiom,
% 0.30/0.57      ! [A4: set_complex] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.57       => ? [Xs3: list_complex] :
% 0.30/0.57            ( ( set_complex2 @ Xs3 )
% 0.30/0.57            = A4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_list
% 0.30/0.57  thf(fact_3321_finite__M__bounded__by__nat,axiom,
% 0.30/0.57      ! [P: nat > $o,I: nat] :
% 0.30/0.57        ( finite_finite_nat
% 0.30/0.57        @ ( collect_nat
% 0.30/0.57          @ ^ [K2: nat] :
% 0.30/0.57              ( ( P @ K2 )
% 0.30/0.57              & ( ord_less_nat @ K2 @ I ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_M_bounded_by_nat
% 0.30/0.57  thf(fact_3322_finite__less__ub,axiom,
% 0.30/0.57      ! [F: nat > nat,U: nat] :
% 0.30/0.57        ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 0.30/0.57       => ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_less_ub
% 0.30/0.57  thf(fact_3323_finite__lists__length__eq,axiom,
% 0.30/0.57      ! [A4: set_complex,N: nat] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.57       => ( finite8712137658972009173omplex
% 0.30/0.57          @ ( collect_list_complex
% 0.30/0.57            @ ^ [Xs: list_complex] :
% 0.30/0.57                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ( size_s3451745648224563538omplex @ Xs )
% 0.30/0.57                  = N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_eq
% 0.30/0.57  thf(fact_3324_finite__lists__length__eq,axiom,
% 0.30/0.57      ! [A4: set_VEBT_VEBT,N: nat] :
% 0.30/0.57        ( ( finite5795047828879050333T_VEBT @ A4 )
% 0.30/0.57       => ( finite3004134309566078307T_VEBT
% 0.30/0.57          @ ( collec5608196760682091941T_VEBT
% 0.30/0.57            @ ^ [Xs: list_VEBT_VEBT] :
% 0.30/0.57                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.57                  = N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_eq
% 0.30/0.57  thf(fact_3325_finite__lists__length__eq,axiom,
% 0.30/0.57      ! [A4: set_o,N: nat] :
% 0.30/0.57        ( ( finite_finite_o @ A4 )
% 0.30/0.57       => ( finite_finite_list_o
% 0.30/0.57          @ ( collect_list_o
% 0.30/0.57            @ ^ [Xs: list_o] :
% 0.30/0.57                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ( size_size_list_o @ Xs )
% 0.30/0.57                  = N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_eq
% 0.30/0.57  thf(fact_3326_finite__lists__length__eq,axiom,
% 0.30/0.57      ! [A4: set_nat,N: nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( finite8100373058378681591st_nat
% 0.30/0.57          @ ( collect_list_nat
% 0.30/0.57            @ ^ [Xs: list_nat] :
% 0.30/0.57                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ( size_size_list_nat @ Xs )
% 0.30/0.57                  = N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_eq
% 0.30/0.57  thf(fact_3327_finite__lists__length__eq,axiom,
% 0.30/0.57      ! [A4: set_int,N: nat] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( finite3922522038869484883st_int
% 0.30/0.57          @ ( collect_list_int
% 0.30/0.57            @ ^ [Xs: list_int] :
% 0.30/0.57                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ( size_size_list_int @ Xs )
% 0.30/0.57                  = N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_eq
% 0.30/0.57  thf(fact_3328_infinite__Icc,axiom,
% 0.30/0.57      ! [A: rat,B2: rat] :
% 0.30/0.57        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.57       => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_Icc
% 0.30/0.57  thf(fact_3329_infinite__Icc,axiom,
% 0.30/0.57      ! [A: real,B2: real] :
% 0.30/0.57        ( ( ord_less_real @ A @ B2 )
% 0.30/0.57       => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_Icc
% 0.30/0.57  thf(fact_3330_finite__lists__length__le,axiom,
% 0.30/0.57      ! [A4: set_complex,N: nat] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.57       => ( finite8712137658972009173omplex
% 0.30/0.57          @ ( collect_list_complex
% 0.30/0.57            @ ^ [Xs: list_complex] :
% 0.30/0.57                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_le
% 0.30/0.57  thf(fact_3331_finite__lists__length__le,axiom,
% 0.30/0.57      ! [A4: set_VEBT_VEBT,N: nat] :
% 0.30/0.57        ( ( finite5795047828879050333T_VEBT @ A4 )
% 0.30/0.57       => ( finite3004134309566078307T_VEBT
% 0.30/0.57          @ ( collec5608196760682091941T_VEBT
% 0.30/0.57            @ ^ [Xs: list_VEBT_VEBT] :
% 0.30/0.57                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_le
% 0.30/0.57  thf(fact_3332_finite__lists__length__le,axiom,
% 0.30/0.57      ! [A4: set_o,N: nat] :
% 0.30/0.57        ( ( finite_finite_o @ A4 )
% 0.30/0.57       => ( finite_finite_list_o
% 0.30/0.57          @ ( collect_list_o
% 0.30/0.57            @ ^ [Xs: list_o] :
% 0.30/0.57                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_le
% 0.30/0.57  thf(fact_3333_finite__lists__length__le,axiom,
% 0.30/0.57      ! [A4: set_nat,N: nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( finite8100373058378681591st_nat
% 0.30/0.57          @ ( collect_list_nat
% 0.30/0.57            @ ^ [Xs: list_nat] :
% 0.30/0.57                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_le
% 0.30/0.57  thf(fact_3334_finite__lists__length__le,axiom,
% 0.30/0.57      ! [A4: set_int,N: nat] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( finite3922522038869484883st_int
% 0.30/0.57          @ ( collect_list_int
% 0.30/0.57            @ ^ [Xs: list_int] :
% 0.30/0.57                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
% 0.30/0.57                & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_lists_length_le
% 0.30/0.57  thf(fact_3335_old_Oprod_Oexhaust,axiom,
% 0.30/0.57      ! [Y3: product_prod_int_int] :
% 0.30/0.57        ~ ! [A3: int,B3: int] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( product_Pair_int_int @ A3 @ B3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.exhaust
% 0.30/0.57  thf(fact_3336_old_Oprod_Oexhaust,axiom,
% 0.30/0.57      ! [Y3: produc8763457246119570046nteger] :
% 0.30/0.57        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc6137756002093451184nteger @ A3 @ B3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.exhaust
% 0.30/0.57  thf(fact_3337_old_Oprod_Oexhaust,axiom,
% 0.30/0.57      ! [Y3: produc1908205239877642774nteger] :
% 0.30/0.57        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc8603105652947943368nteger @ A3 @ B3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.exhaust
% 0.30/0.57  thf(fact_3338_old_Oprod_Oexhaust,axiom,
% 0.30/0.57      ! [Y3: produc2285326912895808259nt_int] :
% 0.30/0.57        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.exhaust
% 0.30/0.57  thf(fact_3339_old_Oprod_Oexhaust,axiom,
% 0.30/0.57      ! [Y3: produc7773217078559923341nt_int] :
% 0.30/0.57        ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % old.prod.exhaust
% 0.30/0.57  thf(fact_3340_surj__pair,axiom,
% 0.30/0.57      ! [P4: product_prod_int_int] :
% 0.30/0.57      ? [X5: int,Y4: int] :
% 0.30/0.57        ( P4
% 0.30/0.57        = ( product_Pair_int_int @ X5 @ Y4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % surj_pair
% 0.30/0.57  thf(fact_3341_surj__pair,axiom,
% 0.30/0.57      ! [P4: produc8763457246119570046nteger] :
% 0.30/0.57      ? [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.57        ( P4
% 0.30/0.57        = ( produc6137756002093451184nteger @ X5 @ Y4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % surj_pair
% 0.30/0.57  thf(fact_3342_surj__pair,axiom,
% 0.30/0.57      ! [P4: produc1908205239877642774nteger] :
% 0.30/0.57      ? [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.57        ( P4
% 0.30/0.57        = ( produc8603105652947943368nteger @ X5 @ Y4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % surj_pair
% 0.30/0.57  thf(fact_3343_surj__pair,axiom,
% 0.30/0.57      ! [P4: produc2285326912895808259nt_int] :
% 0.30/0.57      ? [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.57        ( P4
% 0.30/0.57        = ( produc5700946648718959541nt_int @ X5 @ Y4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % surj_pair
% 0.30/0.57  thf(fact_3344_surj__pair,axiom,
% 0.30/0.57      ! [P4: produc7773217078559923341nt_int] :
% 0.30/0.57      ? [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.57        ( P4
% 0.30/0.57        = ( produc4305682042979456191nt_int @ X5 @ Y4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % surj_pair
% 0.30/0.57  thf(fact_3345_prod__cases,axiom,
% 0.30/0.57      ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 0.30/0.57        ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
% 0.30/0.57       => ( P @ P4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases
% 0.30/0.57  thf(fact_3346_prod__cases,axiom,
% 0.30/0.57      ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
% 0.30/0.57        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.30/0.57       => ( P @ P4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases
% 0.30/0.57  thf(fact_3347_prod__cases,axiom,
% 0.30/0.57      ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
% 0.30/0.57        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 0.30/0.57       => ( P @ P4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases
% 0.30/0.57  thf(fact_3348_prod__cases,axiom,
% 0.30/0.57      ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
% 0.30/0.57        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 0.30/0.57       => ( P @ P4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases
% 0.30/0.57  thf(fact_3349_prod__cases,axiom,
% 0.30/0.57      ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
% 0.30/0.57        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.30/0.57       => ( P @ P4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases
% 0.30/0.57  thf(fact_3350_Pair__inject,axiom,
% 0.30/0.57      ! [A: int,B2: int,A2: int,B: int] :
% 0.30/0.57        ( ( ( product_Pair_int_int @ A @ B2 )
% 0.30/0.57          = ( product_Pair_int_int @ A2 @ B ) )
% 0.30/0.57       => ~ ( ( A = A2 )
% 0.30/0.57           => ( B2 != B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Pair_inject
% 0.30/0.57  thf(fact_3351_Pair__inject,axiom,
% 0.30/0.57      ! [A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.57        ( ( ( produc6137756002093451184nteger @ A @ B2 )
% 0.30/0.57          = ( produc6137756002093451184nteger @ A2 @ B ) )
% 0.30/0.57       => ~ ( ( A = A2 )
% 0.30/0.57           => ( B2 != B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Pair_inject
% 0.30/0.57  thf(fact_3352_Pair__inject,axiom,
% 0.30/0.57      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.57        ( ( ( produc8603105652947943368nteger @ A @ B2 )
% 0.30/0.57          = ( produc8603105652947943368nteger @ A2 @ B ) )
% 0.30/0.57       => ~ ( ( A = A2 )
% 0.30/0.57           => ( B2 != B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Pair_inject
% 0.30/0.57  thf(fact_3353_Pair__inject,axiom,
% 0.30/0.57      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.57        ( ( ( produc5700946648718959541nt_int @ A @ B2 )
% 0.30/0.57          = ( produc5700946648718959541nt_int @ A2 @ B ) )
% 0.30/0.57       => ~ ( ( A = A2 )
% 0.30/0.57           => ( B2 != B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Pair_inject
% 0.30/0.57  thf(fact_3354_Pair__inject,axiom,
% 0.30/0.57      ! [A: int > option6357759511663192854e_term,B2: product_prod_int_int,A2: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.57        ( ( ( produc4305682042979456191nt_int @ A @ B2 )
% 0.30/0.57          = ( produc4305682042979456191nt_int @ A2 @ B ) )
% 0.30/0.57       => ~ ( ( A = A2 )
% 0.30/0.57           => ( B2 != B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Pair_inject
% 0.30/0.57  thf(fact_3355_subset__eq__atLeast0__atMost__finite,axiom,
% 0.30/0.57      ! [N4: set_nat,N: nat] :
% 0.30/0.57        ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.57       => ( finite_finite_nat @ N4 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % subset_eq_atLeast0_atMost_finite
% 0.30/0.57  thf(fact_3356_prod__cases3,axiom,
% 0.30/0.57      ! [Y3: produc8763457246119570046nteger] :
% 0.30/0.57        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C2: code_integer] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases3
% 0.30/0.57  thf(fact_3357_prod__cases3,axiom,
% 0.30/0.57      ! [Y3: produc1908205239877642774nteger] :
% 0.30/0.57        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C2: code_integer] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases3
% 0.30/0.57  thf(fact_3358_prod__cases3,axiom,
% 0.30/0.57      ! [Y3: produc2285326912895808259nt_int] :
% 0.30/0.57        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases3
% 0.30/0.57  thf(fact_3359_prod__cases3,axiom,
% 0.30/0.57      ! [Y3: produc7773217078559923341nt_int] :
% 0.30/0.57        ~ ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] :
% 0.30/0.57            ( Y3
% 0.30/0.57           != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_cases3
% 0.30/0.57  thf(fact_3360_prod__induct3,axiom,
% 0.30/0.57      ! [P: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
% 0.30/0.57        ( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) )
% 0.30/0.57       => ( P @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_induct3
% 0.30/0.57  thf(fact_3361_prod__induct3,axiom,
% 0.30/0.57      ! [P: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
% 0.30/0.57        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) )
% 0.30/0.57       => ( P @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_induct3
% 0.30/0.57  thf(fact_3362_prod__induct3,axiom,
% 0.30/0.57      ! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
% 0.30/0.57        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
% 0.30/0.57       => ( P @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_induct3
% 0.30/0.57  thf(fact_3363_prod__induct3,axiom,
% 0.30/0.57      ! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
% 0.30/0.57        ( ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
% 0.30/0.57       => ( P @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod_induct3
% 0.30/0.57  thf(fact_3364_finite__Collect__le__nat,axiom,
% 0.30/0.57      ! [K: nat] :
% 0.30/0.57        ( finite_finite_nat
% 0.30/0.57        @ ( collect_nat
% 0.30/0.57          @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_le_nat
% 0.30/0.57  thf(fact_3365_finite__Collect__less__nat,axiom,
% 0.30/0.57      ! [K: nat] :
% 0.30/0.57        ( finite_finite_nat
% 0.30/0.57        @ ( collect_nat
% 0.30/0.57          @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_less_nat
% 0.30/0.57  thf(fact_3366_finite__Collect__subsets,axiom,
% 0.30/0.57      ! [A4: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( finite1152437895449049373et_nat
% 0.30/0.57          @ ( collect_set_nat
% 0.30/0.57            @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_subsets
% 0.30/0.57  thf(fact_3367_finite__Collect__subsets,axiom,
% 0.30/0.57      ! [A4: set_complex] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.57       => ( finite6551019134538273531omplex
% 0.30/0.57          @ ( collect_set_complex
% 0.30/0.57            @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_subsets
% 0.30/0.57  thf(fact_3368_finite__Collect__subsets,axiom,
% 0.30/0.57      ! [A4: set_int] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( finite6197958912794628473et_int
% 0.30/0.57          @ ( collect_set_int
% 0.30/0.57            @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_subsets
% 0.30/0.57  thf(fact_3369_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: real > $o,Q: real > real > $o] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [X: real] :
% 0.30/0.57                ? [Y: real] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: real] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_real
% 0.30/0.57                  @ ( collect_real
% 0.30/0.57                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3370_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: real > $o,Q: nat > real > $o] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [X: nat] :
% 0.30/0.57                ? [Y: real] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: real] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_nat
% 0.30/0.57                  @ ( collect_nat
% 0.30/0.57                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3371_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: real > $o,Q: int > real > $o] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [X: int] :
% 0.30/0.57                ? [Y: real] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: real] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_int
% 0.30/0.57                  @ ( collect_int
% 0.30/0.57                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3372_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: real > $o,Q: complex > real > $o] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [X: complex] :
% 0.30/0.57                ? [Y: real] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: real] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite3207457112153483333omplex
% 0.30/0.57                  @ ( collect_complex
% 0.30/0.57                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3373_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: nat > $o,Q: real > nat > $o] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [X: real] :
% 0.30/0.57                ? [Y: nat] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: nat] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_real
% 0.30/0.57                  @ ( collect_real
% 0.30/0.57                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3374_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: nat > $o,Q: nat > nat > $o] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [X: nat] :
% 0.30/0.57                ? [Y: nat] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: nat] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_nat
% 0.30/0.57                  @ ( collect_nat
% 0.30/0.57                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3375_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: nat > $o,Q: int > nat > $o] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [X: int] :
% 0.30/0.57                ? [Y: nat] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: nat] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_int
% 0.30/0.57                  @ ( collect_int
% 0.30/0.57                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3376_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: nat > $o,Q: complex > nat > $o] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [X: complex] :
% 0.30/0.57                ? [Y: nat] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: nat] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite3207457112153483333omplex
% 0.30/0.57                  @ ( collect_complex
% 0.30/0.57                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3377_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: int > $o,Q: real > int > $o] :
% 0.30/0.57        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [X: real] :
% 0.30/0.57                ? [Y: int] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: int] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_real
% 0.30/0.57                  @ ( collect_real
% 0.30/0.57                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3378_finite__Collect__bounded__ex,axiom,
% 0.30/0.57      ! [P: int > $o,Q: nat > int > $o] :
% 0.30/0.57        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [X: nat] :
% 0.30/0.57                ? [Y: int] :
% 0.30/0.57                  ( ( P @ Y )
% 0.30/0.57                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.57          = ( ! [Y: int] :
% 0.30/0.57                ( ( P @ Y )
% 0.30/0.57               => ( finite_finite_nat
% 0.30/0.57                  @ ( collect_nat
% 0.30/0.57                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_Collect_bounded_ex
% 0.30/0.57  thf(fact_3379_finite__roots__unity,axiom,
% 0.30/0.57      ! [N: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.57       => ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [Z2: real] :
% 0.30/0.57                ( ( power_power_real @ Z2 @ N )
% 0.30/0.57                = one_one_real ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_roots_unity
% 0.30/0.57  thf(fact_3380_finite__roots__unity,axiom,
% 0.30/0.57      ! [N: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.57       => ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [Z2: complex] :
% 0.30/0.57                ( ( power_power_complex @ Z2 @ N )
% 0.30/0.57                = one_one_complex ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_roots_unity
% 0.30/0.57  thf(fact_3381_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 0.30/0.57        ( ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [I4: real] :
% 0.30/0.57                ( ( member_real @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_complex ) ) ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_complex ) ) ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3382_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 0.30/0.57        ( ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [I4: nat] :
% 0.30/0.57                ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_complex ) ) ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_complex ) ) ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3383_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 0.30/0.57        ( ( finite_finite_int
% 0.30/0.57          @ ( collect_int
% 0.30/0.57            @ ^ [I4: int] :
% 0.30/0.57                ( ( member_int @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_complex ) ) ) )
% 0.30/0.57       => ( ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_complex ) ) ) )
% 0.30/0.57         => ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3384_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 0.30/0.57        ( ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [I4: complex] :
% 0.30/0.57                ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_complex ) ) ) )
% 0.30/0.57       => ( ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_complex ) ) ) )
% 0.30/0.57         => ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3385_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 0.30/0.57        ( ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [I4: real] :
% 0.30/0.57                ( ( member_real @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_real ) ) ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_real ) ) ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3386_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 0.30/0.57        ( ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [I4: nat] :
% 0.30/0.57                ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_real ) ) ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_real ) ) ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3387_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 0.30/0.57        ( ( finite_finite_int
% 0.30/0.57          @ ( collect_int
% 0.30/0.57            @ ^ [I4: int] :
% 0.30/0.57                ( ( member_int @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_real ) ) ) )
% 0.30/0.57       => ( ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_real ) ) ) )
% 0.30/0.57         => ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3388_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 0.30/0.57        ( ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [I4: complex] :
% 0.30/0.57                ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_real ) ) ) )
% 0.30/0.57       => ( ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_real ) ) ) )
% 0.30/0.57         => ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3389_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 0.30/0.57        ( ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [I4: real] :
% 0.30/0.57                ( ( member_real @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_rat ) ) ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_rat ) ) ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_rat ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3390_prod_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 0.30/0.57        ( ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [I4: nat] :
% 0.30/0.57                ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != one_one_rat ) ) ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != one_one_rat ) ) ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( times_times_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != one_one_rat ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % prod.finite_Collect_op
% 0.30/0.57  thf(fact_3391_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 0.30/0.57        ( ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [I4: real] :
% 0.30/0.57                ( ( member_real @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_complex ) ) ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_complex ) ) ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3392_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 0.30/0.57        ( ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [I4: nat] :
% 0.30/0.57                ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_complex ) ) ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_complex ) ) ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3393_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 0.30/0.57        ( ( finite_finite_int
% 0.30/0.57          @ ( collect_int
% 0.30/0.57            @ ^ [I4: int] :
% 0.30/0.57                ( ( member_int @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_complex ) ) ) )
% 0.30/0.57       => ( ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_complex ) ) ) )
% 0.30/0.57         => ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3394_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 0.30/0.57        ( ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [I4: complex] :
% 0.30/0.57                ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_complex ) ) ) )
% 0.30/0.57       => ( ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_complex ) ) ) )
% 0.30/0.57         => ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3395_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 0.30/0.57        ( ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [I4: real] :
% 0.30/0.57                ( ( member_real @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_real ) ) ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_real ) ) ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3396_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 0.30/0.57        ( ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [I4: nat] :
% 0.30/0.57                ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_real ) ) ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_real ) ) ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3397_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 0.30/0.57        ( ( finite_finite_int
% 0.30/0.57          @ ( collect_int
% 0.30/0.57            @ ^ [I4: int] :
% 0.30/0.57                ( ( member_int @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_real ) ) ) )
% 0.30/0.57       => ( ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_real ) ) ) )
% 0.30/0.57         => ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [I4: int] :
% 0.30/0.57                  ( ( member_int @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3398_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 0.30/0.57        ( ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [I4: complex] :
% 0.30/0.57                ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_real ) ) ) )
% 0.30/0.57       => ( ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_real ) ) ) )
% 0.30/0.57         => ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [I4: complex] :
% 0.30/0.57                  ( ( member_complex @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3399_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 0.30/0.57        ( ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [I4: real] :
% 0.30/0.57                ( ( member_real @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_rat ) ) ) )
% 0.30/0.57       => ( ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_rat ) ) ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [I4: real] :
% 0.30/0.57                  ( ( member_real @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3400_sum_Ofinite__Collect__op,axiom,
% 0.30/0.57      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 0.30/0.57        ( ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [I4: nat] :
% 0.30/0.57                ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                & ( ( X3 @ I4 )
% 0.30/0.57                 != zero_zero_rat ) ) ) )
% 0.30/0.57       => ( ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( Y3 @ I4 )
% 0.30/0.57                   != zero_zero_rat ) ) ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [I4: nat] :
% 0.30/0.57                  ( ( member_nat @ I4 @ I5 )
% 0.30/0.57                  & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 0.30/0.57                   != zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % sum.finite_Collect_op
% 0.30/0.57  thf(fact_3401_finite__interval__int1,axiom,
% 0.30/0.57      ! [A: int,B2: int] :
% 0.30/0.57        ( finite_finite_int
% 0.30/0.57        @ ( collect_int
% 0.30/0.57          @ ^ [I4: int] :
% 0.30/0.57              ( ( ord_less_eq_int @ A @ I4 )
% 0.30/0.57              & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_interval_int1
% 0.30/0.57  thf(fact_3402_finite__interval__int4,axiom,
% 0.30/0.57      ! [A: int,B2: int] :
% 0.30/0.57        ( finite_finite_int
% 0.30/0.57        @ ( collect_int
% 0.30/0.57          @ ^ [I4: int] :
% 0.30/0.57              ( ( ord_less_int @ A @ I4 )
% 0.30/0.57              & ( ord_less_int @ I4 @ B2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_interval_int4
% 0.30/0.57  thf(fact_3403_finite__interval__int3,axiom,
% 0.30/0.57      ! [A: int,B2: int] :
% 0.30/0.57        ( finite_finite_int
% 0.30/0.57        @ ( collect_int
% 0.30/0.57          @ ^ [I4: int] :
% 0.30/0.57              ( ( ord_less_int @ A @ I4 )
% 0.30/0.57              & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_interval_int3
% 0.30/0.57  thf(fact_3404_finite__interval__int2,axiom,
% 0.30/0.57      ! [A: int,B2: int] :
% 0.30/0.57        ( finite_finite_int
% 0.30/0.57        @ ( collect_int
% 0.30/0.57          @ ^ [I4: int] :
% 0.30/0.57              ( ( ord_less_eq_int @ A @ I4 )
% 0.30/0.57              & ( ord_less_int @ I4 @ B2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_interval_int2
% 0.30/0.57  thf(fact_3405_finite__maxlen,axiom,
% 0.30/0.57      ! [M7: set_list_VEBT_VEBT] :
% 0.30/0.57        ( ( finite3004134309566078307T_VEBT @ M7 )
% 0.30/0.57       => ? [N3: nat] :
% 0.30/0.57          ! [X6: list_VEBT_VEBT] :
% 0.30/0.57            ( ( member2936631157270082147T_VEBT @ X6 @ M7 )
% 0.30/0.57           => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X6 ) @ N3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_maxlen
% 0.30/0.57  thf(fact_3406_finite__maxlen,axiom,
% 0.30/0.57      ! [M7: set_list_o] :
% 0.30/0.57        ( ( finite_finite_list_o @ M7 )
% 0.30/0.57       => ? [N3: nat] :
% 0.30/0.57          ! [X6: list_o] :
% 0.30/0.57            ( ( member_list_o @ X6 @ M7 )
% 0.30/0.57           => ( ord_less_nat @ ( size_size_list_o @ X6 ) @ N3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_maxlen
% 0.30/0.57  thf(fact_3407_finite__maxlen,axiom,
% 0.30/0.57      ! [M7: set_list_nat] :
% 0.30/0.57        ( ( finite8100373058378681591st_nat @ M7 )
% 0.30/0.57       => ? [N3: nat] :
% 0.30/0.57          ! [X6: list_nat] :
% 0.30/0.57            ( ( member_list_nat @ X6 @ M7 )
% 0.30/0.57           => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_maxlen
% 0.30/0.57  thf(fact_3408_finite__has__minimal2,axiom,
% 0.30/0.57      ! [A4: set_real,A: real] :
% 0.30/0.57        ( ( finite_finite_real @ A4 )
% 0.30/0.57       => ( ( member_real @ A @ A4 )
% 0.30/0.57         => ? [X5: real] :
% 0.30/0.57              ( ( member_real @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_real @ X5 @ A )
% 0.30/0.57              & ! [Xa: real] :
% 0.30/0.57                  ( ( member_real @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_real @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal2
% 0.30/0.57  thf(fact_3409_finite__has__minimal2,axiom,
% 0.30/0.57      ! [A4: set_set_int,A: set_int] :
% 0.30/0.57        ( ( finite6197958912794628473et_int @ A4 )
% 0.30/0.57       => ( ( member_set_int @ A @ A4 )
% 0.30/0.57         => ? [X5: set_int] :
% 0.30/0.57              ( ( member_set_int @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_set_int @ X5 @ A )
% 0.30/0.57              & ! [Xa: set_int] :
% 0.30/0.57                  ( ( member_set_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal2
% 0.30/0.57  thf(fact_3410_finite__has__minimal2,axiom,
% 0.30/0.57      ! [A4: set_rat,A: rat] :
% 0.30/0.57        ( ( finite_finite_rat @ A4 )
% 0.30/0.57       => ( ( member_rat @ A @ A4 )
% 0.30/0.57         => ? [X5: rat] :
% 0.30/0.57              ( ( member_rat @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_rat @ X5 @ A )
% 0.30/0.57              & ! [Xa: rat] :
% 0.30/0.57                  ( ( member_rat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal2
% 0.30/0.57  thf(fact_3411_finite__has__minimal2,axiom,
% 0.30/0.57      ! [A4: set_num,A: num] :
% 0.30/0.57        ( ( finite_finite_num @ A4 )
% 0.30/0.57       => ( ( member_num @ A @ A4 )
% 0.30/0.57         => ? [X5: num] :
% 0.30/0.57              ( ( member_num @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_num @ X5 @ A )
% 0.30/0.57              & ! [Xa: num] :
% 0.30/0.57                  ( ( member_num @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_num @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal2
% 0.30/0.57  thf(fact_3412_finite__has__minimal2,axiom,
% 0.30/0.57      ! [A4: set_nat,A: nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( ( member_nat @ A @ A4 )
% 0.30/0.57         => ? [X5: nat] :
% 0.30/0.57              ( ( member_nat @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_nat @ X5 @ A )
% 0.30/0.57              & ! [Xa: nat] :
% 0.30/0.57                  ( ( member_nat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal2
% 0.30/0.57  thf(fact_3413_finite__has__minimal2,axiom,
% 0.30/0.57      ! [A4: set_int,A: int] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( ( member_int @ A @ A4 )
% 0.30/0.57         => ? [X5: int] :
% 0.30/0.57              ( ( member_int @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_int @ X5 @ A )
% 0.30/0.57              & ! [Xa: int] :
% 0.30/0.57                  ( ( member_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_int @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal2
% 0.30/0.57  thf(fact_3414_finite__has__maximal2,axiom,
% 0.30/0.57      ! [A4: set_real,A: real] :
% 0.30/0.57        ( ( finite_finite_real @ A4 )
% 0.30/0.57       => ( ( member_real @ A @ A4 )
% 0.30/0.57         => ? [X5: real] :
% 0.30/0.57              ( ( member_real @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_real @ A @ X5 )
% 0.30/0.57              & ! [Xa: real] :
% 0.30/0.57                  ( ( member_real @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_real @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal2
% 0.30/0.57  thf(fact_3415_finite__has__maximal2,axiom,
% 0.30/0.57      ! [A4: set_set_int,A: set_int] :
% 0.30/0.57        ( ( finite6197958912794628473et_int @ A4 )
% 0.30/0.57       => ( ( member_set_int @ A @ A4 )
% 0.30/0.57         => ? [X5: set_int] :
% 0.30/0.57              ( ( member_set_int @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_set_int @ A @ X5 )
% 0.30/0.57              & ! [Xa: set_int] :
% 0.30/0.57                  ( ( member_set_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal2
% 0.30/0.57  thf(fact_3416_finite__has__maximal2,axiom,
% 0.30/0.57      ! [A4: set_rat,A: rat] :
% 0.30/0.57        ( ( finite_finite_rat @ A4 )
% 0.30/0.57       => ( ( member_rat @ A @ A4 )
% 0.30/0.57         => ? [X5: rat] :
% 0.30/0.57              ( ( member_rat @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_rat @ A @ X5 )
% 0.30/0.57              & ! [Xa: rat] :
% 0.30/0.57                  ( ( member_rat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal2
% 0.30/0.57  thf(fact_3417_finite__has__maximal2,axiom,
% 0.30/0.57      ! [A4: set_num,A: num] :
% 0.30/0.57        ( ( finite_finite_num @ A4 )
% 0.30/0.57       => ( ( member_num @ A @ A4 )
% 0.30/0.57         => ? [X5: num] :
% 0.30/0.57              ( ( member_num @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_num @ A @ X5 )
% 0.30/0.57              & ! [Xa: num] :
% 0.30/0.57                  ( ( member_num @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_num @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal2
% 0.30/0.57  thf(fact_3418_finite__has__maximal2,axiom,
% 0.30/0.57      ! [A4: set_nat,A: nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( ( member_nat @ A @ A4 )
% 0.30/0.57         => ? [X5: nat] :
% 0.30/0.57              ( ( member_nat @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_nat @ A @ X5 )
% 0.30/0.57              & ! [Xa: nat] :
% 0.30/0.57                  ( ( member_nat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal2
% 0.30/0.57  thf(fact_3419_finite__has__maximal2,axiom,
% 0.30/0.57      ! [A4: set_int,A: int] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( ( member_int @ A @ A4 )
% 0.30/0.57         => ? [X5: int] :
% 0.30/0.57              ( ( member_int @ X5 @ A4 )
% 0.30/0.57              & ( ord_less_eq_int @ A @ X5 )
% 0.30/0.57              & ! [Xa: int] :
% 0.30/0.57                  ( ( member_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_int @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal2
% 0.30/0.57  thf(fact_3420_rev__finite__subset,axiom,
% 0.30/0.57      ! [B5: set_nat,A4: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ B5 )
% 0.30/0.57       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.57         => ( finite_finite_nat @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % rev_finite_subset
% 0.30/0.57  thf(fact_3421_rev__finite__subset,axiom,
% 0.30/0.57      ! [B5: set_complex,A4: set_complex] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.57       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.57         => ( finite3207457112153483333omplex @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % rev_finite_subset
% 0.30/0.57  thf(fact_3422_rev__finite__subset,axiom,
% 0.30/0.57      ! [B5: set_int,A4: set_int] :
% 0.30/0.57        ( ( finite_finite_int @ B5 )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.57         => ( finite_finite_int @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % rev_finite_subset
% 0.30/0.57  thf(fact_3423_infinite__super,axiom,
% 0.30/0.57      ! [S2: set_nat,T3: set_nat] :
% 0.30/0.57        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.57       => ( ~ ( finite_finite_nat @ S2 )
% 0.30/0.57         => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_super
% 0.30/0.57  thf(fact_3424_infinite__super,axiom,
% 0.30/0.57      ! [S2: set_complex,T3: set_complex] :
% 0.30/0.57        ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.57       => ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.30/0.57         => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_super
% 0.30/0.57  thf(fact_3425_infinite__super,axiom,
% 0.30/0.57      ! [S2: set_int,T3: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.57       => ( ~ ( finite_finite_int @ S2 )
% 0.30/0.57         => ~ ( finite_finite_int @ T3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % infinite_super
% 0.30/0.57  thf(fact_3426_finite__subset,axiom,
% 0.30/0.57      ! [A4: set_nat,B5: set_nat] :
% 0.30/0.57        ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.57       => ( ( finite_finite_nat @ B5 )
% 0.30/0.57         => ( finite_finite_nat @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_subset
% 0.30/0.57  thf(fact_3427_finite__subset,axiom,
% 0.30/0.57      ! [A4: set_complex,B5: set_complex] :
% 0.30/0.57        ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.57       => ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.57         => ( finite3207457112153483333omplex @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_subset
% 0.30/0.57  thf(fact_3428_finite__subset,axiom,
% 0.30/0.57      ! [A4: set_int,B5: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.57       => ( ( finite_finite_int @ B5 )
% 0.30/0.57         => ( finite_finite_int @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_subset
% 0.30/0.57  thf(fact_3429_finite__psubset__induct,axiom,
% 0.30/0.57      ! [A4: set_nat,P: set_nat > $o] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( ! [A7: set_nat] :
% 0.30/0.57              ( ( finite_finite_nat @ A7 )
% 0.30/0.57             => ( ! [B7: set_nat] :
% 0.30/0.57                    ( ( ord_less_set_nat @ B7 @ A7 )
% 0.30/0.57                   => ( P @ B7 ) )
% 0.30/0.57               => ( P @ A7 ) ) )
% 0.30/0.57         => ( P @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_psubset_induct
% 0.30/0.57  thf(fact_3430_finite__psubset__induct,axiom,
% 0.30/0.57      ! [A4: set_int,P: set_int > $o] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( ! [A7: set_int] :
% 0.30/0.57              ( ( finite_finite_int @ A7 )
% 0.30/0.57             => ( ! [B7: set_int] :
% 0.30/0.57                    ( ( ord_less_set_int @ B7 @ A7 )
% 0.30/0.57                   => ( P @ B7 ) )
% 0.30/0.57               => ( P @ A7 ) ) )
% 0.30/0.57         => ( P @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_psubset_induct
% 0.30/0.57  thf(fact_3431_finite__psubset__induct,axiom,
% 0.30/0.57      ! [A4: set_complex,P: set_complex > $o] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.57       => ( ! [A7: set_complex] :
% 0.30/0.57              ( ( finite3207457112153483333omplex @ A7 )
% 0.30/0.57             => ( ! [B7: set_complex] :
% 0.30/0.57                    ( ( ord_less_set_complex @ B7 @ A7 )
% 0.30/0.57                   => ( P @ B7 ) )
% 0.30/0.57               => ( P @ A7 ) ) )
% 0.30/0.57         => ( P @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_psubset_induct
% 0.30/0.57  thf(fact_3432_finite__image__set,axiom,
% 0.30/0.57      ! [P: real > $o,F: real > real] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [Uu3: real] :
% 0.30/0.57              ? [X: real] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3433_finite__image__set,axiom,
% 0.30/0.57      ! [P: real > $o,F: real > nat] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [Uu3: nat] :
% 0.30/0.57              ? [X: real] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3434_finite__image__set,axiom,
% 0.30/0.57      ! [P: real > $o,F: real > int] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( finite_finite_int
% 0.30/0.57          @ ( collect_int
% 0.30/0.57            @ ^ [Uu3: int] :
% 0.30/0.57              ? [X: real] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3435_finite__image__set,axiom,
% 0.30/0.57      ! [P: real > $o,F: real > complex] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [Uu3: complex] :
% 0.30/0.57              ? [X: real] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3436_finite__image__set,axiom,
% 0.30/0.57      ! [P: nat > $o,F: nat > real] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [Uu3: real] :
% 0.30/0.57              ? [X: nat] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3437_finite__image__set,axiom,
% 0.30/0.57      ! [P: nat > $o,F: nat > nat] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [Uu3: nat] :
% 0.30/0.57              ? [X: nat] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3438_finite__image__set,axiom,
% 0.30/0.57      ! [P: nat > $o,F: nat > int] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( finite_finite_int
% 0.30/0.57          @ ( collect_int
% 0.30/0.57            @ ^ [Uu3: int] :
% 0.30/0.57              ? [X: nat] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3439_finite__image__set,axiom,
% 0.30/0.57      ! [P: nat > $o,F: nat > complex] :
% 0.30/0.57        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.57       => ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [Uu3: complex] :
% 0.30/0.57              ? [X: nat] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3440_finite__image__set,axiom,
% 0.30/0.57      ! [P: int > $o,F: int > real] :
% 0.30/0.57        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.57       => ( finite_finite_real
% 0.30/0.57          @ ( collect_real
% 0.30/0.57            @ ^ [Uu3: real] :
% 0.30/0.57              ? [X: int] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3441_finite__image__set,axiom,
% 0.30/0.57      ! [P: int > $o,F: int > nat] :
% 0.30/0.57        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.57       => ( finite_finite_nat
% 0.30/0.57          @ ( collect_nat
% 0.30/0.57            @ ^ [Uu3: nat] :
% 0.30/0.57              ? [X: int] :
% 0.30/0.57                ( ( Uu3
% 0.30/0.57                  = ( F @ X ) )
% 0.30/0.57                & ( P @ X ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set
% 0.30/0.57  thf(fact_3442_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: real > $o,F: real > real > real] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [Uu3: real] :
% 0.30/0.57                ? [X: real,Y: real] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3443_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: real > $o,F: real > real > nat] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [Uu3: nat] :
% 0.30/0.57                ? [X: real,Y: real] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3444_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: real > $o,F: real > real > int] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.57         => ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [Uu3: int] :
% 0.30/0.57                ? [X: real,Y: real] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3445_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: real > $o,F: real > real > complex] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.57         => ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [Uu3: complex] :
% 0.30/0.57                ? [X: real,Y: real] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3446_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [Uu3: real] :
% 0.30/0.57                ? [X: real,Y: nat] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3447_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [Uu3: nat] :
% 0.30/0.57                ? [X: real,Y: nat] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3448_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: nat > $o,F: real > nat > int] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.57         => ( finite_finite_int
% 0.30/0.57            @ ( collect_int
% 0.30/0.57              @ ^ [Uu3: int] :
% 0.30/0.57                ? [X: real,Y: nat] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3449_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.57         => ( finite3207457112153483333omplex
% 0.30/0.57            @ ( collect_complex
% 0.30/0.57              @ ^ [Uu3: complex] :
% 0.30/0.57                ? [X: real,Y: nat] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3450_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: int > $o,F: real > int > real] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 0.30/0.57         => ( finite_finite_real
% 0.30/0.57            @ ( collect_real
% 0.30/0.57              @ ^ [Uu3: real] :
% 0.30/0.57                ? [X: real,Y: int] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3451_finite__image__set2,axiom,
% 0.30/0.57      ! [P: real > $o,Q: int > $o,F: real > int > nat] :
% 0.30/0.57        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.57       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 0.30/0.57         => ( finite_finite_nat
% 0.30/0.57            @ ( collect_nat
% 0.30/0.57              @ ^ [Uu3: nat] :
% 0.30/0.57                ? [X: real,Y: int] :
% 0.30/0.57                  ( ( Uu3
% 0.30/0.57                    = ( F @ X @ Y ) )
% 0.30/0.57                  & ( P @ X )
% 0.30/0.57                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_image_set2
% 0.30/0.57  thf(fact_3452_finite__has__minimal,axiom,
% 0.30/0.57      ! [A4: set_real] :
% 0.30/0.57        ( ( finite_finite_real @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_real )
% 0.30/0.57         => ? [X5: real] :
% 0.30/0.57              ( ( member_real @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: real] :
% 0.30/0.57                  ( ( member_real @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_real @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal
% 0.30/0.57  thf(fact_3453_finite__has__minimal,axiom,
% 0.30/0.57      ! [A4: set_set_int] :
% 0.30/0.57        ( ( finite6197958912794628473et_int @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_set_int )
% 0.30/0.57         => ? [X5: set_int] :
% 0.30/0.57              ( ( member_set_int @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: set_int] :
% 0.30/0.57                  ( ( member_set_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal
% 0.30/0.57  thf(fact_3454_finite__has__minimal,axiom,
% 0.30/0.57      ! [A4: set_rat] :
% 0.30/0.57        ( ( finite_finite_rat @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_rat )
% 0.30/0.57         => ? [X5: rat] :
% 0.30/0.57              ( ( member_rat @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: rat] :
% 0.30/0.57                  ( ( member_rat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal
% 0.30/0.57  thf(fact_3455_finite__has__minimal,axiom,
% 0.30/0.57      ! [A4: set_num] :
% 0.30/0.57        ( ( finite_finite_num @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_num )
% 0.30/0.57         => ? [X5: num] :
% 0.30/0.57              ( ( member_num @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: num] :
% 0.30/0.57                  ( ( member_num @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_num @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal
% 0.30/0.57  thf(fact_3456_finite__has__minimal,axiom,
% 0.30/0.57      ! [A4: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_nat )
% 0.30/0.57         => ? [X5: nat] :
% 0.30/0.57              ( ( member_nat @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: nat] :
% 0.30/0.57                  ( ( member_nat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal
% 0.30/0.57  thf(fact_3457_finite__has__minimal,axiom,
% 0.30/0.57      ! [A4: set_int] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_int )
% 0.30/0.57         => ? [X5: int] :
% 0.30/0.57              ( ( member_int @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: int] :
% 0.30/0.57                  ( ( member_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_int @ Xa @ X5 )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_minimal
% 0.30/0.57  thf(fact_3458_finite__has__maximal,axiom,
% 0.30/0.57      ! [A4: set_real] :
% 0.30/0.57        ( ( finite_finite_real @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_real )
% 0.30/0.57         => ? [X5: real] :
% 0.30/0.57              ( ( member_real @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: real] :
% 0.30/0.57                  ( ( member_real @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_real @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal
% 0.30/0.57  thf(fact_3459_finite__has__maximal,axiom,
% 0.30/0.57      ! [A4: set_set_int] :
% 0.30/0.57        ( ( finite6197958912794628473et_int @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_set_int )
% 0.30/0.57         => ? [X5: set_int] :
% 0.30/0.57              ( ( member_set_int @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: set_int] :
% 0.30/0.57                  ( ( member_set_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal
% 0.30/0.57  thf(fact_3460_finite__has__maximal,axiom,
% 0.30/0.57      ! [A4: set_rat] :
% 0.30/0.57        ( ( finite_finite_rat @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_rat )
% 0.30/0.57         => ? [X5: rat] :
% 0.30/0.57              ( ( member_rat @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: rat] :
% 0.30/0.57                  ( ( member_rat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal
% 0.30/0.57  thf(fact_3461_finite__has__maximal,axiom,
% 0.30/0.57      ! [A4: set_num] :
% 0.30/0.57        ( ( finite_finite_num @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_num )
% 0.30/0.57         => ? [X5: num] :
% 0.30/0.57              ( ( member_num @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: num] :
% 0.30/0.57                  ( ( member_num @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_num @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal
% 0.30/0.57  thf(fact_3462_finite__has__maximal,axiom,
% 0.30/0.57      ! [A4: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_nat )
% 0.30/0.57         => ? [X5: nat] :
% 0.30/0.57              ( ( member_nat @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: nat] :
% 0.30/0.57                  ( ( member_nat @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal
% 0.30/0.57  thf(fact_3463_finite__has__maximal,axiom,
% 0.30/0.57      ! [A4: set_int] :
% 0.30/0.57        ( ( finite_finite_int @ A4 )
% 0.30/0.57       => ( ( A4 != bot_bot_set_int )
% 0.30/0.57         => ? [X5: int] :
% 0.30/0.57              ( ( member_int @ X5 @ A4 )
% 0.30/0.57              & ! [Xa: int] :
% 0.30/0.57                  ( ( member_int @ Xa @ A4 )
% 0.30/0.57                 => ( ( ord_less_eq_int @ X5 @ Xa )
% 0.30/0.57                   => ( X5 = Xa ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_has_maximal
% 0.30/0.57  thf(fact_3464_finite__nth__roots,axiom,
% 0.30/0.57      ! [N: nat,C: complex] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.57       => ( finite3207457112153483333omplex
% 0.30/0.57          @ ( collect_complex
% 0.30/0.57            @ ^ [Z2: complex] :
% 0.30/0.57                ( ( power_power_complex @ Z2 @ N )
% 0.30/0.57                = C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_nth_roots
% 0.30/0.57  thf(fact_3465_insert__simp__excp,axiom,
% 0.30/0.57      ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X3: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ X3 @ Mi )
% 0.30/0.57         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.57           => ( ( X3 != Ma )
% 0.30/0.57             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.57                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_simp_excp
% 0.30/0.57  thf(fact_3466_insert__simp__norm,axiom,
% 0.30/0.57      ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ Mi @ X3 )
% 0.30/0.57         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.57           => ( ( X3 != Ma )
% 0.30/0.57             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.57                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X3 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( 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 ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_simp_norm
% 0.30/0.57  thf(fact_3467_gcd__nat__induct,axiom,
% 0.30/0.57      ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.30/0.57        ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 0.30/0.57       => ( ! [M4: nat,N3: nat] :
% 0.30/0.57              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.57             => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
% 0.30/0.57               => ( P @ M4 @ N3 ) ) )
% 0.30/0.57         => ( P @ M @ N ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % gcd_nat_induct
% 0.30/0.57  thf(fact_3468_concat__bit__Suc,axiom,
% 0.30/0.57      ! [N: nat,K: int,L: int] :
% 0.30/0.57        ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.30/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % concat_bit_Suc
% 0.30/0.57  thf(fact_3469_order__refl,axiom,
% 0.30/0.57      ! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).
% 0.30/0.57  
% 0.30/0.57  % order_refl
% 0.30/0.57  thf(fact_3470_order__refl,axiom,
% 0.30/0.57      ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).
% 0.30/0.57  
% 0.30/0.57  % order_refl
% 0.30/0.57  thf(fact_3471_order__refl,axiom,
% 0.30/0.57      ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% 0.30/0.57  
% 0.30/0.57  % order_refl
% 0.30/0.57  thf(fact_3472_order__refl,axiom,
% 0.30/0.57      ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 0.30/0.57  
% 0.30/0.57  % order_refl
% 0.30/0.57  thf(fact_3473_order__refl,axiom,
% 0.30/0.57      ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% 0.30/0.57  
% 0.30/0.57  % order_refl
% 0.30/0.57  thf(fact_3474_dual__order_Orefl,axiom,
% 0.30/0.57      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.refl
% 0.30/0.57  thf(fact_3475_dual__order_Orefl,axiom,
% 0.30/0.57      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.refl
% 0.30/0.57  thf(fact_3476_dual__order_Orefl,axiom,
% 0.30/0.57      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.refl
% 0.30/0.57  thf(fact_3477_dual__order_Orefl,axiom,
% 0.30/0.57      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.refl
% 0.30/0.57  thf(fact_3478_dual__order_Orefl,axiom,
% 0.30/0.57      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.refl
% 0.30/0.57  thf(fact_3479_length__list__update,axiom,
% 0.30/0.57      ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 0.30/0.57        ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) )
% 0.30/0.57        = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % length_list_update
% 0.30/0.57  thf(fact_3480_length__list__update,axiom,
% 0.30/0.57      ! [Xs2: list_o,I: nat,X3: $o] :
% 0.30/0.57        ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X3 ) )
% 0.30/0.57        = ( size_size_list_o @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % length_list_update
% 0.30/0.57  thf(fact_3481_length__list__update,axiom,
% 0.30/0.57      ! [Xs2: list_nat,I: nat,X3: nat] :
% 0.30/0.57        ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) )
% 0.30/0.57        = ( size_size_list_nat @ Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % length_list_update
% 0.30/0.57  thf(fact_3482_max__Suc__Suc,axiom,
% 0.30/0.57      ! [M: nat,N: nat] :
% 0.30/0.57        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.57        = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_Suc_Suc
% 0.30/0.57  thf(fact_3483_max__0R,axiom,
% 0.30/0.57      ! [N: nat] :
% 0.30/0.57        ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.30/0.57        = N ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0R
% 0.30/0.57  thf(fact_3484_max__0L,axiom,
% 0.30/0.57      ! [N: nat] :
% 0.30/0.57        ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.30/0.57        = N ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0L
% 0.30/0.57  thf(fact_3485_max__nat_Oright__neutral,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % max_nat.right_neutral
% 0.30/0.57  thf(fact_3486_max__nat_Oneutr__eq__iff,axiom,
% 0.30/0.57      ! [A: nat,B2: nat] :
% 0.30/0.57        ( ( zero_zero_nat
% 0.30/0.57          = ( ord_max_nat @ A @ B2 ) )
% 0.30/0.57        = ( ( A = zero_zero_nat )
% 0.30/0.57          & ( B2 = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_nat.neutr_eq_iff
% 0.30/0.57  thf(fact_3487_max__nat_Oleft__neutral,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % max_nat.left_neutral
% 0.30/0.57  thf(fact_3488_max__nat_Oeq__neutr__iff,axiom,
% 0.30/0.57      ! [A: nat,B2: nat] :
% 0.30/0.57        ( ( ( ord_max_nat @ A @ B2 )
% 0.30/0.57          = zero_zero_nat )
% 0.30/0.57        = ( ( A = zero_zero_nat )
% 0.30/0.57          & ( B2 = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_nat.eq_neutr_iff
% 0.30/0.57  thf(fact_3489_nth__list__update__neq,axiom,
% 0.30/0.57      ! [I: nat,J: nat,Xs2: list_int,X3: int] :
% 0.30/0.57        ( ( I != J )
% 0.30/0.57       => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.57          = ( nth_int @ Xs2 @ J ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_neq
% 0.30/0.57  thf(fact_3490_nth__list__update__neq,axiom,
% 0.30/0.57      ! [I: nat,J: nat,Xs2: list_nat,X3: nat] :
% 0.30/0.57        ( ( I != J )
% 0.30/0.57       => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.57          = ( nth_nat @ Xs2 @ J ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_neq
% 0.30/0.57  thf(fact_3491_nth__list__update__neq,axiom,
% 0.30/0.57      ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.30/0.57        ( ( I != J )
% 0.30/0.57       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.57          = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_neq
% 0.30/0.57  thf(fact_3492_list__update__id,axiom,
% 0.30/0.57      ! [Xs2: list_int,I: nat] :
% 0.30/0.57        ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
% 0.30/0.57        = Xs2 ) ).
% 0.30/0.57  
% 0.30/0.57  % list_update_id
% 0.30/0.57  thf(fact_3493_list__update__id,axiom,
% 0.30/0.57      ! [Xs2: list_nat,I: nat] :
% 0.30/0.57        ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
% 0.30/0.57        = Xs2 ) ).
% 0.30/0.57  
% 0.30/0.57  % list_update_id
% 0.30/0.57  thf(fact_3494_list__update__id,axiom,
% 0.30/0.57      ! [Xs2: list_VEBT_VEBT,I: nat] :
% 0.30/0.57        ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
% 0.30/0.57        = Xs2 ) ).
% 0.30/0.57  
% 0.30/0.57  % list_update_id
% 0.30/0.57  thf(fact_3495_concat__bit__0,axiom,
% 0.30/0.57      ! [K: int,L: int] :
% 0.30/0.57        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.30/0.57        = L ) ).
% 0.30/0.57  
% 0.30/0.57  % concat_bit_0
% 0.30/0.57  thf(fact_3496_max__number__of_I1_J,axiom,
% 0.30/0.57      ! [U: num,V: num] :
% 0.30/0.57        ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57            = ( numera1916890842035813515d_enat @ V ) ) )
% 0.30/0.57        & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.30/0.57            = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_number_of(1)
% 0.30/0.57  thf(fact_3497_max__number__of_I1_J,axiom,
% 0.30/0.57      ! [U: num,V: num] :
% 0.30/0.57        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57            = ( numeral_numeral_real @ V ) ) )
% 0.30/0.57        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.57            = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_number_of(1)
% 0.30/0.57  thf(fact_3498_max__number__of_I1_J,axiom,
% 0.30/0.57      ! [U: num,V: num] :
% 0.30/0.57        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57            = ( numeral_numeral_rat @ V ) ) )
% 0.30/0.57        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.57            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_number_of(1)
% 0.30/0.57  thf(fact_3499_max__number__of_I1_J,axiom,
% 0.30/0.57      ! [U: num,V: num] :
% 0.30/0.57        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57            = ( numeral_numeral_nat @ V ) ) )
% 0.30/0.57        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.57            = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_number_of(1)
% 0.30/0.57  thf(fact_3500_max__number__of_I1_J,axiom,
% 0.30/0.57      ! [U: num,V: num] :
% 0.30/0.57        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57            = ( numeral_numeral_int @ V ) ) )
% 0.30/0.57        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.57            = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_number_of(1)
% 0.30/0.57  thf(fact_3501_max__0__1_I4_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ zero_z5237406670263579293d_enat )
% 0.30/0.57        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(4)
% 0.30/0.57  thf(fact_3502_max__0__1_I4_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ zero_zero_real )
% 0.30/0.57        = ( numeral_numeral_real @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(4)
% 0.30/0.57  thf(fact_3503_max__0__1_I4_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ zero_zero_rat )
% 0.30/0.57        = ( numeral_numeral_rat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(4)
% 0.30/0.57  thf(fact_3504_max__0__1_I4_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ zero_zero_nat )
% 0.30/0.57        = ( numeral_numeral_nat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(4)
% 0.30/0.57  thf(fact_3505_max__0__1_I4_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ zero_zero_int )
% 0.30/0.57        = ( numeral_numeral_int @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(4)
% 0.30/0.57  thf(fact_3506_max__0__1_I3_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
% 0.30/0.57        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(3)
% 0.30/0.57  thf(fact_3507_max__0__1_I3_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_real @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(3)
% 0.30/0.57  thf(fact_3508_max__0__1_I3_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_rat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(3)
% 0.30/0.57  thf(fact_3509_max__0__1_I3_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_nat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(3)
% 0.30/0.57  thf(fact_3510_max__0__1_I3_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_int @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(3)
% 0.30/0.57  thf(fact_3511_max__0__1_I1_J,axiom,
% 0.30/0.57      ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.30/0.57      = one_one_real ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(1)
% 0.30/0.57  thf(fact_3512_max__0__1_I1_J,axiom,
% 0.30/0.57      ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.30/0.57      = one_one_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(1)
% 0.30/0.57  thf(fact_3513_max__0__1_I1_J,axiom,
% 0.30/0.57      ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.30/0.57      = one_one_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(1)
% 0.30/0.57  thf(fact_3514_max__0__1_I1_J,axiom,
% 0.30/0.57      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 0.30/0.57      = one_on7984719198319812577d_enat ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(1)
% 0.30/0.57  thf(fact_3515_max__0__1_I1_J,axiom,
% 0.30/0.57      ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.30/0.57      = one_one_int ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(1)
% 0.30/0.57  thf(fact_3516_max__0__1_I2_J,axiom,
% 0.30/0.57      ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.30/0.57      = one_one_real ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(2)
% 0.30/0.57  thf(fact_3517_max__0__1_I2_J,axiom,
% 0.30/0.57      ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.30/0.57      = one_one_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(2)
% 0.30/0.57  thf(fact_3518_max__0__1_I2_J,axiom,
% 0.30/0.57      ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.30/0.57      = one_one_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(2)
% 0.30/0.57  thf(fact_3519_max__0__1_I2_J,axiom,
% 0.30/0.57      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 0.30/0.57      = one_on7984719198319812577d_enat ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(2)
% 0.30/0.57  thf(fact_3520_max__0__1_I2_J,axiom,
% 0.30/0.57      ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.30/0.57      = one_one_int ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(2)
% 0.30/0.57  thf(fact_3521_max__0__1_I6_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat )
% 0.30/0.57        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(6)
% 0.30/0.57  thf(fact_3522_max__0__1_I6_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ one_one_real )
% 0.30/0.57        = ( numeral_numeral_real @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(6)
% 0.30/0.57  thf(fact_3523_max__0__1_I6_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat )
% 0.30/0.57        = ( numeral_numeral_rat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(6)
% 0.30/0.57  thf(fact_3524_max__0__1_I6_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat )
% 0.30/0.57        = ( numeral_numeral_nat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(6)
% 0.30/0.57  thf(fact_3525_max__0__1_I6_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ one_one_int )
% 0.30/0.57        = ( numeral_numeral_int @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(6)
% 0.30/0.57  thf(fact_3526_max__0__1_I5_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
% 0.30/0.57        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(5)
% 0.30/0.57  thf(fact_3527_max__0__1_I5_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_real @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(5)
% 0.30/0.57  thf(fact_3528_max__0__1_I5_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_rat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(5)
% 0.30/0.57  thf(fact_3529_max__0__1_I5_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_nat @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(5)
% 0.30/0.57  thf(fact_3530_max__0__1_I5_J,axiom,
% 0.30/0.57      ! [X3: num] :
% 0.30/0.57        ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
% 0.30/0.57        = ( numeral_numeral_int @ X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_0_1(5)
% 0.30/0.57  thf(fact_3531_list__update__beyond,axiom,
% 0.30/0.57      ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
% 0.30/0.57       => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 )
% 0.30/0.57          = Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % list_update_beyond
% 0.30/0.57  thf(fact_3532_list__update__beyond,axiom,
% 0.30/0.57      ! [Xs2: list_o,I: nat,X3: $o] :
% 0.30/0.57        ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
% 0.30/0.57       => ( ( list_update_o @ Xs2 @ I @ X3 )
% 0.30/0.57          = Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % list_update_beyond
% 0.30/0.57  thf(fact_3533_list__update__beyond,axiom,
% 0.30/0.57      ! [Xs2: list_nat,I: nat,X3: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
% 0.30/0.57       => ( ( list_update_nat @ Xs2 @ I @ X3 )
% 0.30/0.57          = Xs2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % list_update_beyond
% 0.30/0.57  thf(fact_3534_concat__bit__nonnegative__iff,axiom,
% 0.30/0.57      ! [N: nat,K: int,L: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 0.30/0.57        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 0.30/0.57  
% 0.30/0.57  % concat_bit_nonnegative_iff
% 0.30/0.57  thf(fact_3535_concat__bit__negative__iff,axiom,
% 0.30/0.57      ! [N: nat,K: int,L: int] :
% 0.30/0.57        ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.30/0.57        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.30/0.57  
% 0.30/0.57  % concat_bit_negative_iff
% 0.30/0.57  thf(fact_3536_nth__list__update__eq,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_int,X3: int] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57       => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ I )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_eq
% 0.30/0.57  thf(fact_3537_nth__list__update__eq,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ I )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_eq
% 0.30/0.57  thf(fact_3538_nth__list__update__eq,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_o,X3: $o] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57       => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X3 ) @ I )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_eq
% 0.30/0.57  thf(fact_3539_nth__list__update__eq,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_nat,X3: nat] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57       => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ I )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nth_list_update_eq
% 0.30/0.57  thf(fact_3540_set__swap,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_int,J: nat] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57         => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
% 0.30/0.57            = ( set_int2 @ Xs2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_swap
% 0.30/0.57  thf(fact_3541_set__swap,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57         => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
% 0.30/0.57            = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_swap
% 0.30/0.57  thf(fact_3542_set__swap,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_o,J: nat] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57         => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
% 0.30/0.57            = ( set_o2 @ Xs2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_swap
% 0.30/0.57  thf(fact_3543_set__swap,axiom,
% 0.30/0.57      ! [I: nat,Xs2: list_nat,J: nat] :
% 0.30/0.57        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57         => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
% 0.30/0.57            = ( set_nat2 @ Xs2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_swap
% 0.30/0.57  thf(fact_3544_max__def,axiom,
% 0.30/0.57      ( ord_ma741700101516333627d_enat
% 0.30/0.57      = ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def
% 0.30/0.57  thf(fact_3545_max__def,axiom,
% 0.30/0.57      ( ord_max_set_int
% 0.30/0.57      = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def
% 0.30/0.57  thf(fact_3546_max__def,axiom,
% 0.30/0.57      ( ord_max_rat
% 0.30/0.57      = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def
% 0.30/0.57  thf(fact_3547_max__def,axiom,
% 0.30/0.57      ( ord_max_num
% 0.30/0.57      = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def
% 0.30/0.57  thf(fact_3548_max__def,axiom,
% 0.30/0.57      ( ord_max_nat
% 0.30/0.57      = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def
% 0.30/0.57  thf(fact_3549_max__def,axiom,
% 0.30/0.57      ( ord_max_int
% 0.30/0.57      = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def
% 0.30/0.57  thf(fact_3550_max__absorb1,axiom,
% 0.30/0.57      ! [Y3: extended_enat,X3: extended_enat] :
% 0.30/0.57        ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
% 0.30/0.57       => ( ( ord_ma741700101516333627d_enat @ X3 @ Y3 )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb1
% 0.30/0.57  thf(fact_3551_max__absorb1,axiom,
% 0.30/0.57      ! [Y3: set_int,X3: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.30/0.57       => ( ( ord_max_set_int @ X3 @ Y3 )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb1
% 0.30/0.57  thf(fact_3552_max__absorb1,axiom,
% 0.30/0.57      ! [Y3: rat,X3: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.57       => ( ( ord_max_rat @ X3 @ Y3 )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb1
% 0.30/0.57  thf(fact_3553_max__absorb1,axiom,
% 0.30/0.57      ! [Y3: num,X3: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.30/0.57       => ( ( ord_max_num @ X3 @ Y3 )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb1
% 0.30/0.57  thf(fact_3554_max__absorb1,axiom,
% 0.30/0.57      ! [Y3: nat,X3: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.57       => ( ( ord_max_nat @ X3 @ Y3 )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb1
% 0.30/0.57  thf(fact_3555_max__absorb1,axiom,
% 0.30/0.57      ! [Y3: int,X3: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.30/0.57       => ( ( ord_max_int @ X3 @ Y3 )
% 0.30/0.57          = X3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb1
% 0.30/0.57  thf(fact_3556_max__absorb2,axiom,
% 0.30/0.57      ! [X3: extended_enat,Y3: extended_enat] :
% 0.30/0.57        ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_ma741700101516333627d_enat @ X3 @ Y3 )
% 0.30/0.57          = Y3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb2
% 0.30/0.57  thf(fact_3557_max__absorb2,axiom,
% 0.30/0.57      ! [X3: set_int,Y3: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_max_set_int @ X3 @ Y3 )
% 0.30/0.57          = Y3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb2
% 0.30/0.57  thf(fact_3558_max__absorb2,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_max_rat @ X3 @ Y3 )
% 0.30/0.57          = Y3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb2
% 0.30/0.57  thf(fact_3559_max__absorb2,axiom,
% 0.30/0.57      ! [X3: num,Y3: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_max_num @ X3 @ Y3 )
% 0.30/0.57          = Y3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb2
% 0.30/0.57  thf(fact_3560_max__absorb2,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_max_nat @ X3 @ Y3 )
% 0.30/0.57          = Y3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb2
% 0.30/0.57  thf(fact_3561_max__absorb2,axiom,
% 0.30/0.57      ! [X3: int,Y3: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_max_int @ X3 @ Y3 )
% 0.30/0.57          = Y3 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_absorb2
% 0.30/0.57  thf(fact_3562_max__add__distrib__left,axiom,
% 0.30/0.57      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.57        ( ( plus_plus_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_real @ ( plus_plus_real @ X3 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_left
% 0.30/0.57  thf(fact_3563_max__add__distrib__left,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.57        ( ( plus_plus_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_left
% 0.30/0.57  thf(fact_3564_max__add__distrib__left,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( ord_max_nat @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_left
% 0.30/0.57  thf(fact_3565_max__add__distrib__left,axiom,
% 0.30/0.57      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_int @ ( plus_plus_int @ X3 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_left
% 0.30/0.57  thf(fact_3566_max__add__distrib__right,axiom,
% 0.30/0.57      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.57        ( ( plus_plus_real @ X3 @ ( ord_max_real @ Y3 @ Z ) )
% 0.30/0.57        = ( ord_max_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( plus_plus_real @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_right
% 0.30/0.57  thf(fact_3567_max__add__distrib__right,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.57        ( ( plus_plus_rat @ X3 @ ( ord_max_rat @ Y3 @ Z ) )
% 0.30/0.57        = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( plus_plus_rat @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_right
% 0.30/0.57  thf(fact_3568_max__add__distrib__right,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ X3 @ ( ord_max_nat @ Y3 @ Z ) )
% 0.30/0.57        = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( plus_plus_nat @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_right
% 0.30/0.57  thf(fact_3569_max__add__distrib__right,axiom,
% 0.30/0.57      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.57        ( ( plus_plus_int @ X3 @ ( ord_max_int @ Y3 @ Z ) )
% 0.30/0.57        = ( ord_max_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( plus_plus_int @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_add_distrib_right
% 0.30/0.57  thf(fact_3570_max__diff__distrib__left,axiom,
% 0.30/0.57      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.57        ( ( minus_minus_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_real @ ( minus_minus_real @ X3 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_diff_distrib_left
% 0.30/0.57  thf(fact_3571_max__diff__distrib__left,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_rat @ ( minus_minus_rat @ X3 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_diff_distrib_left
% 0.30/0.57  thf(fact_3572_max__diff__distrib__left,axiom,
% 0.30/0.57      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.57        ( ( minus_minus_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
% 0.30/0.57        = ( ord_max_int @ ( minus_minus_int @ X3 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_diff_distrib_left
% 0.30/0.57  thf(fact_3573_nat__add__max__left,axiom,
% 0.30/0.57      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.30/0.57        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_add_max_left
% 0.30/0.57  thf(fact_3574_nat__add__max__right,axiom,
% 0.30/0.57      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.30/0.57        = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_add_max_right
% 0.30/0.57  thf(fact_3575_nat__mult__max__right,axiom,
% 0.30/0.57      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.57        ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.30/0.57        = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_mult_max_right
% 0.30/0.57  thf(fact_3576_nat__mult__max__left,axiom,
% 0.30/0.57      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.57        ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.30/0.57        = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_mult_max_left
% 0.30/0.57  thf(fact_3577_max__def__raw,axiom,
% 0.30/0.57      ( ord_ma741700101516333627d_enat
% 0.30/0.57      = ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def_raw
% 0.30/0.57  thf(fact_3578_max__def__raw,axiom,
% 0.30/0.57      ( ord_max_set_int
% 0.30/0.57      = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def_raw
% 0.30/0.57  thf(fact_3579_max__def__raw,axiom,
% 0.30/0.57      ( ord_max_rat
% 0.30/0.57      = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def_raw
% 0.30/0.57  thf(fact_3580_max__def__raw,axiom,
% 0.30/0.57      ( ord_max_num
% 0.30/0.57      = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def_raw
% 0.30/0.57  thf(fact_3581_max__def__raw,axiom,
% 0.30/0.57      ( ord_max_nat
% 0.30/0.57      = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def_raw
% 0.30/0.57  thf(fact_3582_max__def__raw,axiom,
% 0.30/0.57      ( ord_max_int
% 0.30/0.57      = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % max_def_raw
% 0.30/0.57  thf(fact_3583_concat__bit__assoc,axiom,
% 0.30/0.57      ! [N: nat,K: int,M: nat,L: int,R: int] :
% 0.30/0.57        ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R ) )
% 0.30/0.57        = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% 0.30/0.57  
% 0.30/0.57  % concat_bit_assoc
% 0.30/0.57  thf(fact_3584_nat__minus__add__max,axiom,
% 0.30/0.57      ! [N: nat,M: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.30/0.57        = ( ord_max_nat @ N @ M ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_minus_add_max
% 0.30/0.57  thf(fact_3585_set__update__subsetI,axiom,
% 0.30/0.57      ! [Xs2: list_real,A4: set_real,X3: real,I: nat] :
% 0.30/0.57        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
% 0.30/0.57       => ( ( member_real @ X3 @ A4 )
% 0.30/0.57         => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_subsetI
% 0.30/0.57  thf(fact_3586_set__update__subsetI,axiom,
% 0.30/0.57      ! [Xs2: list_complex,A4: set_complex,X3: complex,I: nat] :
% 0.30/0.57        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
% 0.30/0.57       => ( ( member_complex @ X3 @ A4 )
% 0.30/0.57         => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_subsetI
% 0.30/0.57  thf(fact_3587_set__update__subsetI,axiom,
% 0.30/0.57      ! [Xs2: list_nat,A4: set_nat,X3: nat,I: nat] :
% 0.30/0.57        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
% 0.30/0.57       => ( ( member_nat @ X3 @ A4 )
% 0.30/0.57         => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_subsetI
% 0.30/0.57  thf(fact_3588_set__update__subsetI,axiom,
% 0.30/0.57      ! [Xs2: list_VEBT_VEBT,A4: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
% 0.30/0.57        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
% 0.30/0.57       => ( ( member_VEBT_VEBT @ X3 @ A4 )
% 0.30/0.57         => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_subsetI
% 0.30/0.57  thf(fact_3589_set__update__subsetI,axiom,
% 0.30/0.57      ! [Xs2: list_int,A4: set_int,X3: int,I: nat] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
% 0.30/0.57       => ( ( member_int @ X3 @ A4 )
% 0.30/0.57         => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_subsetI
% 0.30/0.57  thf(fact_3590_set__update__memI,axiom,
% 0.30/0.57      ! [N: nat,Xs2: list_real,X3: real] :
% 0.30/0.57        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.30/0.57       => ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_memI
% 0.30/0.57  thf(fact_3591_set__update__memI,axiom,
% 0.30/0.57      ! [N: nat,Xs2: list_complex,X3: complex] :
% 0.30/0.57        ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.30/0.57       => ( member_complex @ X3 @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_memI
% 0.30/0.57  thf(fact_3592_set__update__memI,axiom,
% 0.30/0.57      ! [N: nat,Xs2: list_int,X3: int] :
% 0.30/0.57        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.57       => ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_memI
% 0.30/0.57  thf(fact_3593_set__update__memI,axiom,
% 0.30/0.57      ! [N: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.57       => ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_memI
% 0.30/0.57  thf(fact_3594_set__update__memI,axiom,
% 0.30/0.57      ! [N: nat,Xs2: list_o,X3: $o] :
% 0.30/0.57        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.57       => ( member_o @ X3 @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_memI
% 0.30/0.57  thf(fact_3595_set__update__memI,axiom,
% 0.30/0.57      ! [N: nat,Xs2: list_nat,X3: nat] :
% 0.30/0.57        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.57       => ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X3 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_update_memI
% 0.30/0.57  thf(fact_3596_nle__le,axiom,
% 0.30/0.57      ! [A: rat,B2: rat] :
% 0.30/0.57        ( ( ~ ( ord_less_eq_rat @ A @ B2 ) )
% 0.30/0.57        = ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.57          & ( B2 != A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nle_le
% 0.30/0.57  thf(fact_3597_nle__le,axiom,
% 0.30/0.57      ! [A: num,B2: num] :
% 0.30/0.57        ( ( ~ ( ord_less_eq_num @ A @ B2 ) )
% 0.30/0.57        = ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.57          & ( B2 != A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nle_le
% 0.30/0.57  thf(fact_3598_nle__le,axiom,
% 0.30/0.57      ! [A: nat,B2: nat] :
% 0.30/0.57        ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
% 0.30/0.57        = ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.57          & ( B2 != A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nle_le
% 0.30/0.57  thf(fact_3599_nle__le,axiom,
% 0.30/0.57      ! [A: int,B2: int] :
% 0.30/0.57        ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
% 0.30/0.57        = ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.57          & ( B2 != A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nle_le
% 0.30/0.57  thf(fact_3600_le__cases3,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.57        ( ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.57         => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 0.30/0.57       => ( ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.57           => ~ ( ord_less_eq_rat @ X3 @ Z ) )
% 0.30/0.57         => ( ( ( ord_less_eq_rat @ X3 @ Z )
% 0.30/0.57             => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 0.30/0.57           => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 0.30/0.57               => ~ ( ord_less_eq_rat @ Y3 @ X3 ) )
% 0.30/0.57             => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.30/0.57                 => ~ ( ord_less_eq_rat @ Z @ X3 ) )
% 0.30/0.57               => ~ ( ( ord_less_eq_rat @ Z @ X3 )
% 0.30/0.57                   => ~ ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_cases3
% 0.30/0.57  thf(fact_3601_le__cases3,axiom,
% 0.30/0.57      ! [X3: num,Y3: num,Z: num] :
% 0.30/0.57        ( ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.57         => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 0.30/0.57       => ( ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.30/0.57           => ~ ( ord_less_eq_num @ X3 @ Z ) )
% 0.30/0.57         => ( ( ( ord_less_eq_num @ X3 @ Z )
% 0.30/0.57             => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 0.30/0.57           => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 0.30/0.57               => ~ ( ord_less_eq_num @ Y3 @ X3 ) )
% 0.30/0.57             => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 0.30/0.57                 => ~ ( ord_less_eq_num @ Z @ X3 ) )
% 0.30/0.57               => ~ ( ( ord_less_eq_num @ Z @ X3 )
% 0.30/0.57                   => ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_cases3
% 0.30/0.57  thf(fact_3602_le__cases3,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.57        ( ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.57         => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 0.30/0.57       => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.57           => ~ ( ord_less_eq_nat @ X3 @ Z ) )
% 0.30/0.57         => ( ( ( ord_less_eq_nat @ X3 @ Z )
% 0.30/0.57             => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 0.30/0.57           => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 0.30/0.57               => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
% 0.30/0.57             => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.30/0.57                 => ~ ( ord_less_eq_nat @ Z @ X3 ) )
% 0.30/0.57               => ~ ( ( ord_less_eq_nat @ Z @ X3 )
% 0.30/0.57                   => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_cases3
% 0.30/0.57  thf(fact_3603_le__cases3,axiom,
% 0.30/0.57      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.57        ( ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.57         => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 0.30/0.57       => ( ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.30/0.57           => ~ ( ord_less_eq_int @ X3 @ Z ) )
% 0.30/0.57         => ( ( ( ord_less_eq_int @ X3 @ Z )
% 0.30/0.57             => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 0.30/0.57           => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 0.30/0.57               => ~ ( ord_less_eq_int @ Y3 @ X3 ) )
% 0.30/0.57             => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.57                 => ~ ( ord_less_eq_int @ Z @ X3 ) )
% 0.30/0.57               => ~ ( ( ord_less_eq_int @ Z @ X3 )
% 0.30/0.57                   => ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_cases3
% 0.30/0.57  thf(fact_3604_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.57            ( ( ord_less_eq_set_int @ X @ Y )
% 0.30/0.57            & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_class.order_eq_iff
% 0.30/0.57  thf(fact_3605_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [X: rat,Y: rat] :
% 0.30/0.57            ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.57            & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_class.order_eq_iff
% 0.30/0.57  thf(fact_3606_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [X: num,Y: num] :
% 0.30/0.57            ( ( ord_less_eq_num @ X @ Y )
% 0.30/0.57            & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_class.order_eq_iff
% 0.30/0.57  thf(fact_3607_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [X: nat,Y: nat] :
% 0.30/0.57            ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.57            & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_class.order_eq_iff
% 0.30/0.57  thf(fact_3608_order__class_Oorder__eq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [X: int,Y: int] :
% 0.30/0.57            ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.57            & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_class.order_eq_iff
% 0.30/0.57  thf(fact_3609_ord__eq__le__trans,axiom,
% 0.30/0.57      ! [A: set_int,B2: set_int,C: set_int] :
% 0.30/0.57        ( ( A = B2 )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_eq_le_trans
% 0.30/0.57  thf(fact_3610_ord__eq__le__trans,axiom,
% 0.30/0.57      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.57        ( ( A = B2 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_eq_le_trans
% 0.30/0.57  thf(fact_3611_ord__eq__le__trans,axiom,
% 0.30/0.57      ! [A: num,B2: num,C: num] :
% 0.30/0.57        ( ( A = B2 )
% 0.30/0.57       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_eq_le_trans
% 0.30/0.57  thf(fact_3612_ord__eq__le__trans,axiom,
% 0.30/0.57      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.57        ( ( A = B2 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_eq_le_trans
% 0.30/0.57  thf(fact_3613_ord__eq__le__trans,axiom,
% 0.30/0.57      ! [A: int,B2: int,C: int] :
% 0.30/0.57        ( ( A = B2 )
% 0.30/0.57       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_eq_le_trans
% 0.30/0.57  thf(fact_3614_ord__le__eq__trans,axiom,
% 0.30/0.57      ! [A: set_int,B2: set_int,C: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.57       => ( ( B2 = C )
% 0.30/0.57         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_le_eq_trans
% 0.30/0.57  thf(fact_3615_ord__le__eq__trans,axiom,
% 0.30/0.57      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.57       => ( ( B2 = C )
% 0.30/0.57         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_le_eq_trans
% 0.30/0.57  thf(fact_3616_ord__le__eq__trans,axiom,
% 0.30/0.57      ! [A: num,B2: num,C: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.57       => ( ( B2 = C )
% 0.30/0.57         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_le_eq_trans
% 0.30/0.57  thf(fact_3617_ord__le__eq__trans,axiom,
% 0.30/0.57      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.57       => ( ( B2 = C )
% 0.30/0.57         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_le_eq_trans
% 0.30/0.57  thf(fact_3618_ord__le__eq__trans,axiom,
% 0.30/0.57      ! [A: int,B2: int,C: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.57       => ( ( B2 = C )
% 0.30/0.57         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % ord_le_eq_trans
% 0.30/0.57  thf(fact_3619_order__antisym,axiom,
% 0.30/0.57      ! [X3: set_int,Y3: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.30/0.57         => ( X3 = Y3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_antisym
% 0.30/0.57  thf(fact_3620_order__antisym,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.57         => ( X3 = Y3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_antisym
% 0.30/0.57  thf(fact_3621_order__antisym,axiom,
% 0.30/0.57      ! [X3: num,Y3: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.30/0.57         => ( X3 = Y3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_antisym
% 0.30/0.57  thf(fact_3622_order__antisym,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.57         => ( X3 = Y3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_antisym
% 0.30/0.57  thf(fact_3623_order__antisym,axiom,
% 0.30/0.57      ! [X3: int,Y3: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.30/0.57         => ( X3 = Y3 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_antisym
% 0.30/0.57  thf(fact_3624_order_Otrans,axiom,
% 0.30/0.57      ! [A: set_int,B2: set_int,C: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order.trans
% 0.30/0.57  thf(fact_3625_order_Otrans,axiom,
% 0.30/0.57      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order.trans
% 0.30/0.57  thf(fact_3626_order_Otrans,axiom,
% 0.30/0.57      ! [A: num,B2: num,C: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order.trans
% 0.30/0.57  thf(fact_3627_order_Otrans,axiom,
% 0.30/0.57      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order.trans
% 0.30/0.57  thf(fact_3628_order_Otrans,axiom,
% 0.30/0.57      ! [A: int,B2: int,C: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.57         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order.trans
% 0.30/0.57  thf(fact_3629_order__trans,axiom,
% 0.30/0.57      ! [X3: set_int,Y3: set_int,Z: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.30/0.57         => ( ord_less_eq_set_int @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_trans
% 0.30/0.57  thf(fact_3630_order__trans,axiom,
% 0.30/0.57      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.30/0.57         => ( ord_less_eq_rat @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_trans
% 0.30/0.57  thf(fact_3631_order__trans,axiom,
% 0.30/0.57      ! [X3: num,Y3: num,Z: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.30/0.57         => ( ord_less_eq_num @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_trans
% 0.30/0.57  thf(fact_3632_order__trans,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.30/0.57         => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_trans
% 0.30/0.57  thf(fact_3633_order__trans,axiom,
% 0.30/0.57      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.57         => ( ord_less_eq_int @ X3 @ Z ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % order_trans
% 0.30/0.57  thf(fact_3634_linorder__wlog,axiom,
% 0.30/0.57      ! [P: rat > rat > $o,A: rat,B2: rat] :
% 0.30/0.57        ( ! [A3: rat,B3: rat] :
% 0.30/0.57            ( ( ord_less_eq_rat @ A3 @ B3 )
% 0.30/0.57           => ( P @ A3 @ B3 ) )
% 0.30/0.57       => ( ! [A3: rat,B3: rat] :
% 0.30/0.57              ( ( P @ B3 @ A3 )
% 0.30/0.57             => ( P @ A3 @ B3 ) )
% 0.30/0.57         => ( P @ A @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % linorder_wlog
% 0.30/0.57  thf(fact_3635_linorder__wlog,axiom,
% 0.30/0.57      ! [P: num > num > $o,A: num,B2: num] :
% 0.30/0.57        ( ! [A3: num,B3: num] :
% 0.30/0.57            ( ( ord_less_eq_num @ A3 @ B3 )
% 0.30/0.57           => ( P @ A3 @ B3 ) )
% 0.30/0.57       => ( ! [A3: num,B3: num] :
% 0.30/0.57              ( ( P @ B3 @ A3 )
% 0.30/0.57             => ( P @ A3 @ B3 ) )
% 0.30/0.57         => ( P @ A @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % linorder_wlog
% 0.30/0.57  thf(fact_3636_linorder__wlog,axiom,
% 0.30/0.57      ! [P: nat > nat > $o,A: nat,B2: nat] :
% 0.30/0.57        ( ! [A3: nat,B3: nat] :
% 0.30/0.57            ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.30/0.57           => ( P @ A3 @ B3 ) )
% 0.30/0.57       => ( ! [A3: nat,B3: nat] :
% 0.30/0.57              ( ( P @ B3 @ A3 )
% 0.30/0.57             => ( P @ A3 @ B3 ) )
% 0.30/0.57         => ( P @ A @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % linorder_wlog
% 0.30/0.57  thf(fact_3637_linorder__wlog,axiom,
% 0.30/0.57      ! [P: int > int > $o,A: int,B2: int] :
% 0.30/0.57        ( ! [A3: int,B3: int] :
% 0.30/0.57            ( ( ord_less_eq_int @ A3 @ B3 )
% 0.30/0.57           => ( P @ A3 @ B3 ) )
% 0.30/0.57       => ( ! [A3: int,B3: int] :
% 0.30/0.57              ( ( P @ B3 @ A3 )
% 0.30/0.57             => ( P @ A3 @ B3 ) )
% 0.30/0.57         => ( P @ A @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % linorder_wlog
% 0.30/0.57  thf(fact_3638_dual__order_Oeq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [A5: set_int,B4: set_int] :
% 0.30/0.57            ( ( ord_less_eq_set_int @ B4 @ A5 )
% 0.30/0.57            & ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.eq_iff
% 0.30/0.57  thf(fact_3639_dual__order_Oeq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.57            ( ( ord_less_eq_rat @ B4 @ A5 )
% 0.30/0.57            & ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.eq_iff
% 0.30/0.57  thf(fact_3640_dual__order_Oeq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [A5: num,B4: num] :
% 0.30/0.57            ( ( ord_less_eq_num @ B4 @ A5 )
% 0.30/0.57            & ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.eq_iff
% 0.30/0.57  thf(fact_3641_dual__order_Oeq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.57            ( ( ord_less_eq_nat @ B4 @ A5 )
% 0.30/0.57            & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.eq_iff
% 0.30/0.57  thf(fact_3642_dual__order_Oeq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [A5: int,B4: int] :
% 0.30/0.57            ( ( ord_less_eq_int @ B4 @ A5 )
% 0.30/0.57            & ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.eq_iff
% 0.30/0.57  thf(fact_3643_dual__order_Oantisym,axiom,
% 0.30/0.57      ! [B2: set_int,A: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.antisym
% 0.30/0.57  thf(fact_3644_dual__order_Oantisym,axiom,
% 0.30/0.57      ! [B2: rat,A: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.antisym
% 0.30/0.57  thf(fact_3645_dual__order_Oantisym,axiom,
% 0.30/0.57      ! [B2: num,A: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.antisym
% 0.30/0.57  thf(fact_3646_dual__order_Oantisym,axiom,
% 0.30/0.57      ! [B2: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.antisym
% 0.30/0.57  thf(fact_3647_dual__order_Oantisym,axiom,
% 0.30/0.57      ! [B2: int,A: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.antisym
% 0.30/0.57  thf(fact_3648_dual__order_Otrans,axiom,
% 0.30/0.57      ! [B2: set_int,A: set_int,C: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ C @ B2 )
% 0.30/0.57         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.trans
% 0.30/0.57  thf(fact_3649_dual__order_Otrans,axiom,
% 0.30/0.57      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_rat @ C @ B2 )
% 0.30/0.57         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.trans
% 0.30/0.57  thf(fact_3650_dual__order_Otrans,axiom,
% 0.30/0.57      ! [B2: num,A: num,C: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_num @ C @ B2 )
% 0.30/0.57         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.trans
% 0.30/0.57  thf(fact_3651_dual__order_Otrans,axiom,
% 0.30/0.57      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_nat @ C @ B2 )
% 0.30/0.57         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.trans
% 0.30/0.57  thf(fact_3652_dual__order_Otrans,axiom,
% 0.30/0.57      ! [B2: int,A: int,C: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.57       => ( ( ord_less_eq_int @ C @ B2 )
% 0.30/0.57         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dual_order.trans
% 0.30/0.57  thf(fact_3653_antisym,axiom,
% 0.30/0.57      ! [A: set_int,B2: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_set_int @ B2 @ A )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % antisym
% 0.30/0.57  thf(fact_3654_antisym,axiom,
% 0.30/0.57      ! [A: rat,B2: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % antisym
% 0.30/0.57  thf(fact_3655_antisym,axiom,
% 0.30/0.57      ! [A: num,B2: num] :
% 0.30/0.57        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % antisym
% 0.30/0.57  thf(fact_3656_antisym,axiom,
% 0.30/0.57      ! [A: nat,B2: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % antisym
% 0.30/0.57  thf(fact_3657_antisym,axiom,
% 0.30/0.57      ! [A: int,B2: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.57       => ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.57         => ( A = B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % antisym
% 0.30/0.57  thf(fact_3658_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.57      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 0.30/0.57      = ( ^ [A5: set_int,B4: set_int] :
% 0.30/0.57            ( ( ord_less_eq_set_int @ A5 @ B4 )
% 0.30/0.58            & ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Orderings.order_eq_iff
% 0.30/0.58  thf(fact_3659_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.58      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 0.30/0.58      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ A5 @ B4 )
% 0.30/0.58            & ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Orderings.order_eq_iff
% 0.30/0.58  thf(fact_3660_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.58      ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 0.30/0.58      = ( ^ [A5: num,B4: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ A5 @ B4 )
% 0.30/0.58            & ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Orderings.order_eq_iff
% 0.30/0.58  thf(fact_3661_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.58      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.58      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ A5 @ B4 )
% 0.30/0.58            & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Orderings.order_eq_iff
% 0.30/0.58  thf(fact_3662_Orderings_Oorder__eq__iff,axiom,
% 0.30/0.58      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.58      = ( ^ [A5: int,B4: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ A5 @ B4 )
% 0.30/0.58            & ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Orderings.order_eq_iff
% 0.30/0.58  thf(fact_3663_order__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3664_order__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3665_order__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3666_order__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: int > rat,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.58         => ( ! [X5: int,Y4: int] :
% 0.30/0.58                ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3667_order__subst1,axiom,
% 0.30/0.58      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3668_order__subst1,axiom,
% 0.30/0.58      ! [A: num,F: num > num,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3669_order__subst1,axiom,
% 0.30/0.58      ! [A: num,F: nat > num,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3670_order__subst1,axiom,
% 0.30/0.58      ! [A: num,F: int > num,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.58         => ( ! [X5: int,Y4: int] :
% 0.30/0.58                ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3671_order__subst1,axiom,
% 0.30/0.58      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3672_order__subst1,axiom,
% 0.30/0.58      ! [A: nat,F: num > nat,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst1
% 0.30/0.58  thf(fact_3673_order__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3674_order__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3675_order__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3676_order__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3677_order__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3678_order__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3679_order__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3680_order__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3681_order__subst2,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3682_order__subst2,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,F: nat > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_subst2
% 0.30/0.58  thf(fact_3683_order__eq__refl,axiom,
% 0.30/0.58      ! [X3: set_int,Y3: set_int] :
% 0.30/0.58        ( ( X3 = Y3 )
% 0.30/0.58       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_eq_refl
% 0.30/0.58  thf(fact_3684_order__eq__refl,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( X3 = Y3 )
% 0.30/0.58       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_eq_refl
% 0.30/0.58  thf(fact_3685_order__eq__refl,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( X3 = Y3 )
% 0.30/0.58       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_eq_refl
% 0.30/0.58  thf(fact_3686_order__eq__refl,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( X3 = Y3 )
% 0.30/0.58       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_eq_refl
% 0.30/0.58  thf(fact_3687_order__eq__refl,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( X3 = Y3 )
% 0.30/0.58       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_eq_refl
% 0.30/0.58  thf(fact_3688_linorder__linear,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.58        | ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_linear
% 0.30/0.58  thf(fact_3689_linorder__linear,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.58        | ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_linear
% 0.30/0.58  thf(fact_3690_linorder__linear,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.58        | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_linear
% 0.30/0.58  thf(fact_3691_linorder__linear,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.58        | ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_linear
% 0.30/0.58  thf(fact_3692_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3693_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3694_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3695_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: int,F: rat > int,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3696_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3697_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: num,F: num > num,B2: num,C: num] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3698_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: nat,F: num > nat,B2: num,C: num] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3699_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: int,F: num > int,B2: num,C: num] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3700_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3701_ord__eq__le__subst,axiom,
% 0.30/0.58      ! [A: num,F: nat > num,B2: nat,C: nat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_le_subst
% 0.30/0.58  thf(fact_3702_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3703_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3704_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3705_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3706_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3707_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3708_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3709_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3710_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3711_ord__le__eq__subst,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,F: nat > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_le_eq_subst
% 0.30/0.58  thf(fact_3712_linorder__le__cases,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ~ ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_le_cases
% 0.30/0.58  thf(fact_3713_linorder__le__cases,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ~ ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_le_cases
% 0.30/0.58  thf(fact_3714_linorder__le__cases,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_le_cases
% 0.30/0.58  thf(fact_3715_linorder__le__cases,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ~ ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_le_cases
% 0.30/0.58  thf(fact_3716_order__antisym__conv,axiom,
% 0.30/0.58      ! [Y3: set_int,X3: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_antisym_conv
% 0.30/0.58  thf(fact_3717_order__antisym__conv,axiom,
% 0.30/0.58      ! [Y3: rat,X3: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_antisym_conv
% 0.30/0.58  thf(fact_3718_order__antisym__conv,axiom,
% 0.30/0.58      ! [Y3: num,X3: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.30/0.58       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_antisym_conv
% 0.30/0.58  thf(fact_3719_order__antisym__conv,axiom,
% 0.30/0.58      ! [Y3: nat,X3: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_antisym_conv
% 0.30/0.58  thf(fact_3720_order__antisym__conv,axiom,
% 0.30/0.58      ! [Y3: int,X3: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.30/0.58       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_antisym_conv
% 0.30/0.58  thf(fact_3721_nth__list__update,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_int,J: nat,X3: int] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.58       => ( ( ( I = J )
% 0.30/0.58           => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58              = X3 ) )
% 0.30/0.58          & ( ( I != J )
% 0.30/0.58           => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58              = ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nth_list_update
% 0.30/0.58  thf(fact_3722_nth__list__update,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.58       => ( ( ( I = J )
% 0.30/0.58           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58              = X3 ) )
% 0.30/0.58          & ( ( I != J )
% 0.30/0.58           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58              = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nth_list_update
% 0.30/0.58  thf(fact_3723_nth__list__update,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_o,X3: $o,J: nat] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.58       => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58          = ( ( ( I = J )
% 0.30/0.58             => X3 )
% 0.30/0.58            & ( ( I != J )
% 0.30/0.58             => ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nth_list_update
% 0.30/0.58  thf(fact_3724_nth__list__update,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_nat,J: nat,X3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.58       => ( ( ( I = J )
% 0.30/0.58           => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58              = X3 ) )
% 0.30/0.58          & ( ( I != J )
% 0.30/0.58           => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
% 0.30/0.58              = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nth_list_update
% 0.30/0.58  thf(fact_3725_list__update__same__conv,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_int,X3: int] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.58       => ( ( ( list_update_int @ Xs2 @ I @ X3 )
% 0.30/0.58            = Xs2 )
% 0.30/0.58          = ( ( nth_int @ Xs2 @ I )
% 0.30/0.58            = X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % list_update_same_conv
% 0.30/0.58  thf(fact_3726_list__update__same__conv,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.58       => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 )
% 0.30/0.58            = Xs2 )
% 0.30/0.58          = ( ( nth_VEBT_VEBT @ Xs2 @ I )
% 0.30/0.58            = X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % list_update_same_conv
% 0.30/0.58  thf(fact_3727_list__update__same__conv,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_o,X3: $o] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.58       => ( ( ( list_update_o @ Xs2 @ I @ X3 )
% 0.30/0.58            = Xs2 )
% 0.30/0.58          = ( ( nth_o @ Xs2 @ I )
% 0.30/0.58            = X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % list_update_same_conv
% 0.30/0.58  thf(fact_3728_list__update__same__conv,axiom,
% 0.30/0.58      ! [I: nat,Xs2: list_nat,X3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.30/0.58       => ( ( ( list_update_nat @ Xs2 @ I @ X3 )
% 0.30/0.58            = Xs2 )
% 0.30/0.58          = ( ( nth_nat @ Xs2 @ I )
% 0.30/0.58            = X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % list_update_same_conv
% 0.30/0.58  thf(fact_3729_lt__ex,axiom,
% 0.30/0.58      ! [X3: real] :
% 0.30/0.58      ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % lt_ex
% 0.30/0.58  thf(fact_3730_lt__ex,axiom,
% 0.30/0.58      ! [X3: rat] :
% 0.30/0.58      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % lt_ex
% 0.30/0.58  thf(fact_3731_lt__ex,axiom,
% 0.30/0.58      ! [X3: int] :
% 0.30/0.58      ? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % lt_ex
% 0.30/0.58  thf(fact_3732_gt__ex,axiom,
% 0.30/0.58      ! [X3: real] :
% 0.30/0.58      ? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% 0.30/0.58  
% 0.30/0.58  % gt_ex
% 0.30/0.58  thf(fact_3733_gt__ex,axiom,
% 0.30/0.58      ! [X3: rat] :
% 0.30/0.58      ? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).
% 0.30/0.58  
% 0.30/0.58  % gt_ex
% 0.30/0.58  thf(fact_3734_gt__ex,axiom,
% 0.30/0.58      ! [X3: nat] :
% 0.30/0.58      ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% 0.30/0.58  
% 0.30/0.58  % gt_ex
% 0.30/0.58  thf(fact_3735_gt__ex,axiom,
% 0.30/0.58      ! [X3: int] :
% 0.30/0.58      ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% 0.30/0.58  
% 0.30/0.58  % gt_ex
% 0.30/0.58  thf(fact_3736_dense,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ? [Z4: real] :
% 0.30/0.58            ( ( ord_less_real @ X3 @ Z4 )
% 0.30/0.58            & ( ord_less_real @ Z4 @ Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense
% 0.30/0.58  thf(fact_3737_dense,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ? [Z4: rat] :
% 0.30/0.58            ( ( ord_less_rat @ X3 @ Z4 )
% 0.30/0.58            & ( ord_less_rat @ Z4 @ Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense
% 0.30/0.58  thf(fact_3738_less__imp__neq,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_imp_neq
% 0.30/0.58  thf(fact_3739_less__imp__neq,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_imp_neq
% 0.30/0.58  thf(fact_3740_less__imp__neq,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_imp_neq
% 0.30/0.58  thf(fact_3741_less__imp__neq,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_imp_neq
% 0.30/0.58  thf(fact_3742_less__imp__neq,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_imp_neq
% 0.30/0.58  thf(fact_3743_order_Oasym,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_real @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.asym
% 0.30/0.58  thf(fact_3744_order_Oasym,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.asym
% 0.30/0.58  thf(fact_3745_order_Oasym,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_num @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.asym
% 0.30/0.58  thf(fact_3746_order_Oasym,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.asym
% 0.30/0.58  thf(fact_3747_order_Oasym,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_int @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.asym
% 0.30/0.58  thf(fact_3748_ord__eq__less__trans,axiom,
% 0.30/0.58      ! [A: real,B2: real,C: real] :
% 0.30/0.58        ( ( A = B2 )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_trans
% 0.30/0.58  thf(fact_3749_ord__eq__less__trans,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( A = B2 )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_trans
% 0.30/0.58  thf(fact_3750_ord__eq__less__trans,axiom,
% 0.30/0.58      ! [A: num,B2: num,C: num] :
% 0.30/0.58        ( ( A = B2 )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_trans
% 0.30/0.58  thf(fact_3751_ord__eq__less__trans,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.58        ( ( A = B2 )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_trans
% 0.30/0.58  thf(fact_3752_ord__eq__less__trans,axiom,
% 0.30/0.58      ! [A: int,B2: int,C: int] :
% 0.30/0.58        ( ( A = B2 )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_trans
% 0.30/0.58  thf(fact_3753_ord__less__eq__trans,axiom,
% 0.30/0.58      ! [A: real,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( B2 = C )
% 0.30/0.58         => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_trans
% 0.30/0.58  thf(fact_3754_ord__less__eq__trans,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( B2 = C )
% 0.30/0.58         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_trans
% 0.30/0.58  thf(fact_3755_ord__less__eq__trans,axiom,
% 0.30/0.58      ! [A: num,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ( ( B2 = C )
% 0.30/0.58         => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_trans
% 0.30/0.58  thf(fact_3756_ord__less__eq__trans,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ( ( B2 = C )
% 0.30/0.58         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_trans
% 0.30/0.58  thf(fact_3757_ord__less__eq__trans,axiom,
% 0.30/0.58      ! [A: int,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ( ( B2 = C )
% 0.30/0.58         => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_trans
% 0.30/0.58  thf(fact_3758_less__induct,axiom,
% 0.30/0.58      ! [P: nat > $o,A: nat] :
% 0.30/0.58        ( ! [X5: nat] :
% 0.30/0.58            ( ! [Y5: nat] :
% 0.30/0.58                ( ( ord_less_nat @ Y5 @ X5 )
% 0.30/0.58               => ( P @ Y5 ) )
% 0.30/0.58           => ( P @ X5 ) )
% 0.30/0.58       => ( P @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_induct
% 0.30/0.58  thf(fact_3759_antisym__conv3,axiom,
% 0.30/0.58      ! [Y3: real,X3: real] :
% 0.30/0.58        ( ~ ( ord_less_real @ Y3 @ X3 )
% 0.30/0.58       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv3
% 0.30/0.58  thf(fact_3760_antisym__conv3,axiom,
% 0.30/0.58      ! [Y3: rat,X3: rat] :
% 0.30/0.58        ( ~ ( ord_less_rat @ Y3 @ X3 )
% 0.30/0.58       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv3
% 0.30/0.58  thf(fact_3761_antisym__conv3,axiom,
% 0.30/0.58      ! [Y3: num,X3: num] :
% 0.30/0.58        ( ~ ( ord_less_num @ Y3 @ X3 )
% 0.30/0.58       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv3
% 0.30/0.58  thf(fact_3762_antisym__conv3,axiom,
% 0.30/0.58      ! [Y3: nat,X3: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ Y3 @ X3 )
% 0.30/0.58       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv3
% 0.30/0.58  thf(fact_3763_antisym__conv3,axiom,
% 0.30/0.58      ! [Y3: int,X3: int] :
% 0.30/0.58        ( ~ ( ord_less_int @ Y3 @ X3 )
% 0.30/0.58       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv3
% 0.30/0.58  thf(fact_3764_linorder__cases,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( X3 != Y3 )
% 0.30/0.58         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_cases
% 0.30/0.58  thf(fact_3765_linorder__cases,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( X3 != Y3 )
% 0.30/0.58         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_cases
% 0.30/0.58  thf(fact_3766_linorder__cases,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( X3 != Y3 )
% 0.30/0.58         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_cases
% 0.30/0.58  thf(fact_3767_linorder__cases,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( X3 != Y3 )
% 0.30/0.58         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_cases
% 0.30/0.58  thf(fact_3768_linorder__cases,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( X3 != Y3 )
% 0.30/0.58         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_cases
% 0.30/0.58  thf(fact_3769_dual__order_Oasym,axiom,
% 0.30/0.58      ! [B2: real,A: real] :
% 0.30/0.58        ( ( ord_less_real @ B2 @ A )
% 0.30/0.58       => ~ ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.asym
% 0.30/0.58  thf(fact_3770_dual__order_Oasym,axiom,
% 0.30/0.58      ! [B2: rat,A: rat] :
% 0.30/0.58        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.58       => ~ ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.asym
% 0.30/0.58  thf(fact_3771_dual__order_Oasym,axiom,
% 0.30/0.58      ! [B2: num,A: num] :
% 0.30/0.58        ( ( ord_less_num @ B2 @ A )
% 0.30/0.58       => ~ ( ord_less_num @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.asym
% 0.30/0.58  thf(fact_3772_dual__order_Oasym,axiom,
% 0.30/0.58      ! [B2: nat,A: nat] :
% 0.30/0.58        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.58       => ~ ( ord_less_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.asym
% 0.30/0.58  thf(fact_3773_dual__order_Oasym,axiom,
% 0.30/0.58      ! [B2: int,A: int] :
% 0.30/0.58        ( ( ord_less_int @ B2 @ A )
% 0.30/0.58       => ~ ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.asym
% 0.30/0.58  thf(fact_3774_dual__order_Oirrefl,axiom,
% 0.30/0.58      ! [A: real] :
% 0.30/0.58        ~ ( ord_less_real @ A @ A ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.irrefl
% 0.30/0.58  thf(fact_3775_dual__order_Oirrefl,axiom,
% 0.30/0.58      ! [A: rat] :
% 0.30/0.58        ~ ( ord_less_rat @ A @ A ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.irrefl
% 0.30/0.58  thf(fact_3776_dual__order_Oirrefl,axiom,
% 0.30/0.58      ! [A: num] :
% 0.30/0.58        ~ ( ord_less_num @ A @ A ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.irrefl
% 0.30/0.58  thf(fact_3777_dual__order_Oirrefl,axiom,
% 0.30/0.58      ! [A: nat] :
% 0.30/0.58        ~ ( ord_less_nat @ A @ A ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.irrefl
% 0.30/0.58  thf(fact_3778_dual__order_Oirrefl,axiom,
% 0.30/0.58      ! [A: int] :
% 0.30/0.58        ~ ( ord_less_int @ A @ A ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.irrefl
% 0.30/0.58  thf(fact_3779_exists__least__iff,axiom,
% 0.30/0.58      ( ( ^ [P2: nat > $o] :
% 0.30/0.58          ? [X4: nat] : ( P2 @ X4 ) )
% 0.30/0.58      = ( ^ [P3: nat > $o] :
% 0.30/0.58          ? [N2: nat] :
% 0.30/0.58            ( ( P3 @ N2 )
% 0.30/0.58            & ! [M3: nat] :
% 0.30/0.58                ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.58               => ~ ( P3 @ M3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % exists_least_iff
% 0.30/0.58  thf(fact_3780_linorder__less__wlog,axiom,
% 0.30/0.58      ! [P: real > real > $o,A: real,B2: real] :
% 0.30/0.58        ( ! [A3: real,B3: real] :
% 0.30/0.58            ( ( ord_less_real @ A3 @ B3 )
% 0.30/0.58           => ( P @ A3 @ B3 ) )
% 0.30/0.58       => ( ! [A3: real] : ( P @ A3 @ A3 )
% 0.30/0.58         => ( ! [A3: real,B3: real] :
% 0.30/0.58                ( ( P @ B3 @ A3 )
% 0.30/0.58               => ( P @ A3 @ B3 ) )
% 0.30/0.58           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_wlog
% 0.30/0.58  thf(fact_3781_linorder__less__wlog,axiom,
% 0.30/0.58      ! [P: rat > rat > $o,A: rat,B2: rat] :
% 0.30/0.58        ( ! [A3: rat,B3: rat] :
% 0.30/0.58            ( ( ord_less_rat @ A3 @ B3 )
% 0.30/0.58           => ( P @ A3 @ B3 ) )
% 0.30/0.58       => ( ! [A3: rat] : ( P @ A3 @ A3 )
% 0.30/0.58         => ( ! [A3: rat,B3: rat] :
% 0.30/0.58                ( ( P @ B3 @ A3 )
% 0.30/0.58               => ( P @ A3 @ B3 ) )
% 0.30/0.58           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_wlog
% 0.30/0.58  thf(fact_3782_linorder__less__wlog,axiom,
% 0.30/0.58      ! [P: num > num > $o,A: num,B2: num] :
% 0.30/0.58        ( ! [A3: num,B3: num] :
% 0.30/0.58            ( ( ord_less_num @ A3 @ B3 )
% 0.30/0.58           => ( P @ A3 @ B3 ) )
% 0.30/0.58       => ( ! [A3: num] : ( P @ A3 @ A3 )
% 0.30/0.58         => ( ! [A3: num,B3: num] :
% 0.30/0.58                ( ( P @ B3 @ A3 )
% 0.30/0.58               => ( P @ A3 @ B3 ) )
% 0.30/0.58           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_wlog
% 0.30/0.58  thf(fact_3783_linorder__less__wlog,axiom,
% 0.30/0.58      ! [P: nat > nat > $o,A: nat,B2: nat] :
% 0.30/0.58        ( ! [A3: nat,B3: nat] :
% 0.30/0.58            ( ( ord_less_nat @ A3 @ B3 )
% 0.30/0.58           => ( P @ A3 @ B3 ) )
% 0.30/0.58       => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 0.30/0.58         => ( ! [A3: nat,B3: nat] :
% 0.30/0.58                ( ( P @ B3 @ A3 )
% 0.30/0.58               => ( P @ A3 @ B3 ) )
% 0.30/0.58           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_wlog
% 0.30/0.58  thf(fact_3784_linorder__less__wlog,axiom,
% 0.30/0.58      ! [P: int > int > $o,A: int,B2: int] :
% 0.30/0.58        ( ! [A3: int,B3: int] :
% 0.30/0.58            ( ( ord_less_int @ A3 @ B3 )
% 0.30/0.58           => ( P @ A3 @ B3 ) )
% 0.30/0.58       => ( ! [A3: int] : ( P @ A3 @ A3 )
% 0.30/0.58         => ( ! [A3: int,B3: int] :
% 0.30/0.58                ( ( P @ B3 @ A3 )
% 0.30/0.58               => ( P @ A3 @ B3 ) )
% 0.30/0.58           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_wlog
% 0.30/0.58  thf(fact_3785_order_Ostrict__trans,axiom,
% 0.30/0.58      ! [A: real,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans
% 0.30/0.58  thf(fact_3786_order_Ostrict__trans,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans
% 0.30/0.58  thf(fact_3787_order_Ostrict__trans,axiom,
% 0.30/0.58      ! [A: num,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans
% 0.30/0.58  thf(fact_3788_order_Ostrict__trans,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans
% 0.30/0.58  thf(fact_3789_order_Ostrict__trans,axiom,
% 0.30/0.58      ! [A: int,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans
% 0.30/0.58  thf(fact_3790_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.30/0.58        = ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.58          | ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_less_iff_gr_or_eq
% 0.30/0.58  thf(fact_3791_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.30/0.58        = ( ( ord_less_rat @ Y3 @ X3 )
% 0.30/0.58          | ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_less_iff_gr_or_eq
% 0.30/0.58  thf(fact_3792_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.30/0.58        = ( ( ord_less_num @ Y3 @ X3 )
% 0.30/0.58          | ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_less_iff_gr_or_eq
% 0.30/0.58  thf(fact_3793_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.30/0.58        = ( ( ord_less_nat @ Y3 @ X3 )
% 0.30/0.58          | ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_less_iff_gr_or_eq
% 0.30/0.58  thf(fact_3794_not__less__iff__gr__or__eq,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.30/0.58        = ( ( ord_less_int @ Y3 @ X3 )
% 0.30/0.58          | ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_less_iff_gr_or_eq
% 0.30/0.58  thf(fact_3795_dual__order_Ostrict__trans,axiom,
% 0.30/0.58      ! [B2: real,A: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_real @ C @ B2 )
% 0.30/0.58         => ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans
% 0.30/0.58  thf(fact_3796_dual__order_Ostrict__trans,axiom,
% 0.30/0.58      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_rat @ C @ B2 )
% 0.30/0.58         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans
% 0.30/0.58  thf(fact_3797_dual__order_Ostrict__trans,axiom,
% 0.30/0.58      ! [B2: num,A: num,C: num] :
% 0.30/0.58        ( ( ord_less_num @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_num @ C @ B2 )
% 0.30/0.58         => ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans
% 0.30/0.58  thf(fact_3798_dual__order_Ostrict__trans,axiom,
% 0.30/0.58      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_nat @ C @ B2 )
% 0.30/0.58         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans
% 0.30/0.58  thf(fact_3799_dual__order_Ostrict__trans,axiom,
% 0.30/0.58      ! [B2: int,A: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_int @ C @ B2 )
% 0.30/0.58         => ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans
% 0.30/0.58  thf(fact_3800_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3801_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3802_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3803_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3804_order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3805_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [B2: real,A: real] :
% 0.30/0.58        ( ( ord_less_real @ B2 @ A )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3806_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [B2: rat,A: rat] :
% 0.30/0.58        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3807_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [B2: num,A: num] :
% 0.30/0.58        ( ( ord_less_num @ B2 @ A )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3808_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [B2: nat,A: nat] :
% 0.30/0.58        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3809_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [B2: int,A: int] :
% 0.30/0.58        ( ( ord_less_int @ B2 @ A )
% 0.30/0.58       => ( A != B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_not_eq
% 0.30/0.58  thf(fact_3810_linorder__neqE,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neqE
% 0.30/0.58  thf(fact_3811_linorder__neqE,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neqE
% 0.30/0.58  thf(fact_3812_linorder__neqE,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58       => ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neqE
% 0.30/0.58  thf(fact_3813_linorder__neqE,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neqE
% 0.30/0.58  thf(fact_3814_linorder__neqE,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neqE
% 0.30/0.58  thf(fact_3815_order__less__asym,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym
% 0.30/0.58  thf(fact_3816_order__less__asym,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym
% 0.30/0.58  thf(fact_3817_order__less__asym,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym
% 0.30/0.58  thf(fact_3818_order__less__asym,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym
% 0.30/0.58  thf(fact_3819_order__less__asym,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym
% 0.30/0.58  thf(fact_3820_linorder__neq__iff,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58        = ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58          | ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neq_iff
% 0.30/0.58  thf(fact_3821_linorder__neq__iff,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58        = ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58          | ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neq_iff
% 0.30/0.58  thf(fact_3822_linorder__neq__iff,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58        = ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58          | ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neq_iff
% 0.30/0.58  thf(fact_3823_linorder__neq__iff,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58        = ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58          | ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neq_iff
% 0.30/0.58  thf(fact_3824_linorder__neq__iff,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( X3 != Y3 )
% 0.30/0.58        = ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58          | ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_neq_iff
% 0.30/0.58  thf(fact_3825_order__less__asym_H,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_real @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym'
% 0.30/0.58  thf(fact_3826_order__less__asym_H,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym'
% 0.30/0.58  thf(fact_3827_order__less__asym_H,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_num @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym'
% 0.30/0.58  thf(fact_3828_order__less__asym_H,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym'
% 0.30/0.58  thf(fact_3829_order__less__asym_H,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ~ ( ord_less_int @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_asym'
% 0.30/0.58  thf(fact_3830_order__less__trans,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_real @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_real @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_trans
% 0.30/0.58  thf(fact_3831_order__less__trans,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_rat @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_rat @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_trans
% 0.30/0.58  thf(fact_3832_order__less__trans,axiom,
% 0.30/0.58      ! [X3: num,Y3: num,Z: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_num @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_num @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_trans
% 0.30/0.58  thf(fact_3833_order__less__trans,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_nat @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_nat @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_trans
% 0.30/0.58  thf(fact_3834_order__less__trans,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_int @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_trans
% 0.30/0.58  thf(fact_3835_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: real,F: real > real,B2: real,C: real] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3836_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: rat,F: real > rat,B2: real,C: real] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3837_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: num,F: real > num,B2: real,C: real] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3838_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: nat,F: real > nat,B2: real,C: real] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3839_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: int,F: real > int,B2: real,C: real] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3840_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3841_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3842_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3843_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3844_ord__eq__less__subst,axiom,
% 0.30/0.58      ! [A: int,F: rat > int,B2: rat,C: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_eq_less_subst
% 0.30/0.58  thf(fact_3845_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3846_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3847_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > num,C: num] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3848_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3849_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > int,C: int] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3850_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3851_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3852_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3853_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3854_ord__less__eq__subst,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ( F @ B2 )
% 0.30/0.58            = C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ord_less_eq_subst
% 0.30/0.58  thf(fact_3855_order__less__irrefl,axiom,
% 0.30/0.58      ! [X3: real] :
% 0.30/0.58        ~ ( ord_less_real @ X3 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_irrefl
% 0.30/0.58  thf(fact_3856_order__less__irrefl,axiom,
% 0.30/0.58      ! [X3: rat] :
% 0.30/0.58        ~ ( ord_less_rat @ X3 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_irrefl
% 0.30/0.58  thf(fact_3857_order__less__irrefl,axiom,
% 0.30/0.58      ! [X3: num] :
% 0.30/0.58        ~ ( ord_less_num @ X3 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_irrefl
% 0.30/0.58  thf(fact_3858_order__less__irrefl,axiom,
% 0.30/0.58      ! [X3: nat] :
% 0.30/0.58        ~ ( ord_less_nat @ X3 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_irrefl
% 0.30/0.58  thf(fact_3859_order__less__irrefl,axiom,
% 0.30/0.58      ! [X3: int] :
% 0.30/0.58        ~ ( ord_less_int @ X3 @ X3 ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_irrefl
% 0.30/0.58  thf(fact_3860_order__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: real > real,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3861_order__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3862_order__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: num > real,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3863_order__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: nat > real,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3864_order__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: int > real,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ! [X5: int,Y4: int] :
% 0.30/0.58                ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3865_order__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: real > rat,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3866_order__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3867_order__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3868_order__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3869_order__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: int > rat,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ! [X5: int,Y4: int] :
% 0.30/0.58                ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst1
% 0.30/0.58  thf(fact_3870_order__less__subst2,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3871_order__less__subst2,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3872_order__less__subst2,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > num,C: num] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3873_order__less__subst2,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3874_order__less__subst2,axiom,
% 0.30/0.58      ! [A: real,B2: real,F: real > int,C: int] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3875_order__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3876_order__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3877_order__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3878_order__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3879_order__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_subst2
% 0.30/0.58  thf(fact_3880_order__less__not__sym,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_not_sym
% 0.30/0.58  thf(fact_3881_order__less__not__sym,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_not_sym
% 0.30/0.58  thf(fact_3882_order__less__not__sym,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_not_sym
% 0.30/0.58  thf(fact_3883_order__less__not__sym,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_not_sym
% 0.30/0.58  thf(fact_3884_order__less__not__sym,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_not_sym
% 0.30/0.58  thf(fact_3885_order__less__imp__triv,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,P: $o] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.58         => P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_triv
% 0.30/0.58  thf(fact_3886_order__less__imp__triv,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,P: $o] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_rat @ Y3 @ X3 )
% 0.30/0.58         => P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_triv
% 0.30/0.58  thf(fact_3887_order__less__imp__triv,axiom,
% 0.30/0.58      ! [X3: num,Y3: num,P: $o] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_num @ Y3 @ X3 )
% 0.30/0.58         => P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_triv
% 0.30/0.58  thf(fact_3888_order__less__imp__triv,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat,P: $o] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_nat @ Y3 @ X3 )
% 0.30/0.58         => P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_triv
% 0.30/0.58  thf(fact_3889_order__less__imp__triv,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,P: $o] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_int @ Y3 @ X3 )
% 0.30/0.58         => P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_triv
% 0.30/0.58  thf(fact_3890_linorder__less__linear,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58        | ( X3 = Y3 )
% 0.30/0.58        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_linear
% 0.30/0.58  thf(fact_3891_linorder__less__linear,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58        | ( X3 = Y3 )
% 0.30/0.58        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_linear
% 0.30/0.58  thf(fact_3892_linorder__less__linear,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58        | ( X3 = Y3 )
% 0.30/0.58        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_linear
% 0.30/0.58  thf(fact_3893_linorder__less__linear,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58        | ( X3 = Y3 )
% 0.30/0.58        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_linear
% 0.30/0.58  thf(fact_3894_linorder__less__linear,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58        | ( X3 = Y3 )
% 0.30/0.58        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_less_linear
% 0.30/0.58  thf(fact_3895_order__less__imp__not__eq,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq
% 0.30/0.58  thf(fact_3896_order__less__imp__not__eq,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq
% 0.30/0.58  thf(fact_3897_order__less__imp__not__eq,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq
% 0.30/0.58  thf(fact_3898_order__less__imp__not__eq,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq
% 0.30/0.58  thf(fact_3899_order__less__imp__not__eq,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( X3 != Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq
% 0.30/0.58  thf(fact_3900_order__less__imp__not__eq2,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( Y3 != X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq2
% 0.30/0.58  thf(fact_3901_order__less__imp__not__eq2,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( Y3 != X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq2
% 0.30/0.58  thf(fact_3902_order__less__imp__not__eq2,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( Y3 != X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq2
% 0.30/0.58  thf(fact_3903_order__less__imp__not__eq2,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( Y3 != X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq2
% 0.30/0.58  thf(fact_3904_order__less__imp__not__eq2,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( Y3 != X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_eq2
% 0.30/0.58  thf(fact_3905_order__less__imp__not__less,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_less
% 0.30/0.58  thf(fact_3906_order__less__imp__not__less,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_less
% 0.30/0.58  thf(fact_3907_order__less__imp__not__less,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_less
% 0.30/0.58  thf(fact_3908_order__less__imp__not__less,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_less
% 0.30/0.58  thf(fact_3909_order__less__imp__not__less,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_not_less
% 0.30/0.58  thf(fact_3910_small__lazy_H_Ocases,axiom,
% 0.30/0.58      ! [X3: product_prod_int_int] :
% 0.30/0.58        ~ ! [D3: int,I3: int] :
% 0.30/0.58            ( X3
% 0.30/0.58           != ( product_Pair_int_int @ D3 @ I3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % small_lazy'.cases
% 0.30/0.58  thf(fact_3911_exhaustive__int_H_Ocases,axiom,
% 0.30/0.58      ! [X3: produc7773217078559923341nt_int] :
% 0.30/0.58        ~ ! [F2: int > option6357759511663192854e_term,D3: int,I3: int] :
% 0.30/0.58            ( X3
% 0.30/0.58           != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % exhaustive_int'.cases
% 0.30/0.58  thf(fact_3912_full__exhaustive__int_H_Ocases,axiom,
% 0.30/0.58      ! [X3: produc2285326912895808259nt_int] :
% 0.30/0.58        ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I3: int] :
% 0.30/0.58            ( X3
% 0.30/0.58           != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % full_exhaustive_int'.cases
% 0.30/0.58  thf(fact_3913_leD,axiom,
% 0.30/0.58      ! [Y3: real,X3: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ Y3 @ X3 )
% 0.30/0.58       => ~ ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leD
% 0.30/0.58  thf(fact_3914_leD,axiom,
% 0.30/0.58      ! [Y3: set_int,X3: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.30/0.58       => ~ ( ord_less_set_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leD
% 0.30/0.58  thf(fact_3915_leD,axiom,
% 0.30/0.58      ! [Y3: rat,X3: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.58       => ~ ( ord_less_rat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leD
% 0.30/0.58  thf(fact_3916_leD,axiom,
% 0.30/0.58      ! [Y3: num,X3: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.30/0.58       => ~ ( ord_less_num @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leD
% 0.30/0.58  thf(fact_3917_leD,axiom,
% 0.30/0.58      ! [Y3: nat,X3: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.58       => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leD
% 0.30/0.58  thf(fact_3918_leD,axiom,
% 0.30/0.58      ! [Y3: int,X3: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.30/0.58       => ~ ( ord_less_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leD
% 0.30/0.58  thf(fact_3919_leI,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leI
% 0.30/0.58  thf(fact_3920_leI,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leI
% 0.30/0.58  thf(fact_3921_leI,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leI
% 0.30/0.58  thf(fact_3922_leI,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leI
% 0.30/0.58  thf(fact_3923_leI,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % leI
% 0.30/0.58  thf(fact_3924_nless__le,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( ~ ( ord_less_real @ A @ B2 ) )
% 0.30/0.58        = ( ~ ( ord_less_eq_real @ A @ B2 )
% 0.30/0.58          | ( A = B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nless_le
% 0.30/0.58  thf(fact_3925_nless__le,axiom,
% 0.30/0.58      ! [A: set_int,B2: set_int] :
% 0.30/0.58        ( ( ~ ( ord_less_set_int @ A @ B2 ) )
% 0.30/0.58        = ( ~ ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.58          | ( A = B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nless_le
% 0.30/0.58  thf(fact_3926_nless__le,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( ~ ( ord_less_rat @ A @ B2 ) )
% 0.30/0.58        = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58          | ( A = B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nless_le
% 0.30/0.58  thf(fact_3927_nless__le,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( ~ ( ord_less_num @ A @ B2 ) )
% 0.30/0.58        = ( ~ ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58          | ( A = B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nless_le
% 0.30/0.58  thf(fact_3928_nless__le,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( ~ ( ord_less_nat @ A @ B2 ) )
% 0.30/0.58        = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58          | ( A = B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nless_le
% 0.30/0.58  thf(fact_3929_nless__le,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( ~ ( ord_less_int @ A @ B2 ) )
% 0.30/0.58        = ( ~ ( ord_less_eq_int @ A @ B2 )
% 0.30/0.58          | ( A = B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nless_le
% 0.30/0.58  thf(fact_3930_antisym__conv1,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv1
% 0.30/0.58  thf(fact_3931_antisym__conv1,axiom,
% 0.30/0.58      ! [X3: set_int,Y3: set_int] :
% 0.30/0.58        ( ~ ( ord_less_set_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv1
% 0.30/0.58  thf(fact_3932_antisym__conv1,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv1
% 0.30/0.58  thf(fact_3933_antisym__conv1,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv1
% 0.30/0.58  thf(fact_3934_antisym__conv1,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv1
% 0.30/0.58  thf(fact_3935_antisym__conv1,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv1
% 0.30/0.58  thf(fact_3936_antisym__conv2,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv2
% 0.30/0.58  thf(fact_3937_antisym__conv2,axiom,
% 0.30/0.58      ! [X3: set_int,Y3: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ~ ( ord_less_set_int @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv2
% 0.30/0.58  thf(fact_3938_antisym__conv2,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv2
% 0.30/0.58  thf(fact_3939_antisym__conv2,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv2
% 0.30/0.58  thf(fact_3940_antisym__conv2,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv2
% 0.30/0.58  thf(fact_3941_antisym__conv2,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.30/0.58          = ( X3 = Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % antisym_conv2
% 0.30/0.58  thf(fact_3942_dense__ge,axiom,
% 0.30/0.58      ! [Z: real,Y3: real] :
% 0.30/0.58        ( ! [X5: real] :
% 0.30/0.58            ( ( ord_less_real @ Z @ X5 )
% 0.30/0.58           => ( ord_less_eq_real @ Y3 @ X5 ) )
% 0.30/0.58       => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_ge
% 0.30/0.58  thf(fact_3943_dense__ge,axiom,
% 0.30/0.58      ! [Z: rat,Y3: rat] :
% 0.30/0.58        ( ! [X5: rat] :
% 0.30/0.58            ( ( ord_less_rat @ Z @ X5 )
% 0.30/0.58           => ( ord_less_eq_rat @ Y3 @ X5 ) )
% 0.30/0.58       => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_ge
% 0.30/0.58  thf(fact_3944_dense__le,axiom,
% 0.30/0.58      ! [Y3: real,Z: real] :
% 0.30/0.58        ( ! [X5: real] :
% 0.30/0.58            ( ( ord_less_real @ X5 @ Y3 )
% 0.30/0.58           => ( ord_less_eq_real @ X5 @ Z ) )
% 0.30/0.58       => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_le
% 0.30/0.58  thf(fact_3945_dense__le,axiom,
% 0.30/0.58      ! [Y3: rat,Z: rat] :
% 0.30/0.58        ( ! [X5: rat] :
% 0.30/0.58            ( ( ord_less_rat @ X5 @ Y3 )
% 0.30/0.58           => ( ord_less_eq_rat @ X5 @ Z ) )
% 0.30/0.58       => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_le
% 0.30/0.58  thf(fact_3946_less__le__not__le,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [X: real,Y: real] :
% 0.30/0.58            ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.58            & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_le_not_le
% 0.30/0.58  thf(fact_3947_less__le__not__le,axiom,
% 0.30/0.58      ( ord_less_set_int
% 0.30/0.58      = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.58            ( ( ord_less_eq_set_int @ X @ Y )
% 0.30/0.58            & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_le_not_le
% 0.30/0.58  thf(fact_3948_less__le__not__le,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [X: rat,Y: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.58            & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_le_not_le
% 0.30/0.58  thf(fact_3949_less__le__not__le,axiom,
% 0.30/0.58      ( ord_less_num
% 0.30/0.58      = ( ^ [X: num,Y: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ X @ Y )
% 0.30/0.58            & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_le_not_le
% 0.30/0.58  thf(fact_3950_less__le__not__le,axiom,
% 0.30/0.58      ( ord_less_nat
% 0.30/0.58      = ( ^ [X: nat,Y: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.58            & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_le_not_le
% 0.30/0.58  thf(fact_3951_less__le__not__le,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [X: int,Y: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.58            & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_le_not_le
% 0.30/0.58  thf(fact_3952_not__le__imp__less,axiom,
% 0.30/0.58      ! [Y3: real,X3: real] :
% 0.30/0.58        ( ~ ( ord_less_eq_real @ Y3 @ X3 )
% 0.30/0.58       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_le_imp_less
% 0.30/0.58  thf(fact_3953_not__le__imp__less,axiom,
% 0.30/0.58      ! [Y3: rat,X3: rat] :
% 0.30/0.58        ( ~ ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.58       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_le_imp_less
% 0.30/0.58  thf(fact_3954_not__le__imp__less,axiom,
% 0.30/0.58      ! [Y3: num,X3: num] :
% 0.30/0.58        ( ~ ( ord_less_eq_num @ Y3 @ X3 )
% 0.30/0.58       => ( ord_less_num @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_le_imp_less
% 0.30/0.58  thf(fact_3955_not__le__imp__less,axiom,
% 0.30/0.58      ! [Y3: nat,X3: nat] :
% 0.30/0.58        ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.58       => ( ord_less_nat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_le_imp_less
% 0.30/0.58  thf(fact_3956_not__le__imp__less,axiom,
% 0.30/0.58      ! [Y3: int,X3: int] :
% 0.30/0.58        ( ~ ( ord_less_eq_int @ Y3 @ X3 )
% 0.30/0.58       => ( ord_less_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % not_le_imp_less
% 0.30/0.58  thf(fact_3957_order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_real
% 0.30/0.58      = ( ^ [A5: real,B4: real] :
% 0.30/0.58            ( ( ord_less_real @ A5 @ B4 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.order_iff_strict
% 0.30/0.58  thf(fact_3958_order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_set_int
% 0.30/0.58      = ( ^ [A5: set_int,B4: set_int] :
% 0.30/0.58            ( ( ord_less_set_int @ A5 @ B4 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.order_iff_strict
% 0.30/0.58  thf(fact_3959_order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_rat
% 0.30/0.58      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.58            ( ( ord_less_rat @ A5 @ B4 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.order_iff_strict
% 0.30/0.58  thf(fact_3960_order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_num
% 0.30/0.58      = ( ^ [A5: num,B4: num] :
% 0.30/0.58            ( ( ord_less_num @ A5 @ B4 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.order_iff_strict
% 0.30/0.58  thf(fact_3961_order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_nat
% 0.30/0.58      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.58            ( ( ord_less_nat @ A5 @ B4 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.order_iff_strict
% 0.30/0.58  thf(fact_3962_order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_int
% 0.30/0.58      = ( ^ [A5: int,B4: int] :
% 0.30/0.58            ( ( ord_less_int @ A5 @ B4 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.order_iff_strict
% 0.30/0.58  thf(fact_3963_order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [A5: real,B4: real] :
% 0.30/0.58            ( ( ord_less_eq_real @ A5 @ B4 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_order
% 0.30/0.58  thf(fact_3964_order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_set_int
% 0.30/0.58      = ( ^ [A5: set_int,B4: set_int] :
% 0.30/0.58            ( ( ord_less_eq_set_int @ A5 @ B4 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_order
% 0.30/0.58  thf(fact_3965_order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ A5 @ B4 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_order
% 0.30/0.58  thf(fact_3966_order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_num
% 0.30/0.58      = ( ^ [A5: num,B4: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ A5 @ B4 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_order
% 0.30/0.58  thf(fact_3967_order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_nat
% 0.30/0.58      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ A5 @ B4 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_order
% 0.30/0.58  thf(fact_3968_order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [A5: int,B4: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ A5 @ B4 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_order
% 0.30/0.58  thf(fact_3969_order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: real,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans1
% 0.30/0.58  thf(fact_3970_order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: set_int,B2: set_int,C: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_set_int @ B2 @ C )
% 0.30/0.58         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans1
% 0.30/0.58  thf(fact_3971_order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans1
% 0.30/0.58  thf(fact_3972_order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: num,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans1
% 0.30/0.58  thf(fact_3973_order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans1
% 0.30/0.58  thf(fact_3974_order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: int,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans1
% 0.30/0.58  thf(fact_3975_order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: real,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_real @ B2 @ C )
% 0.30/0.58         => ( ord_less_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans2
% 0.30/0.58  thf(fact_3976_order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: set_int,B2: set_int,C: set_int] :
% 0.30/0.58        ( ( ord_less_set_int @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ B2 @ C )
% 0.30/0.58         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans2
% 0.30/0.58  thf(fact_3977_order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.58         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans2
% 0.30/0.58  thf(fact_3978_order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: num,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.58         => ( ord_less_num @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans2
% 0.30/0.58  thf(fact_3979_order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ B2 @ C )
% 0.30/0.58         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans2
% 0.30/0.58  thf(fact_3980_order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: int,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_eq_int @ B2 @ C )
% 0.30/0.58         => ( ord_less_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_trans2
% 0.30/0.58  thf(fact_3981_order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [A5: real,B4: real] :
% 0.30/0.58            ( ( ord_less_eq_real @ A5 @ B4 )
% 0.30/0.58            & ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_not
% 0.30/0.58  thf(fact_3982_order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_set_int
% 0.30/0.58      = ( ^ [A5: set_int,B4: set_int] :
% 0.30/0.58            ( ( ord_less_eq_set_int @ A5 @ B4 )
% 0.30/0.58            & ~ ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_not
% 0.30/0.58  thf(fact_3983_order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ A5 @ B4 )
% 0.30/0.58            & ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_not
% 0.30/0.58  thf(fact_3984_order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_num
% 0.30/0.58      = ( ^ [A5: num,B4: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ A5 @ B4 )
% 0.30/0.58            & ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_not
% 0.30/0.58  thf(fact_3985_order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_nat
% 0.30/0.58      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ A5 @ B4 )
% 0.30/0.58            & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_not
% 0.30/0.58  thf(fact_3986_order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [A5: int,B4: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ A5 @ B4 )
% 0.30/0.58            & ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_iff_not
% 0.30/0.58  thf(fact_3987_dense__ge__bounded,axiom,
% 0.30/0.58      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ Z @ X3 )
% 0.30/0.58       => ( ! [W2: real] :
% 0.30/0.58              ( ( ord_less_real @ Z @ W2 )
% 0.30/0.58             => ( ( ord_less_real @ W2 @ X3 )
% 0.30/0.58               => ( ord_less_eq_real @ Y3 @ W2 ) ) )
% 0.30/0.58         => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_ge_bounded
% 0.30/0.58  thf(fact_3988_dense__ge__bounded,axiom,
% 0.30/0.58      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ Z @ X3 )
% 0.30/0.58       => ( ! [W2: rat] :
% 0.30/0.58              ( ( ord_less_rat @ Z @ W2 )
% 0.30/0.58             => ( ( ord_less_rat @ W2 @ X3 )
% 0.30/0.58               => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
% 0.30/0.58         => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_ge_bounded
% 0.30/0.58  thf(fact_3989_dense__le__bounded,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ! [W2: real] :
% 0.30/0.58              ( ( ord_less_real @ X3 @ W2 )
% 0.30/0.58             => ( ( ord_less_real @ W2 @ Y3 )
% 0.30/0.58               => ( ord_less_eq_real @ W2 @ Z ) ) )
% 0.30/0.58         => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_le_bounded
% 0.30/0.58  thf(fact_3990_dense__le__bounded,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ! [W2: rat] :
% 0.30/0.58              ( ( ord_less_rat @ X3 @ W2 )
% 0.30/0.58             => ( ( ord_less_rat @ W2 @ Y3 )
% 0.30/0.58               => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 0.30/0.58         => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dense_le_bounded
% 0.30/0.58  thf(fact_3991_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_real
% 0.30/0.58      = ( ^ [B4: real,A5: real] :
% 0.30/0.58            ( ( ord_less_real @ B4 @ A5 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.order_iff_strict
% 0.30/0.58  thf(fact_3992_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_set_int
% 0.30/0.58      = ( ^ [B4: set_int,A5: set_int] :
% 0.30/0.58            ( ( ord_less_set_int @ B4 @ A5 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.order_iff_strict
% 0.30/0.58  thf(fact_3993_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_rat
% 0.30/0.58      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.58            ( ( ord_less_rat @ B4 @ A5 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.order_iff_strict
% 0.30/0.58  thf(fact_3994_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_num
% 0.30/0.58      = ( ^ [B4: num,A5: num] :
% 0.30/0.58            ( ( ord_less_num @ B4 @ A5 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.order_iff_strict
% 0.30/0.58  thf(fact_3995_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_nat
% 0.30/0.58      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.58            ( ( ord_less_nat @ B4 @ A5 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.order_iff_strict
% 0.30/0.58  thf(fact_3996_dual__order_Oorder__iff__strict,axiom,
% 0.30/0.58      ( ord_less_eq_int
% 0.30/0.58      = ( ^ [B4: int,A5: int] :
% 0.30/0.58            ( ( ord_less_int @ B4 @ A5 )
% 0.30/0.58            | ( A5 = B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.order_iff_strict
% 0.30/0.58  thf(fact_3997_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [B4: real,A5: real] :
% 0.30/0.58            ( ( ord_less_eq_real @ B4 @ A5 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_order
% 0.30/0.58  thf(fact_3998_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_set_int
% 0.30/0.58      = ( ^ [B4: set_int,A5: set_int] :
% 0.30/0.58            ( ( ord_less_eq_set_int @ B4 @ A5 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_order
% 0.30/0.58  thf(fact_3999_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ B4 @ A5 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_order
% 0.30/0.58  thf(fact_4000_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_num
% 0.30/0.58      = ( ^ [B4: num,A5: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ B4 @ A5 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_order
% 0.30/0.58  thf(fact_4001_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_nat
% 0.30/0.58      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ B4 @ A5 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_order
% 0.30/0.58  thf(fact_4002_dual__order_Ostrict__iff__order,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [B4: int,A5: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ B4 @ A5 )
% 0.30/0.58            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_order
% 0.30/0.58  thf(fact_4003_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [B2: real,A: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_real @ C @ B2 )
% 0.30/0.58         => ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans1
% 0.30/0.58  thf(fact_4004_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [B2: set_int,A: set_int,C: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_set_int @ C @ B2 )
% 0.30/0.58         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans1
% 0.30/0.58  thf(fact_4005_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_rat @ C @ B2 )
% 0.30/0.58         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans1
% 0.30/0.58  thf(fact_4006_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [B2: num,A: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_num @ C @ B2 )
% 0.30/0.58         => ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans1
% 0.30/0.58  thf(fact_4007_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_nat @ C @ B2 )
% 0.30/0.58         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans1
% 0.30/0.58  thf(fact_4008_dual__order_Ostrict__trans1,axiom,
% 0.30/0.58      ! [B2: int,A: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_int @ C @ B2 )
% 0.30/0.58         => ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans1
% 0.30/0.58  thf(fact_4009_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [B2: real,A: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_eq_real @ C @ B2 )
% 0.30/0.58         => ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans2
% 0.30/0.58  thf(fact_4010_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [B2: set_int,A: set_int,C: set_int] :
% 0.30/0.58        ( ( ord_less_set_int @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ C @ B2 )
% 0.30/0.58         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans2
% 0.30/0.58  thf(fact_4011_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_eq_rat @ C @ B2 )
% 0.30/0.58         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans2
% 0.30/0.58  thf(fact_4012_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [B2: num,A: num,C: num] :
% 0.30/0.58        ( ( ord_less_num @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_eq_num @ C @ B2 )
% 0.30/0.58         => ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans2
% 0.30/0.58  thf(fact_4013_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_eq_nat @ C @ B2 )
% 0.30/0.58         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans2
% 0.30/0.58  thf(fact_4014_dual__order_Ostrict__trans2,axiom,
% 0.30/0.58      ! [B2: int,A: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ B2 @ A )
% 0.30/0.58       => ( ( ord_less_eq_int @ C @ B2 )
% 0.30/0.58         => ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_trans2
% 0.30/0.58  thf(fact_4015_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [B4: real,A5: real] :
% 0.30/0.58            ( ( ord_less_eq_real @ B4 @ A5 )
% 0.30/0.58            & ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_not
% 0.30/0.58  thf(fact_4016_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_set_int
% 0.30/0.58      = ( ^ [B4: set_int,A5: set_int] :
% 0.30/0.58            ( ( ord_less_eq_set_int @ B4 @ A5 )
% 0.30/0.58            & ~ ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_not
% 0.30/0.58  thf(fact_4017_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ B4 @ A5 )
% 0.30/0.58            & ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_not
% 0.30/0.58  thf(fact_4018_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_num
% 0.30/0.58      = ( ^ [B4: num,A5: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ B4 @ A5 )
% 0.30/0.58            & ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_not
% 0.30/0.58  thf(fact_4019_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_nat
% 0.30/0.58      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ B4 @ A5 )
% 0.30/0.58            & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_not
% 0.30/0.58  thf(fact_4020_dual__order_Ostrict__iff__not,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [B4: int,A5: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ B4 @ A5 )
% 0.30/0.58            & ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_iff_not
% 0.30/0.58  thf(fact_4021_order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B2 )
% 0.30/0.58       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_order
% 0.30/0.58  thf(fact_4022_order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: set_int,B2: set_int] :
% 0.30/0.58        ( ( ord_less_set_int @ A @ B2 )
% 0.30/0.58       => ( ord_less_eq_set_int @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_order
% 0.30/0.58  thf(fact_4023_order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.58       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_order
% 0.30/0.58  thf(fact_4024_order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( ord_less_num @ A @ B2 )
% 0.30/0.58       => ( ord_less_eq_num @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_order
% 0.30/0.58  thf(fact_4025_order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.58       => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_order
% 0.30/0.58  thf(fact_4026_order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B2 )
% 0.30/0.58       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order.strict_implies_order
% 0.30/0.58  thf(fact_4027_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [B2: real,A: real] :
% 0.30/0.58        ( ( ord_less_real @ B2 @ A )
% 0.30/0.58       => ( ord_less_eq_real @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_order
% 0.30/0.58  thf(fact_4028_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [B2: set_int,A: set_int] :
% 0.30/0.58        ( ( ord_less_set_int @ B2 @ A )
% 0.30/0.58       => ( ord_less_eq_set_int @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_order
% 0.30/0.58  thf(fact_4029_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [B2: rat,A: rat] :
% 0.30/0.58        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.58       => ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_order
% 0.30/0.58  thf(fact_4030_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [B2: num,A: num] :
% 0.30/0.58        ( ( ord_less_num @ B2 @ A )
% 0.30/0.58       => ( ord_less_eq_num @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_order
% 0.30/0.58  thf(fact_4031_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [B2: nat,A: nat] :
% 0.30/0.58        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.58       => ( ord_less_eq_nat @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_order
% 0.30/0.58  thf(fact_4032_dual__order_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [B2: int,A: int] :
% 0.30/0.58        ( ( ord_less_int @ B2 @ A )
% 0.30/0.58       => ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dual_order.strict_implies_order
% 0.30/0.58  thf(fact_4033_order__le__less,axiom,
% 0.30/0.58      ( ord_less_eq_real
% 0.30/0.58      = ( ^ [X: real,Y: real] :
% 0.30/0.58            ( ( ord_less_real @ X @ Y )
% 0.30/0.58            | ( X = Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less
% 0.30/0.58  thf(fact_4034_order__le__less,axiom,
% 0.30/0.58      ( ord_less_eq_set_int
% 0.30/0.58      = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.58            ( ( ord_less_set_int @ X @ Y )
% 0.30/0.58            | ( X = Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less
% 0.30/0.58  thf(fact_4035_order__le__less,axiom,
% 0.30/0.58      ( ord_less_eq_rat
% 0.30/0.58      = ( ^ [X: rat,Y: rat] :
% 0.30/0.58            ( ( ord_less_rat @ X @ Y )
% 0.30/0.58            | ( X = Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less
% 0.30/0.58  thf(fact_4036_order__le__less,axiom,
% 0.30/0.58      ( ord_less_eq_num
% 0.30/0.58      = ( ^ [X: num,Y: num] :
% 0.30/0.58            ( ( ord_less_num @ X @ Y )
% 0.30/0.58            | ( X = Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less
% 0.30/0.58  thf(fact_4037_order__le__less,axiom,
% 0.30/0.58      ( ord_less_eq_nat
% 0.30/0.58      = ( ^ [X: nat,Y: nat] :
% 0.30/0.58            ( ( ord_less_nat @ X @ Y )
% 0.30/0.58            | ( X = Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less
% 0.30/0.58  thf(fact_4038_order__le__less,axiom,
% 0.30/0.58      ( ord_less_eq_int
% 0.30/0.58      = ( ^ [X: int,Y: int] :
% 0.30/0.58            ( ( ord_less_int @ X @ Y )
% 0.30/0.58            | ( X = Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less
% 0.30/0.58  thf(fact_4039_order__less__le,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [X: real,Y: real] :
% 0.30/0.58            ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.58            & ( X != Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le
% 0.30/0.58  thf(fact_4040_order__less__le,axiom,
% 0.30/0.58      ( ord_less_set_int
% 0.30/0.58      = ( ^ [X: set_int,Y: set_int] :
% 0.30/0.58            ( ( ord_less_eq_set_int @ X @ Y )
% 0.30/0.58            & ( X != Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le
% 0.30/0.58  thf(fact_4041_order__less__le,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [X: rat,Y: rat] :
% 0.30/0.58            ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.58            & ( X != Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le
% 0.30/0.58  thf(fact_4042_order__less__le,axiom,
% 0.30/0.58      ( ord_less_num
% 0.30/0.58      = ( ^ [X: num,Y: num] :
% 0.30/0.58            ( ( ord_less_eq_num @ X @ Y )
% 0.30/0.58            & ( X != Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le
% 0.30/0.58  thf(fact_4043_order__less__le,axiom,
% 0.30/0.58      ( ord_less_nat
% 0.30/0.58      = ( ^ [X: nat,Y: nat] :
% 0.30/0.58            ( ( ord_less_eq_nat @ X @ Y )
% 0.30/0.58            & ( X != Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le
% 0.30/0.58  thf(fact_4044_order__less__le,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [X: int,Y: int] :
% 0.30/0.58            ( ( ord_less_eq_int @ X @ Y )
% 0.30/0.58            & ( X != Y ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le
% 0.30/0.58  thf(fact_4045_linorder__not__le,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ~ ( ord_less_eq_real @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_le
% 0.30/0.58  thf(fact_4046_linorder__not__le,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ~ ( ord_less_eq_rat @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_le
% 0.30/0.58  thf(fact_4047_linorder__not__le,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_le
% 0.30/0.58  thf(fact_4048_linorder__not__le,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_le
% 0.30/0.58  thf(fact_4049_linorder__not__le,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_le
% 0.30/0.58  thf(fact_4050_linorder__not__less,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_less
% 0.30/0.58  thf(fact_4051_linorder__not__less,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_less
% 0.30/0.58  thf(fact_4052_linorder__not__less,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_less
% 0.30/0.58  thf(fact_4053_linorder__not__less,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_less
% 0.30/0.58  thf(fact_4054_linorder__not__less,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.30/0.58        = ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linorder_not_less
% 0.30/0.58  thf(fact_4055_order__less__imp__le,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_le
% 0.30/0.58  thf(fact_4056_order__less__imp__le,axiom,
% 0.30/0.58      ! [X3: set_int,Y3: set_int] :
% 0.30/0.58        ( ( ord_less_set_int @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_le
% 0.30/0.58  thf(fact_4057_order__less__imp__le,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_le
% 0.30/0.58  thf(fact_4058_order__less__imp__le,axiom,
% 0.30/0.58      ! [X3: num,Y3: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_le
% 0.30/0.58  thf(fact_4059_order__less__imp__le,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_le
% 0.30/0.58  thf(fact_4060_order__less__imp__le,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_imp_le
% 0.30/0.58  thf(fact_4061_order__le__neq__trans,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.58       => ( ( A != B2 )
% 0.30/0.58         => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_neq_trans
% 0.30/0.58  thf(fact_4062_order__le__neq__trans,axiom,
% 0.30/0.58      ! [A: set_int,B2: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.58       => ( ( A != B2 )
% 0.30/0.58         => ( ord_less_set_int @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_neq_trans
% 0.30/0.58  thf(fact_4063_order__le__neq__trans,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( A != B2 )
% 0.30/0.58         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_neq_trans
% 0.30/0.58  thf(fact_4064_order__le__neq__trans,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( A != B2 )
% 0.30/0.58         => ( ord_less_num @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_neq_trans
% 0.30/0.58  thf(fact_4065_order__le__neq__trans,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58       => ( ( A != B2 )
% 0.30/0.58         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_neq_trans
% 0.30/0.58  thf(fact_4066_order__le__neq__trans,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.58       => ( ( A != B2 )
% 0.30/0.58         => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_neq_trans
% 0.30/0.58  thf(fact_4067_order__neq__le__trans,axiom,
% 0.30/0.58      ! [A: real,B2: real] :
% 0.30/0.58        ( ( A != B2 )
% 0.30/0.58       => ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.58         => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_neq_le_trans
% 0.30/0.58  thf(fact_4068_order__neq__le__trans,axiom,
% 0.30/0.58      ! [A: set_int,B2: set_int] :
% 0.30/0.58        ( ( A != B2 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ A @ B2 )
% 0.30/0.58         => ( ord_less_set_int @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_neq_le_trans
% 0.30/0.58  thf(fact_4069_order__neq__le__trans,axiom,
% 0.30/0.58      ! [A: rat,B2: rat] :
% 0.30/0.58        ( ( A != B2 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_neq_le_trans
% 0.30/0.58  thf(fact_4070_order__neq__le__trans,axiom,
% 0.30/0.58      ! [A: num,B2: num] :
% 0.30/0.58        ( ( A != B2 )
% 0.30/0.58       => ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58         => ( ord_less_num @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_neq_le_trans
% 0.30/0.58  thf(fact_4071_order__neq__le__trans,axiom,
% 0.30/0.58      ! [A: nat,B2: nat] :
% 0.30/0.58        ( ( A != B2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.58         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_neq_le_trans
% 0.30/0.58  thf(fact_4072_order__neq__le__trans,axiom,
% 0.30/0.58      ! [A: int,B2: int] :
% 0.30/0.58        ( ( A != B2 )
% 0.30/0.58       => ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.58         => ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_neq_le_trans
% 0.30/0.58  thf(fact_4073_order__le__less__trans,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_real @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_real @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_trans
% 0.30/0.58  thf(fact_4074_order__le__less__trans,axiom,
% 0.30/0.58      ! [X3: set_int,Y3: set_int,Z: set_int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_set_int @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_set_int @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_trans
% 0.30/0.58  thf(fact_4075_order__le__less__trans,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_rat @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_rat @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_trans
% 0.30/0.58  thf(fact_4076_order__le__less__trans,axiom,
% 0.30/0.58      ! [X3: num,Y3: num,Z: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_num @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_num @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_trans
% 0.30/0.58  thf(fact_4077_order__le__less__trans,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_nat @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_nat @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_trans
% 0.30/0.58  thf(fact_4078_order__le__less__trans,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_int @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_trans
% 0.30/0.58  thf(fact_4079_order__less__le__trans,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.58        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_real @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_real @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le_trans
% 0.30/0.58  thf(fact_4080_order__less__le__trans,axiom,
% 0.30/0.58      ! [X3: set_int,Y3: set_int,Z: set_int] :
% 0.30/0.58        ( ( ord_less_set_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_set_int @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le_trans
% 0.30/0.58  thf(fact_4081_order__less__le__trans,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.58        ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_rat @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le_trans
% 0.30/0.58  thf(fact_4082_order__less__le__trans,axiom,
% 0.30/0.58      ! [X3: num,Y3: num,Z: num] :
% 0.30/0.58        ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_num @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le_trans
% 0.30/0.58  thf(fact_4083_order__less__le__trans,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.58        ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_nat @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le_trans
% 0.30/0.58  thf(fact_4084_order__less__le__trans,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.58        ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.58         => ( ord_less_int @ X3 @ Z ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_less_le_trans
% 0.30/0.58  thf(fact_4085_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: real > real,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4086_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4087_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: num > real,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4088_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: nat > real,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4089_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: real,F: int > real,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ! [X5: int,Y4: int] :
% 0.30/0.58                ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4090_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: real > rat,B2: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_real @ B2 @ C )
% 0.30/0.58         => ( ! [X5: real,Y4: real] :
% 0.30/0.58                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4091_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_rat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4092_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_num @ B2 @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4093_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_nat @ B2 @ C )
% 0.30/0.58         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.58                ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4094_order__le__less__subst1,axiom,
% 0.30/0.58      ! [A: rat,F: int > rat,B2: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 0.30/0.58       => ( ( ord_less_int @ B2 @ C )
% 0.30/0.58         => ( ! [X5: int,Y4: int] :
% 0.30/0.58                ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst1
% 0.30/0.58  thf(fact_4095_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4096_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4097_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4098_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4099_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.58                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4100_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4101_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4102_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > num,C: num] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4103_order__le__less__subst2,axiom,
% 0.30/0.58      ! [A: num,B2: num,F: num > nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.58       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 0.30/0.58         => ( ! [X5: num,Y4: num] :
% 0.30/0.58                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.58               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.58           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % order_le_less_subst2
% 0.30/0.58  thf(fact_4104_order__le__less__subst2,axiom,
% 0.30/0.59      ! [A: num,B2: num,F: num > int,C: int] :
% 0.30/0.59        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_less_subst2
% 0.30/0.59  thf(fact_4105_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 0.30/0.59        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4106_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4107_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 0.30/0.59        ( ( ord_less_num @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4108_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 0.30/0.59        ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4109_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: int,F: rat > int,B2: rat,C: rat] :
% 0.30/0.59        ( ( ord_less_int @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_rat @ B2 @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4110_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: real,F: num > real,B2: num,C: num] :
% 0.30/0.59        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4111_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: rat,F: num > rat,B2: num,C: num] :
% 0.30/0.59        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4112_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: num,F: num > num,B2: num,C: num] :
% 0.30/0.59        ( ( ord_less_num @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4113_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: nat,F: num > nat,B2: num,C: num] :
% 0.30/0.59        ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4114_order__less__le__subst1,axiom,
% 0.30/0.59      ! [A: int,F: num > int,B2: num,C: num] :
% 0.30/0.59        ( ( ord_less_int @ A @ ( F @ B2 ) )
% 0.30/0.59       => ( ( ord_less_eq_num @ B2 @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst1
% 0.30/0.59  thf(fact_4115_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: real,B2: real,F: real > real,C: real] :
% 0.30/0.59        ( ( ord_less_real @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: real,Y4: real] :
% 0.30/0.59                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4116_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 0.30/0.59        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4117_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: num,B2: num,F: num > real,C: real] :
% 0.30/0.59        ( ( ord_less_num @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4118_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,F: nat > real,C: real] :
% 0.30/0.59        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.59                ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4119_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: int,B2: int,F: int > real,C: real] :
% 0.30/0.59        ( ( ord_less_int @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: int,Y4: int] :
% 0.30/0.59                ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4120_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: real,B2: real,F: real > rat,C: rat] :
% 0.30/0.59        ( ( ord_less_real @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: real,Y4: real] :
% 0.30/0.59                ( ( ord_less_real @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4121_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: rat,Y4: rat] :
% 0.30/0.59                ( ( ord_less_rat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4122_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: num,B2: num,F: num > rat,C: rat] :
% 0.30/0.59        ( ( ord_less_num @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: num,Y4: num] :
% 0.30/0.59                ( ( ord_less_num @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4123_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 0.30/0.59        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: nat,Y4: nat] :
% 0.30/0.59                ( ( ord_less_nat @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4124_order__less__le__subst2,axiom,
% 0.30/0.59      ! [A: int,B2: int,F: int > rat,C: rat] :
% 0.30/0.59        ( ( ord_less_int @ A @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 0.30/0.59         => ( ! [X5: int,Y4: int] :
% 0.30/0.59                ( ( ord_less_int @ X5 @ Y4 )
% 0.30/0.59               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.30/0.59           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_less_le_subst2
% 0.30/0.59  thf(fact_4125_linorder__le__less__linear,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.59        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % linorder_le_less_linear
% 0.30/0.59  thf(fact_4126_linorder__le__less__linear,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.59        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % linorder_le_less_linear
% 0.30/0.59  thf(fact_4127_linorder__le__less__linear,axiom,
% 0.30/0.59      ! [X3: num,Y3: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.59        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % linorder_le_less_linear
% 0.30/0.59  thf(fact_4128_linorder__le__less__linear,axiom,
% 0.30/0.59      ! [X3: nat,Y3: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.59        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % linorder_le_less_linear
% 0.30/0.59  thf(fact_4129_linorder__le__less__linear,axiom,
% 0.30/0.59      ! [X3: int,Y3: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.59        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % linorder_le_less_linear
% 0.30/0.59  thf(fact_4130_order__le__imp__less__or__eq,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.59       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.59          | ( X3 = Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_imp_less_or_eq
% 0.30/0.59  thf(fact_4131_order__le__imp__less__or__eq,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_set_int @ X3 @ Y3 )
% 0.30/0.59          | ( X3 = Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_imp_less_or_eq
% 0.30/0.59  thf(fact_4132_order__le__imp__less__or__eq,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.59       => ( ( ord_less_rat @ X3 @ Y3 )
% 0.30/0.59          | ( X3 = Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_imp_less_or_eq
% 0.30/0.59  thf(fact_4133_order__le__imp__less__or__eq,axiom,
% 0.30/0.59      ! [X3: num,Y3: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.30/0.59       => ( ( ord_less_num @ X3 @ Y3 )
% 0.30/0.59          | ( X3 = Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_imp_less_or_eq
% 0.30/0.59  thf(fact_4134_order__le__imp__less__or__eq,axiom,
% 0.30/0.59      ! [X3: nat,Y3: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.30/0.59       => ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.59          | ( X3 = Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_imp_less_or_eq
% 0.30/0.59  thf(fact_4135_order__le__imp__less__or__eq,axiom,
% 0.30/0.59      ! [X3: int,Y3: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.59       => ( ( ord_less_int @ X3 @ Y3 )
% 0.30/0.59          | ( X3 = Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % order_le_imp_less_or_eq
% 0.30/0.59  thf(fact_4136_bot_Oextremum,axiom,
% 0.30/0.59      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum
% 0.30/0.59  thf(fact_4137_bot_Oextremum,axiom,
% 0.30/0.59      ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum
% 0.30/0.59  thf(fact_4138_bot_Oextremum,axiom,
% 0.30/0.59      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum
% 0.30/0.59  thf(fact_4139_bot_Oextremum,axiom,
% 0.30/0.59      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum
% 0.30/0.59  thf(fact_4140_bot_Oextremum,axiom,
% 0.30/0.59      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum
% 0.30/0.59  thf(fact_4141_bot_Oextremum__unique,axiom,
% 0.30/0.59      ! [A: set_nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.30/0.59        = ( A = bot_bot_set_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_unique
% 0.30/0.59  thf(fact_4142_bot_Oextremum__unique,axiom,
% 0.30/0.59      ! [A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 0.30/0.59        = ( A = bot_bo4199563552545308370d_enat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_unique
% 0.30/0.59  thf(fact_4143_bot_Oextremum__unique,axiom,
% 0.30/0.59      ! [A: set_real] :
% 0.30/0.59        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.30/0.59        = ( A = bot_bot_set_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_unique
% 0.30/0.59  thf(fact_4144_bot_Oextremum__unique,axiom,
% 0.30/0.59      ! [A: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.30/0.59        = ( A = bot_bot_set_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_unique
% 0.30/0.59  thf(fact_4145_bot_Oextremum__unique,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.30/0.59        = ( A = bot_bot_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_unique
% 0.30/0.59  thf(fact_4146_bot_Oextremum__uniqueI,axiom,
% 0.30/0.59      ! [A: set_nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.30/0.59       => ( A = bot_bot_set_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_uniqueI
% 0.30/0.59  thf(fact_4147_bot_Oextremum__uniqueI,axiom,
% 0.30/0.59      ! [A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 0.30/0.59       => ( A = bot_bo4199563552545308370d_enat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_uniqueI
% 0.30/0.59  thf(fact_4148_bot_Oextremum__uniqueI,axiom,
% 0.30/0.59      ! [A: set_real] :
% 0.30/0.59        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.30/0.59       => ( A = bot_bot_set_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_uniqueI
% 0.30/0.59  thf(fact_4149_bot_Oextremum__uniqueI,axiom,
% 0.30/0.59      ! [A: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.30/0.59       => ( A = bot_bot_set_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_uniqueI
% 0.30/0.59  thf(fact_4150_bot_Oextremum__uniqueI,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.30/0.59       => ( A = bot_bot_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_uniqueI
% 0.30/0.59  thf(fact_4151_bot_Oextremum__strict,axiom,
% 0.30/0.59      ! [A: set_nat] :
% 0.30/0.59        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_strict
% 0.30/0.59  thf(fact_4152_bot_Oextremum__strict,axiom,
% 0.30/0.59      ! [A: extended_enat] :
% 0.30/0.59        ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_strict
% 0.30/0.59  thf(fact_4153_bot_Oextremum__strict,axiom,
% 0.30/0.59      ! [A: set_int] :
% 0.30/0.59        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_strict
% 0.30/0.59  thf(fact_4154_bot_Oextremum__strict,axiom,
% 0.30/0.59      ! [A: set_real] :
% 0.30/0.59        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_strict
% 0.30/0.59  thf(fact_4155_bot_Oextremum__strict,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.extremum_strict
% 0.30/0.59  thf(fact_4156_bot_Onot__eq__extremum,axiom,
% 0.30/0.59      ! [A: set_nat] :
% 0.30/0.59        ( ( A != bot_bot_set_nat )
% 0.30/0.59        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.not_eq_extremum
% 0.30/0.59  thf(fact_4157_bot_Onot__eq__extremum,axiom,
% 0.30/0.59      ! [A: extended_enat] :
% 0.30/0.59        ( ( A != bot_bo4199563552545308370d_enat )
% 0.30/0.59        = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.not_eq_extremum
% 0.30/0.59  thf(fact_4158_bot_Onot__eq__extremum,axiom,
% 0.30/0.59      ! [A: set_int] :
% 0.30/0.59        ( ( A != bot_bot_set_int )
% 0.30/0.59        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.not_eq_extremum
% 0.30/0.59  thf(fact_4159_bot_Onot__eq__extremum,axiom,
% 0.30/0.59      ! [A: set_real] :
% 0.30/0.59        ( ( A != bot_bot_set_real )
% 0.30/0.59        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.not_eq_extremum
% 0.30/0.59  thf(fact_4160_bot_Onot__eq__extremum,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( A != bot_bot_nat )
% 0.30/0.59        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bot.not_eq_extremum
% 0.30/0.59  thf(fact_4161_Euclid__induct,axiom,
% 0.30/0.59      ! [P: nat > nat > $o,A: nat,B2: nat] :
% 0.30/0.59        ( ! [A3: nat,B3: nat] :
% 0.30/0.59            ( ( P @ A3 @ B3 )
% 0.30/0.59            = ( P @ B3 @ A3 ) )
% 0.30/0.59       => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
% 0.30/0.59         => ( ! [A3: nat,B3: nat] :
% 0.30/0.59                ( ( P @ A3 @ B3 )
% 0.30/0.59               => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
% 0.30/0.59           => ( P @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Euclid_induct
% 0.30/0.59  thf(fact_4162_vebt__insert_Osimps_I5_J,axiom,
% 0.30/0.59      ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.30/0.59        ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.59        = ( if_VEBT_VEBT
% 0.30/0.59          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.59            & ~ ( ( X3 = Mi )
% 0.30/0.59                | ( X3 = Ma ) ) )
% 0.30/0.59          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ X3 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.30/0.59          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % vebt_insert.simps(5)
% 0.30/0.59  thf(fact_4163_vebt__insert_Oelims,axiom,
% 0.30/0.59      ! [X3: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.30/0.59        ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
% 0.30/0.59          = Y3 )
% 0.30/0.59       => ( ! [A3: $o,B3: $o] :
% 0.30/0.59              ( ( X3
% 0.30/0.59                = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.59             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.59                   => ( Y3
% 0.30/0.59                      = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.30/0.59                  & ( ( Xa2 != zero_zero_nat )
% 0.30/0.59                   => ( ( ( Xa2 = one_one_nat )
% 0.30/0.59                       => ( Y3
% 0.30/0.59                          = ( vEBT_Leaf @ A3 @ $true ) ) )
% 0.30/0.59                      & ( ( Xa2 != one_one_nat )
% 0.30/0.59                       => ( Y3
% 0.30/0.59                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
% 0.30/0.59         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.59                ( ( X3
% 0.30/0.59                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.30/0.59               => ( Y3
% 0.30/0.59                 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
% 0.30/0.59           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.59                  ( ( X3
% 0.30/0.59                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.30/0.59                 => ( Y3
% 0.30/0.59                   != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
% 0.30/0.59             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.59                    ( ( X3
% 0.30/0.59                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.59                   => ( Y3
% 0.30/0.59                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.30/0.59               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.59                      ( ( X3
% 0.30/0.59                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.59                     => ( Y3
% 0.30/0.59                       != ( if_VEBT_VEBT
% 0.30/0.59                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.59                            & ~ ( ( Xa2 = Mi2 )
% 0.30/0.59                                | ( Xa2 = Ma2 ) ) )
% 0.30/0.59                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.30/0.59                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % vebt_insert.elims
% 0.30/0.59  thf(fact_4164_vebt__insert_Opelims,axiom,
% 0.30/0.59      ! [X3: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.30/0.59        ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
% 0.30/0.59          = Y3 )
% 0.30/0.59       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.59         => ( ! [A3: $o,B3: $o] :
% 0.30/0.59                ( ( X3
% 0.30/0.59                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.59               => ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.59                     => ( Y3
% 0.30/0.59                        = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.30/0.59                    & ( ( Xa2 != zero_zero_nat )
% 0.30/0.59                     => ( ( ( Xa2 = one_one_nat )
% 0.30/0.59                         => ( Y3
% 0.30/0.59                            = ( vEBT_Leaf @ A3 @ $true ) ) )
% 0.30/0.59                        & ( ( Xa2 != one_one_nat )
% 0.30/0.59                         => ( Y3
% 0.30/0.59                            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
% 0.30/0.59                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 0.30/0.59           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.59                  ( ( X3
% 0.30/0.59                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.30/0.59                 => ( ( Y3
% 0.30/0.59                      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.30/0.59                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa2 ) ) ) )
% 0.30/0.59             => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.30/0.59                    ( ( X3
% 0.30/0.59                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.30/0.59                   => ( ( Y3
% 0.30/0.59                        = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.30/0.59                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
% 0.30/0.59               => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.59                      ( ( X3
% 0.30/0.59                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.59                     => ( ( Y3
% 0.30/0.59                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.59                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
% 0.30/0.59                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.59                        ( ( X3
% 0.30/0.59                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.59                       => ( ( Y3
% 0.30/0.59                            = ( if_VEBT_VEBT
% 0.30/0.59                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.59                                & ~ ( ( Xa2 = Mi2 )
% 0.30/0.59                                    | ( Xa2 = Ma2 ) ) )
% 0.30/0.59                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.30/0.59                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.30/0.59                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % vebt_insert.pelims
% 0.30/0.59  thf(fact_4165_max__less__iff__conj,axiom,
% 0.30/0.59      ! [X3: extended_enat,Y3: extended_enat,Z: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) @ Z )
% 0.30/0.59        = ( ( ord_le72135733267957522d_enat @ X3 @ Z )
% 0.30/0.59          & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max_less_iff_conj
% 0.30/0.59  thf(fact_4166_max__less__iff__conj,axiom,
% 0.30/0.59      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.59        ( ( ord_less_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
% 0.30/0.59        = ( ( ord_less_real @ X3 @ Z )
% 0.30/0.59          & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max_less_iff_conj
% 0.30/0.59  thf(fact_4167_max__less__iff__conj,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
% 0.30/0.59        = ( ( ord_less_rat @ X3 @ Z )
% 0.30/0.59          & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max_less_iff_conj
% 0.30/0.59  thf(fact_4168_max__less__iff__conj,axiom,
% 0.30/0.59      ! [X3: num,Y3: num,Z: num] :
% 0.30/0.59        ( ( ord_less_num @ ( ord_max_num @ X3 @ Y3 ) @ Z )
% 0.30/0.59        = ( ( ord_less_num @ X3 @ Z )
% 0.30/0.59          & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max_less_iff_conj
% 0.30/0.59  thf(fact_4169_max__less__iff__conj,axiom,
% 0.30/0.59      ! [X3: nat,Y3: nat,Z: nat] :
% 0.30/0.59        ( ( ord_less_nat @ ( ord_max_nat @ X3 @ Y3 ) @ Z )
% 0.30/0.59        = ( ( ord_less_nat @ X3 @ Z )
% 0.30/0.59          & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max_less_iff_conj
% 0.30/0.59  thf(fact_4170_max__less__iff__conj,axiom,
% 0.30/0.59      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.59        ( ( ord_less_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
% 0.30/0.59        = ( ( ord_less_int @ X3 @ Z )
% 0.30/0.59          & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max_less_iff_conj
% 0.30/0.59  thf(fact_4171_max_Oabsorb4,axiom,
% 0.30/0.59      ! [A: extended_enat,B2: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ A @ B2 )
% 0.30/0.59       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb4
% 0.30/0.59  thf(fact_4172_max_Oabsorb4,axiom,
% 0.30/0.59      ! [A: real,B2: real] :
% 0.30/0.59        ( ( ord_less_real @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_real @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb4
% 0.30/0.59  thf(fact_4173_max_Oabsorb4,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_rat @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb4
% 0.30/0.59  thf(fact_4174_max_Oabsorb4,axiom,
% 0.30/0.59      ! [A: num,B2: num] :
% 0.30/0.59        ( ( ord_less_num @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_num @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb4
% 0.30/0.59  thf(fact_4175_max_Oabsorb4,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( ord_less_nat @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_nat @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb4
% 0.30/0.59  thf(fact_4176_max_Oabsorb4,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( ord_less_int @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_int @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb4
% 0.30/0.59  thf(fact_4177_max_Oabsorb3,axiom,
% 0.30/0.59      ! [B2: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ B2 @ A )
% 0.30/0.59       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb3
% 0.30/0.59  thf(fact_4178_max_Oabsorb3,axiom,
% 0.30/0.59      ! [B2: real,A: real] :
% 0.30/0.59        ( ( ord_less_real @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_real @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb3
% 0.30/0.59  thf(fact_4179_max_Oabsorb3,axiom,
% 0.30/0.59      ! [B2: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_rat @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb3
% 0.30/0.59  thf(fact_4180_max_Oabsorb3,axiom,
% 0.30/0.59      ! [B2: num,A: num] :
% 0.30/0.59        ( ( ord_less_num @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_num @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb3
% 0.30/0.59  thf(fact_4181_max_Oabsorb3,axiom,
% 0.30/0.59      ! [B2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_nat @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_nat @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb3
% 0.30/0.59  thf(fact_4182_max_Oabsorb3,axiom,
% 0.30/0.59      ! [B2: int,A: int] :
% 0.30/0.59        ( ( ord_less_int @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_int @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb3
% 0.30/0.59  thf(fact_4183_max_Oabsorb1,axiom,
% 0.30/0.59      ! [B2: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 0.30/0.59       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb1
% 0.30/0.59  thf(fact_4184_max_Oabsorb1,axiom,
% 0.30/0.59      ! [B2: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_rat @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb1
% 0.30/0.59  thf(fact_4185_max_Oabsorb1,axiom,
% 0.30/0.59      ! [B2: num,A: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_num @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb1
% 0.30/0.59  thf(fact_4186_max_Oabsorb1,axiom,
% 0.30/0.59      ! [B2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_nat @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb1
% 0.30/0.59  thf(fact_4187_max_Oabsorb1,axiom,
% 0.30/0.59      ! [B2: int,A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.59       => ( ( ord_max_int @ A @ B2 )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb1
% 0.30/0.59  thf(fact_4188_max_Oabsorb2,axiom,
% 0.30/0.59      ! [A: extended_enat,B2: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ A @ B2 )
% 0.30/0.59       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb2
% 0.30/0.59  thf(fact_4189_max_Oabsorb2,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_rat @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb2
% 0.30/0.59  thf(fact_4190_max_Oabsorb2,axiom,
% 0.30/0.59      ! [A: num,B2: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_num @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb2
% 0.30/0.59  thf(fact_4191_max_Oabsorb2,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_nat @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb2
% 0.30/0.59  thf(fact_4192_max_Oabsorb2,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.59       => ( ( ord_max_int @ A @ B2 )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb2
% 0.30/0.59  thf(fact_4193_max_Obounded__iff,axiom,
% 0.30/0.59      ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
% 0.30/0.59        = ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 0.30/0.59          & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.bounded_iff
% 0.30/0.59  thf(fact_4194_max_Obounded__iff,axiom,
% 0.30/0.59      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A )
% 0.30/0.59        = ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.59          & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.bounded_iff
% 0.30/0.59  thf(fact_4195_max_Obounded__iff,axiom,
% 0.30/0.59      ! [B2: num,C: num,A: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A )
% 0.30/0.59        = ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.59          & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.bounded_iff
% 0.30/0.59  thf(fact_4196_max_Obounded__iff,axiom,
% 0.30/0.59      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
% 0.30/0.59        = ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.59          & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.bounded_iff
% 0.30/0.59  thf(fact_4197_max_Obounded__iff,axiom,
% 0.30/0.59      ! [B2: int,C: int,A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
% 0.30/0.59        = ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.59          & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.bounded_iff
% 0.30/0.59  thf(fact_4198_dbl__simps_I3_J,axiom,
% 0.30/0.59      ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.30/0.59      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(3)
% 0.30/0.59  thf(fact_4199_dbl__simps_I3_J,axiom,
% 0.30/0.59      ( ( neg_numeral_dbl_real @ one_one_real )
% 0.30/0.59      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(3)
% 0.30/0.59  thf(fact_4200_dbl__simps_I3_J,axiom,
% 0.30/0.59      ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.30/0.59      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(3)
% 0.30/0.59  thf(fact_4201_dbl__simps_I3_J,axiom,
% 0.30/0.59      ( ( neg_numeral_dbl_int @ one_one_int )
% 0.30/0.59      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(3)
% 0.30/0.59  thf(fact_4202_max__enat__simps_I3_J,axiom,
% 0.30/0.59      ! [Q2: extended_enat] :
% 0.30/0.59        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.30/0.59        = Q2 ) ).
% 0.30/0.59  
% 0.30/0.59  % max_enat_simps(3)
% 0.30/0.59  thf(fact_4203_max__enat__simps_I2_J,axiom,
% 0.30/0.59      ! [Q2: extended_enat] :
% 0.30/0.59        ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.30/0.59        = Q2 ) ).
% 0.30/0.59  
% 0.30/0.59  % max_enat_simps(2)
% 0.30/0.59  thf(fact_4204_dbl__simps_I2_J,axiom,
% 0.30/0.59      ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 0.30/0.59      = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(2)
% 0.30/0.59  thf(fact_4205_dbl__simps_I2_J,axiom,
% 0.30/0.59      ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.30/0.59      = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(2)
% 0.30/0.59  thf(fact_4206_dbl__simps_I2_J,axiom,
% 0.30/0.59      ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.30/0.59      = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(2)
% 0.30/0.59  thf(fact_4207_dbl__simps_I2_J,axiom,
% 0.30/0.59      ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.30/0.59      = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(2)
% 0.30/0.59  thf(fact_4208_dbl__simps_I5_J,axiom,
% 0.30/0.59      ! [K: num] :
% 0.30/0.59        ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.30/0.59        = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(5)
% 0.30/0.59  thf(fact_4209_dbl__simps_I5_J,axiom,
% 0.30/0.59      ! [K: num] :
% 0.30/0.59        ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.30/0.59        = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(5)
% 0.30/0.59  thf(fact_4210_dbl__simps_I5_J,axiom,
% 0.30/0.59      ! [K: num] :
% 0.30/0.59        ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.59        = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(5)
% 0.30/0.59  thf(fact_4211_dbl__simps_I5_J,axiom,
% 0.30/0.59      ! [K: num] :
% 0.30/0.59        ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.59        = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_simps(5)
% 0.30/0.59  thf(fact_4212_bot__nat__def,axiom,
% 0.30/0.59      bot_bot_nat = zero_zero_nat ).
% 0.30/0.59  
% 0.30/0.59  % bot_nat_def
% 0.30/0.59  thf(fact_4213_bot__enat__def,axiom,
% 0.30/0.59      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.30/0.59  
% 0.30/0.59  % bot_enat_def
% 0.30/0.59  thf(fact_4214_dbl__def,axiom,
% 0.30/0.59      ( neg_numeral_dbl_real
% 0.30/0.59      = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_def
% 0.30/0.59  thf(fact_4215_dbl__def,axiom,
% 0.30/0.59      ( neg_numeral_dbl_rat
% 0.30/0.59      = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_def
% 0.30/0.59  thf(fact_4216_dbl__def,axiom,
% 0.30/0.59      ( neg_numeral_dbl_int
% 0.30/0.59      = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dbl_def
% 0.30/0.59  thf(fact_4217_max_OcoboundedI2,axiom,
% 0.30/0.59      ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ C @ B2 )
% 0.30/0.59       => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI2
% 0.30/0.59  thf(fact_4218_max_OcoboundedI2,axiom,
% 0.30/0.59      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ C @ B2 )
% 0.30/0.59       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI2
% 0.30/0.59  thf(fact_4219_max_OcoboundedI2,axiom,
% 0.30/0.59      ! [C: num,B2: num,A: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ C @ B2 )
% 0.30/0.59       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI2
% 0.30/0.59  thf(fact_4220_max_OcoboundedI2,axiom,
% 0.30/0.59      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ C @ B2 )
% 0.30/0.59       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI2
% 0.30/0.59  thf(fact_4221_max_OcoboundedI2,axiom,
% 0.30/0.59      ! [C: int,B2: int,A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ C @ B2 )
% 0.30/0.59       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI2
% 0.30/0.59  thf(fact_4222_max_OcoboundedI1,axiom,
% 0.30/0.59      ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.30/0.59       => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI1
% 0.30/0.59  thf(fact_4223_max_OcoboundedI1,axiom,
% 0.30/0.59      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.59       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI1
% 0.30/0.59  thf(fact_4224_max_OcoboundedI1,axiom,
% 0.30/0.59      ! [C: num,A: num,B2: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ C @ A )
% 0.30/0.59       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI1
% 0.30/0.59  thf(fact_4225_max_OcoboundedI1,axiom,
% 0.30/0.59      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.59       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI1
% 0.30/0.59  thf(fact_4226_max_OcoboundedI1,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ C @ A )
% 0.30/0.59       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.coboundedI1
% 0.30/0.59  thf(fact_4227_max_Oabsorb__iff2,axiom,
% 0.30/0.59      ( ord_le2932123472753598470d_enat
% 0.30/0.59      = ( ^ [A5: extended_enat,B4: extended_enat] :
% 0.30/0.59            ( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
% 0.30/0.59            = B4 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff2
% 0.30/0.59  thf(fact_4228_max_Oabsorb__iff2,axiom,
% 0.30/0.59      ( ord_less_eq_rat
% 0.30/0.59      = ( ^ [A5: rat,B4: rat] :
% 0.30/0.59            ( ( ord_max_rat @ A5 @ B4 )
% 0.30/0.59            = B4 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff2
% 0.30/0.59  thf(fact_4229_max_Oabsorb__iff2,axiom,
% 0.30/0.59      ( ord_less_eq_num
% 0.30/0.59      = ( ^ [A5: num,B4: num] :
% 0.30/0.59            ( ( ord_max_num @ A5 @ B4 )
% 0.30/0.59            = B4 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff2
% 0.30/0.59  thf(fact_4230_max_Oabsorb__iff2,axiom,
% 0.30/0.59      ( ord_less_eq_nat
% 0.30/0.59      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.59            ( ( ord_max_nat @ A5 @ B4 )
% 0.30/0.59            = B4 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff2
% 0.30/0.59  thf(fact_4231_max_Oabsorb__iff2,axiom,
% 0.30/0.59      ( ord_less_eq_int
% 0.30/0.59      = ( ^ [A5: int,B4: int] :
% 0.30/0.59            ( ( ord_max_int @ A5 @ B4 )
% 0.30/0.59            = B4 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff2
% 0.30/0.59  thf(fact_4232_max_Oabsorb__iff1,axiom,
% 0.30/0.59      ( ord_le2932123472753598470d_enat
% 0.30/0.59      = ( ^ [B4: extended_enat,A5: extended_enat] :
% 0.30/0.59            ( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
% 0.30/0.59            = A5 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff1
% 0.30/0.59  thf(fact_4233_max_Oabsorb__iff1,axiom,
% 0.30/0.59      ( ord_less_eq_rat
% 0.30/0.59      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.59            ( ( ord_max_rat @ A5 @ B4 )
% 0.30/0.59            = A5 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff1
% 0.30/0.59  thf(fact_4234_max_Oabsorb__iff1,axiom,
% 0.30/0.59      ( ord_less_eq_num
% 0.30/0.59      = ( ^ [B4: num,A5: num] :
% 0.30/0.59            ( ( ord_max_num @ A5 @ B4 )
% 0.30/0.59            = A5 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff1
% 0.30/0.59  thf(fact_4235_max_Oabsorb__iff1,axiom,
% 0.30/0.59      ( ord_less_eq_nat
% 0.30/0.59      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.59            ( ( ord_max_nat @ A5 @ B4 )
% 0.30/0.59            = A5 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff1
% 0.30/0.59  thf(fact_4236_max_Oabsorb__iff1,axiom,
% 0.30/0.59      ( ord_less_eq_int
% 0.30/0.59      = ( ^ [B4: int,A5: int] :
% 0.30/0.59            ( ( ord_max_int @ A5 @ B4 )
% 0.30/0.59            = A5 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.absorb_iff1
% 0.30/0.59  thf(fact_4237_le__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: extended_enat,X3: extended_enat,Y3: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_le2932123472753598470d_enat @ Z @ X3 )
% 0.30/0.59          | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_max_iff_disj
% 0.30/0.59  thf(fact_4238_le__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_eq_rat @ Z @ X3 )
% 0.30/0.59          | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_max_iff_disj
% 0.30/0.59  thf(fact_4239_le__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: num,X3: num,Y3: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_eq_num @ Z @ X3 )
% 0.30/0.59          | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_max_iff_disj
% 0.30/0.59  thf(fact_4240_le__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: nat,X3: nat,Y3: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_eq_nat @ Z @ X3 )
% 0.30/0.59          | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_max_iff_disj
% 0.30/0.59  thf(fact_4241_le__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: int,X3: int,Y3: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_eq_int @ Z @ X3 )
% 0.30/0.59          | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_max_iff_disj
% 0.30/0.59  thf(fact_4242_max_Ocobounded2,axiom,
% 0.30/0.59      ! [B2: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B2 @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded2
% 0.30/0.59  thf(fact_4243_max_Ocobounded2,axiom,
% 0.30/0.59      ! [B2: rat,A: rat] : ( ord_less_eq_rat @ B2 @ ( ord_max_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded2
% 0.30/0.59  thf(fact_4244_max_Ocobounded2,axiom,
% 0.30/0.59      ! [B2: num,A: num] : ( ord_less_eq_num @ B2 @ ( ord_max_num @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded2
% 0.30/0.59  thf(fact_4245_max_Ocobounded2,axiom,
% 0.30/0.59      ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( ord_max_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded2
% 0.30/0.59  thf(fact_4246_max_Ocobounded2,axiom,
% 0.30/0.59      ! [B2: int,A: int] : ( ord_less_eq_int @ B2 @ ( ord_max_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded2
% 0.30/0.59  thf(fact_4247_max_Ocobounded1,axiom,
% 0.30/0.59      ! [A: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded1
% 0.30/0.59  thf(fact_4248_max_Ocobounded1,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded1
% 0.30/0.59  thf(fact_4249_max_Ocobounded1,axiom,
% 0.30/0.59      ! [A: num,B2: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded1
% 0.30/0.59  thf(fact_4250_max_Ocobounded1,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded1
% 0.30/0.59  thf(fact_4251_max_Ocobounded1,axiom,
% 0.30/0.59      ! [A: int,B2: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.cobounded1
% 0.30/0.59  thf(fact_4252_max_Oorder__iff,axiom,
% 0.30/0.59      ( ord_le2932123472753598470d_enat
% 0.30/0.59      = ( ^ [B4: extended_enat,A5: extended_enat] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( ord_ma741700101516333627d_enat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.order_iff
% 0.30/0.59  thf(fact_4253_max_Oorder__iff,axiom,
% 0.30/0.59      ( ord_less_eq_rat
% 0.30/0.59      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( ord_max_rat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.order_iff
% 0.30/0.59  thf(fact_4254_max_Oorder__iff,axiom,
% 0.30/0.59      ( ord_less_eq_num
% 0.30/0.59      = ( ^ [B4: num,A5: num] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( ord_max_num @ A5 @ B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.order_iff
% 0.30/0.59  thf(fact_4255_max_Oorder__iff,axiom,
% 0.30/0.59      ( ord_less_eq_nat
% 0.30/0.59      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( ord_max_nat @ A5 @ B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.order_iff
% 0.30/0.59  thf(fact_4256_max_Oorder__iff,axiom,
% 0.30/0.59      ( ord_less_eq_int
% 0.30/0.59      = ( ^ [B4: int,A5: int] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( ord_max_int @ A5 @ B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.order_iff
% 0.30/0.59  thf(fact_4257_max_OboundedI,axiom,
% 0.30/0.59      ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 0.30/0.59       => ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.30/0.59         => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedI
% 0.30/0.59  thf(fact_4258_max_OboundedI,axiom,
% 0.30/0.59      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.59       => ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.59         => ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedI
% 0.30/0.59  thf(fact_4259_max_OboundedI,axiom,
% 0.30/0.59      ! [B2: num,A: num,C: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.59       => ( ( ord_less_eq_num @ C @ A )
% 0.30/0.59         => ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedI
% 0.30/0.59  thf(fact_4260_max_OboundedI,axiom,
% 0.30/0.59      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.59       => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.59         => ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedI
% 0.30/0.59  thf(fact_4261_max_OboundedI,axiom,
% 0.30/0.59      ! [B2: int,A: int,C: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.59       => ( ( ord_less_eq_int @ C @ A )
% 0.30/0.59         => ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedI
% 0.30/0.59  thf(fact_4262_max_OboundedE,axiom,
% 0.30/0.59      ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 0.30/0.59           => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedE
% 0.30/0.59  thf(fact_4263_max_OboundedE,axiom,
% 0.30/0.59      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedE
% 0.30/0.59  thf(fact_4264_max_OboundedE,axiom,
% 0.30/0.59      ! [B2: num,C: num,A: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedE
% 0.30/0.59  thf(fact_4265_max_OboundedE,axiom,
% 0.30/0.59      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedE
% 0.30/0.59  thf(fact_4266_max_OboundedE,axiom,
% 0.30/0.59      ! [B2: int,C: int,A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.boundedE
% 0.30/0.59  thf(fact_4267_max_OorderI,axiom,
% 0.30/0.59      ! [A: extended_enat,B2: extended_enat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( ord_ma741700101516333627d_enat @ A @ B2 ) )
% 0.30/0.59       => ( ord_le2932123472753598470d_enat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderI
% 0.30/0.59  thf(fact_4268_max_OorderI,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( ord_max_rat @ A @ B2 ) )
% 0.30/0.59       => ( ord_less_eq_rat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderI
% 0.30/0.59  thf(fact_4269_max_OorderI,axiom,
% 0.30/0.59      ! [A: num,B2: num] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( ord_max_num @ A @ B2 ) )
% 0.30/0.59       => ( ord_less_eq_num @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderI
% 0.30/0.59  thf(fact_4270_max_OorderI,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( ord_max_nat @ A @ B2 ) )
% 0.30/0.59       => ( ord_less_eq_nat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderI
% 0.30/0.59  thf(fact_4271_max_OorderI,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( ord_max_int @ A @ B2 ) )
% 0.30/0.59       => ( ord_less_eq_int @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderI
% 0.30/0.59  thf(fact_4272_max_OorderE,axiom,
% 0.30/0.59      ! [B2: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 0.30/0.59       => ( A
% 0.30/0.59          = ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderE
% 0.30/0.59  thf(fact_4273_max_OorderE,axiom,
% 0.30/0.59      ! [B2: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ B2 @ A )
% 0.30/0.59       => ( A
% 0.30/0.59          = ( ord_max_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderE
% 0.30/0.59  thf(fact_4274_max_OorderE,axiom,
% 0.30/0.59      ! [B2: num,A: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ B2 @ A )
% 0.30/0.59       => ( A
% 0.30/0.59          = ( ord_max_num @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderE
% 0.30/0.59  thf(fact_4275_max_OorderE,axiom,
% 0.30/0.59      ! [B2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ B2 @ A )
% 0.30/0.59       => ( A
% 0.30/0.59          = ( ord_max_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderE
% 0.30/0.59  thf(fact_4276_max_OorderE,axiom,
% 0.30/0.59      ! [B2: int,A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ B2 @ A )
% 0.30/0.59       => ( A
% 0.30/0.59          = ( ord_max_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.orderE
% 0.30/0.59  thf(fact_4277_max_Omono,axiom,
% 0.30/0.59      ! [C: extended_enat,A: extended_enat,D: extended_enat,B2: extended_enat] :
% 0.30/0.59        ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.30/0.59       => ( ( ord_le2932123472753598470d_enat @ D @ B2 )
% 0.30/0.59         => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.mono
% 0.30/0.59  thf(fact_4278_max_Omono,axiom,
% 0.30/0.59      ! [C: rat,A: rat,D: rat,B2: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.59       => ( ( ord_less_eq_rat @ D @ B2 )
% 0.30/0.59         => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.mono
% 0.30/0.59  thf(fact_4279_max_Omono,axiom,
% 0.30/0.59      ! [C: num,A: num,D: num,B2: num] :
% 0.30/0.59        ( ( ord_less_eq_num @ C @ A )
% 0.30/0.59       => ( ( ord_less_eq_num @ D @ B2 )
% 0.30/0.59         => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.mono
% 0.30/0.59  thf(fact_4280_max_Omono,axiom,
% 0.30/0.59      ! [C: nat,A: nat,D: nat,B2: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.59       => ( ( ord_less_eq_nat @ D @ B2 )
% 0.30/0.59         => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.mono
% 0.30/0.59  thf(fact_4281_max_Omono,axiom,
% 0.30/0.59      ! [C: int,A: int,D: int,B2: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ C @ A )
% 0.30/0.59       => ( ( ord_less_eq_int @ D @ B2 )
% 0.30/0.59         => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.mono
% 0.30/0.59  thf(fact_4282_less__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: extended_enat,X3: extended_enat,Y3: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_le72135733267957522d_enat @ Z @ X3 )
% 0.30/0.59          | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_max_iff_disj
% 0.30/0.59  thf(fact_4283_less__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.59        ( ( ord_less_real @ Z @ ( ord_max_real @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_real @ Z @ X3 )
% 0.30/0.59          | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_max_iff_disj
% 0.30/0.59  thf(fact_4284_less__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.59        ( ( ord_less_rat @ Z @ ( ord_max_rat @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_rat @ Z @ X3 )
% 0.30/0.59          | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_max_iff_disj
% 0.30/0.59  thf(fact_4285_less__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: num,X3: num,Y3: num] :
% 0.30/0.59        ( ( ord_less_num @ Z @ ( ord_max_num @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_num @ Z @ X3 )
% 0.30/0.59          | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_max_iff_disj
% 0.30/0.59  thf(fact_4286_less__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: nat,X3: nat,Y3: nat] :
% 0.30/0.59        ( ( ord_less_nat @ Z @ ( ord_max_nat @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_nat @ Z @ X3 )
% 0.30/0.59          | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_max_iff_disj
% 0.30/0.59  thf(fact_4287_less__max__iff__disj,axiom,
% 0.30/0.59      ! [Z: int,X3: int,Y3: int] :
% 0.30/0.59        ( ( ord_less_int @ Z @ ( ord_max_int @ X3 @ Y3 ) )
% 0.30/0.59        = ( ( ord_less_int @ Z @ X3 )
% 0.30/0.59          | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_max_iff_disj
% 0.30/0.59  thf(fact_4288_max_Ostrict__boundedE,axiom,
% 0.30/0.59      ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_le72135733267957522d_enat @ B2 @ A )
% 0.30/0.59           => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_boundedE
% 0.30/0.59  thf(fact_4289_max_Ostrict__boundedE,axiom,
% 0.30/0.59      ! [B2: real,C: real,A: real] :
% 0.30/0.59        ( ( ord_less_real @ ( ord_max_real @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_real @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_boundedE
% 0.30/0.59  thf(fact_4290_max_Ostrict__boundedE,axiom,
% 0.30/0.59      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( ord_max_rat @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_rat @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_boundedE
% 0.30/0.59  thf(fact_4291_max_Ostrict__boundedE,axiom,
% 0.30/0.59      ! [B2: num,C: num,A: num] :
% 0.30/0.59        ( ( ord_less_num @ ( ord_max_num @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_num @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_boundedE
% 0.30/0.59  thf(fact_4292_max_Ostrict__boundedE,axiom,
% 0.30/0.59      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_nat @ ( ord_max_nat @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_nat @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_boundedE
% 0.30/0.59  thf(fact_4293_max_Ostrict__boundedE,axiom,
% 0.30/0.59      ! [B2: int,C: int,A: int] :
% 0.30/0.59        ( ( ord_less_int @ ( ord_max_int @ B2 @ C ) @ A )
% 0.30/0.59       => ~ ( ( ord_less_int @ B2 @ A )
% 0.30/0.59           => ~ ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_boundedE
% 0.30/0.59  thf(fact_4294_max_Ostrict__order__iff,axiom,
% 0.30/0.59      ( ord_le72135733267957522d_enat
% 0.30/0.59      = ( ^ [B4: extended_enat,A5: extended_enat] :
% 0.30/0.59            ( ( A5
% 0.30/0.59              = ( ord_ma741700101516333627d_enat @ A5 @ B4 ) )
% 0.30/0.59            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_order_iff
% 0.30/0.59  thf(fact_4295_max_Ostrict__order__iff,axiom,
% 0.30/0.59      ( ord_less_real
% 0.30/0.59      = ( ^ [B4: real,A5: real] :
% 0.30/0.59            ( ( A5
% 0.30/0.59              = ( ord_max_real @ A5 @ B4 ) )
% 0.30/0.59            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_order_iff
% 0.30/0.59  thf(fact_4296_max_Ostrict__order__iff,axiom,
% 0.30/0.59      ( ord_less_rat
% 0.30/0.59      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.59            ( ( A5
% 0.30/0.59              = ( ord_max_rat @ A5 @ B4 ) )
% 0.30/0.59            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_order_iff
% 0.30/0.59  thf(fact_4297_max_Ostrict__order__iff,axiom,
% 0.30/0.59      ( ord_less_num
% 0.30/0.59      = ( ^ [B4: num,A5: num] :
% 0.30/0.59            ( ( A5
% 0.30/0.59              = ( ord_max_num @ A5 @ B4 ) )
% 0.30/0.59            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_order_iff
% 0.30/0.59  thf(fact_4298_max_Ostrict__order__iff,axiom,
% 0.30/0.59      ( ord_less_nat
% 0.30/0.59      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.59            ( ( A5
% 0.30/0.59              = ( ord_max_nat @ A5 @ B4 ) )
% 0.30/0.59            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_order_iff
% 0.30/0.59  thf(fact_4299_max_Ostrict__order__iff,axiom,
% 0.30/0.59      ( ord_less_int
% 0.30/0.59      = ( ^ [B4: int,A5: int] :
% 0.30/0.59            ( ( A5
% 0.30/0.59              = ( ord_max_int @ A5 @ B4 ) )
% 0.30/0.59            & ( A5 != B4 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_order_iff
% 0.30/0.59  thf(fact_4300_max_Ostrict__coboundedI1,axiom,
% 0.30/0.59      ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ C @ A )
% 0.30/0.59       => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI1
% 0.30/0.59  thf(fact_4301_max_Ostrict__coboundedI1,axiom,
% 0.30/0.59      ! [C: real,A: real,B2: real] :
% 0.30/0.59        ( ( ord_less_real @ C @ A )
% 0.30/0.59       => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI1
% 0.30/0.59  thf(fact_4302_max_Ostrict__coboundedI1,axiom,
% 0.30/0.59      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.59        ( ( ord_less_rat @ C @ A )
% 0.30/0.59       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI1
% 0.30/0.59  thf(fact_4303_max_Ostrict__coboundedI1,axiom,
% 0.30/0.59      ! [C: num,A: num,B2: num] :
% 0.30/0.59        ( ( ord_less_num @ C @ A )
% 0.30/0.59       => ( ord_less_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI1
% 0.30/0.59  thf(fact_4304_max_Ostrict__coboundedI1,axiom,
% 0.30/0.59      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( ord_less_nat @ C @ A )
% 0.30/0.59       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI1
% 0.30/0.59  thf(fact_4305_max_Ostrict__coboundedI1,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( ord_less_int @ C @ A )
% 0.30/0.59       => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI1
% 0.30/0.59  thf(fact_4306_max_Ostrict__coboundedI2,axiom,
% 0.30/0.59      ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
% 0.30/0.59        ( ( ord_le72135733267957522d_enat @ C @ B2 )
% 0.30/0.59       => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI2
% 0.30/0.59  thf(fact_4307_max_Ostrict__coboundedI2,axiom,
% 0.30/0.59      ! [C: real,B2: real,A: real] :
% 0.30/0.59        ( ( ord_less_real @ C @ B2 )
% 0.30/0.59       => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI2
% 0.30/0.59  thf(fact_4308_max_Ostrict__coboundedI2,axiom,
% 0.30/0.59      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ C @ B2 )
% 0.30/0.59       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI2
% 0.30/0.59  thf(fact_4309_max_Ostrict__coboundedI2,axiom,
% 0.30/0.59      ! [C: num,B2: num,A: num] :
% 0.30/0.59        ( ( ord_less_num @ C @ B2 )
% 0.30/0.59       => ( ord_less_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI2
% 0.30/0.59  thf(fact_4310_max_Ostrict__coboundedI2,axiom,
% 0.30/0.59      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_nat @ C @ B2 )
% 0.30/0.59       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI2
% 0.30/0.59  thf(fact_4311_max_Ostrict__coboundedI2,axiom,
% 0.30/0.59      ! [C: int,B2: int,A: int] :
% 0.30/0.59        ( ( ord_less_int @ C @ B2 )
% 0.30/0.59       => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % max.strict_coboundedI2
% 0.30/0.59  thf(fact_4312_even__succ__mod__exp,axiom,
% 0.30/0.59      ! [A: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_mod_exp
% 0.30/0.59  thf(fact_4313_even__succ__mod__exp,axiom,
% 0.30/0.59      ! [A: int,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_mod_exp
% 0.30/0.59  thf(fact_4314_even__succ__mod__exp,axiom,
% 0.30/0.59      ! [A: code_integer,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_mod_exp
% 0.30/0.59  thf(fact_4315_option_Osize__gen_I2_J,axiom,
% 0.30/0.59      ! [X3: nat > nat,X2: nat] :
% 0.30/0.59        ( ( size_option_nat @ X3 @ ( some_nat @ X2 ) )
% 0.30/0.59        = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % option.size_gen(2)
% 0.30/0.59  thf(fact_4316_option_Osize__gen_I2_J,axiom,
% 0.30/0.59      ! [X3: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
% 0.30/0.59        ( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X2 ) )
% 0.30/0.59        = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % option.size_gen(2)
% 0.30/0.59  thf(fact_4317_option_Osize__gen_I2_J,axiom,
% 0.30/0.59      ! [X3: num > nat,X2: num] :
% 0.30/0.59        ( ( size_option_num @ X3 @ ( some_num @ X2 ) )
% 0.30/0.59        = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % option.size_gen(2)
% 0.30/0.59  thf(fact_4318_even__succ__div__exp,axiom,
% 0.30/0.59      ! [A: code_integer,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_exp
% 0.30/0.59  thf(fact_4319_even__succ__div__exp,axiom,
% 0.30/0.59      ! [A: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_exp
% 0.30/0.59  thf(fact_4320_even__succ__div__exp,axiom,
% 0.30/0.59      ! [A: int,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.59            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_exp
% 0.30/0.59  thf(fact_4321_signed__take__bit__Suc,axiom,
% 0.30/0.59      ! [N: nat,A: code_integer] :
% 0.30/0.59        ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.30/0.59        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_Suc
% 0.30/0.59  thf(fact_4322_signed__take__bit__Suc,axiom,
% 0.30/0.59      ! [N: nat,A: int] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.30/0.59        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_Suc
% 0.30/0.59  thf(fact_4323_infinite__growing,axiom,
% 0.30/0.59      ! [X9: set_real] :
% 0.30/0.59        ( ( X9 != bot_bot_set_real )
% 0.30/0.59       => ( ! [X5: real] :
% 0.30/0.59              ( ( member_real @ X5 @ X9 )
% 0.30/0.59             => ? [Xa: real] :
% 0.30/0.59                  ( ( member_real @ Xa @ X9 )
% 0.30/0.59                  & ( ord_less_real @ X5 @ Xa ) ) )
% 0.30/0.59         => ~ ( finite_finite_real @ X9 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_growing
% 0.30/0.59  thf(fact_4324_infinite__growing,axiom,
% 0.30/0.59      ! [X9: set_rat] :
% 0.30/0.59        ( ( X9 != bot_bot_set_rat )
% 0.30/0.59       => ( ! [X5: rat] :
% 0.30/0.59              ( ( member_rat @ X5 @ X9 )
% 0.30/0.59             => ? [Xa: rat] :
% 0.30/0.59                  ( ( member_rat @ Xa @ X9 )
% 0.30/0.59                  & ( ord_less_rat @ X5 @ Xa ) ) )
% 0.30/0.59         => ~ ( finite_finite_rat @ X9 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_growing
% 0.30/0.59  thf(fact_4325_infinite__growing,axiom,
% 0.30/0.59      ! [X9: set_num] :
% 0.30/0.59        ( ( X9 != bot_bot_set_num )
% 0.30/0.59       => ( ! [X5: num] :
% 0.30/0.59              ( ( member_num @ X5 @ X9 )
% 0.30/0.59             => ? [Xa: num] :
% 0.30/0.59                  ( ( member_num @ Xa @ X9 )
% 0.30/0.59                  & ( ord_less_num @ X5 @ Xa ) ) )
% 0.30/0.59         => ~ ( finite_finite_num @ X9 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_growing
% 0.30/0.59  thf(fact_4326_infinite__growing,axiom,
% 0.30/0.59      ! [X9: set_nat] :
% 0.30/0.59        ( ( X9 != bot_bot_set_nat )
% 0.30/0.59       => ( ! [X5: nat] :
% 0.30/0.59              ( ( member_nat @ X5 @ X9 )
% 0.30/0.59             => ? [Xa: nat] :
% 0.30/0.59                  ( ( member_nat @ Xa @ X9 )
% 0.30/0.59                  & ( ord_less_nat @ X5 @ Xa ) ) )
% 0.30/0.59         => ~ ( finite_finite_nat @ X9 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_growing
% 0.30/0.59  thf(fact_4327_infinite__growing,axiom,
% 0.30/0.59      ! [X9: set_int] :
% 0.30/0.59        ( ( X9 != bot_bot_set_int )
% 0.30/0.59       => ( ! [X5: int] :
% 0.30/0.59              ( ( member_int @ X5 @ X9 )
% 0.30/0.59             => ? [Xa: int] :
% 0.30/0.59                  ( ( member_int @ Xa @ X9 )
% 0.30/0.59                  & ( ord_less_int @ X5 @ Xa ) ) )
% 0.30/0.59         => ~ ( finite_finite_int @ X9 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_growing
% 0.30/0.59  thf(fact_4328_nat__dvd__1__iff__1,axiom,
% 0.30/0.59      ! [M: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.30/0.59        = ( M = one_one_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nat_dvd_1_iff_1
% 0.30/0.59  thf(fact_4329_dvd__0__left__iff,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.30/0.59        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left_iff
% 0.30/0.59  thf(fact_4330_dvd__0__left__iff,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.30/0.59        = ( A = zero_zero_complex ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left_iff
% 0.30/0.59  thf(fact_4331_dvd__0__left__iff,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.30/0.59        = ( A = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left_iff
% 0.30/0.59  thf(fact_4332_dvd__0__left__iff,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.30/0.59        = ( A = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left_iff
% 0.30/0.59  thf(fact_4333_dvd__0__left__iff,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.59        = ( A = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left_iff
% 0.30/0.59  thf(fact_4334_dvd__0__left__iff,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.30/0.59        = ( A = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left_iff
% 0.30/0.59  thf(fact_4335_dvd__0__right,axiom,
% 0.30/0.59      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_right
% 0.30/0.59  thf(fact_4336_dvd__0__right,axiom,
% 0.30/0.59      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_right
% 0.30/0.59  thf(fact_4337_dvd__0__right,axiom,
% 0.30/0.59      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_right
% 0.30/0.59  thf(fact_4338_dvd__0__right,axiom,
% 0.30/0.59      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_right
% 0.30/0.59  thf(fact_4339_dvd__0__right,axiom,
% 0.30/0.59      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_right
% 0.30/0.59  thf(fact_4340_dvd__0__right,axiom,
% 0.30/0.59      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_right
% 0.30/0.59  thf(fact_4341_dvd__add__triv__left__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_left_iff
% 0.30/0.59  thf(fact_4342_dvd__add__triv__left__iff,axiom,
% 0.30/0.59      ! [A: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_left_iff
% 0.30/0.59  thf(fact_4343_dvd__add__triv__left__iff,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_left_iff
% 0.30/0.59  thf(fact_4344_dvd__add__triv__left__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_left_iff
% 0.30/0.59  thf(fact_4345_dvd__add__triv__left__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_left_iff
% 0.30/0.59  thf(fact_4346_dvd__add__triv__right__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ A ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_right_iff
% 0.30/0.59  thf(fact_4347_dvd__add__triv__right__iff,axiom,
% 0.30/0.59      ! [A: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 0.30/0.59        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_right_iff
% 0.30/0.59  thf(fact_4348_dvd__add__triv__right__iff,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 0.30/0.59        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_right_iff
% 0.30/0.59  thf(fact_4349_dvd__add__triv__right__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 0.30/0.59        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_right_iff
% 0.30/0.59  thf(fact_4350_dvd__add__triv__right__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 0.30/0.59        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_triv_right_iff
% 0.30/0.59  thf(fact_4351_dvd__1__iff__1,axiom,
% 0.30/0.59      ! [M: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.59        = ( M
% 0.30/0.59          = ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_1_iff_1
% 0.30/0.59  thf(fact_4352_dvd__1__left,axiom,
% 0.30/0.59      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_1_left
% 0.30/0.59  thf(fact_4353_div__dvd__div,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.59         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.30/0.59            = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_dvd_div
% 0.30/0.59  thf(fact_4354_div__dvd__div,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.59         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.30/0.59            = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_dvd_div
% 0.30/0.59  thf(fact_4355_div__dvd__div,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ A @ C )
% 0.30/0.59         => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.30/0.59            = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_dvd_div
% 0.30/0.59  thf(fact_4356_nat__mult__dvd__cancel__disj,axiom,
% 0.30/0.59      ! [K: nat,M: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.59        = ( ( K = zero_zero_nat )
% 0.30/0.59          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nat_mult_dvd_cancel_disj
% 0.30/0.59  thf(fact_4357_signed__take__bit__of__0,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_of_0
% 0.30/0.59  thf(fact_4358_dvd__times__right__cancel__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.30/0.59          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_times_right_cancel_iff
% 0.30/0.59  thf(fact_4359_dvd__times__right__cancel__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( A != zero_zero_nat )
% 0.30/0.59       => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
% 0.30/0.59          = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_times_right_cancel_iff
% 0.30/0.59  thf(fact_4360_dvd__times__right__cancel__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( A != zero_zero_int )
% 0.30/0.59       => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
% 0.30/0.59          = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_times_right_cancel_iff
% 0.30/0.59  thf(fact_4361_dvd__times__left__cancel__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.30/0.59          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_times_left_cancel_iff
% 0.30/0.59  thf(fact_4362_dvd__times__left__cancel__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( A != zero_zero_nat )
% 0.30/0.59       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
% 0.30/0.59          = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_times_left_cancel_iff
% 0.30/0.59  thf(fact_4363_dvd__times__left__cancel__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( A != zero_zero_int )
% 0.30/0.59       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
% 0.30/0.59          = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_times_left_cancel_iff
% 0.30/0.59  thf(fact_4364_dvd__mult__cancel__right,axiom,
% 0.30/0.59      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.59        = ( ( C = zero_z3403309356797280102nteger )
% 0.30/0.59          | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_right
% 0.30/0.59  thf(fact_4365_dvd__mult__cancel__right,axiom,
% 0.30/0.59      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.59        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
% 0.30/0.59        = ( ( C = zero_zero_complex )
% 0.30/0.59          | ( dvd_dvd_complex @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_right
% 0.30/0.59  thf(fact_4366_dvd__mult__cancel__right,axiom,
% 0.30/0.59      ! [A: real,C: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 0.30/0.59        = ( ( C = zero_zero_real )
% 0.30/0.59          | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_right
% 0.30/0.59  thf(fact_4367_dvd__mult__cancel__right,axiom,
% 0.30/0.59      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 0.30/0.59        = ( ( C = zero_zero_rat )
% 0.30/0.59          | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_right
% 0.30/0.59  thf(fact_4368_dvd__mult__cancel__right,axiom,
% 0.30/0.59      ! [A: int,C: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 0.30/0.59        = ( ( C = zero_zero_int )
% 0.30/0.59          | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_right
% 0.30/0.59  thf(fact_4369_dvd__mult__cancel__left,axiom,
% 0.30/0.59      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 0.30/0.59        = ( ( C = zero_z3403309356797280102nteger )
% 0.30/0.59          | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_left
% 0.30/0.59  thf(fact_4370_dvd__mult__cancel__left,axiom,
% 0.30/0.59      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.59        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 0.30/0.59        = ( ( C = zero_zero_complex )
% 0.30/0.59          | ( dvd_dvd_complex @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_left
% 0.30/0.59  thf(fact_4371_dvd__mult__cancel__left,axiom,
% 0.30/0.59      ! [C: real,A: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 0.30/0.59        = ( ( C = zero_zero_real )
% 0.30/0.59          | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_left
% 0.30/0.59  thf(fact_4372_dvd__mult__cancel__left,axiom,
% 0.30/0.59      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 0.30/0.59        = ( ( C = zero_zero_rat )
% 0.30/0.59          | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_left
% 0.30/0.59  thf(fact_4373_dvd__mult__cancel__left,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.59        = ( ( C = zero_zero_int )
% 0.30/0.59          | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_cancel_left
% 0.30/0.59  thf(fact_4374_unit__prod,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_prod
% 0.30/0.59  thf(fact_4375_unit__prod,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.59         => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_prod
% 0.30/0.59  thf(fact_4376_unit__prod,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.59         => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_prod
% 0.30/0.59  thf(fact_4377_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_right_iff
% 0.30/0.59  thf(fact_4378_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
% 0.30/0.59        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_right_iff
% 0.30/0.59  thf(fact_4379_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A ) ) )
% 0.30/0.59        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_right_iff
% 0.30/0.59  thf(fact_4380_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
% 0.30/0.59        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_right_iff
% 0.30/0.59  thf(fact_4381_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
% 0.30/0.59        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_right_iff
% 0.30/0.59  thf(fact_4382_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.59      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_left_iff
% 0.30/0.59  thf(fact_4383_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.59      ! [A: real,C: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_left_iff
% 0.30/0.59  thf(fact_4384_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.59      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_left_iff
% 0.30/0.59  thf(fact_4385_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.59      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_left_iff
% 0.30/0.59  thf(fact_4386_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.59      ! [A: int,C: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_times_triv_left_iff
% 0.30/0.59  thf(fact_4387_dvd__mult__div__cancel,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ A ) )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_div_cancel
% 0.30/0.59  thf(fact_4388_dvd__mult__div__cancel,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_div_cancel
% 0.30/0.59  thf(fact_4389_dvd__mult__div__cancel,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_div_cancel
% 0.30/0.59  thf(fact_4390_dvd__div__mult__self,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_mult_self
% 0.30/0.59  thf(fact_4391_dvd__div__mult__self,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_mult_self
% 0.30/0.59  thf(fact_4392_dvd__div__mult__self,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_mult_self
% 0.30/0.59  thf(fact_4393_unit__div__1__div__1,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59       => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_1_div_1
% 0.30/0.59  thf(fact_4394_unit__div__1__div__1,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59       => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_1_div_1
% 0.30/0.59  thf(fact_4395_unit__div__1__div__1,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59       => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_1_div_1
% 0.30/0.59  thf(fact_4396_unit__div__1__unit,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_1_unit
% 0.30/0.59  thf(fact_4397_unit__div__1__unit,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59       => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_1_unit
% 0.30/0.59  thf(fact_4398_unit__div__1__unit,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59       => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_1_unit
% 0.30/0.59  thf(fact_4399_unit__div,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div
% 0.30/0.59  thf(fact_4400_unit__div,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.59         => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div
% 0.30/0.59  thf(fact_4401_unit__div,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.59         => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div
% 0.30/0.59  thf(fact_4402_div__add,axiom,
% 0.30/0.59      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.30/0.59            = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_add
% 0.30/0.59  thf(fact_4403_div__add,axiom,
% 0.30/0.59      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.59         => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.59            = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_add
% 0.30/0.59  thf(fact_4404_div__add,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59         => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.59            = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_add
% 0.30/0.59  thf(fact_4405_div__diff,axiom,
% 0.30/0.59      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59         => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 0.30/0.59            = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_diff
% 0.30/0.59  thf(fact_4406_div__diff,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59         => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 0.30/0.59            = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_diff
% 0.30/0.59  thf(fact_4407_dvd__imp__mod__0,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( modulo_modulo_nat @ B2 @ A )
% 0.30/0.59          = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_imp_mod_0
% 0.30/0.59  thf(fact_4408_dvd__imp__mod__0,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( modulo_modulo_int @ B2 @ A )
% 0.30/0.59          = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_imp_mod_0
% 0.30/0.59  thf(fact_4409_dvd__imp__mod__0,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( modulo364778990260209775nteger @ B2 @ A )
% 0.30/0.59          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_imp_mod_0
% 0.30/0.59  thf(fact_4410_signed__take__bit__Suc__1,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.30/0.59        = one_one_int ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_Suc_1
% 0.30/0.59  thf(fact_4411_signed__take__bit__numeral__of__1,axiom,
% 0.30/0.59      ! [K: num] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.30/0.59        = one_one_int ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_numeral_of_1
% 0.30/0.59  thf(fact_4412_unit__mult__div__div,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59       => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.30/0.59          = ( divide6298287555418463151nteger @ B2 @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_mult_div_div
% 0.30/0.59  thf(fact_4413_unit__mult__div__div,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59       => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.30/0.59          = ( divide_divide_nat @ B2 @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_mult_div_div
% 0.30/0.59  thf(fact_4414_unit__mult__div__div,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59       => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
% 0.30/0.59          = ( divide_divide_int @ B2 @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_mult_div_div
% 0.30/0.59  thf(fact_4415_unit__div__mult__self,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.59       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_mult_self
% 0.30/0.59  thf(fact_4416_unit__div__mult__self,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.59       => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_mult_self
% 0.30/0.59  thf(fact_4417_unit__div__mult__self,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.59       => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 0.30/0.59          = B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_div_mult_self
% 0.30/0.59  thf(fact_4418_even__Suc__Suc__iff,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.30/0.59        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_Suc_Suc_iff
% 0.30/0.59  thf(fact_4419_even__Suc,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_Suc
% 0.30/0.59  thf(fact_4420_pow__divides__pow__iff,axiom,
% 0.30/0.59      ! [N: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 0.30/0.59          = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % pow_divides_pow_iff
% 0.30/0.59  thf(fact_4421_pow__divides__pow__iff,axiom,
% 0.30/0.59      ! [N: nat,A: int,B2: int] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.59       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 0.30/0.59          = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % pow_divides_pow_iff
% 0.30/0.59  thf(fact_4422_even__mult__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.30/0.59        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mult_iff
% 0.30/0.59  thf(fact_4423_even__mult__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
% 0.30/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mult_iff
% 0.30/0.59  thf(fact_4424_even__mult__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
% 0.30/0.59        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mult_iff
% 0.30/0.59  thf(fact_4425_even__add,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.30/0.59        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_add
% 0.30/0.59  thf(fact_4426_even__add,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_add
% 0.30/0.59  thf(fact_4427_even__add,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.59        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_add
% 0.30/0.59  thf(fact_4428_odd__add,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) )
% 0.30/0.59        = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59         != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_add
% 0.30/0.59  thf(fact_4429_odd__add,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
% 0.30/0.59        = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59         != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_add
% 0.30/0.59  thf(fact_4430_odd__add,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
% 0.30/0.59        = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.59         != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_add
% 0.30/0.59  thf(fact_4431_even__mod__2__iff,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.59        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mod_2_iff
% 0.30/0.59  thf(fact_4432_even__mod__2__iff,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.59        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mod_2_iff
% 0.30/0.59  thf(fact_4433_even__mod__2__iff,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mod_2_iff
% 0.30/0.59  thf(fact_4434_even__Suc__div__two,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_Suc_div_two
% 0.30/0.59  thf(fact_4435_odd__Suc__div__two,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_Suc_div_two
% 0.30/0.59  thf(fact_4436_signed__take__bit__Suc__bit0,axiom,
% 0.30/0.59      ! [N: nat,K: num] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.30/0.59        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_Suc_bit0
% 0.30/0.59  thf(fact_4437_zero__le__power__eq__numeral,axiom,
% 0.30/0.59      ! [A: real,W: num] :
% 0.30/0.59        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_le_power_eq_numeral
% 0.30/0.59  thf(fact_4438_zero__le__power__eq__numeral,axiom,
% 0.30/0.59      ! [A: rat,W: num] :
% 0.30/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_le_power_eq_numeral
% 0.30/0.59  thf(fact_4439_zero__le__power__eq__numeral,axiom,
% 0.30/0.59      ! [A: int,W: num] :
% 0.30/0.59        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_le_power_eq_numeral
% 0.30/0.59  thf(fact_4440_power__less__zero__eq__numeral,axiom,
% 0.30/0.59      ! [A: real,W: num] :
% 0.30/0.59        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_less_zero_eq_numeral
% 0.30/0.59  thf(fact_4441_power__less__zero__eq__numeral,axiom,
% 0.30/0.59      ! [A: rat,W: num] :
% 0.30/0.59        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_less_zero_eq_numeral
% 0.30/0.59  thf(fact_4442_power__less__zero__eq__numeral,axiom,
% 0.30/0.59      ! [A: int,W: num] :
% 0.30/0.59        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_less_zero_eq_numeral
% 0.30/0.59  thf(fact_4443_power__less__zero__eq,axiom,
% 0.30/0.59      ! [A: real,N: nat] :
% 0.30/0.59        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_less_zero_eq
% 0.30/0.59  thf(fact_4444_power__less__zero__eq,axiom,
% 0.30/0.59      ! [A: rat,N: nat] :
% 0.30/0.59        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_less_zero_eq
% 0.30/0.59  thf(fact_4445_power__less__zero__eq,axiom,
% 0.30/0.59      ! [A: int,N: nat] :
% 0.30/0.59        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_less_zero_eq
% 0.30/0.59  thf(fact_4446_even__plus__one__iff,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.30/0.59        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_plus_one_iff
% 0.30/0.59  thf(fact_4447_even__plus__one__iff,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.30/0.59        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_plus_one_iff
% 0.30/0.59  thf(fact_4448_even__plus__one__iff,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.30/0.59        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_plus_one_iff
% 0.30/0.59  thf(fact_4449_even__diff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_diff
% 0.30/0.59  thf(fact_4450_even__diff,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_diff
% 0.30/0.59  thf(fact_4451_odd__Suc__minus__one,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.59          = N ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_Suc_minus_one
% 0.30/0.59  thf(fact_4452_even__diff__nat,axiom,
% 0.30/0.59      ! [M: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.59        = ( ( ord_less_nat @ M @ N )
% 0.30/0.59          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_diff_nat
% 0.30/0.59  thf(fact_4453_zero__less__power__eq__numeral,axiom,
% 0.30/0.59      ! [A: real,W: num] :
% 0.30/0.59        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.59        = ( ( ( numeral_numeral_nat @ W )
% 0.30/0.59            = zero_zero_nat )
% 0.30/0.59          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( A != zero_zero_real ) )
% 0.30/0.59          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_less_power_eq_numeral
% 0.30/0.59  thf(fact_4454_zero__less__power__eq__numeral,axiom,
% 0.30/0.59      ! [A: rat,W: num] :
% 0.30/0.59        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.59        = ( ( ( numeral_numeral_nat @ W )
% 0.30/0.59            = zero_zero_nat )
% 0.30/0.59          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( A != zero_zero_rat ) )
% 0.30/0.59          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_less_power_eq_numeral
% 0.30/0.59  thf(fact_4455_zero__less__power__eq__numeral,axiom,
% 0.30/0.59      ! [A: int,W: num] :
% 0.30/0.59        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.30/0.59        = ( ( ( numeral_numeral_nat @ W )
% 0.30/0.59            = zero_zero_nat )
% 0.30/0.59          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( A != zero_zero_int ) )
% 0.30/0.59          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_less_power_eq_numeral
% 0.30/0.59  thf(fact_4456_even__succ__div__2,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_2
% 0.30/0.59  thf(fact_4457_even__succ__div__2,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_2
% 0.30/0.59  thf(fact_4458_even__succ__div__2,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_2
% 0.30/0.59  thf(fact_4459_even__succ__div__two,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_two
% 0.30/0.59  thf(fact_4460_even__succ__div__two,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_two
% 0.30/0.59  thf(fact_4461_even__succ__div__two,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_succ_div_two
% 0.30/0.59  thf(fact_4462_odd__succ__div__two,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_succ_div_two
% 0.30/0.59  thf(fact_4463_odd__succ__div__two,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_succ_div_two
% 0.30/0.59  thf(fact_4464_odd__succ__div__two,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59          = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_succ_div_two
% 0.30/0.59  thf(fact_4465_even__power,axiom,
% 0.30/0.59      ! [A: code_integer,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.30/0.59        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_power
% 0.30/0.59  thf(fact_4466_even__power,axiom,
% 0.30/0.59      ! [A: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.30/0.59        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_power
% 0.30/0.59  thf(fact_4467_even__power,axiom,
% 0.30/0.59      ! [A: int,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.30/0.59        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_power
% 0.30/0.59  thf(fact_4468_odd__two__times__div__two__nat,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.59       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.59          = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_two_times_div_two_nat
% 0.30/0.59  thf(fact_4469_odd__two__times__div__two__succ,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_two_times_div_two_succ
% 0.30/0.59  thf(fact_4470_odd__two__times__div__two__succ,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_two_times_div_two_succ
% 0.30/0.59  thf(fact_4471_odd__two__times__div__two__succ,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.59       => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % odd_two_times_div_two_succ
% 0.30/0.59  thf(fact_4472_power__le__zero__eq__numeral,axiom,
% 0.30/0.59      ! [A: real,W: num] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.30/0.59        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.59            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_le_zero_eq_numeral
% 0.30/0.59  thf(fact_4473_power__le__zero__eq__numeral,axiom,
% 0.30/0.59      ! [A: rat,W: num] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.30/0.59        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.59            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_le_zero_eq_numeral
% 0.30/0.59  thf(fact_4474_power__le__zero__eq__numeral,axiom,
% 0.30/0.59      ! [A: int,W: num] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.30/0.59        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.59            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.59              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_le_zero_eq_numeral
% 0.30/0.59  thf(fact_4475_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.30/0.59        = ( N = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_parity_class.even_mask_iff
% 0.30/0.59  thf(fact_4476_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.30/0.59        = ( N = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_parity_class.even_mask_iff
% 0.30/0.59  thf(fact_4477_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.59      ! [N: nat] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.30/0.59        = ( N = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_parity_class.even_mask_iff
% 0.30/0.59  thf(fact_4478_dvd__trans,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ B2 @ C )
% 0.30/0.59         => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_trans
% 0.30/0.59  thf(fact_4479_dvd__trans,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ B2 @ C )
% 0.30/0.59         => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_trans
% 0.30/0.59  thf(fact_4480_dvd__trans,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ B2 @ C )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_trans
% 0.30/0.59  thf(fact_4481_dvd__refl,axiom,
% 0.30/0.59      ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_refl
% 0.30/0.59  thf(fact_4482_dvd__refl,axiom,
% 0.30/0.59      ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_refl
% 0.30/0.59  thf(fact_4483_dvd__refl,axiom,
% 0.30/0.59      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_refl
% 0.30/0.59  thf(fact_4484_dvd__0__left,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.30/0.59       => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left
% 0.30/0.59  thf(fact_4485_dvd__0__left,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.30/0.59       => ( A = zero_zero_complex ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left
% 0.30/0.59  thf(fact_4486_dvd__0__left,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.30/0.59       => ( A = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left
% 0.30/0.59  thf(fact_4487_dvd__0__left,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.30/0.59       => ( A = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left
% 0.30/0.59  thf(fact_4488_dvd__0__left,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.59       => ( A = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left
% 0.30/0.59  thf(fact_4489_dvd__0__left,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.30/0.59       => ( A = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_0_left
% 0.30/0.59  thf(fact_4490_dvd__productE,axiom,
% 0.30/0.59      ! [P4: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B2 ) )
% 0.30/0.59       => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.59              ( ( P4
% 0.30/0.59                = ( times_times_nat @ X5 @ Y4 ) )
% 0.30/0.59             => ( ( dvd_dvd_nat @ X5 @ A )
% 0.30/0.59               => ~ ( dvd_dvd_nat @ Y4 @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_productE
% 0.30/0.59  thf(fact_4491_dvd__productE,axiom,
% 0.30/0.59      ! [P4: int,A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B2 ) )
% 0.30/0.59       => ~ ! [X5: int,Y4: int] :
% 0.30/0.59              ( ( P4
% 0.30/0.59                = ( times_times_int @ X5 @ Y4 ) )
% 0.30/0.59             => ( ( dvd_dvd_int @ X5 @ A )
% 0.30/0.59               => ~ ( dvd_dvd_int @ Y4 @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_productE
% 0.30/0.59  thf(fact_4492_division__decomp,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.59       => ? [B8: nat,C5: nat] :
% 0.30/0.59            ( ( A
% 0.30/0.59              = ( times_times_nat @ B8 @ C5 ) )
% 0.30/0.59            & ( dvd_dvd_nat @ B8 @ B2 )
% 0.30/0.59            & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % division_decomp
% 0.30/0.59  thf(fact_4493_division__decomp,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.59       => ? [B8: int,C5: int] :
% 0.30/0.59            ( ( A
% 0.30/0.59              = ( times_times_int @ B8 @ C5 ) )
% 0.30/0.59            & ( dvd_dvd_int @ B8 @ B2 )
% 0.30/0.59            & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % division_decomp
% 0.30/0.59  thf(fact_4494_dvd__triv__right,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_right
% 0.30/0.59  thf(fact_4495_dvd__triv__right,axiom,
% 0.30/0.59      ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_right
% 0.30/0.59  thf(fact_4496_dvd__triv__right,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_right
% 0.30/0.59  thf(fact_4497_dvd__triv__right,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_right
% 0.30/0.59  thf(fact_4498_dvd__triv__right,axiom,
% 0.30/0.59      ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_right
% 0.30/0.59  thf(fact_4499_dvd__mult__right,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_right
% 0.30/0.59  thf(fact_4500_dvd__mult__right,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_real @ B2 @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_right
% 0.30/0.59  thf(fact_4501_dvd__mult__right,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_rat @ B2 @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_right
% 0.30/0.59  thf(fact_4502_dvd__mult__right,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_nat @ B2 @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_right
% 0.30/0.59  thf(fact_4503_dvd__mult__right,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_int @ B2 @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_right
% 0.30/0.59  thf(fact_4504_mult__dvd__mono,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_dvd_mono
% 0.30/0.59  thf(fact_4505_mult__dvd__mono,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_real @ C @ D )
% 0.30/0.59         => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_dvd_mono
% 0.30/0.59  thf(fact_4506_mult__dvd__mono,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_rat @ C @ D )
% 0.30/0.59         => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_dvd_mono
% 0.30/0.59  thf(fact_4507_mult__dvd__mono,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ C @ D )
% 0.30/0.59         => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_dvd_mono
% 0.30/0.59  thf(fact_4508_mult__dvd__mono,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ D )
% 0.30/0.59         => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_dvd_mono
% 0.30/0.59  thf(fact_4509_dvd__triv__left,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_left
% 0.30/0.59  thf(fact_4510_dvd__triv__left,axiom,
% 0.30/0.59      ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_left
% 0.30/0.59  thf(fact_4511_dvd__triv__left,axiom,
% 0.30/0.59      ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_left
% 0.30/0.59  thf(fact_4512_dvd__triv__left,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_left
% 0.30/0.59  thf(fact_4513_dvd__triv__left,axiom,
% 0.30/0.59      ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_triv_left
% 0.30/0.59  thf(fact_4514_dvd__mult__left,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_left
% 0.30/0.59  thf(fact_4515_dvd__mult__left,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_real @ A @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_left
% 0.30/0.59  thf(fact_4516_dvd__mult__left,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_left
% 0.30/0.59  thf(fact_4517_dvd__mult__left,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_left
% 0.30/0.59  thf(fact_4518_dvd__mult__left,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.59       => ( dvd_dvd_int @ A @ C ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult_left
% 0.30/0.59  thf(fact_4519_dvd__mult2,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult2
% 0.30/0.59  thf(fact_4520_dvd__mult2,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ B2 )
% 0.30/0.59       => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult2
% 0.30/0.59  thf(fact_4521_dvd__mult2,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ B2 )
% 0.30/0.59       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult2
% 0.30/0.59  thf(fact_4522_dvd__mult2,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult2
% 0.30/0.59  thf(fact_4523_dvd__mult2,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult2
% 0.30/0.59  thf(fact_4524_dvd__mult,axiom,
% 0.30/0.59      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult
% 0.30/0.59  thf(fact_4525_dvd__mult,axiom,
% 0.30/0.59      ! [A: real,C: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ C )
% 0.30/0.59       => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult
% 0.30/0.59  thf(fact_4526_dvd__mult,axiom,
% 0.30/0.59      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.59       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult
% 0.30/0.59  thf(fact_4527_dvd__mult,axiom,
% 0.30/0.59      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.59       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult
% 0.30/0.59  thf(fact_4528_dvd__mult,axiom,
% 0.30/0.59      ! [A: int,C: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ C )
% 0.30/0.59       => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mult
% 0.30/0.59  thf(fact_4529_dvd__def,axiom,
% 0.30/0.59      ( dvd_dvd_Code_integer
% 0.30/0.59      = ( ^ [B4: code_integer,A5: code_integer] :
% 0.30/0.59          ? [K2: code_integer] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( times_3573771949741848930nteger @ B4 @ K2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_def
% 0.30/0.59  thf(fact_4530_dvd__def,axiom,
% 0.30/0.59      ( dvd_dvd_real
% 0.30/0.59      = ( ^ [B4: real,A5: real] :
% 0.30/0.59          ? [K2: real] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( times_times_real @ B4 @ K2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_def
% 0.30/0.59  thf(fact_4531_dvd__def,axiom,
% 0.30/0.59      ( dvd_dvd_rat
% 0.30/0.59      = ( ^ [B4: rat,A5: rat] :
% 0.30/0.59          ? [K2: rat] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( times_times_rat @ B4 @ K2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_def
% 0.30/0.59  thf(fact_4532_dvd__def,axiom,
% 0.30/0.59      ( dvd_dvd_nat
% 0.30/0.59      = ( ^ [B4: nat,A5: nat] :
% 0.30/0.59          ? [K2: nat] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( times_times_nat @ B4 @ K2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_def
% 0.30/0.59  thf(fact_4533_dvd__def,axiom,
% 0.30/0.59      ( dvd_dvd_int
% 0.30/0.59      = ( ^ [B4: int,A5: int] :
% 0.30/0.59          ? [K2: int] :
% 0.30/0.59            ( A5
% 0.30/0.59            = ( times_times_int @ B4 @ K2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_def
% 0.30/0.59  thf(fact_4534_dvdI,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,K: code_integer] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( times_3573771949741848930nteger @ B2 @ K ) )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdI
% 0.30/0.59  thf(fact_4535_dvdI,axiom,
% 0.30/0.59      ! [A: real,B2: real,K: real] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( times_times_real @ B2 @ K ) )
% 0.30/0.59       => ( dvd_dvd_real @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdI
% 0.30/0.59  thf(fact_4536_dvdI,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,K: rat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( times_times_rat @ B2 @ K ) )
% 0.30/0.59       => ( dvd_dvd_rat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdI
% 0.30/0.59  thf(fact_4537_dvdI,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,K: nat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( times_times_nat @ B2 @ K ) )
% 0.30/0.59       => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdI
% 0.30/0.59  thf(fact_4538_dvdI,axiom,
% 0.30/0.59      ! [A: int,B2: int,K: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( times_times_int @ B2 @ K ) )
% 0.30/0.59       => ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdI
% 0.30/0.59  thf(fact_4539_dvdE,axiom,
% 0.30/0.59      ! [B2: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.59       => ~ ! [K3: code_integer] :
% 0.30/0.59              ( A
% 0.30/0.59             != ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdE
% 0.30/0.59  thf(fact_4540_dvdE,axiom,
% 0.30/0.59      ! [B2: real,A: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ B2 @ A )
% 0.30/0.59       => ~ ! [K3: real] :
% 0.30/0.59              ( A
% 0.30/0.59             != ( times_times_real @ B2 @ K3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdE
% 0.30/0.59  thf(fact_4541_dvdE,axiom,
% 0.30/0.59      ! [B2: rat,A: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ B2 @ A )
% 0.30/0.59       => ~ ! [K3: rat] :
% 0.30/0.59              ( A
% 0.30/0.59             != ( times_times_rat @ B2 @ K3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdE
% 0.30/0.59  thf(fact_4542_dvdE,axiom,
% 0.30/0.59      ! [B2: nat,A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.59       => ~ ! [K3: nat] :
% 0.30/0.59              ( A
% 0.30/0.59             != ( times_times_nat @ B2 @ K3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdE
% 0.30/0.59  thf(fact_4543_dvdE,axiom,
% 0.30/0.59      ! [B2: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.59       => ~ ! [K3: int] :
% 0.30/0.59              ( A
% 0.30/0.59             != ( times_times_int @ B2 @ K3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvdE
% 0.30/0.59  thf(fact_4544_dvd__unit__imp__unit,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_unit_imp_unit
% 0.30/0.59  thf(fact_4545_dvd__unit__imp__unit,axiom,
% 0.30/0.59      ! [A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.59         => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_unit_imp_unit
% 0.30/0.59  thf(fact_4546_dvd__unit__imp__unit,axiom,
% 0.30/0.59      ! [A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.59         => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_unit_imp_unit
% 0.30/0.59  thf(fact_4547_unit__imp__dvd,axiom,
% 0.30/0.59      ! [B2: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_imp_dvd
% 0.30/0.59  thf(fact_4548_unit__imp__dvd,axiom,
% 0.30/0.59      ! [B2: nat,A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.59       => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_imp_dvd
% 0.30/0.59  thf(fact_4549_unit__imp__dvd,axiom,
% 0.30/0.59      ! [B2: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.59       => ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % unit_imp_dvd
% 0.30/0.59  thf(fact_4550_one__dvd,axiom,
% 0.30/0.59      ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % one_dvd
% 0.30/0.59  thf(fact_4551_one__dvd,axiom,
% 0.30/0.59      ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % one_dvd
% 0.30/0.59  thf(fact_4552_one__dvd,axiom,
% 0.30/0.59      ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % one_dvd
% 0.30/0.59  thf(fact_4553_one__dvd,axiom,
% 0.30/0.59      ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % one_dvd
% 0.30/0.59  thf(fact_4554_one__dvd,axiom,
% 0.30/0.59      ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % one_dvd
% 0.30/0.59  thf(fact_4555_one__dvd,axiom,
% 0.30/0.59      ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.30/0.59  
% 0.30/0.59  % one_dvd
% 0.30/0.59  thf(fact_4556_dvd__add,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add
% 0.30/0.59  thf(fact_4557_dvd__add,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_real @ A @ C )
% 0.30/0.59         => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add
% 0.30/0.59  thf(fact_4558_dvd__add,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.59         => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add
% 0.30/0.59  thf(fact_4559_dvd__add,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.59         => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add
% 0.30/0.59  thf(fact_4560_dvd__add,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ A @ C )
% 0.30/0.59         => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add
% 0.30/0.59  thf(fact_4561_dvd__add__left__iff,axiom,
% 0.30/0.59      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_left_iff
% 0.30/0.59  thf(fact_4562_dvd__add__left__iff,axiom,
% 0.30/0.59      ! [A: real,C: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ C )
% 0.30/0.59       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_left_iff
% 0.30/0.59  thf(fact_4563_dvd__add__left__iff,axiom,
% 0.30/0.59      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.59       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_left_iff
% 0.30/0.59  thf(fact_4564_dvd__add__left__iff,axiom,
% 0.30/0.59      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.59       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_left_iff
% 0.30/0.59  thf(fact_4565_dvd__add__left__iff,axiom,
% 0.30/0.59      ! [A: int,C: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ C )
% 0.30/0.59       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_left_iff
% 0.30/0.59  thf(fact_4566_dvd__add__right__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_right_iff
% 0.30/0.59  thf(fact_4567_dvd__add__right__iff,axiom,
% 0.30/0.59      ! [A: real,B2: real,C: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_right_iff
% 0.30/0.59  thf(fact_4568_dvd__add__right__iff,axiom,
% 0.30/0.59      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_right_iff
% 0.30/0.59  thf(fact_4569_dvd__add__right__iff,axiom,
% 0.30/0.59      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_right_iff
% 0.30/0.59  thf(fact_4570_dvd__add__right__iff,axiom,
% 0.30/0.59      ! [A: int,B2: int,C: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 0.30/0.59          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_add_right_iff
% 0.30/0.59  thf(fact_4571_dvd__diff,axiom,
% 0.30/0.59      ! [X3: code_integer,Y3: code_integer,Z: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ X3 @ Z )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ X3 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff
% 0.30/0.59  thf(fact_4572_dvd__diff,axiom,
% 0.30/0.59      ! [X3: complex,Y3: complex,Z: complex] :
% 0.30/0.59        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 0.30/0.59       => ( ( dvd_dvd_complex @ X3 @ Z )
% 0.30/0.59         => ( dvd_dvd_complex @ X3 @ ( minus_minus_complex @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff
% 0.30/0.59  thf(fact_4573_dvd__diff,axiom,
% 0.30/0.59      ! [X3: real,Y3: real,Z: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ X3 @ Y3 )
% 0.30/0.59       => ( ( dvd_dvd_real @ X3 @ Z )
% 0.30/0.59         => ( dvd_dvd_real @ X3 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff
% 0.30/0.59  thf(fact_4574_dvd__diff,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat,Z: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ X3 @ Y3 )
% 0.30/0.59       => ( ( dvd_dvd_rat @ X3 @ Z )
% 0.30/0.59         => ( dvd_dvd_rat @ X3 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff
% 0.30/0.59  thf(fact_4575_dvd__diff,axiom,
% 0.30/0.59      ! [X3: int,Y3: int,Z: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ X3 @ Y3 )
% 0.30/0.59       => ( ( dvd_dvd_int @ X3 @ Z )
% 0.30/0.59         => ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff
% 0.30/0.59  thf(fact_4576_dvd__diff__commute,axiom,
% 0.30/0.59      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff_commute
% 0.30/0.59  thf(fact_4577_dvd__diff__commute,axiom,
% 0.30/0.59      ! [A: int,C: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 0.30/0.59        = ( dvd_dvd_int @ A @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff_commute
% 0.30/0.59  thf(fact_4578_div__div__div__same,axiom,
% 0.30/0.59      ! [D: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ D @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.59         => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
% 0.30/0.59            = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_div_div_same
% 0.30/0.59  thf(fact_4579_div__div__div__same,axiom,
% 0.30/0.59      ! [D: nat,B2: nat,A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ D @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.59         => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
% 0.30/0.59            = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_div_div_same
% 0.30/0.59  thf(fact_4580_div__div__div__same,axiom,
% 0.30/0.59      ! [D: int,B2: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ D @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.59         => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
% 0.30/0.59            = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_div_div_same
% 0.30/0.59  thf(fact_4581_dvd__div__eq__cancel,axiom,
% 0.30/0.59      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.30/0.59          = ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59         => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59           => ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_cancel
% 0.30/0.59  thf(fact_4582_dvd__div__eq__cancel,axiom,
% 0.30/0.59      ! [A: complex,C: complex,B2: complex] :
% 0.30/0.59        ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.59          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.59       => ( ( dvd_dvd_complex @ C @ A )
% 0.30/0.59         => ( ( dvd_dvd_complex @ C @ B2 )
% 0.30/0.59           => ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_cancel
% 0.30/0.59  thf(fact_4583_dvd__div__eq__cancel,axiom,
% 0.30/0.59      ! [A: real,C: real,B2: real] :
% 0.30/0.59        ( ( ( divide_divide_real @ A @ C )
% 0.30/0.59          = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.59       => ( ( dvd_dvd_real @ C @ A )
% 0.30/0.59         => ( ( dvd_dvd_real @ C @ B2 )
% 0.30/0.59           => ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_cancel
% 0.30/0.59  thf(fact_4584_dvd__div__eq__cancel,axiom,
% 0.30/0.59      ! [A: rat,C: rat,B2: rat] :
% 0.30/0.59        ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.59          = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.59       => ( ( dvd_dvd_rat @ C @ A )
% 0.30/0.59         => ( ( dvd_dvd_rat @ C @ B2 )
% 0.30/0.59           => ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_cancel
% 0.30/0.59  thf(fact_4585_dvd__div__eq__cancel,axiom,
% 0.30/0.59      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.59        ( ( ( divide_divide_nat @ A @ C )
% 0.30/0.59          = ( divide_divide_nat @ B2 @ C ) )
% 0.30/0.59       => ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.59         => ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.59           => ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_cancel
% 0.30/0.59  thf(fact_4586_dvd__div__eq__cancel,axiom,
% 0.30/0.59      ! [A: int,C: int,B2: int] :
% 0.30/0.59        ( ( ( divide_divide_int @ A @ C )
% 0.30/0.59          = ( divide_divide_int @ B2 @ C ) )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59         => ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59           => ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_cancel
% 0.30/0.59  thf(fact_4587_dvd__div__eq__iff,axiom,
% 0.30/0.59      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59         => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.30/0.59              = ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.30/0.59            = ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_iff
% 0.30/0.59  thf(fact_4588_dvd__div__eq__iff,axiom,
% 0.30/0.59      ! [C: complex,A: complex,B2: complex] :
% 0.30/0.59        ( ( dvd_dvd_complex @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_complex @ C @ B2 )
% 0.30/0.59         => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.59              = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.59            = ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_iff
% 0.30/0.59  thf(fact_4589_dvd__div__eq__iff,axiom,
% 0.30/0.59      ! [C: real,A: real,B2: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_real @ C @ B2 )
% 0.30/0.59         => ( ( ( divide_divide_real @ A @ C )
% 0.30/0.59              = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.59            = ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_iff
% 0.30/0.59  thf(fact_4590_dvd__div__eq__iff,axiom,
% 0.30/0.59      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_rat @ C @ B2 )
% 0.30/0.59         => ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.59              = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.59            = ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_iff
% 0.30/0.59  thf(fact_4591_dvd__div__eq__iff,axiom,
% 0.30/0.59      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.59         => ( ( ( divide_divide_nat @ A @ C )
% 0.30/0.59              = ( divide_divide_nat @ B2 @ C ) )
% 0.30/0.59            = ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_iff
% 0.30/0.59  thf(fact_4592_dvd__div__eq__iff,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59         => ( ( ( divide_divide_int @ A @ C )
% 0.30/0.59              = ( divide_divide_int @ B2 @ C ) )
% 0.30/0.59            = ( A = B2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_eq_iff
% 0.30/0.59  thf(fact_4593_dvd__power__same,axiom,
% 0.30/0.59      ! [X3: code_integer,Y3: code_integer,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 0.30/0.59       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_power_same
% 0.30/0.59  thf(fact_4594_dvd__power__same,axiom,
% 0.30/0.59      ! [X3: nat,Y3: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 0.30/0.59       => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_power_same
% 0.30/0.59  thf(fact_4595_dvd__power__same,axiom,
% 0.30/0.59      ! [X3: real,Y3: real,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_real @ X3 @ Y3 )
% 0.30/0.59       => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_power_same
% 0.30/0.59  thf(fact_4596_dvd__power__same,axiom,
% 0.30/0.59      ! [X3: int,Y3: int,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_int @ X3 @ Y3 )
% 0.30/0.59       => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_power_same
% 0.30/0.59  thf(fact_4597_dvd__power__same,axiom,
% 0.30/0.59      ! [X3: complex,Y3: complex,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 0.30/0.59       => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_power_same
% 0.30/0.59  thf(fact_4598_mod__mod__cancel,axiom,
% 0.30/0.59      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.59       => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
% 0.30/0.59          = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mod_mod_cancel
% 0.30/0.59  thf(fact_4599_mod__mod__cancel,axiom,
% 0.30/0.59      ! [C: int,B2: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59       => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
% 0.30/0.59          = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mod_mod_cancel
% 0.30/0.59  thf(fact_4600_mod__mod__cancel,axiom,
% 0.30/0.59      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59       => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C )
% 0.30/0.59          = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mod_mod_cancel
% 0.30/0.59  thf(fact_4601_dvd__mod,axiom,
% 0.30/0.59      ! [K: nat,M: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.59       => ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.59         => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod
% 0.30/0.59  thf(fact_4602_dvd__mod,axiom,
% 0.30/0.59      ! [K: int,M: int,N: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ K @ M )
% 0.30/0.59       => ( ( dvd_dvd_int @ K @ N )
% 0.30/0.59         => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod
% 0.30/0.59  thf(fact_4603_dvd__mod,axiom,
% 0.30/0.59      ! [K: code_integer,M: code_integer,N: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ K @ M )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ K @ N )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod
% 0.30/0.59  thf(fact_4604_dvd__mod__imp__dvd,axiom,
% 0.30/0.59      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.59       => ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.59         => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod_imp_dvd
% 0.30/0.59  thf(fact_4605_dvd__mod__imp__dvd,axiom,
% 0.30/0.59      ! [C: int,A: int,B2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59         => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod_imp_dvd
% 0.30/0.59  thf(fact_4606_dvd__mod__imp__dvd,axiom,
% 0.30/0.59      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59         => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod_imp_dvd
% 0.30/0.59  thf(fact_4607_dvd__mod__iff,axiom,
% 0.30/0.59      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 0.30/0.59          = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod_iff
% 0.30/0.59  thf(fact_4608_dvd__mod__iff,axiom,
% 0.30/0.59      ! [C: int,B2: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 0.30/0.59          = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod_iff
% 0.30/0.59  thf(fact_4609_dvd__mod__iff,axiom,
% 0.30/0.59      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.59       => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 0.30/0.59          = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_mod_iff
% 0.30/0.59  thf(fact_4610_signed__take__bit__mult,axiom,
% 0.30/0.59      ! [N: nat,K: int,L: int] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.30/0.59        = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_mult
% 0.30/0.59  thf(fact_4611_signed__take__bit__add,axiom,
% 0.30/0.59      ! [N: nat,K: int,L: int] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.30/0.59        = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_add
% 0.30/0.59  thf(fact_4612_dvd__diff__nat,axiom,
% 0.30/0.59      ! [K: nat,M: nat,N: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.59       => ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.59         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_diff_nat
% 0.30/0.59  thf(fact_4613_signed__take__bit__diff,axiom,
% 0.30/0.59      ! [N: nat,K: int,L: int] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 0.30/0.59        = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_diff
% 0.30/0.59  thf(fact_4614_subset__divisors__dvd,axiom,
% 0.30/0.59      ! [A: complex,B2: complex] :
% 0.30/0.59        ( ( ord_le211207098394363844omplex
% 0.30/0.59          @ ( collect_complex
% 0.30/0.59            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 0.30/0.59          @ ( collect_complex
% 0.30/0.59            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B2 ) ) )
% 0.30/0.59        = ( dvd_dvd_complex @ A @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_divisors_dvd
% 0.30/0.59  thf(fact_4615_subset__divisors__dvd,axiom,
% 0.30/0.59      ! [A: real,B2: real] :
% 0.30/0.59        ( ( ord_less_eq_set_real
% 0.30/0.60          @ ( collect_real
% 0.30/0.60            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 0.30/0.60          @ ( collect_real
% 0.30/0.60            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
% 0.30/0.60        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_divisors_dvd
% 0.30/0.60  thf(fact_4616_subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( ord_less_eq_set_nat
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
% 0.30/0.60        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_divisors_dvd
% 0.30/0.60  thf(fact_4617_subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( ord_le7084787975880047091nteger
% 0.30/0.60          @ ( collect_Code_integer
% 0.30/0.60            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 0.30/0.60          @ ( collect_Code_integer
% 0.30/0.60            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B2 ) ) )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_divisors_dvd
% 0.30/0.60  thf(fact_4618_subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( ord_less_eq_set_int
% 0.30/0.60          @ ( collect_int
% 0.30/0.60            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 0.30/0.60          @ ( collect_int
% 0.30/0.60            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
% 0.30/0.60        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_divisors_dvd
% 0.30/0.60  thf(fact_4619_strict__subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( ord_less_set_complex
% 0.30/0.60          @ ( collect_complex
% 0.30/0.60            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 0.30/0.60          @ ( collect_complex
% 0.30/0.60            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B2 ) ) )
% 0.30/0.60        = ( ( dvd_dvd_complex @ A @ B2 )
% 0.30/0.60          & ~ ( dvd_dvd_complex @ B2 @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % strict_subset_divisors_dvd
% 0.30/0.60  thf(fact_4620_strict__subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( ord_less_set_real
% 0.30/0.60          @ ( collect_real
% 0.30/0.60            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 0.30/0.60          @ ( collect_real
% 0.30/0.60            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
% 0.30/0.60        = ( ( dvd_dvd_real @ A @ B2 )
% 0.30/0.60          & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % strict_subset_divisors_dvd
% 0.30/0.60  thf(fact_4621_strict__subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( ord_less_set_nat
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.60          & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % strict_subset_divisors_dvd
% 0.30/0.60  thf(fact_4622_strict__subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( ord_less_set_int
% 0.30/0.60          @ ( collect_int
% 0.30/0.60            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 0.30/0.60          @ ( collect_int
% 0.30/0.60            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
% 0.30/0.60        = ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.60          & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % strict_subset_divisors_dvd
% 0.30/0.60  thf(fact_4623_strict__subset__divisors__dvd,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( ord_le1307284697595431911nteger
% 0.30/0.60          @ ( collect_Code_integer
% 0.30/0.60            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 0.30/0.60          @ ( collect_Code_integer
% 0.30/0.60            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B2 ) ) )
% 0.30/0.60        = ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.60          & ~ ( dvd_dvd_Code_integer @ B2 @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % strict_subset_divisors_dvd
% 0.30/0.60  thf(fact_4624_even__signed__take__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_signed_take_bit_iff
% 0.30/0.60  thf(fact_4625_even__signed__take__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.30/0.60        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_signed_take_bit_iff
% 0.30/0.60  thf(fact_4626_not__is__unit__0,axiom,
% 0.30/0.60      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.30/0.60  
% 0.30/0.60  % not_is_unit_0
% 0.30/0.60  thf(fact_4627_not__is__unit__0,axiom,
% 0.30/0.60      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % not_is_unit_0
% 0.30/0.60  thf(fact_4628_not__is__unit__0,axiom,
% 0.30/0.60      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.30/0.60  
% 0.30/0.60  % not_is_unit_0
% 0.30/0.60  thf(fact_4629_pinf_I9_J,axiom,
% 0.30/0.60      ! [D: code_integer,S: code_integer] :
% 0.30/0.60      ? [Z4: code_integer] :
% 0.30/0.60      ! [X6: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ Z4 @ X6 )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(9)
% 0.30/0.60  thf(fact_4630_pinf_I9_J,axiom,
% 0.30/0.60      ! [D: real,S: real] :
% 0.30/0.60      ? [Z4: real] :
% 0.30/0.60      ! [X6: real] :
% 0.30/0.60        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.60       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(9)
% 0.30/0.60  thf(fact_4631_pinf_I9_J,axiom,
% 0.30/0.60      ! [D: rat,S: rat] :
% 0.30/0.60      ? [Z4: rat] :
% 0.30/0.60      ! [X6: rat] :
% 0.30/0.60        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.60       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(9)
% 0.30/0.60  thf(fact_4632_pinf_I9_J,axiom,
% 0.30/0.60      ! [D: nat,S: nat] :
% 0.30/0.60      ? [Z4: nat] :
% 0.30/0.60      ! [X6: nat] :
% 0.30/0.60        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.60       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(9)
% 0.30/0.60  thf(fact_4633_pinf_I9_J,axiom,
% 0.30/0.60      ! [D: int,S: int] :
% 0.30/0.60      ? [Z4: int] :
% 0.30/0.60      ! [X6: int] :
% 0.30/0.60        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.60       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(9)
% 0.30/0.60  thf(fact_4634_pinf_I10_J,axiom,
% 0.30/0.60      ! [D: code_integer,S: code_integer] :
% 0.30/0.60      ? [Z4: code_integer] :
% 0.30/0.60      ! [X6: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ Z4 @ X6 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(10)
% 0.30/0.60  thf(fact_4635_pinf_I10_J,axiom,
% 0.30/0.60      ! [D: real,S: real] :
% 0.30/0.60      ? [Z4: real] :
% 0.30/0.60      ! [X6: real] :
% 0.30/0.60        ( ( ord_less_real @ Z4 @ X6 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(10)
% 0.30/0.60  thf(fact_4636_pinf_I10_J,axiom,
% 0.30/0.60      ! [D: rat,S: rat] :
% 0.30/0.60      ? [Z4: rat] :
% 0.30/0.60      ! [X6: rat] :
% 0.30/0.60        ( ( ord_less_rat @ Z4 @ X6 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(10)
% 0.30/0.60  thf(fact_4637_pinf_I10_J,axiom,
% 0.30/0.60      ! [D: nat,S: nat] :
% 0.30/0.60      ? [Z4: nat] :
% 0.30/0.60      ! [X6: nat] :
% 0.30/0.60        ( ( ord_less_nat @ Z4 @ X6 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(10)
% 0.30/0.60  thf(fact_4638_pinf_I10_J,axiom,
% 0.30/0.60      ! [D: int,S: int] :
% 0.30/0.60      ? [Z4: int] :
% 0.30/0.60      ! [X6: int] :
% 0.30/0.60        ( ( ord_less_int @ Z4 @ X6 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pinf(10)
% 0.30/0.60  thf(fact_4639_minf_I9_J,axiom,
% 0.30/0.60      ! [D: code_integer,S: code_integer] :
% 0.30/0.60      ? [Z4: code_integer] :
% 0.30/0.60      ! [X6: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ X6 @ Z4 )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(9)
% 0.30/0.60  thf(fact_4640_minf_I9_J,axiom,
% 0.30/0.60      ! [D: real,S: real] :
% 0.30/0.60      ? [Z4: real] :
% 0.30/0.60      ! [X6: real] :
% 0.30/0.60        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.60       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(9)
% 0.30/0.60  thf(fact_4641_minf_I9_J,axiom,
% 0.30/0.60      ! [D: rat,S: rat] :
% 0.30/0.60      ? [Z4: rat] :
% 0.30/0.60      ! [X6: rat] :
% 0.30/0.60        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.60       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(9)
% 0.30/0.60  thf(fact_4642_minf_I9_J,axiom,
% 0.30/0.60      ! [D: nat,S: nat] :
% 0.30/0.60      ? [Z4: nat] :
% 0.30/0.60      ! [X6: nat] :
% 0.30/0.60        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.60       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(9)
% 0.30/0.60  thf(fact_4643_minf_I9_J,axiom,
% 0.30/0.60      ! [D: int,S: int] :
% 0.30/0.60      ? [Z4: int] :
% 0.30/0.60      ! [X6: int] :
% 0.30/0.60        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.60       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) )
% 0.30/0.60          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(9)
% 0.30/0.60  thf(fact_4644_minf_I10_J,axiom,
% 0.30/0.60      ! [D: code_integer,S: code_integer] :
% 0.30/0.60      ? [Z4: code_integer] :
% 0.30/0.60      ! [X6: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ X6 @ Z4 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(10)
% 0.30/0.60  thf(fact_4645_minf_I10_J,axiom,
% 0.30/0.60      ! [D: real,S: real] :
% 0.30/0.60      ? [Z4: real] :
% 0.30/0.60      ! [X6: real] :
% 0.30/0.60        ( ( ord_less_real @ X6 @ Z4 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(10)
% 0.30/0.60  thf(fact_4646_minf_I10_J,axiom,
% 0.30/0.60      ! [D: rat,S: rat] :
% 0.30/0.60      ? [Z4: rat] :
% 0.30/0.60      ! [X6: rat] :
% 0.30/0.60        ( ( ord_less_rat @ X6 @ Z4 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(10)
% 0.30/0.60  thf(fact_4647_minf_I10_J,axiom,
% 0.30/0.60      ! [D: nat,S: nat] :
% 0.30/0.60      ? [Z4: nat] :
% 0.30/0.60      ! [X6: nat] :
% 0.30/0.60        ( ( ord_less_nat @ X6 @ Z4 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(10)
% 0.30/0.60  thf(fact_4648_minf_I10_J,axiom,
% 0.30/0.60      ! [D: int,S: int] :
% 0.30/0.60      ? [Z4: int] :
% 0.30/0.60      ! [X6: int] :
% 0.30/0.60        ( ( ord_less_int @ X6 @ Z4 )
% 0.30/0.60       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) )
% 0.30/0.60          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minf(10)
% 0.30/0.60  thf(fact_4649_dvd__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.60       => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.30/0.60            = zero_z3403309356797280102nteger )
% 0.30/0.60          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_0_iff
% 0.30/0.60  thf(fact_4650_dvd__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: complex,A: complex] :
% 0.30/0.60        ( ( dvd_dvd_complex @ B2 @ A )
% 0.30/0.60       => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.30/0.60            = zero_zero_complex )
% 0.30/0.60          = ( A = zero_zero_complex ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_0_iff
% 0.30/0.60  thf(fact_4651_dvd__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: real,A: real] :
% 0.30/0.60        ( ( dvd_dvd_real @ B2 @ A )
% 0.30/0.60       => ( ( ( divide_divide_real @ A @ B2 )
% 0.30/0.60            = zero_zero_real )
% 0.30/0.60          = ( A = zero_zero_real ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_0_iff
% 0.30/0.60  thf(fact_4652_dvd__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: rat,A: rat] :
% 0.30/0.60        ( ( dvd_dvd_rat @ B2 @ A )
% 0.30/0.60       => ( ( ( divide_divide_rat @ A @ B2 )
% 0.30/0.60            = zero_zero_rat )
% 0.30/0.60          = ( A = zero_zero_rat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_0_iff
% 0.30/0.60  thf(fact_4653_dvd__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.60       => ( ( ( divide_divide_nat @ A @ B2 )
% 0.30/0.60            = zero_zero_nat )
% 0.30/0.60          = ( A = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_0_iff
% 0.30/0.60  thf(fact_4654_dvd__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.60       => ( ( ( divide_divide_int @ A @ B2 )
% 0.30/0.60            = zero_zero_int )
% 0.30/0.60          = ( A = zero_zero_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_0_iff
% 0.30/0.60  thf(fact_4655_unit__mult__right__cancel,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60       => ( ( ( times_3573771949741848930nteger @ B2 @ A )
% 0.30/0.60            = ( times_3573771949741848930nteger @ C @ A ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_mult_right_cancel
% 0.30/0.60  thf(fact_4656_unit__mult__right__cancel,axiom,
% 0.30/0.60      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60       => ( ( ( times_times_nat @ B2 @ A )
% 0.30/0.60            = ( times_times_nat @ C @ A ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_mult_right_cancel
% 0.30/0.60  thf(fact_4657_unit__mult__right__cancel,axiom,
% 0.30/0.60      ! [A: int,B2: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60       => ( ( ( times_times_int @ B2 @ A )
% 0.30/0.60            = ( times_times_int @ C @ A ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_mult_right_cancel
% 0.30/0.60  thf(fact_4658_unit__mult__left__cancel,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60       => ( ( ( times_3573771949741848930nteger @ A @ B2 )
% 0.30/0.60            = ( times_3573771949741848930nteger @ A @ C ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_mult_left_cancel
% 0.30/0.60  thf(fact_4659_unit__mult__left__cancel,axiom,
% 0.30/0.60      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60       => ( ( ( times_times_nat @ A @ B2 )
% 0.30/0.60            = ( times_times_nat @ A @ C ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_mult_left_cancel
% 0.30/0.60  thf(fact_4660_unit__mult__left__cancel,axiom,
% 0.30/0.60      ! [A: int,B2: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60       => ( ( ( times_times_int @ A @ B2 )
% 0.30/0.60            = ( times_times_int @ A @ C ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_mult_left_cancel
% 0.30/0.60  thf(fact_4661_mult__unit__dvd__iff_H,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_unit_dvd_iff'
% 0.30/0.60  thf(fact_4662_mult__unit__dvd__iff_H,axiom,
% 0.30/0.60      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_unit_dvd_iff'
% 0.30/0.60  thf(fact_4663_mult__unit__dvd__iff_H,axiom,
% 0.30/0.60      ! [A: int,B2: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_unit_dvd_iff'
% 0.30/0.60  thf(fact_4664_dvd__mult__unit__iff_H,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_unit_iff'
% 0.30/0.60  thf(fact_4665_dvd__mult__unit__iff_H,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.60          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_unit_iff'
% 0.30/0.60  thf(fact_4666_dvd__mult__unit__iff_H,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.60          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_unit_iff'
% 0.30/0.60  thf(fact_4667_mult__unit__dvd__iff,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_unit_dvd_iff
% 0.30/0.60  thf(fact_4668_mult__unit__dvd__iff,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_unit_dvd_iff
% 0.30/0.60  thf(fact_4669_mult__unit__dvd__iff,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_unit_dvd_iff
% 0.30/0.60  thf(fact_4670_dvd__mult__unit__iff,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_unit_iff
% 0.30/0.60  thf(fact_4671_dvd__mult__unit__iff,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
% 0.30/0.60          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_unit_iff
% 0.30/0.60  thf(fact_4672_dvd__mult__unit__iff,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
% 0.30/0.60          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_unit_iff
% 0.30/0.60  thf(fact_4673_is__unit__mult__iff,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer )
% 0.30/0.60        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60          & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_mult_iff
% 0.30/0.60  thf(fact_4674_is__unit__mult__iff,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
% 0.30/0.60        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60          & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_mult_iff
% 0.30/0.60  thf(fact_4675_is__unit__mult__iff,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
% 0.30/0.60        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60          & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_mult_iff
% 0.30/0.60  thf(fact_4676_div__mult__div__if__dvd,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.30/0.60         => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.30/0.60            = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_div_if_dvd
% 0.30/0.60  thf(fact_4677_div__mult__div__if__dvd,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,D: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.60       => ( ( dvd_dvd_nat @ D @ C )
% 0.30/0.60         => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
% 0.30/0.60            = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_div_if_dvd
% 0.30/0.60  thf(fact_4678_div__mult__div__if__dvd,axiom,
% 0.30/0.60      ! [B2: int,A: int,D: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.60       => ( ( dvd_dvd_int @ D @ C )
% 0.30/0.60         => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
% 0.30/0.60            = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_div_if_dvd
% 0.30/0.60  thf(fact_4679_dvd__mult__imp__div,axiom,
% 0.30/0.60      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 )
% 0.30/0.60       => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_imp_div
% 0.30/0.60  thf(fact_4680_dvd__mult__imp__div,axiom,
% 0.30/0.60      ! [A: nat,C: nat,B2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
% 0.30/0.60       => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_imp_div
% 0.30/0.60  thf(fact_4681_dvd__mult__imp__div,axiom,
% 0.30/0.60      ! [A: int,C: int,B2: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
% 0.30/0.60       => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_imp_div
% 0.30/0.60  thf(fact_4682_dvd__div__mult2__eq,axiom,
% 0.30/0.60      ! [B2: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A )
% 0.30/0.60       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.60          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_mult2_eq
% 0.30/0.60  thf(fact_4683_dvd__div__mult2__eq,axiom,
% 0.30/0.60      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
% 0.30/0.60       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.60          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_mult2_eq
% 0.30/0.60  thf(fact_4684_dvd__div__mult2__eq,axiom,
% 0.30/0.60      ! [B2: int,C: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
% 0.30/0.60       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.60          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_mult2_eq
% 0.30/0.60  thf(fact_4685_div__div__eq__right,axiom,
% 0.30/0.60      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.60         => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.30/0.60            = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_div_eq_right
% 0.30/0.60  thf(fact_4686_div__div__eq__right,axiom,
% 0.30/0.60      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.60       => ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.60         => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.30/0.60            = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_div_eq_right
% 0.30/0.60  thf(fact_4687_div__div__eq__right,axiom,
% 0.30/0.60      ! [C: int,B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.60       => ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.60         => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.30/0.60            = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_div_eq_right
% 0.30/0.60  thf(fact_4688_div__mult__swap,axiom,
% 0.30/0.60      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.60       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.30/0.60          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_swap
% 0.30/0.60  thf(fact_4689_div__mult__swap,axiom,
% 0.30/0.60      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.60       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.30/0.60          = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_swap
% 0.30/0.60  thf(fact_4690_div__mult__swap,axiom,
% 0.30/0.60      ! [C: int,B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.60       => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.30/0.60          = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_swap
% 0.30/0.60  thf(fact_4691_dvd__div__mult,axiom,
% 0.30/0.60      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.60       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A )
% 0.30/0.60          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_mult
% 0.30/0.60  thf(fact_4692_dvd__div__mult,axiom,
% 0.30/0.60      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.60       => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
% 0.30/0.60          = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_mult
% 0.30/0.60  thf(fact_4693_dvd__div__mult,axiom,
% 0.30/0.60      ! [C: int,B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.60       => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
% 0.30/0.60          = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_mult
% 0.30/0.60  thf(fact_4694_dvd__div__unit__iff,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B2 ) )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_unit_iff
% 0.30/0.60  thf(fact_4695_dvd__div__unit__iff,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
% 0.30/0.60          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_unit_iff
% 0.30/0.60  thf(fact_4696_dvd__div__unit__iff,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
% 0.30/0.60          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_unit_iff
% 0.30/0.60  thf(fact_4697_div__unit__dvd__iff,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_unit_dvd_iff
% 0.30/0.60  thf(fact_4698_div__unit__dvd__iff,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_unit_dvd_iff
% 0.30/0.60  thf(fact_4699_div__unit__dvd__iff,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 0.30/0.60          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_unit_dvd_iff
% 0.30/0.60  thf(fact_4700_unit__div__cancel,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60       => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 0.30/0.60            = ( divide6298287555418463151nteger @ C @ A ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_cancel
% 0.30/0.60  thf(fact_4701_unit__div__cancel,axiom,
% 0.30/0.60      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60       => ( ( ( divide_divide_nat @ B2 @ A )
% 0.30/0.60            = ( divide_divide_nat @ C @ A ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_cancel
% 0.30/0.60  thf(fact_4702_unit__div__cancel,axiom,
% 0.30/0.60      ! [A: int,B2: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60       => ( ( ( divide_divide_int @ B2 @ A )
% 0.30/0.60            = ( divide_divide_int @ C @ A ) )
% 0.30/0.60          = ( B2 = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_cancel
% 0.30/0.60  thf(fact_4703_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.60      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.60       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.30/0.60          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_plus_div_distrib_dvd_left
% 0.30/0.60  thf(fact_4704_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.60      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.60       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.60          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_plus_div_distrib_dvd_left
% 0.30/0.60  thf(fact_4705_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.60      ! [C: int,A: int,B2: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.60       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.60          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_plus_div_distrib_dvd_left
% 0.30/0.60  thf(fact_4706_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.60      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.60       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 0.30/0.60          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_plus_div_distrib_dvd_right
% 0.30/0.60  thf(fact_4707_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.60      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.60       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 0.30/0.60          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_plus_div_distrib_dvd_right
% 0.30/0.60  thf(fact_4708_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.60      ! [C: int,B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.60       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 0.30/0.60          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_plus_div_distrib_dvd_right
% 0.30/0.60  thf(fact_4709_div__power,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.60       => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ N )
% 0.30/0.60          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_power
% 0.30/0.60  thf(fact_4710_div__power,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.60       => ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
% 0.30/0.60          = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_power
% 0.30/0.60  thf(fact_4711_div__power,axiom,
% 0.30/0.60      ! [B2: int,A: int,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.60       => ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
% 0.30/0.60          = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_power
% 0.30/0.60  thf(fact_4712_mod__eq__0__iff__dvd,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ A @ B2 )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60        = ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_eq_0_iff_dvd
% 0.30/0.60  thf(fact_4713_mod__eq__0__iff__dvd,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_eq_0_iff_dvd
% 0.30/0.60  thf(fact_4714_mod__eq__0__iff__dvd,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.30/0.60          = zero_z3403309356797280102nteger )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_eq_0_iff_dvd
% 0.30/0.60  thf(fact_4715_dvd__eq__mod__eq__0,axiom,
% 0.30/0.60      ( dvd_dvd_nat
% 0.30/0.60      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.60            ( ( modulo_modulo_nat @ B4 @ A5 )
% 0.30/0.60            = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_eq_mod_eq_0
% 0.30/0.60  thf(fact_4716_dvd__eq__mod__eq__0,axiom,
% 0.30/0.60      ( dvd_dvd_int
% 0.30/0.60      = ( ^ [A5: int,B4: int] :
% 0.30/0.60            ( ( modulo_modulo_int @ B4 @ A5 )
% 0.30/0.60            = zero_zero_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_eq_mod_eq_0
% 0.30/0.60  thf(fact_4717_dvd__eq__mod__eq__0,axiom,
% 0.30/0.60      ( dvd_dvd_Code_integer
% 0.30/0.60      = ( ^ [A5: code_integer,B4: code_integer] :
% 0.30/0.60            ( ( modulo364778990260209775nteger @ B4 @ A5 )
% 0.30/0.60            = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_eq_mod_eq_0
% 0.30/0.60  thf(fact_4718_mod__0__imp__dvd,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ A @ B2 )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60       => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_0_imp_dvd
% 0.30/0.60  thf(fact_4719_mod__0__imp__dvd,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60       => ( dvd_dvd_int @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_0_imp_dvd
% 0.30/0.60  thf(fact_4720_mod__0__imp__dvd,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.30/0.60          = zero_z3403309356797280102nteger )
% 0.30/0.60       => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_0_imp_dvd
% 0.30/0.60  thf(fact_4721_dvd__power__le,axiom,
% 0.30/0.60      ! [X3: code_integer,Y3: code_integer,N: nat,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_le
% 0.30/0.60  thf(fact_4722_dvd__power__le,axiom,
% 0.30/0.60      ! [X3: nat,Y3: nat,N: nat,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60         => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_le
% 0.30/0.60  thf(fact_4723_dvd__power__le,axiom,
% 0.30/0.60      ! [X3: real,Y3: real,N: nat,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_real @ X3 @ Y3 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60         => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_le
% 0.30/0.60  thf(fact_4724_dvd__power__le,axiom,
% 0.30/0.60      ! [X3: int,Y3: int,N: nat,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ X3 @ Y3 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60         => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_le
% 0.30/0.60  thf(fact_4725_dvd__power__le,axiom,
% 0.30/0.60      ! [X3: complex,Y3: complex,N: nat,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60         => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_le
% 0.30/0.60  thf(fact_4726_power__le__dvd,axiom,
% 0.30/0.60      ! [A: code_integer,N: nat,B2: code_integer,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B2 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_dvd
% 0.30/0.60  thf(fact_4727_power__le__dvd,axiom,
% 0.30/0.60      ! [A: nat,N: nat,B2: nat,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_dvd
% 0.30/0.60  thf(fact_4728_power__le__dvd,axiom,
% 0.30/0.60      ! [A: real,N: nat,B2: real,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B2 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_dvd
% 0.30/0.60  thf(fact_4729_power__le__dvd,axiom,
% 0.30/0.60      ! [A: int,N: nat,B2: int,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_dvd
% 0.30/0.60  thf(fact_4730_power__le__dvd,axiom,
% 0.30/0.60      ! [A: complex,N: nat,B2: complex,M: nat] :
% 0.30/0.60        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B2 )
% 0.30/0.60       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_dvd
% 0.30/0.60  thf(fact_4731_le__imp__power__dvd,axiom,
% 0.30/0.60      ! [M: nat,N: nat,A: code_integer] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_imp_power_dvd
% 0.30/0.60  thf(fact_4732_le__imp__power__dvd,axiom,
% 0.30/0.60      ! [M: nat,N: nat,A: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_imp_power_dvd
% 0.30/0.60  thf(fact_4733_le__imp__power__dvd,axiom,
% 0.30/0.60      ! [M: nat,N: nat,A: real] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_imp_power_dvd
% 0.30/0.60  thf(fact_4734_le__imp__power__dvd,axiom,
% 0.30/0.60      ! [M: nat,N: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_imp_power_dvd
% 0.30/0.60  thf(fact_4735_le__imp__power__dvd,axiom,
% 0.30/0.60      ! [M: nat,N: nat,A: complex] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_imp_power_dvd
% 0.30/0.60  thf(fact_4736_mod__eq__dvd__iff,axiom,
% 0.30/0.60      ! [A: int,C: int,B2: int] :
% 0.30/0.60        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.60          = ( modulo_modulo_int @ B2 @ C ) )
% 0.30/0.60        = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_eq_dvd_iff
% 0.30/0.60  thf(fact_4737_mod__eq__dvd__iff,axiom,
% 0.30/0.60      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.60          = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_eq_dvd_iff
% 0.30/0.60  thf(fact_4738_dvd__minus__mod,axiom,
% 0.30/0.60      ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_mod
% 0.30/0.60  thf(fact_4739_dvd__minus__mod,axiom,
% 0.30/0.60      ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_mod
% 0.30/0.60  thf(fact_4740_dvd__minus__mod,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_mod
% 0.30/0.60  thf(fact_4741_dvd__pos__nat,axiom,
% 0.30/0.60      ! [N: nat,M: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60       => ( ( dvd_dvd_nat @ M @ N )
% 0.30/0.60         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_pos_nat
% 0.30/0.60  thf(fact_4742_nat__dvd__not__less,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.60       => ( ( ord_less_nat @ M @ N )
% 0.30/0.60         => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % nat_dvd_not_less
% 0.30/0.60  thf(fact_4743_dvd__minus__self,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.60        = ( ( ord_less_nat @ N @ M )
% 0.30/0.60          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_self
% 0.30/0.60  thf(fact_4744_zdvd__antisym__nonneg,axiom,
% 0.30/0.60      ! [M: int,N: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.30/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.30/0.60         => ( ( dvd_dvd_int @ M @ N )
% 0.30/0.60           => ( ( dvd_dvd_int @ N @ M )
% 0.30/0.60             => ( M = N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_antisym_nonneg
% 0.30/0.60  thf(fact_4745_less__eq__dvd__minus,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60       => ( ( dvd_dvd_nat @ M @ N )
% 0.30/0.60          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_eq_dvd_minus
% 0.30/0.60  thf(fact_4746_dvd__diffD1,axiom,
% 0.30/0.60      ! [K: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.60       => ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.60         => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60           => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_diffD1
% 0.30/0.60  thf(fact_4747_dvd__diffD,axiom,
% 0.30/0.60      ! [K: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.60       => ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.60         => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_diffD
% 0.30/0.60  thf(fact_4748_zdvd__not__zless,axiom,
% 0.30/0.60      ! [M: int,N: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.60       => ( ( ord_less_int @ M @ N )
% 0.30/0.60         => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_not_zless
% 0.30/0.60  thf(fact_4749_zdvd__mono,axiom,
% 0.30/0.60      ! [K: int,M: int,T: int] :
% 0.30/0.60        ( ( K != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ M @ T )
% 0.30/0.60          = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_mono
% 0.30/0.60  thf(fact_4750_zdvd__mult__cancel,axiom,
% 0.30/0.60      ! [K: int,M: int,N: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.30/0.60       => ( ( K != zero_zero_int )
% 0.30/0.60         => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_mult_cancel
% 0.30/0.60  thf(fact_4751_bezout__add__nat,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60      ? [D3: nat,X5: nat,Y4: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ D3 @ A )
% 0.30/0.60        & ( dvd_dvd_nat @ D3 @ B2 )
% 0.30/0.60        & ( ( ( times_times_nat @ A @ X5 )
% 0.30/0.60            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) )
% 0.30/0.60          | ( ( times_times_nat @ B2 @ X5 )
% 0.30/0.60            = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bezout_add_nat
% 0.30/0.60  thf(fact_4752_bezout__lemma__nat,axiom,
% 0.30/0.60      ! [D: nat,A: nat,B2: nat,X3: nat,Y3: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.60       => ( ( dvd_dvd_nat @ D @ B2 )
% 0.30/0.60         => ( ( ( ( times_times_nat @ A @ X3 )
% 0.30/0.60                = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D ) )
% 0.30/0.60              | ( ( times_times_nat @ B2 @ X3 )
% 0.30/0.60                = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 0.30/0.60           => ? [X5: nat,Y4: nat] :
% 0.30/0.60                ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.60                & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.60                & ( ( ( times_times_nat @ A @ X5 )
% 0.30/0.60                    = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y4 ) @ D ) )
% 0.30/0.60                  | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X5 )
% 0.30/0.60                    = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bezout_lemma_nat
% 0.30/0.60  thf(fact_4753_bezout1__nat,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60      ? [D3: nat,X5: nat,Y4: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ D3 @ A )
% 0.30/0.60        & ( dvd_dvd_nat @ D3 @ B2 )
% 0.30/0.60        & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 0.30/0.60            = D3 )
% 0.30/0.60          | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.30/0.60            = D3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bezout1_nat
% 0.30/0.60  thf(fact_4754_zdvd__period,axiom,
% 0.30/0.60      ! [A: int,D: int,X3: int,T: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ D )
% 0.30/0.60       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.60          = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_period
% 0.30/0.60  thf(fact_4755_zdvd__reduce,axiom,
% 0.30/0.60      ! [K: int,N: int,M: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.30/0.60        = ( dvd_dvd_int @ K @ N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_reduce
% 0.30/0.60  thf(fact_4756_div2__even__ext__nat,axiom,
% 0.30/0.60      ! [X3: nat,Y3: nat] :
% 0.30/0.60        ( ( ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60          = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.60       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.60            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 0.30/0.60         => ( X3 = Y3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div2_even_ext_nat
% 0.30/0.60  thf(fact_4757_unit__dvdE,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.60           => ! [C2: code_integer] :
% 0.30/0.60                ( B2
% 0.30/0.60               != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_dvdE
% 0.30/0.60  thf(fact_4758_unit__dvdE,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60       => ~ ( ( A != zero_zero_nat )
% 0.30/0.60           => ! [C2: nat] :
% 0.30/0.60                ( B2
% 0.30/0.60               != ( times_times_nat @ A @ C2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_dvdE
% 0.30/0.60  thf(fact_4759_unit__dvdE,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60       => ~ ( ( A != zero_zero_int )
% 0.30/0.60           => ! [C2: int] :
% 0.30/0.60                ( B2
% 0.30/0.60               != ( times_times_int @ A @ C2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_dvdE
% 0.30/0.60  thf(fact_4760_unity__coeff__ex,axiom,
% 0.30/0.60      ! [P: code_integer > $o,L: code_integer] :
% 0.30/0.60        ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
% 0.30/0.60        = ( ? [X: code_integer] :
% 0.30/0.60              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 0.30/0.60              & ( P @ X ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unity_coeff_ex
% 0.30/0.60  thf(fact_4761_unity__coeff__ex,axiom,
% 0.30/0.60      ! [P: complex > $o,L: complex] :
% 0.30/0.60        ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
% 0.30/0.60        = ( ? [X: complex] :
% 0.30/0.60              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
% 0.30/0.60              & ( P @ X ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unity_coeff_ex
% 0.30/0.60  thf(fact_4762_unity__coeff__ex,axiom,
% 0.30/0.60      ! [P: real > $o,L: real] :
% 0.30/0.60        ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
% 0.30/0.60        = ( ? [X: real] :
% 0.30/0.60              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
% 0.30/0.60              & ( P @ X ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unity_coeff_ex
% 0.30/0.60  thf(fact_4763_unity__coeff__ex,axiom,
% 0.30/0.60      ! [P: rat > $o,L: rat] :
% 0.30/0.60        ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
% 0.30/0.60        = ( ? [X: rat] :
% 0.30/0.60              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 0.30/0.60              & ( P @ X ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unity_coeff_ex
% 0.30/0.60  thf(fact_4764_unity__coeff__ex,axiom,
% 0.30/0.60      ! [P: nat > $o,L: nat] :
% 0.30/0.60        ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
% 0.30/0.60        = ( ? [X: nat] :
% 0.30/0.60              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 0.30/0.60              & ( P @ X ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unity_coeff_ex
% 0.30/0.60  thf(fact_4765_unity__coeff__ex,axiom,
% 0.30/0.60      ! [P: int > $o,L: int] :
% 0.30/0.60        ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
% 0.30/0.60        = ( ? [X: int] :
% 0.30/0.60              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
% 0.30/0.60              & ( P @ X ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unity_coeff_ex
% 0.30/0.60  thf(fact_4766_dvd__div__eq__mult,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 0.30/0.60        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.60         => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 0.30/0.60              = C )
% 0.30/0.60            = ( B2
% 0.30/0.60              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_mult
% 0.30/0.60  thf(fact_4767_dvd__div__eq__mult,axiom,
% 0.30/0.60      ! [A: nat,B2: nat,C: nat] :
% 0.30/0.60        ( ( A != zero_zero_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.60         => ( ( ( divide_divide_nat @ B2 @ A )
% 0.30/0.60              = C )
% 0.30/0.60            = ( B2
% 0.30/0.60              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_mult
% 0.30/0.60  thf(fact_4768_dvd__div__eq__mult,axiom,
% 0.30/0.60      ! [A: int,B2: int,C: int] :
% 0.30/0.60        ( ( A != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.60         => ( ( ( divide_divide_int @ B2 @ A )
% 0.30/0.60              = C )
% 0.30/0.60            = ( B2
% 0.30/0.60              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_eq_mult
% 0.30/0.60  thf(fact_4769_div__dvd__iff__mult,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( B2 != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.60         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 0.30/0.60            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_dvd_iff_mult
% 0.30/0.60  thf(fact_4770_div__dvd__iff__mult,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( B2 != zero_zero_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.60         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 0.30/0.60            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_dvd_iff_mult
% 0.30/0.60  thf(fact_4771_div__dvd__iff__mult,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( B2 != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.60         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 0.30/0.60            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_dvd_iff_mult
% 0.30/0.60  thf(fact_4772_dvd__div__iff__mult,axiom,
% 0.30/0.60      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( C != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 0.30/0.60         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.30/0.60            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_iff_mult
% 0.30/0.60  thf(fact_4773_dvd__div__iff__mult,axiom,
% 0.30/0.60      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.60        ( ( C != zero_zero_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ C @ B2 )
% 0.30/0.60         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.30/0.60            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_iff_mult
% 0.30/0.60  thf(fact_4774_dvd__div__iff__mult,axiom,
% 0.30/0.60      ! [C: int,B2: int,A: int] :
% 0.30/0.60        ( ( C != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ C @ B2 )
% 0.30/0.60         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.30/0.60            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_iff_mult
% 0.30/0.60  thf(fact_4775_dvd__div__div__eq__mult,axiom,
% 0.30/0.60      ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 0.30/0.60        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( C != zero_z3403309356797280102nteger )
% 0.30/0.60         => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 0.30/0.60           => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.30/0.60             => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 0.30/0.60                  = ( divide6298287555418463151nteger @ D @ C ) )
% 0.30/0.60                = ( ( times_3573771949741848930nteger @ B2 @ C )
% 0.30/0.60                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_div_eq_mult
% 0.30/0.60  thf(fact_4776_dvd__div__div__eq__mult,axiom,
% 0.30/0.60      ! [A: nat,C: nat,B2: nat,D: nat] :
% 0.30/0.60        ( ( A != zero_zero_nat )
% 0.30/0.60       => ( ( C != zero_zero_nat )
% 0.30/0.60         => ( ( dvd_dvd_nat @ A @ B2 )
% 0.30/0.60           => ( ( dvd_dvd_nat @ C @ D )
% 0.30/0.60             => ( ( ( divide_divide_nat @ B2 @ A )
% 0.30/0.60                  = ( divide_divide_nat @ D @ C ) )
% 0.30/0.60                = ( ( times_times_nat @ B2 @ C )
% 0.30/0.60                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_div_eq_mult
% 0.30/0.60  thf(fact_4777_dvd__div__div__eq__mult,axiom,
% 0.30/0.60      ! [A: int,C: int,B2: int,D: int] :
% 0.30/0.60        ( ( A != zero_zero_int )
% 0.30/0.60       => ( ( C != zero_zero_int )
% 0.30/0.60         => ( ( dvd_dvd_int @ A @ B2 )
% 0.30/0.60           => ( ( dvd_dvd_int @ C @ D )
% 0.30/0.60             => ( ( ( divide_divide_int @ B2 @ A )
% 0.30/0.60                  = ( divide_divide_int @ D @ C ) )
% 0.30/0.60                = ( ( times_times_int @ B2 @ C )
% 0.30/0.60                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_div_div_eq_mult
% 0.30/0.60  thf(fact_4778_unit__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.30/0.60            = zero_z3403309356797280102nteger )
% 0.30/0.60          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_eq_0_iff
% 0.30/0.60  thf(fact_4779_unit__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( ( divide_divide_nat @ A @ B2 )
% 0.30/0.60            = zero_zero_nat )
% 0.30/0.60          = ( A = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_eq_0_iff
% 0.30/0.60  thf(fact_4780_unit__div__eq__0__iff,axiom,
% 0.30/0.60      ! [B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( ( divide_divide_int @ A @ B2 )
% 0.30/0.60            = zero_zero_int )
% 0.30/0.60          = ( A = zero_zero_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_eq_0_iff
% 0.30/0.60  thf(fact_4781_even__numeral,axiom,
% 0.30/0.60      ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_numeral
% 0.30/0.60  thf(fact_4782_even__numeral,axiom,
% 0.30/0.60      ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_numeral
% 0.30/0.60  thf(fact_4783_even__numeral,axiom,
% 0.30/0.60      ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_numeral
% 0.30/0.60  thf(fact_4784_inf__period_I4_J,axiom,
% 0.30/0.60      ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.30/0.60       => ! [X6: code_integer,K4: code_integer] :
% 0.30/0.60            ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) ) )
% 0.30/0.60            = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(4)
% 0.30/0.60  thf(fact_4785_inf__period_I4_J,axiom,
% 0.30/0.60      ! [D: complex,D4: complex,T: complex] :
% 0.30/0.60        ( ( dvd_dvd_complex @ D @ D4 )
% 0.30/0.60       => ! [X6: complex,K4: complex] :
% 0.30/0.60            ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X6 @ T ) ) )
% 0.30/0.60            = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(4)
% 0.30/0.60  thf(fact_4786_inf__period_I4_J,axiom,
% 0.30/0.60      ! [D: real,D4: real,T: real] :
% 0.30/0.60        ( ( dvd_dvd_real @ D @ D4 )
% 0.30/0.60       => ! [X6: real,K4: real] :
% 0.30/0.60            ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) ) )
% 0.30/0.60            = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(4)
% 0.30/0.60  thf(fact_4787_inf__period_I4_J,axiom,
% 0.30/0.60      ! [D: rat,D4: rat,T: rat] :
% 0.30/0.60        ( ( dvd_dvd_rat @ D @ D4 )
% 0.30/0.60       => ! [X6: rat,K4: rat] :
% 0.30/0.60            ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) ) )
% 0.30/0.60            = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(4)
% 0.30/0.60  thf(fact_4788_inf__period_I4_J,axiom,
% 0.30/0.60      ! [D: int,D4: int,T: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.60       => ! [X6: int,K4: int] :
% 0.30/0.60            ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) ) )
% 0.30/0.60            = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(4)
% 0.30/0.60  thf(fact_4789_inf__period_I3_J,axiom,
% 0.30/0.60      ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.30/0.60       => ! [X6: code_integer,K4: code_integer] :
% 0.30/0.60            ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) )
% 0.30/0.60            = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(3)
% 0.30/0.60  thf(fact_4790_inf__period_I3_J,axiom,
% 0.30/0.60      ! [D: complex,D4: complex,T: complex] :
% 0.30/0.60        ( ( dvd_dvd_complex @ D @ D4 )
% 0.30/0.60       => ! [X6: complex,K4: complex] :
% 0.30/0.60            ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X6 @ T ) )
% 0.30/0.60            = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(3)
% 0.30/0.60  thf(fact_4791_inf__period_I3_J,axiom,
% 0.30/0.60      ! [D: real,D4: real,T: real] :
% 0.30/0.60        ( ( dvd_dvd_real @ D @ D4 )
% 0.30/0.60       => ! [X6: real,K4: real] :
% 0.30/0.60            ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) )
% 0.30/0.60            = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(3)
% 0.30/0.60  thf(fact_4792_inf__period_I3_J,axiom,
% 0.30/0.60      ! [D: rat,D4: rat,T: rat] :
% 0.30/0.60        ( ( dvd_dvd_rat @ D @ D4 )
% 0.30/0.60       => ! [X6: rat,K4: rat] :
% 0.30/0.60            ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) )
% 0.30/0.60            = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(3)
% 0.30/0.60  thf(fact_4793_inf__period_I3_J,axiom,
% 0.30/0.60      ! [D: int,D4: int,T: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ D @ D4 )
% 0.30/0.60       => ! [X6: int,K4: int] :
% 0.30/0.60            ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
% 0.30/0.60            = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % inf_period(3)
% 0.30/0.60  thf(fact_4794_unit__eq__div1,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 0.30/0.60            = C )
% 0.30/0.60          = ( A
% 0.30/0.60            = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_eq_div1
% 0.30/0.60  thf(fact_4795_unit__eq__div1,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( ( divide_divide_nat @ A @ B2 )
% 0.30/0.60            = C )
% 0.30/0.60          = ( A
% 0.30/0.60            = ( times_times_nat @ C @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_eq_div1
% 0.30/0.60  thf(fact_4796_unit__eq__div1,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( ( divide_divide_int @ A @ B2 )
% 0.30/0.60            = C )
% 0.30/0.60          = ( A
% 0.30/0.60            = ( times_times_int @ C @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_eq_div1
% 0.30/0.60  thf(fact_4797_unit__eq__div2,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( A
% 0.30/0.60            = ( divide6298287555418463151nteger @ C @ B2 ) )
% 0.30/0.60          = ( ( times_3573771949741848930nteger @ A @ B2 )
% 0.30/0.60            = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_eq_div2
% 0.30/0.60  thf(fact_4798_unit__eq__div2,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( A
% 0.30/0.60            = ( divide_divide_nat @ C @ B2 ) )
% 0.30/0.60          = ( ( times_times_nat @ A @ B2 )
% 0.30/0.60            = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_eq_div2
% 0.30/0.60  thf(fact_4799_unit__eq__div2,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( A
% 0.30/0.60            = ( divide_divide_int @ C @ B2 ) )
% 0.30/0.60          = ( ( times_times_int @ A @ B2 )
% 0.30/0.60            = C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_eq_div2
% 0.30/0.60  thf(fact_4800_div__mult__unit2,axiom,
% 0.30/0.60      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.60         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.60            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_unit2
% 0.30/0.60  thf(fact_4801_div__mult__unit2,axiom,
% 0.30/0.60      ! [C: nat,B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ B2 @ A )
% 0.30/0.60         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.60            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_unit2
% 0.30/0.60  thf(fact_4802_div__mult__unit2,axiom,
% 0.30/0.60      ! [C: int,B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.60         => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.60            = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_mult_unit2
% 0.30/0.60  thf(fact_4803_unit__div__commute,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 0.30/0.60          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_commute
% 0.30/0.60  thf(fact_4804_unit__div__commute,axiom,
% 0.30/0.60      ! [B2: nat,A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 0.30/0.60          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_commute
% 0.30/0.60  thf(fact_4805_unit__div__commute,axiom,
% 0.30/0.60      ! [B2: int,A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 0.30/0.60          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_commute
% 0.30/0.60  thf(fact_4806_unit__div__mult__swap,axiom,
% 0.30/0.60      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.60       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 0.30/0.60          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_mult_swap
% 0.30/0.60  thf(fact_4807_unit__div__mult__swap,axiom,
% 0.30/0.60      ! [C: nat,A: nat,B2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.60       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 0.30/0.60          = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_mult_swap
% 0.30/0.60  thf(fact_4808_unit__div__mult__swap,axiom,
% 0.30/0.60      ! [C: int,A: int,B2: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.60       => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 0.30/0.60          = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_div_mult_swap
% 0.30/0.60  thf(fact_4809_is__unit__div__mult2__eq,axiom,
% 0.30/0.60      ! [B2: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.60         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 0.30/0.60            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult2_eq
% 0.30/0.60  thf(fact_4810_is__unit__div__mult2__eq,axiom,
% 0.30/0.60      ! [B2: nat,C: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.60         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 0.30/0.60            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult2_eq
% 0.30/0.60  thf(fact_4811_is__unit__div__mult2__eq,axiom,
% 0.30/0.60      ! [B2: int,C: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.60         => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 0.30/0.60            = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult2_eq
% 0.30/0.60  thf(fact_4812_unit__imp__mod__eq__0,axiom,
% 0.30/0.60      ! [B2: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60       => ( ( modulo_modulo_nat @ A @ B2 )
% 0.30/0.60          = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_imp_mod_eq_0
% 0.30/0.60  thf(fact_4813_unit__imp__mod__eq__0,axiom,
% 0.30/0.60      ! [B2: int,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60       => ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.60          = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_imp_mod_eq_0
% 0.30/0.60  thf(fact_4814_unit__imp__mod__eq__0,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60       => ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.30/0.60          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % unit_imp_mod_eq_0
% 0.30/0.60  thf(fact_4815_is__unit__power__iff,axiom,
% 0.30/0.60      ! [A: code_integer,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.30/0.60        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_power_iff
% 0.30/0.60  thf(fact_4816_is__unit__power__iff,axiom,
% 0.30/0.60      ! [A: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.30/0.60        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_power_iff
% 0.30/0.60  thf(fact_4817_is__unit__power__iff,axiom,
% 0.30/0.60      ! [A: int,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.30/0.60        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_power_iff
% 0.30/0.60  thf(fact_4818_dvd__imp__le,axiom,
% 0.30/0.60      ! [K: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ K @ N )
% 0.30/0.60       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60         => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_imp_le
% 0.30/0.60  thf(fact_4819_nat__mult__dvd__cancel1,axiom,
% 0.30/0.60      ! [K: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.60          = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % nat_mult_dvd_cancel1
% 0.30/0.60  thf(fact_4820_dvd__mult__cancel,axiom,
% 0.30/0.60      ! [K: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.30/0.60       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.60         => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_cancel
% 0.30/0.60  thf(fact_4821_bezout__add__strong__nat,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( A != zero_zero_nat )
% 0.30/0.60       => ? [D3: nat,X5: nat,Y4: nat] :
% 0.30/0.60            ( ( dvd_dvd_nat @ D3 @ A )
% 0.30/0.60            & ( dvd_dvd_nat @ D3 @ B2 )
% 0.30/0.60            & ( ( times_times_nat @ A @ X5 )
% 0.30/0.60              = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bezout_add_strong_nat
% 0.30/0.60  thf(fact_4822_zdvd__imp__le,axiom,
% 0.30/0.60      ! [Z: int,N: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ Z @ N )
% 0.30/0.60       => ( ( ord_less_int @ zero_zero_int @ N )
% 0.30/0.60         => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zdvd_imp_le
% 0.30/0.60  thf(fact_4823_mod__greater__zero__iff__not__dvd,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.60        = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_greater_zero_iff_not_dvd
% 0.30/0.60  thf(fact_4824_mod__eq__dvd__iff__nat,axiom,
% 0.30/0.60      ! [N: nat,M: nat,Q2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.60       => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.30/0.60            = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.30/0.60          = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_eq_dvd_iff_nat
% 0.30/0.60  thf(fact_4825_prod__decode__aux_Ocases,axiom,
% 0.30/0.60      ! [X3: product_prod_nat_nat] :
% 0.30/0.60        ~ ! [K3: nat,M4: nat] :
% 0.30/0.60            ( X3
% 0.30/0.60           != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % prod_decode_aux.cases
% 0.30/0.60  thf(fact_4826_finite__divisors__nat,axiom,
% 0.30/0.60      ! [M: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.60       => ( finite_finite_nat
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_divisors_nat
% 0.30/0.60  thf(fact_4827_even__zero,axiom,
% 0.30/0.60      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.30/0.60  
% 0.30/0.60  % even_zero
% 0.30/0.60  thf(fact_4828_even__zero,axiom,
% 0.30/0.60      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.30/0.60  
% 0.30/0.60  % even_zero
% 0.30/0.60  thf(fact_4829_even__zero,axiom,
% 0.30/0.60      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.30/0.60  
% 0.30/0.60  % even_zero
% 0.30/0.60  thf(fact_4830_is__unitE,axiom,
% 0.30/0.60      ! [A: code_integer,C: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.60       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.60           => ! [B3: code_integer] :
% 0.30/0.60                ( ( B3 != zero_z3403309356797280102nteger )
% 0.30/0.60               => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 0.30/0.60                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.30/0.60                      = B3 )
% 0.30/0.60                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 0.30/0.60                        = A )
% 0.30/0.60                     => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 0.30/0.60                          = one_one_Code_integer )
% 0.30/0.60                       => ( ( divide6298287555418463151nteger @ C @ A )
% 0.30/0.60                         != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unitE
% 0.30/0.60  thf(fact_4831_is__unitE,axiom,
% 0.30/0.60      ! [A: nat,C: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.60       => ~ ( ( A != zero_zero_nat )
% 0.30/0.60           => ! [B3: nat] :
% 0.30/0.60                ( ( B3 != zero_zero_nat )
% 0.30/0.60               => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 0.30/0.60                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.30/0.60                      = B3 )
% 0.30/0.60                   => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 0.30/0.60                        = A )
% 0.30/0.60                     => ( ( ( times_times_nat @ A @ B3 )
% 0.30/0.60                          = one_one_nat )
% 0.30/0.60                       => ( ( divide_divide_nat @ C @ A )
% 0.30/0.60                         != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unitE
% 0.30/0.60  thf(fact_4832_is__unitE,axiom,
% 0.30/0.60      ! [A: int,C: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.60       => ~ ( ( A != zero_zero_int )
% 0.30/0.60           => ! [B3: int] :
% 0.30/0.60                ( ( B3 != zero_zero_int )
% 0.30/0.60               => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 0.30/0.60                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.30/0.60                      = B3 )
% 0.30/0.60                   => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 0.30/0.60                        = A )
% 0.30/0.60                     => ( ( ( times_times_int @ A @ B3 )
% 0.30/0.60                          = one_one_int )
% 0.30/0.60                       => ( ( divide_divide_int @ C @ A )
% 0.30/0.60                         != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unitE
% 0.30/0.60  thf(fact_4833_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.30/0.60            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult_cancel_left
% 0.30/0.60  thf(fact_4834_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( A != zero_zero_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
% 0.30/0.60            = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult_cancel_left
% 0.30/0.60  thf(fact_4835_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( A != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
% 0.30/0.60            = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult_cancel_left
% 0.30/0.60  thf(fact_4836_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.30/0.60         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ A ) )
% 0.30/0.60            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult_cancel_right
% 0.30/0.60  thf(fact_4837_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ( A != zero_zero_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.30/0.60         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
% 0.30/0.60            = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult_cancel_right
% 0.30/0.60  thf(fact_4838_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( A != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.30/0.60         => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
% 0.30/0.60            = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % is_unit_div_mult_cancel_right
% 0.30/0.60  thf(fact_4839_evenE,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ~ ! [B3: code_integer] :
% 0.30/0.60              ( A
% 0.30/0.60             != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % evenE
% 0.30/0.60  thf(fact_4840_evenE,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ~ ! [B3: nat] :
% 0.30/0.60              ( A
% 0.30/0.60             != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % evenE
% 0.30/0.60  thf(fact_4841_evenE,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ~ ! [B3: int] :
% 0.30/0.60              ( A
% 0.30/0.60             != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % evenE
% 0.30/0.60  thf(fact_4842_odd__one,axiom,
% 0.30/0.60      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_one
% 0.30/0.60  thf(fact_4843_odd__one,axiom,
% 0.30/0.60      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_one
% 0.30/0.60  thf(fact_4844_odd__one,axiom,
% 0.30/0.60      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_one
% 0.30/0.60  thf(fact_4845_odd__even__add,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.60         => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_even_add
% 0.30/0.60  thf(fact_4846_odd__even__add,axiom,
% 0.30/0.60      ! [A: nat,B2: nat] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.60         => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_even_add
% 0.30/0.60  thf(fact_4847_odd__even__add,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.60         => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_even_add
% 0.30/0.60  thf(fact_4848_bit__eq__rec,axiom,
% 0.30/0.60      ( ( ^ [Y6: code_integer,Z5: code_integer] : ( Y6 = Z5 ) )
% 0.30/0.60      = ( ^ [A5: code_integer,B4: code_integer] :
% 0.30/0.60            ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 )
% 0.30/0.60              = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
% 0.30/0.60            & ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60              = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_eq_rec
% 0.30/0.60  thf(fact_4849_bit__eq__rec,axiom,
% 0.30/0.60      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.60      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.60            ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
% 0.30/0.60              = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
% 0.30/0.60            & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60              = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_eq_rec
% 0.30/0.60  thf(fact_4850_bit__eq__rec,axiom,
% 0.30/0.60      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.60      = ( ^ [A5: int,B4: int] :
% 0.30/0.60            ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
% 0.30/0.60              = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
% 0.30/0.60            & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60              = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_eq_rec
% 0.30/0.60  thf(fact_4851_dvd__power__iff,axiom,
% 0.30/0.60      ! [X3: code_integer,M: nat,N: nat] :
% 0.30/0.60        ( ( X3 != zero_z3403309356797280102nteger )
% 0.30/0.60       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M ) @ ( power_8256067586552552935nteger @ X3 @ N ) )
% 0.30/0.60          = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
% 0.30/0.60            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_iff
% 0.30/0.60  thf(fact_4852_dvd__power__iff,axiom,
% 0.30/0.60      ! [X3: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( X3 != zero_zero_nat )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M ) @ ( power_power_nat @ X3 @ N ) )
% 0.30/0.60          = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
% 0.30/0.60            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_iff
% 0.30/0.60  thf(fact_4853_dvd__power__iff,axiom,
% 0.30/0.60      ! [X3: int,M: nat,N: nat] :
% 0.30/0.60        ( ( X3 != zero_zero_int )
% 0.30/0.60       => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ N ) )
% 0.30/0.60          = ( ( dvd_dvd_int @ X3 @ one_one_int )
% 0.30/0.60            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_iff
% 0.30/0.60  thf(fact_4854_dvd__power,axiom,
% 0.30/0.60      ! [N: nat,X3: code_integer] :
% 0.30/0.60        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          | ( X3 = one_one_Code_integer ) )
% 0.30/0.60       => ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power
% 0.30/0.60  thf(fact_4855_dvd__power,axiom,
% 0.30/0.60      ! [N: nat,X3: rat] :
% 0.30/0.60        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          | ( X3 = one_one_rat ) )
% 0.30/0.60       => ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power
% 0.30/0.60  thf(fact_4856_dvd__power,axiom,
% 0.30/0.60      ! [N: nat,X3: nat] :
% 0.30/0.60        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          | ( X3 = one_one_nat ) )
% 0.30/0.60       => ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power
% 0.30/0.60  thf(fact_4857_dvd__power,axiom,
% 0.30/0.60      ! [N: nat,X3: real] :
% 0.30/0.60        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          | ( X3 = one_one_real ) )
% 0.30/0.60       => ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power
% 0.30/0.60  thf(fact_4858_dvd__power,axiom,
% 0.30/0.60      ! [N: nat,X3: int] :
% 0.30/0.60        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          | ( X3 = one_one_int ) )
% 0.30/0.60       => ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power
% 0.30/0.60  thf(fact_4859_dvd__power,axiom,
% 0.30/0.60      ! [N: nat,X3: complex] :
% 0.30/0.60        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          | ( X3 = one_one_complex ) )
% 0.30/0.60       => ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power
% 0.30/0.60  thf(fact_4860_even__even__mod__4__iff,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_even_mod_4_iff
% 0.30/0.60  thf(fact_4861_dvd__mult__cancel2,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.30/0.60          = ( N = one_one_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_cancel2
% 0.30/0.60  thf(fact_4862_dvd__mult__cancel1,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.30/0.60          = ( N = one_one_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_mult_cancel1
% 0.30/0.60  thf(fact_4863_dvd__minus__add,axiom,
% 0.30/0.60      ! [Q2: nat,N: nat,R: nat,M: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ Q2 @ N )
% 0.30/0.60       => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 0.30/0.60         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 0.30/0.60            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_add
% 0.30/0.60  thf(fact_4864_power__dvd__imp__le,axiom,
% 0.30/0.60      ! [I: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.30/0.60       => ( ( ord_less_nat @ one_one_nat @ I )
% 0.30/0.60         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_dvd_imp_le
% 0.30/0.60  thf(fact_4865_mod__nat__eqI,axiom,
% 0.30/0.60      ! [R: nat,N: nat,M: nat] :
% 0.30/0.60        ( ( ord_less_nat @ R @ N )
% 0.30/0.60       => ( ( ord_less_eq_nat @ R @ M )
% 0.30/0.60         => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
% 0.30/0.60           => ( ( modulo_modulo_nat @ M @ N )
% 0.30/0.60              = R ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_nat_eqI
% 0.30/0.60  thf(fact_4866_mod__int__pos__iff,axiom,
% 0.30/0.60      ! [K: int,L: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.30/0.60        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.60          | ( ( L = zero_zero_int )
% 0.30/0.60            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.30/0.60          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_int_pos_iff
% 0.30/0.60  thf(fact_4867_even__two__times__div__two,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_two_times_div_two
% 0.30/0.60  thf(fact_4868_even__two__times__div__two,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_two_times_div_two
% 0.30/0.60  thf(fact_4869_even__two__times__div__two,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_two_times_div_two
% 0.30/0.60  thf(fact_4870_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60          = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_iff_mod_2_eq_zero
% 0.30/0.60  thf(fact_4871_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60          = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_iff_mod_2_eq_zero
% 0.30/0.60  thf(fact_4872_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_iff_mod_2_eq_zero
% 0.30/0.60  thf(fact_4873_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.60        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60          = one_one_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_iff_mod_2_eq_one
% 0.30/0.60  thf(fact_4874_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.60        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60          = one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_iff_mod_2_eq_one
% 0.30/0.60  thf(fact_4875_odd__iff__mod__2__eq__one,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.60        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60          = one_one_Code_integer ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_iff_mod_2_eq_one
% 0.30/0.60  thf(fact_4876_power__mono__odd,axiom,
% 0.30/0.60      ! [N: nat,A: real,B2: real] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.60         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_mono_odd
% 0.30/0.60  thf(fact_4877_power__mono__odd,axiom,
% 0.30/0.60      ! [N: nat,A: rat,B2: rat] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.60         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_mono_odd
% 0.30/0.60  thf(fact_4878_power__mono__odd,axiom,
% 0.30/0.60      ! [N: nat,A: int,B2: int] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.60         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_mono_odd
% 0.30/0.60  thf(fact_4879_odd__pos,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_pos
% 0.30/0.60  thf(fact_4880_dvd__power__iff__le,axiom,
% 0.30/0.60      ! [K: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.60       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.30/0.60          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_power_iff_le
% 0.30/0.60  thf(fact_4881_signed__take__bit__int__less__exp,axiom,
% 0.30/0.60      ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_less_exp
% 0.30/0.60  thf(fact_4882_even__unset__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_unset_bit_iff
% 0.30/0.60  thf(fact_4883_even__unset__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_unset_bit_iff
% 0.30/0.60  thf(fact_4884_even__unset__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_unset_bit_iff
% 0.30/0.60  thf(fact_4885_even__set__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60          & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_set_bit_iff
% 0.30/0.60  thf(fact_4886_even__set__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60          & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_set_bit_iff
% 0.30/0.60  thf(fact_4887_even__set__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60          & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_set_bit_iff
% 0.30/0.60  thf(fact_4888_even__flip__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_flip_bit_iff
% 0.30/0.60  thf(fact_4889_even__flip__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_flip_bit_iff
% 0.30/0.60  thf(fact_4890_even__flip__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_flip_bit_iff
% 0.30/0.60  thf(fact_4891_even__diff__iff,axiom,
% 0.30/0.60      ! [K: int,L: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 0.30/0.60        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_diff_iff
% 0.30/0.60  thf(fact_4892_oddE,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ~ ! [B3: code_integer] :
% 0.30/0.60              ( A
% 0.30/0.60             != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % oddE
% 0.30/0.60  thf(fact_4893_oddE,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ~ ! [B3: nat] :
% 0.30/0.60              ( A
% 0.30/0.60             != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % oddE
% 0.30/0.60  thf(fact_4894_oddE,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ~ ! [B3: int] :
% 0.30/0.60              ( A
% 0.30/0.60             != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % oddE
% 0.30/0.60  thf(fact_4895_parity__cases,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60           != zero_zero_nat ) )
% 0.30/0.60       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60             != one_one_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % parity_cases
% 0.30/0.60  thf(fact_4896_parity__cases,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60           != zero_zero_int ) )
% 0.30/0.60       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60             != one_one_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % parity_cases
% 0.30/0.60  thf(fact_4897_parity__cases,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60           != zero_z3403309356797280102nteger ) )
% 0.30/0.60       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60             != one_one_Code_integer ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % parity_cases
% 0.30/0.60  thf(fact_4898_mod2__eq__if,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60            = zero_zero_nat ) )
% 0.30/0.60        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60            = one_one_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod2_eq_if
% 0.30/0.60  thf(fact_4899_mod2__eq__if,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60            = zero_zero_int ) )
% 0.30/0.60        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60            = one_one_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod2_eq_if
% 0.30/0.60  thf(fact_4900_mod2__eq__if,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60            = zero_z3403309356797280102nteger ) )
% 0.30/0.60        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60            = one_one_Code_integer ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod2_eq_if
% 0.30/0.60  thf(fact_4901_zero__le__even__power,axiom,
% 0.30/0.60      ! [N: nat,A: real] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_even_power
% 0.30/0.60  thf(fact_4902_zero__le__even__power,axiom,
% 0.30/0.60      ! [N: nat,A: rat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_even_power
% 0.30/0.60  thf(fact_4903_zero__le__even__power,axiom,
% 0.30/0.60      ! [N: nat,A: int] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_even_power
% 0.30/0.60  thf(fact_4904_zero__le__odd__power,axiom,
% 0.30/0.60      ! [N: nat,A: real] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.30/0.60          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_odd_power
% 0.30/0.60  thf(fact_4905_zero__le__odd__power,axiom,
% 0.30/0.60      ! [N: nat,A: rat] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.60          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_odd_power
% 0.30/0.60  thf(fact_4906_zero__le__odd__power,axiom,
% 0.30/0.60      ! [N: nat,A: int] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.30/0.60          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_odd_power
% 0.30/0.60  thf(fact_4907_zero__le__power__eq,axiom,
% 0.30/0.60      ! [A: real,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_power_eq
% 0.30/0.60  thf(fact_4908_zero__le__power__eq,axiom,
% 0.30/0.60      ! [A: rat,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_power_eq
% 0.30/0.60  thf(fact_4909_zero__le__power__eq,axiom,
% 0.30/0.60      ! [A: int,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.30/0.60        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_power_eq
% 0.30/0.60  thf(fact_4910_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.60      ! [K: int,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.30/0.60        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_greater_eq_self_iff
% 0.30/0.60  thf(fact_4911_signed__take__bit__int__less__self__iff,axiom,
% 0.30/0.60      ! [N: nat,K: int] :
% 0.30/0.60        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_less_self_iff
% 0.30/0.60  thf(fact_4912_zero__less__power__eq,axiom,
% 0.30/0.60      ! [A: real,N: nat] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.30/0.60        = ( ( N = zero_zero_nat )
% 0.30/0.60          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( A != zero_zero_real ) )
% 0.30/0.60          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_power_eq
% 0.30/0.60  thf(fact_4913_zero__less__power__eq,axiom,
% 0.30/0.60      ! [A: rat,N: nat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.60        = ( ( N = zero_zero_nat )
% 0.30/0.60          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( A != zero_zero_rat ) )
% 0.30/0.60          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_power_eq
% 0.30/0.60  thf(fact_4914_zero__less__power__eq,axiom,
% 0.30/0.60      ! [A: int,N: nat] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.30/0.60        = ( ( N = zero_zero_nat )
% 0.30/0.60          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( A != zero_zero_int ) )
% 0.30/0.60          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_power_eq
% 0.30/0.60  thf(fact_4915_signed__take__bit__int__less__eq,axiom,
% 0.30/0.60      ! [N: nat,K: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.30/0.60       => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_less_eq
% 0.30/0.60  thf(fact_4916_even__mask__div__iff_H,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mask_div_iff'
% 0.30/0.60  thf(fact_4917_even__mask__div__iff_H,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mask_div_iff'
% 0.30/0.60  thf(fact_4918_even__mask__div__iff_H,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mask_div_iff'
% 0.30/0.60  thf(fact_4919_power__le__zero__eq,axiom,
% 0.30/0.60      ! [A: real,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.30/0.60        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.60            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_zero_eq
% 0.30/0.60  thf(fact_4920_power__le__zero__eq,axiom,
% 0.30/0.60      ! [A: rat,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.30/0.60        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.60            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_zero_eq
% 0.30/0.60  thf(fact_4921_power__le__zero__eq,axiom,
% 0.30/0.60      ! [A: int,N: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.30/0.60        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.60          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.60            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_le_zero_eq
% 0.30/0.60  thf(fact_4922_option_Osize__gen_I1_J,axiom,
% 0.30/0.60      ! [X3: nat > nat] :
% 0.30/0.60        ( ( size_option_nat @ X3 @ none_nat )
% 0.30/0.60        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % option.size_gen(1)
% 0.30/0.60  thf(fact_4923_option_Osize__gen_I1_J,axiom,
% 0.30/0.60      ! [X3: product_prod_nat_nat > nat] :
% 0.30/0.60        ( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
% 0.30/0.60        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % option.size_gen(1)
% 0.30/0.60  thf(fact_4924_option_Osize__gen_I1_J,axiom,
% 0.30/0.60      ! [X3: num > nat] :
% 0.30/0.60        ( ( size_option_num @ X3 @ none_num )
% 0.30/0.60        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % option.size_gen(1)
% 0.30/0.60  thf(fact_4925_even__mod__4__div__2,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = ( suc @ zero_zero_nat ) )
% 0.30/0.60       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mod_4_div_2
% 0.30/0.60  thf(fact_4926_even__mask__div__iff,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            = zero_z3403309356797280102nteger )
% 0.30/0.60          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mask_div_iff
% 0.30/0.60  thf(fact_4927_even__mask__div__iff,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            = zero_zero_nat )
% 0.30/0.60          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mask_div_iff
% 0.30/0.60  thf(fact_4928_even__mask__div__iff,axiom,
% 0.30/0.60      ! [M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            = zero_zero_int )
% 0.30/0.60          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mask_div_iff
% 0.30/0.60  thf(fact_4929_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.60      ! [A: code_integer,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ( ord_less_nat @ N @ M )
% 0.30/0.60          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            = zero_z3403309356797280102nteger )
% 0.30/0.60          | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mult_exp_div_exp_iff
% 0.30/0.60  thf(fact_4930_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.60      ! [A: nat,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ( ord_less_nat @ N @ M )
% 0.30/0.60          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            = zero_zero_nat )
% 0.30/0.60          | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60            & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mult_exp_div_exp_iff
% 0.30/0.60  thf(fact_4931_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.60      ! [A: int,M: nat,N: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60        = ( ( ord_less_nat @ N @ M )
% 0.30/0.60          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.60            = zero_zero_int )
% 0.30/0.60          | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.60            & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_mult_exp_div_exp_iff
% 0.30/0.60  thf(fact_4932_ex__min__if__finite,axiom,
% 0.30/0.60      ! [S2: set_real] :
% 0.30/0.60        ( ( finite_finite_real @ S2 )
% 0.30/0.60       => ( ( S2 != bot_bot_set_real )
% 0.30/0.60         => ? [X5: real] :
% 0.30/0.60              ( ( member_real @ X5 @ S2 )
% 0.30/0.60              & ~ ? [Xa: real] :
% 0.30/0.60                    ( ( member_real @ Xa @ S2 )
% 0.30/0.60                    & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_min_if_finite
% 0.30/0.60  thf(fact_4933_ex__min__if__finite,axiom,
% 0.30/0.60      ! [S2: set_rat] :
% 0.30/0.60        ( ( finite_finite_rat @ S2 )
% 0.30/0.60       => ( ( S2 != bot_bot_set_rat )
% 0.30/0.60         => ? [X5: rat] :
% 0.30/0.60              ( ( member_rat @ X5 @ S2 )
% 0.30/0.60              & ~ ? [Xa: rat] :
% 0.30/0.60                    ( ( member_rat @ Xa @ S2 )
% 0.30/0.60                    & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_min_if_finite
% 0.30/0.60  thf(fact_4934_ex__min__if__finite,axiom,
% 0.30/0.60      ! [S2: set_num] :
% 0.30/0.60        ( ( finite_finite_num @ S2 )
% 0.30/0.60       => ( ( S2 != bot_bot_set_num )
% 0.30/0.60         => ? [X5: num] :
% 0.30/0.60              ( ( member_num @ X5 @ S2 )
% 0.30/0.60              & ~ ? [Xa: num] :
% 0.30/0.60                    ( ( member_num @ Xa @ S2 )
% 0.30/0.60                    & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_min_if_finite
% 0.30/0.60  thf(fact_4935_ex__min__if__finite,axiom,
% 0.30/0.60      ! [S2: set_nat] :
% 0.30/0.60        ( ( finite_finite_nat @ S2 )
% 0.30/0.60       => ( ( S2 != bot_bot_set_nat )
% 0.30/0.60         => ? [X5: nat] :
% 0.30/0.60              ( ( member_nat @ X5 @ S2 )
% 0.30/0.60              & ~ ? [Xa: nat] :
% 0.30/0.60                    ( ( member_nat @ Xa @ S2 )
% 0.30/0.60                    & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_min_if_finite
% 0.30/0.60  thf(fact_4936_ex__min__if__finite,axiom,
% 0.30/0.60      ! [S2: set_int] :
% 0.30/0.60        ( ( finite_finite_int @ S2 )
% 0.30/0.60       => ( ( S2 != bot_bot_set_int )
% 0.30/0.60         => ? [X5: int] :
% 0.30/0.60              ( ( member_int @ X5 @ S2 )
% 0.30/0.60              & ~ ? [Xa: int] :
% 0.30/0.60                    ( ( member_int @ Xa @ S2 )
% 0.30/0.60                    & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_min_if_finite
% 0.30/0.60  thf(fact_4937_triangle__def,axiom,
% 0.30/0.60      ( nat_triangle
% 0.30/0.60      = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % triangle_def
% 0.30/0.60  thf(fact_4938_vebt__buildup_Oelims,axiom,
% 0.30/0.60      ! [X3: nat,Y3: vEBT_VEBT] :
% 0.30/0.60        ( ( ( vEBT_vebt_buildup @ X3 )
% 0.30/0.60          = Y3 )
% 0.30/0.60       => ( ( ( X3 = zero_zero_nat )
% 0.30/0.60           => ( Y3
% 0.30/0.60             != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.60         => ( ( ( X3
% 0.30/0.60                = ( suc @ zero_zero_nat ) )
% 0.30/0.60             => ( Y3
% 0.30/0.60               != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.60           => ~ ! [Va2: nat] :
% 0.30/0.60                  ( ( X3
% 0.30/0.60                    = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.60                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.60                       => ( Y3
% 0.30/0.60                          = ( 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 ) ) ) ) ) ) )
% 0.30/0.60                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.60                       => ( Y3
% 0.30/0.60                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % vebt_buildup.elims
% 0.30/0.60  thf(fact_4939_signed__take__bit__rec,axiom,
% 0.30/0.60      ( bit_ri6519982836138164636nteger
% 0.30/0.60      = ( ^ [N2: nat,A5: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_rec
% 0.30/0.60  thf(fact_4940_signed__take__bit__rec,axiom,
% 0.30/0.60      ( bit_ri631733984087533419it_int
% 0.30/0.60      = ( ^ [N2: nat,A5: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_rec
% 0.30/0.60  thf(fact_4941_flip__bit__0,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.30/0.60        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % flip_bit_0
% 0.30/0.60  thf(fact_4942_flip__bit__0,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.30/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % flip_bit_0
% 0.30/0.60  thf(fact_4943_flip__bit__0,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.30/0.60        = ( 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 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % flip_bit_0
% 0.30/0.60  thf(fact_4944_set__decode__Suc,axiom,
% 0.30/0.60      ! [N: nat,X3: nat] :
% 0.30/0.60        ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X3 ) )
% 0.30/0.60        = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % set_decode_Suc
% 0.30/0.60  thf(fact_4945_diff__shunt__var,axiom,
% 0.30/0.60      ! [X3: set_nat,Y3: set_nat] :
% 0.30/0.60        ( ( ( minus_minus_set_nat @ X3 @ Y3 )
% 0.30/0.60          = bot_bot_set_nat )
% 0.30/0.60        = ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_shunt_var
% 0.30/0.60  thf(fact_4946_diff__shunt__var,axiom,
% 0.30/0.60      ! [X3: set_real,Y3: set_real] :
% 0.30/0.60        ( ( ( minus_minus_set_real @ X3 @ Y3 )
% 0.30/0.60          = bot_bot_set_real )
% 0.30/0.60        = ( ord_less_eq_set_real @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_shunt_var
% 0.30/0.60  thf(fact_4947_diff__shunt__var,axiom,
% 0.30/0.60      ! [X3: set_int,Y3: set_int] :
% 0.30/0.60        ( ( ( minus_minus_set_int @ X3 @ Y3 )
% 0.30/0.60          = bot_bot_set_int )
% 0.30/0.60        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_shunt_var
% 0.30/0.60  thf(fact_4948_add__scale__eq__noteq,axiom,
% 0.30/0.60      ! [R: complex,A: complex,B2: complex,C: complex,D: complex] :
% 0.30/0.60        ( ( R != zero_zero_complex )
% 0.30/0.60       => ( ( ( A = B2 )
% 0.30/0.60            & ( C != D ) )
% 0.30/0.60         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
% 0.30/0.60           != ( plus_plus_complex @ B2 @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_scale_eq_noteq
% 0.30/0.60  thf(fact_4949_add__scale__eq__noteq,axiom,
% 0.30/0.60      ! [R: real,A: real,B2: real,C: real,D: real] :
% 0.30/0.60        ( ( R != zero_zero_real )
% 0.30/0.60       => ( ( ( A = B2 )
% 0.30/0.60            & ( C != D ) )
% 0.30/0.60         => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 0.30/0.60           != ( plus_plus_real @ B2 @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_scale_eq_noteq
% 0.30/0.60  thf(fact_4950_add__scale__eq__noteq,axiom,
% 0.30/0.60      ! [R: rat,A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.60        ( ( R != zero_zero_rat )
% 0.30/0.60       => ( ( ( A = B2 )
% 0.30/0.60            & ( C != D ) )
% 0.30/0.60         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 0.30/0.60           != ( plus_plus_rat @ B2 @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_scale_eq_noteq
% 0.30/0.60  thf(fact_4951_add__scale__eq__noteq,axiom,
% 0.30/0.60      ! [R: nat,A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.60        ( ( R != zero_zero_nat )
% 0.30/0.60       => ( ( ( A = B2 )
% 0.30/0.60            & ( C != D ) )
% 0.30/0.60         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 0.30/0.60           != ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_scale_eq_noteq
% 0.30/0.60  thf(fact_4952_add__scale__eq__noteq,axiom,
% 0.30/0.60      ! [R: int,A: int,B2: int,C: int,D: int] :
% 0.30/0.60        ( ( R != zero_zero_int )
% 0.30/0.60       => ( ( ( A = B2 )
% 0.30/0.60            & ( C != D ) )
% 0.30/0.60         => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 0.30/0.60           != ( plus_plus_int @ B2 @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_scale_eq_noteq
% 0.30/0.60  thf(fact_4953_intind,axiom,
% 0.30/0.60      ! [I: nat,N: nat,P: int > $o,X3: int] :
% 0.30/0.60        ( ( ord_less_nat @ I @ N )
% 0.30/0.60       => ( ( P @ X3 )
% 0.30/0.60         => ( P @ ( nth_int @ ( replicate_int @ N @ X3 ) @ I ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % intind
% 0.30/0.60  thf(fact_4954_intind,axiom,
% 0.30/0.60      ! [I: nat,N: nat,P: nat > $o,X3: nat] :
% 0.30/0.60        ( ( ord_less_nat @ I @ N )
% 0.30/0.60       => ( ( P @ X3 )
% 0.30/0.60         => ( P @ ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % intind
% 0.30/0.60  thf(fact_4955_intind,axiom,
% 0.30/0.60      ! [I: nat,N: nat,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 0.30/0.60        ( ( ord_less_nat @ I @ N )
% 0.30/0.60       => ( ( P @ X3 )
% 0.30/0.60         => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % intind
% 0.30/0.60  thf(fact_4956_add_Oinverse__inverse,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = A ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_inverse
% 0.30/0.60  thf(fact_4957_add_Oinverse__inverse,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = A ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_inverse
% 0.30/0.60  thf(fact_4958_add_Oinverse__inverse,axiom,
% 0.30/0.60      ! [A: complex] :
% 0.30/0.60        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.60        = A ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_inverse
% 0.30/0.60  thf(fact_4959_add_Oinverse__inverse,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = A ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_inverse
% 0.30/0.60  thf(fact_4960_add_Oinverse__inverse,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = A ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_inverse
% 0.30/0.60  thf(fact_4961_neg__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ A )
% 0.30/0.60          = ( uminus_uminus_real @ B2 ) )
% 0.30/0.60        = ( A = B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_iff_equal
% 0.30/0.60  thf(fact_4962_neg__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( ( uminus_uminus_int @ A )
% 0.30/0.60          = ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = ( A = B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_iff_equal
% 0.30/0.60  thf(fact_4963_neg__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60          = ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.60        = ( A = B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_iff_equal
% 0.30/0.60  thf(fact_4964_neg__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60          = ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = ( A = B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_iff_equal
% 0.30/0.60  thf(fact_4965_neg__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60          = ( uminus_uminus_rat @ B2 ) )
% 0.30/0.60        = ( A = B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_iff_equal
% 0.30/0.60  thf(fact_4966_verit__minus__simplify_I4_J,axiom,
% 0.30/0.60      ! [B2: real] :
% 0.30/0.60        ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(4)
% 0.30/0.60  thf(fact_4967_verit__minus__simplify_I4_J,axiom,
% 0.30/0.60      ! [B2: int] :
% 0.30/0.60        ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(4)
% 0.30/0.60  thf(fact_4968_verit__minus__simplify_I4_J,axiom,
% 0.30/0.60      ! [B2: complex] :
% 0.30/0.60        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(4)
% 0.30/0.60  thf(fact_4969_verit__minus__simplify_I4_J,axiom,
% 0.30/0.60      ! [B2: code_integer] :
% 0.30/0.60        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(4)
% 0.30/0.60  thf(fact_4970_verit__minus__simplify_I4_J,axiom,
% 0.30/0.60      ! [B2: rat] :
% 0.30/0.60        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(4)
% 0.30/0.60  thf(fact_4971_Compl__subset__Compl__iff,axiom,
% 0.30/0.60      ! [A4: set_int,B5: set_int] :
% 0.30/0.60        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B5 ) )
% 0.30/0.60        = ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_subset_Compl_iff
% 0.30/0.60  thf(fact_4972_Compl__anti__mono,axiom,
% 0.30/0.60      ! [A4: set_int,B5: set_int] :
% 0.30/0.60        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.60       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_anti_mono
% 0.30/0.60  thf(fact_4973_compl__le__compl__iff,axiom,
% 0.30/0.60      ! [X3: set_int,Y3: set_int] :
% 0.30/0.60        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 0.30/0.60        = ( ord_less_eq_set_int @ Y3 @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % compl_le_compl_iff
% 0.30/0.60  thf(fact_4974_neg__le__iff__le,axiom,
% 0.30/0.60      ! [B2: real,A: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_iff_le
% 0.30/0.60  thf(fact_4975_neg__le__iff__le,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_iff_le
% 0.30/0.60  thf(fact_4976_neg__le__iff__le,axiom,
% 0.30/0.60      ! [B2: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_iff_le
% 0.30/0.60  thf(fact_4977_neg__le__iff__le,axiom,
% 0.30/0.60      ! [B2: int,A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_iff_le
% 0.30/0.60  thf(fact_4978_add_Oinverse__neutral,axiom,
% 0.30/0.60      ( ( uminus_uminus_real @ zero_zero_real )
% 0.30/0.60      = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_neutral
% 0.30/0.60  thf(fact_4979_add_Oinverse__neutral,axiom,
% 0.30/0.60      ( ( uminus_uminus_int @ zero_zero_int )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_neutral
% 0.30/0.60  thf(fact_4980_add_Oinverse__neutral,axiom,
% 0.30/0.60      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.30/0.60      = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_neutral
% 0.30/0.60  thf(fact_4981_add_Oinverse__neutral,axiom,
% 0.30/0.60      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.30/0.60      = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_neutral
% 0.30/0.60  thf(fact_4982_add_Oinverse__neutral,axiom,
% 0.30/0.60      ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.30/0.60      = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % add.inverse_neutral
% 0.30/0.60  thf(fact_4983_neg__0__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( zero_zero_real
% 0.30/0.60          = ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( zero_zero_real = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_equal_iff_equal
% 0.30/0.60  thf(fact_4984_neg__0__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( zero_zero_int
% 0.30/0.60          = ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( zero_zero_int = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_equal_iff_equal
% 0.30/0.60  thf(fact_4985_neg__0__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: complex] :
% 0.30/0.60        ( ( zero_zero_complex
% 0.30/0.60          = ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.60        = ( zero_zero_complex = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_equal_iff_equal
% 0.30/0.60  thf(fact_4986_neg__0__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( zero_z3403309356797280102nteger
% 0.30/0.60          = ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_equal_iff_equal
% 0.30/0.60  thf(fact_4987_neg__0__equal__iff__equal,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( zero_zero_rat
% 0.30/0.60          = ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( zero_zero_rat = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_equal_iff_equal
% 0.30/0.60  thf(fact_4988_neg__equal__0__iff__equal,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ A )
% 0.30/0.60          = zero_zero_real )
% 0.30/0.60        = ( A = zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_0_iff_equal
% 0.30/0.60  thf(fact_4989_neg__equal__0__iff__equal,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ( uminus_uminus_int @ A )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( A = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_0_iff_equal
% 0.30/0.60  thf(fact_4990_neg__equal__0__iff__equal,axiom,
% 0.30/0.60      ! [A: complex] :
% 0.30/0.60        ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.60          = zero_zero_complex )
% 0.30/0.60        = ( A = zero_zero_complex ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_0_iff_equal
% 0.30/0.60  thf(fact_4991_neg__equal__0__iff__equal,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60          = zero_z3403309356797280102nteger )
% 0.30/0.60        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_0_iff_equal
% 0.30/0.60  thf(fact_4992_neg__equal__0__iff__equal,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60          = zero_zero_rat )
% 0.30/0.60        = ( A = zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_0_iff_equal
% 0.30/0.60  thf(fact_4993_equal__neg__zero,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( A
% 0.30/0.60          = ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( A = zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % equal_neg_zero
% 0.30/0.60  thf(fact_4994_equal__neg__zero,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( A
% 0.30/0.60          = ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( A = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % equal_neg_zero
% 0.30/0.60  thf(fact_4995_equal__neg__zero,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( A
% 0.30/0.60          = ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % equal_neg_zero
% 0.30/0.60  thf(fact_4996_equal__neg__zero,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( A
% 0.30/0.60          = ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( A = zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % equal_neg_zero
% 0.30/0.60  thf(fact_4997_neg__equal__zero,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ A )
% 0.30/0.60          = A )
% 0.30/0.60        = ( A = zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_zero
% 0.30/0.60  thf(fact_4998_neg__equal__zero,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ( uminus_uminus_int @ A )
% 0.30/0.60          = A )
% 0.30/0.60        = ( A = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_zero
% 0.30/0.60  thf(fact_4999_neg__equal__zero,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.60          = A )
% 0.30/0.60        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_zero
% 0.30/0.60  thf(fact_5000_neg__equal__zero,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.60          = A )
% 0.30/0.60        = ( A = zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_equal_zero
% 0.30/0.60  thf(fact_5001_neg__less__iff__less,axiom,
% 0.30/0.60      ! [B2: real,A: real] :
% 0.30/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_iff_less
% 0.30/0.60  thf(fact_5002_neg__less__iff__less,axiom,
% 0.30/0.60      ! [B2: int,A: int] :
% 0.30/0.60        ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_iff_less
% 0.30/0.60  thf(fact_5003_neg__less__iff__less,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( ord_le6747313008572928689nteger @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_iff_less
% 0.30/0.60  thf(fact_5004_neg__less__iff__less,axiom,
% 0.30/0.60      ! [B2: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_iff_less
% 0.30/0.60  thf(fact_5005_neg__numeral__eq__iff,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.30/0.60          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60        = ( M = N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_eq_iff
% 0.30/0.60  thf(fact_5006_neg__numeral__eq__iff,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.30/0.60          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60        = ( M = N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_eq_iff
% 0.30/0.60  thf(fact_5007_neg__numeral__eq__iff,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.30/0.60          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60        = ( M = N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_eq_iff
% 0.30/0.60  thf(fact_5008_neg__numeral__eq__iff,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.30/0.60          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60        = ( M = N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_eq_iff
% 0.30/0.60  thf(fact_5009_neg__numeral__eq__iff,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.30/0.60          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60        = ( M = N ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_eq_iff
% 0.30/0.60  thf(fact_5010_mult__minus__left,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.60        = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_left
% 0.30/0.60  thf(fact_5011_mult__minus__left,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.60        = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_left
% 0.30/0.60  thf(fact_5012_mult__minus__left,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_left
% 0.30/0.60  thf(fact_5013_mult__minus__left,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_left
% 0.30/0.60  thf(fact_5014_mult__minus__left,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.60        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_left
% 0.30/0.60  thf(fact_5015_minus__mult__minus,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.60        = ( times_times_real @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mult_minus
% 0.30/0.60  thf(fact_5016_minus__mult__minus,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = ( times_times_int @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mult_minus
% 0.30/0.60  thf(fact_5017_minus__mult__minus,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.60        = ( times_times_complex @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mult_minus
% 0.30/0.60  thf(fact_5018_minus__mult__minus,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mult_minus
% 0.30/0.60  thf(fact_5019_minus__mult__minus,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.60        = ( times_times_rat @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mult_minus
% 0.30/0.60  thf(fact_5020_mult__minus__right,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_right
% 0.30/0.60  thf(fact_5021_mult__minus__right,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_right
% 0.30/0.60  thf(fact_5022_mult__minus__right,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_right
% 0.30/0.60  thf(fact_5023_mult__minus__right,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_right
% 0.30/0.60  thf(fact_5024_mult__minus__right,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus_right
% 0.30/0.60  thf(fact_5025_minus__add__distrib,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.60        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_distrib
% 0.30/0.60  thf(fact_5026_minus__add__distrib,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.60        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_distrib
% 0.30/0.60  thf(fact_5027_minus__add__distrib,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 0.30/0.60        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_distrib
% 0.30/0.60  thf(fact_5028_minus__add__distrib,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.30/0.60        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_distrib
% 0.30/0.60  thf(fact_5029_minus__add__distrib,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.60        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_distrib
% 0.30/0.60  thf(fact_5030_minus__add__cancel,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_cancel
% 0.30/0.60  thf(fact_5031_minus__add__cancel,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_cancel
% 0.30/0.60  thf(fact_5032_minus__add__cancel,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_cancel
% 0.30/0.60  thf(fact_5033_minus__add__cancel,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_cancel
% 0.30/0.60  thf(fact_5034_minus__add__cancel,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_add_cancel
% 0.30/0.60  thf(fact_5035_add__minus__cancel,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % add_minus_cancel
% 0.30/0.60  thf(fact_5036_add__minus__cancel,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % add_minus_cancel
% 0.30/0.60  thf(fact_5037_add__minus__cancel,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % add_minus_cancel
% 0.30/0.60  thf(fact_5038_add__minus__cancel,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % add_minus_cancel
% 0.30/0.60  thf(fact_5039_add__minus__cancel,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) )
% 0.30/0.60        = B2 ) ).
% 0.30/0.60  
% 0.30/0.60  % add_minus_cancel
% 0.30/0.60  thf(fact_5040_minus__diff__eq,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.60        = ( minus_minus_real @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_diff_eq
% 0.30/0.60  thf(fact_5041_minus__diff__eq,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.60        = ( minus_minus_int @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_diff_eq
% 0.30/0.60  thf(fact_5042_minus__diff__eq,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) )
% 0.30/0.60        = ( minus_minus_complex @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_diff_eq
% 0.30/0.60  thf(fact_5043_minus__diff__eq,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 0.30/0.60        = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_diff_eq
% 0.30/0.60  thf(fact_5044_minus__diff__eq,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.30/0.60        = ( minus_minus_rat @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_diff_eq
% 0.30/0.60  thf(fact_5045_div__minus__minus,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = ( divide_divide_int @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_minus_minus
% 0.30/0.60  thf(fact_5046_div__minus__minus,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = ( divide6298287555418463151nteger @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_minus_minus
% 0.30/0.60  thf(fact_5047_dvd__minus__iff,axiom,
% 0.30/0.60      ! [X3: real,Y3: real] :
% 0.30/0.60        ( ( dvd_dvd_real @ X3 @ ( uminus_uminus_real @ Y3 ) )
% 0.30/0.60        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_iff
% 0.30/0.60  thf(fact_5048_dvd__minus__iff,axiom,
% 0.30/0.60      ! [X3: int,Y3: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
% 0.30/0.60        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_iff
% 0.30/0.60  thf(fact_5049_dvd__minus__iff,axiom,
% 0.30/0.60      ! [X3: complex,Y3: complex] :
% 0.30/0.60        ( ( dvd_dvd_complex @ X3 @ ( uminus1482373934393186551omplex @ Y3 ) )
% 0.30/0.60        = ( dvd_dvd_complex @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_iff
% 0.30/0.60  thf(fact_5050_dvd__minus__iff,axiom,
% 0.30/0.60      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ X3 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_iff
% 0.30/0.60  thf(fact_5051_dvd__minus__iff,axiom,
% 0.30/0.60      ! [X3: rat,Y3: rat] :
% 0.30/0.60        ( ( dvd_dvd_rat @ X3 @ ( uminus_uminus_rat @ Y3 ) )
% 0.30/0.60        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dvd_minus_iff
% 0.30/0.60  thf(fact_5052_minus__dvd__iff,axiom,
% 0.30/0.60      ! [X3: real,Y3: real] :
% 0.30/0.60        ( ( dvd_dvd_real @ ( uminus_uminus_real @ X3 ) @ Y3 )
% 0.30/0.60        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_dvd_iff
% 0.30/0.60  thf(fact_5053_minus__dvd__iff,axiom,
% 0.30/0.60      ! [X3: int,Y3: int] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
% 0.30/0.60        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_dvd_iff
% 0.30/0.60  thf(fact_5054_minus__dvd__iff,axiom,
% 0.30/0.60      ! [X3: complex,Y3: complex] :
% 0.30/0.60        ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 )
% 0.30/0.60        = ( dvd_dvd_complex @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_dvd_iff
% 0.30/0.60  thf(fact_5055_minus__dvd__iff,axiom,
% 0.30/0.60      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X3 ) @ Y3 )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_dvd_iff
% 0.30/0.60  thf(fact_5056_minus__dvd__iff,axiom,
% 0.30/0.60      ! [X3: rat,Y3: rat] :
% 0.30/0.60        ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X3 ) @ Y3 )
% 0.30/0.60        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_dvd_iff
% 0.30/0.60  thf(fact_5057_mod__minus__minus,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_minus_minus
% 0.30/0.60  thf(fact_5058_mod__minus__minus,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_minus_minus
% 0.30/0.60  thf(fact_5059_of__bool__less__eq__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.30/0.60        = ( P
% 0.30/0.60         => Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_eq_iff
% 0.30/0.60  thf(fact_5060_of__bool__less__eq__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.30/0.60        = ( P
% 0.30/0.60         => Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_eq_iff
% 0.30/0.60  thf(fact_5061_of__bool__less__eq__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.30/0.60        = ( P
% 0.30/0.60         => Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_eq_iff
% 0.30/0.60  thf(fact_5062_of__bool__less__eq__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.30/0.60        = ( P
% 0.30/0.60         => Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_eq_iff
% 0.30/0.60  thf(fact_5063_of__bool__eq_I1_J,axiom,
% 0.30/0.60      ( ( zero_n1201886186963655149omplex @ $false )
% 0.30/0.60      = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(1)
% 0.30/0.60  thf(fact_5064_of__bool__eq_I1_J,axiom,
% 0.30/0.60      ( ( zero_n3304061248610475627l_real @ $false )
% 0.30/0.60      = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(1)
% 0.30/0.60  thf(fact_5065_of__bool__eq_I1_J,axiom,
% 0.30/0.60      ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.30/0.60      = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(1)
% 0.30/0.60  thf(fact_5066_of__bool__eq_I1_J,axiom,
% 0.30/0.60      ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.30/0.60      = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(1)
% 0.30/0.60  thf(fact_5067_of__bool__eq_I1_J,axiom,
% 0.30/0.60      ( ( zero_n2684676970156552555ol_int @ $false )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(1)
% 0.30/0.60  thf(fact_5068_of__bool__eq_I1_J,axiom,
% 0.30/0.60      ( ( zero_n356916108424825756nteger @ $false )
% 0.30/0.60      = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(1)
% 0.30/0.60  thf(fact_5069_of__bool__eq__0__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.30/0.60          = zero_zero_complex )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_0_iff
% 0.30/0.60  thf(fact_5070_of__bool__eq__0__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.30/0.60          = zero_zero_real )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_0_iff
% 0.30/0.60  thf(fact_5071_of__bool__eq__0__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.30/0.60          = zero_zero_rat )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_0_iff
% 0.30/0.60  thf(fact_5072_of__bool__eq__0__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_0_iff
% 0.30/0.60  thf(fact_5073_of__bool__eq__0__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_0_iff
% 0.30/0.60  thf(fact_5074_of__bool__eq__0__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n356916108424825756nteger @ P )
% 0.30/0.60          = zero_z3403309356797280102nteger )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_0_iff
% 0.30/0.60  thf(fact_5075_real__add__minus__iff,axiom,
% 0.30/0.60      ! [X3: real,A: real] :
% 0.30/0.60        ( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
% 0.30/0.60          = zero_zero_real )
% 0.30/0.60        = ( X3 = A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % real_add_minus_iff
% 0.30/0.60  thf(fact_5076_of__bool__less__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.30/0.60        = ( ~ P
% 0.30/0.60          & Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_iff
% 0.30/0.60  thf(fact_5077_of__bool__less__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.30/0.60        = ( ~ P
% 0.30/0.60          & Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_iff
% 0.30/0.60  thf(fact_5078_of__bool__less__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.30/0.60        = ( ~ P
% 0.30/0.60          & Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_iff
% 0.30/0.60  thf(fact_5079_of__bool__less__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.30/0.60        = ( ~ P
% 0.30/0.60          & Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_iff
% 0.30/0.60  thf(fact_5080_of__bool__less__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.30/0.60        = ( ~ P
% 0.30/0.60          & Q ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_iff
% 0.30/0.60  thf(fact_5081_of__bool__eq_I2_J,axiom,
% 0.30/0.60      ( ( zero_n1201886186963655149omplex @ $true )
% 0.30/0.60      = one_one_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(2)
% 0.30/0.60  thf(fact_5082_of__bool__eq_I2_J,axiom,
% 0.30/0.60      ( ( zero_n3304061248610475627l_real @ $true )
% 0.30/0.60      = one_one_real ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(2)
% 0.30/0.60  thf(fact_5083_of__bool__eq_I2_J,axiom,
% 0.30/0.60      ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.30/0.60      = one_one_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(2)
% 0.30/0.60  thf(fact_5084_of__bool__eq_I2_J,axiom,
% 0.30/0.60      ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.30/0.60      = one_one_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(2)
% 0.30/0.60  thf(fact_5085_of__bool__eq_I2_J,axiom,
% 0.30/0.60      ( ( zero_n2684676970156552555ol_int @ $true )
% 0.30/0.60      = one_one_int ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(2)
% 0.30/0.60  thf(fact_5086_of__bool__eq_I2_J,axiom,
% 0.30/0.60      ( ( zero_n356916108424825756nteger @ $true )
% 0.30/0.60      = one_one_Code_integer ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq(2)
% 0.30/0.60  thf(fact_5087_of__bool__eq__1__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.30/0.60          = one_one_complex )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_1_iff
% 0.30/0.60  thf(fact_5088_of__bool__eq__1__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.30/0.60          = one_one_real )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_1_iff
% 0.30/0.60  thf(fact_5089_of__bool__eq__1__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.30/0.60          = one_one_rat )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_1_iff
% 0.30/0.60  thf(fact_5090_of__bool__eq__1__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.30/0.60          = one_one_nat )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_1_iff
% 0.30/0.60  thf(fact_5091_of__bool__eq__1__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.30/0.60          = one_one_int )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_1_iff
% 0.30/0.60  thf(fact_5092_of__bool__eq__1__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ( zero_n356916108424825756nteger @ P )
% 0.30/0.60          = one_one_Code_integer )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_eq_1_iff
% 0.30/0.60  thf(fact_5093_length__replicate,axiom,
% 0.30/0.60      ! [N: nat,X3: vEBT_VEBT] :
% 0.30/0.60        ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) )
% 0.30/0.60        = N ) ).
% 0.30/0.60  
% 0.30/0.60  % length_replicate
% 0.30/0.60  thf(fact_5094_length__replicate,axiom,
% 0.30/0.60      ! [N: nat,X3: $o] :
% 0.30/0.60        ( ( size_size_list_o @ ( replicate_o @ N @ X3 ) )
% 0.30/0.60        = N ) ).
% 0.30/0.60  
% 0.30/0.60  % length_replicate
% 0.30/0.60  thf(fact_5095_length__replicate,axiom,
% 0.30/0.60      ! [N: nat,X3: nat] :
% 0.30/0.60        ( ( size_size_list_nat @ ( replicate_nat @ N @ X3 ) )
% 0.30/0.60        = N ) ).
% 0.30/0.60  
% 0.30/0.60  % length_replicate
% 0.30/0.60  thf(fact_5096_of__bool__or__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( zero_n2687167440665602831ol_nat
% 0.30/0.60          @ ( P
% 0.30/0.60            | Q ) )
% 0.30/0.60        = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_or_iff
% 0.30/0.60  thf(fact_5097_of__bool__or__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( zero_n2684676970156552555ol_int
% 0.30/0.60          @ ( P
% 0.30/0.60            | Q ) )
% 0.30/0.60        = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_or_iff
% 0.30/0.60  thf(fact_5098_of__bool__or__iff,axiom,
% 0.30/0.60      ! [P: $o,Q: $o] :
% 0.30/0.60        ( ( zero_n356916108424825756nteger
% 0.30/0.60          @ ( P
% 0.30/0.60            | Q ) )
% 0.30/0.60        = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_or_iff
% 0.30/0.60  thf(fact_5099_neg__less__eq__nonneg,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_eq_nonneg
% 0.30/0.60  thf(fact_5100_neg__less__eq__nonneg,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_eq_nonneg
% 0.30/0.60  thf(fact_5101_neg__less__eq__nonneg,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_eq_nonneg
% 0.30/0.60  thf(fact_5102_neg__less__eq__nonneg,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_eq_nonneg
% 0.30/0.60  thf(fact_5103_less__eq__neg__nonpos,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_eq_neg_nonpos
% 0.30/0.60  thf(fact_5104_less__eq__neg__nonpos,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_eq_neg_nonpos
% 0.30/0.60  thf(fact_5105_less__eq__neg__nonpos,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_eq_neg_nonpos
% 0.30/0.60  thf(fact_5106_less__eq__neg__nonpos,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_eq_neg_nonpos
% 0.30/0.60  thf(fact_5107_neg__le__0__iff__le,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/0.60        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_0_iff_le
% 0.30/0.60  thf(fact_5108_neg__le__0__iff__le,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.60        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_0_iff_le
% 0.30/0.60  thf(fact_5109_neg__le__0__iff__le,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.30/0.60        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_0_iff_le
% 0.30/0.60  thf(fact_5110_neg__le__0__iff__le,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_0_iff_le
% 0.30/0.60  thf(fact_5111_neg__0__le__iff__le,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_le_iff_le
% 0.30/0.60  thf(fact_5112_neg__0__le__iff__le,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_le_iff_le
% 0.30/0.60  thf(fact_5113_neg__0__le__iff__le,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_le_iff_le
% 0.30/0.60  thf(fact_5114_neg__0__le__iff__le,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_le_iff_le
% 0.30/0.60  thf(fact_5115_neg__less__0__iff__less,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/0.60        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_0_iff_less
% 0.30/0.60  thf(fact_5116_neg__less__0__iff__less,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_0_iff_less
% 0.30/0.60  thf(fact_5117_neg__less__0__iff__less,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.60        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_0_iff_less
% 0.30/0.60  thf(fact_5118_neg__less__0__iff__less,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.30/0.60        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_0_iff_less
% 0.30/0.60  thf(fact_5119_neg__0__less__iff__less,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_less_iff_less
% 0.30/0.60  thf(fact_5120_neg__0__less__iff__less,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_less_iff_less
% 0.30/0.60  thf(fact_5121_neg__0__less__iff__less,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_less_iff_less
% 0.30/0.60  thf(fact_5122_neg__0__less__iff__less,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_0_less_iff_less
% 0.30/0.60  thf(fact_5123_neg__less__pos,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_pos
% 0.30/0.60  thf(fact_5124_neg__less__pos,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_pos
% 0.30/0.60  thf(fact_5125_neg__less__pos,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_pos
% 0.30/0.60  thf(fact_5126_neg__less__pos,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_pos
% 0.30/0.60  thf(fact_5127_less__neg__neg,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_neg_neg
% 0.30/0.60  thf(fact_5128_less__neg__neg,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_neg_neg
% 0.30/0.60  thf(fact_5129_less__neg__neg,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_neg_neg
% 0.30/0.60  thf(fact_5130_less__neg__neg,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_neg_neg
% 0.30/0.60  thf(fact_5131_ab__left__minus,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.60        = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % ab_left_minus
% 0.30/0.60  thf(fact_5132_ab__left__minus,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % ab_left_minus
% 0.30/0.60  thf(fact_5133_ab__left__minus,axiom,
% 0.30/0.60      ! [A: complex] :
% 0.30/0.60        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.30/0.60        = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % ab_left_minus
% 0.30/0.60  thf(fact_5134_ab__left__minus,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.60        = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % ab_left_minus
% 0.30/0.60  thf(fact_5135_ab__left__minus,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.60        = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % ab_left_minus
% 0.30/0.60  thf(fact_5136_add_Oright__inverse,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.60        = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % add.right_inverse
% 0.30/0.60  thf(fact_5137_add_Oright__inverse,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % add.right_inverse
% 0.30/0.60  thf(fact_5138_add_Oright__inverse,axiom,
% 0.30/0.60      ! [A: complex] :
% 0.30/0.60        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.60        = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % add.right_inverse
% 0.30/0.60  thf(fact_5139_add_Oright__inverse,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.60        = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % add.right_inverse
% 0.30/0.60  thf(fact_5140_add_Oright__inverse,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.60        = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % add.right_inverse
% 0.30/0.60  thf(fact_5141_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60      ! [B2: real] :
% 0.30/0.60        ( ( minus_minus_real @ zero_zero_real @ B2 )
% 0.30/0.60        = ( uminus_uminus_real @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(3)
% 0.30/0.60  thf(fact_5142_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60      ! [B2: int] :
% 0.30/0.60        ( ( minus_minus_int @ zero_zero_int @ B2 )
% 0.30/0.60        = ( uminus_uminus_int @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(3)
% 0.30/0.60  thf(fact_5143_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60      ! [B2: complex] :
% 0.30/0.60        ( ( minus_minus_complex @ zero_zero_complex @ B2 )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(3)
% 0.30/0.60  thf(fact_5144_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60      ! [B2: code_integer] :
% 0.30/0.60        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(3)
% 0.30/0.60  thf(fact_5145_verit__minus__simplify_I3_J,axiom,
% 0.30/0.60      ! [B2: rat] :
% 0.30/0.60        ( ( minus_minus_rat @ zero_zero_rat @ B2 )
% 0.30/0.60        = ( uminus_uminus_rat @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_minus_simplify(3)
% 0.30/0.60  thf(fact_5146_diff__0,axiom,
% 0.30/0.60      ! [A: real] :
% 0.30/0.60        ( ( minus_minus_real @ zero_zero_real @ A )
% 0.30/0.60        = ( uminus_uminus_real @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_0
% 0.30/0.60  thf(fact_5147_diff__0,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( minus_minus_int @ zero_zero_int @ A )
% 0.30/0.60        = ( uminus_uminus_int @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_0
% 0.30/0.60  thf(fact_5148_diff__0,axiom,
% 0.30/0.60      ! [A: complex] :
% 0.30/0.60        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_0
% 0.30/0.60  thf(fact_5149_diff__0,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_0
% 0.30/0.60  thf(fact_5150_diff__0,axiom,
% 0.30/0.60      ! [A: rat] :
% 0.30/0.60        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.30/0.60        = ( uminus_uminus_rat @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_0
% 0.30/0.60  thf(fact_5151_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_simps(3)
% 0.30/0.60  thf(fact_5152_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_simps(3)
% 0.30/0.60  thf(fact_5153_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_simps(3)
% 0.30/0.60  thf(fact_5154_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_simps(3)
% 0.30/0.60  thf(fact_5155_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.60      ! [M: num,N: num] :
% 0.30/0.60        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_simps(3)
% 0.30/0.60  thf(fact_5156_mult__minus1,axiom,
% 0.30/0.60      ! [Z: real] :
% 0.30/0.60        ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.30/0.60        = ( uminus_uminus_real @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1
% 0.30/0.60  thf(fact_5157_mult__minus1,axiom,
% 0.30/0.60      ! [Z: int] :
% 0.30/0.60        ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.30/0.60        = ( uminus_uminus_int @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1
% 0.30/0.60  thf(fact_5158_mult__minus1,axiom,
% 0.30/0.60      ! [Z: complex] :
% 0.30/0.60        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1
% 0.30/0.60  thf(fact_5159_mult__minus1,axiom,
% 0.30/0.60      ! [Z: code_integer] :
% 0.30/0.60        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1
% 0.30/0.60  thf(fact_5160_mult__minus1,axiom,
% 0.30/0.60      ! [Z: rat] :
% 0.30/0.60        ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.30/0.60        = ( uminus_uminus_rat @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1
% 0.30/0.60  thf(fact_5161_mult__minus1__right,axiom,
% 0.30/0.60      ! [Z: real] :
% 0.30/0.60        ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60        = ( uminus_uminus_real @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1_right
% 0.30/0.60  thf(fact_5162_mult__minus1__right,axiom,
% 0.30/0.60      ! [Z: int] :
% 0.30/0.60        ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60        = ( uminus_uminus_int @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1_right
% 0.30/0.60  thf(fact_5163_mult__minus1__right,axiom,
% 0.30/0.60      ! [Z: complex] :
% 0.30/0.60        ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1_right
% 0.30/0.60  thf(fact_5164_mult__minus1__right,axiom,
% 0.30/0.60      ! [Z: code_integer] :
% 0.30/0.60        ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1_right
% 0.30/0.60  thf(fact_5165_mult__minus1__right,axiom,
% 0.30/0.60      ! [Z: rat] :
% 0.30/0.60        ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60        = ( uminus_uminus_rat @ Z ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_minus1_right
% 0.30/0.60  thf(fact_5166_uminus__add__conv__diff,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.60        = ( minus_minus_real @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % uminus_add_conv_diff
% 0.30/0.60  thf(fact_5167_uminus__add__conv__diff,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.60        = ( minus_minus_int @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % uminus_add_conv_diff
% 0.30/0.60  thf(fact_5168_uminus__add__conv__diff,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.30/0.60        = ( minus_minus_complex @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % uminus_add_conv_diff
% 0.30/0.60  thf(fact_5169_uminus__add__conv__diff,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.60        = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % uminus_add_conv_diff
% 0.30/0.60  thf(fact_5170_uminus__add__conv__diff,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.60        = ( minus_minus_rat @ B2 @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % uminus_add_conv_diff
% 0.30/0.60  thf(fact_5171_diff__minus__eq__add,axiom,
% 0.30/0.60      ! [A: real,B2: real] :
% 0.30/0.60        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.60        = ( plus_plus_real @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_minus_eq_add
% 0.30/0.60  thf(fact_5172_diff__minus__eq__add,axiom,
% 0.30/0.60      ! [A: int,B2: int] :
% 0.30/0.60        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.60        = ( plus_plus_int @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_minus_eq_add
% 0.30/0.60  thf(fact_5173_diff__minus__eq__add,axiom,
% 0.30/0.60      ! [A: complex,B2: complex] :
% 0.30/0.60        ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.60        = ( plus_plus_complex @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_minus_eq_add
% 0.30/0.60  thf(fact_5174_diff__minus__eq__add,axiom,
% 0.30/0.60      ! [A: code_integer,B2: code_integer] :
% 0.30/0.60        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.60        = ( plus_p5714425477246183910nteger @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_minus_eq_add
% 0.30/0.60  thf(fact_5175_diff__minus__eq__add,axiom,
% 0.30/0.60      ! [A: rat,B2: rat] :
% 0.30/0.60        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.60        = ( plus_plus_rat @ A @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_minus_eq_add
% 0.30/0.60  thf(fact_5176_divide__minus1,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( divide_divide_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60        = ( uminus_uminus_real @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_minus1
% 0.30/0.60  thf(fact_5177_divide__minus1,axiom,
% 0.30/0.60      ! [X3: complex] :
% 0.30/0.60        ( ( divide1717551699836669952omplex @ X3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_minus1
% 0.30/0.60  thf(fact_5178_divide__minus1,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( divide_divide_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60        = ( uminus_uminus_rat @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_minus1
% 0.30/0.60  thf(fact_5179_div__minus1__right,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60        = ( uminus_uminus_int @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_minus1_right
% 0.30/0.60  thf(fact_5180_div__minus1__right,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_minus1_right
% 0.30/0.60  thf(fact_5181_minus__mod__self1,axiom,
% 0.30/0.60      ! [B2: int,A: int] :
% 0.30/0.60        ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
% 0.30/0.60        = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mod_self1
% 0.30/0.60  thf(fact_5182_minus__mod__self1,axiom,
% 0.30/0.60      ! [B2: code_integer,A: code_integer] :
% 0.30/0.60        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A ) @ B2 )
% 0.30/0.60        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_mod_self1
% 0.30/0.60  thf(fact_5183_zero__less__of__bool__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_of_bool_iff
% 0.30/0.60  thf(fact_5184_zero__less__of__bool__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_of_bool_iff
% 0.30/0.60  thf(fact_5185_zero__less__of__bool__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_of_bool_iff
% 0.30/0.60  thf(fact_5186_zero__less__of__bool__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_of_bool_iff
% 0.30/0.60  thf(fact_5187_zero__less__of__bool__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.30/0.60        = P ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_of_bool_iff
% 0.30/0.60  thf(fact_5188_of__bool__less__one__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_one_iff
% 0.30/0.60  thf(fact_5189_of__bool__less__one__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_one_iff
% 0.30/0.60  thf(fact_5190_of__bool__less__one__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_one_iff
% 0.30/0.60  thf(fact_5191_of__bool__less__one__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_one_iff
% 0.30/0.60  thf(fact_5192_of__bool__less__one__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.30/0.60        = ( ~ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_less_one_iff
% 0.30/0.60  thf(fact_5193_of__bool__not__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.30/0.60        = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_not_iff
% 0.30/0.60  thf(fact_5194_of__bool__not__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.30/0.60        = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_not_iff
% 0.30/0.60  thf(fact_5195_of__bool__not__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.30/0.60        = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_not_iff
% 0.30/0.60  thf(fact_5196_of__bool__not__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.30/0.60        = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_not_iff
% 0.30/0.60  thf(fact_5197_of__bool__not__iff,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( zero_n356916108424825756nteger @ ~ P )
% 0.30/0.60        = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_bool_not_iff
% 0.30/0.60  thf(fact_5198_Suc__0__mod__eq,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.60        = ( zero_n2687167440665602831ol_nat
% 0.30/0.60          @ ( N
% 0.30/0.60           != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_0_mod_eq
% 0.30/0.60  thf(fact_5199_signed__take__bit__of__minus__1,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_of_minus_1
% 0.30/0.60  thf(fact_5200_signed__take__bit__of__minus__1,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_of_minus_1
% 0.30/0.60  thf(fact_5201_Ball__set__replicate,axiom,
% 0.30/0.60      ! [N: nat,A: int,P: int > $o] :
% 0.30/0.60        ( ( ! [X: int] :
% 0.30/0.60              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.30/0.60             => ( P @ X ) ) )
% 0.30/0.60        = ( ( P @ A )
% 0.30/0.60          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Ball_set_replicate
% 0.30/0.60  thf(fact_5202_Ball__set__replicate,axiom,
% 0.30/0.60      ! [N: nat,A: nat,P: nat > $o] :
% 0.30/0.60        ( ( ! [X: nat] :
% 0.30/0.60              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.30/0.60             => ( P @ X ) ) )
% 0.30/0.60        = ( ( P @ A )
% 0.30/0.60          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Ball_set_replicate
% 0.30/0.60  thf(fact_5203_Ball__set__replicate,axiom,
% 0.30/0.60      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.60        ( ( ! [X: vEBT_VEBT] :
% 0.30/0.60              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.30/0.60             => ( P @ X ) ) )
% 0.30/0.60        = ( ( P @ A )
% 0.30/0.60          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Ball_set_replicate
% 0.30/0.60  thf(fact_5204_Bex__set__replicate,axiom,
% 0.30/0.60      ! [N: nat,A: int,P: int > $o] :
% 0.30/0.60        ( ( ? [X: int] :
% 0.30/0.60              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 0.30/0.60              & ( P @ X ) ) )
% 0.30/0.60        = ( ( P @ A )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Bex_set_replicate
% 0.30/0.60  thf(fact_5205_Bex__set__replicate,axiom,
% 0.30/0.60      ! [N: nat,A: nat,P: nat > $o] :
% 0.30/0.60        ( ( ? [X: nat] :
% 0.30/0.60              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 0.30/0.60              & ( P @ X ) ) )
% 0.30/0.60        = ( ( P @ A )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Bex_set_replicate
% 0.30/0.60  thf(fact_5206_Bex__set__replicate,axiom,
% 0.30/0.60      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.60        ( ( ? [X: vEBT_VEBT] :
% 0.30/0.60              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 0.30/0.60              & ( P @ X ) ) )
% 0.30/0.60        = ( ( P @ A )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Bex_set_replicate
% 0.30/0.60  thf(fact_5207_in__set__replicate,axiom,
% 0.30/0.60      ! [X3: real,N: nat,Y3: real] :
% 0.30/0.60        ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ Y3 ) ) )
% 0.30/0.60        = ( ( X3 = Y3 )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % in_set_replicate
% 0.30/0.60  thf(fact_5208_in__set__replicate,axiom,
% 0.30/0.60      ! [X3: complex,N: nat,Y3: complex] :
% 0.30/0.60        ( ( member_complex @ X3 @ ( set_complex2 @ ( replicate_complex @ N @ Y3 ) ) )
% 0.30/0.60        = ( ( X3 = Y3 )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % in_set_replicate
% 0.30/0.60  thf(fact_5209_in__set__replicate,axiom,
% 0.30/0.60      ! [X3: int,N: nat,Y3: int] :
% 0.30/0.60        ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ Y3 ) ) )
% 0.30/0.60        = ( ( X3 = Y3 )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % in_set_replicate
% 0.30/0.60  thf(fact_5210_in__set__replicate,axiom,
% 0.30/0.60      ! [X3: nat,N: nat,Y3: nat] :
% 0.30/0.60        ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ Y3 ) ) )
% 0.30/0.60        = ( ( X3 = Y3 )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % in_set_replicate
% 0.30/0.60  thf(fact_5211_in__set__replicate,axiom,
% 0.30/0.60      ! [X3: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
% 0.30/0.60        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y3 ) ) )
% 0.30/0.60        = ( ( X3 = Y3 )
% 0.30/0.60          & ( N != zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % in_set_replicate
% 0.30/0.60  thf(fact_5212_nth__replicate,axiom,
% 0.30/0.60      ! [I: nat,N: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_nat @ I @ N )
% 0.30/0.60       => ( ( nth_int @ ( replicate_int @ N @ X3 ) @ I )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % nth_replicate
% 0.30/0.60  thf(fact_5213_nth__replicate,axiom,
% 0.30/0.60      ! [I: nat,N: nat,X3: nat] :
% 0.30/0.60        ( ( ord_less_nat @ I @ N )
% 0.30/0.60       => ( ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % nth_replicate
% 0.30/0.60  thf(fact_5214_nth__replicate,axiom,
% 0.30/0.60      ! [I: nat,N: nat,X3: vEBT_VEBT] :
% 0.30/0.60        ( ( ord_less_nat @ I @ N )
% 0.30/0.60       => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % nth_replicate
% 0.30/0.60  thf(fact_5215_dbl__simps_I1_J,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_simps(1)
% 0.30/0.60  thf(fact_5216_dbl__simps_I1_J,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_simps(1)
% 0.30/0.60  thf(fact_5217_dbl__simps_I1_J,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_simps(1)
% 0.30/0.60  thf(fact_5218_dbl__simps_I1_J,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_simps(1)
% 0.30/0.60  thf(fact_5219_dbl__simps_I1_J,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_simps(1)
% 0.30/0.60  thf(fact_5220_triangle__Suc,axiom,
% 0.30/0.60      ! [N: nat] :
% 0.30/0.60        ( ( nat_triangle @ ( suc @ N ) )
% 0.30/0.60        = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % triangle_Suc
% 0.30/0.60  thf(fact_5221_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.30/0.60      = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(8)
% 0.30/0.60  thf(fact_5222_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(8)
% 0.30/0.60  thf(fact_5223_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.30/0.60      = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(8)
% 0.30/0.60  thf(fact_5224_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.30/0.60      = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(8)
% 0.30/0.60  thf(fact_5225_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.60      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.30/0.60      = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(8)
% 0.30/0.60  thf(fact_5226_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60      = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(7)
% 0.30/0.60  thf(fact_5227_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(7)
% 0.30/0.60  thf(fact_5228_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60      = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(7)
% 0.30/0.60  thf(fact_5229_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60      = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(7)
% 0.30/0.60  thf(fact_5230_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.60      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60      = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(7)
% 0.30/0.60  thf(fact_5231_diff__numeral__special_I12_J,axiom,
% 0.30/0.60      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60      = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_numeral_special(12)
% 0.30/0.60  thf(fact_5232_diff__numeral__special_I12_J,axiom,
% 0.30/0.60      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_numeral_special(12)
% 0.30/0.60  thf(fact_5233_diff__numeral__special_I12_J,axiom,
% 0.30/0.60      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60      = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_numeral_special(12)
% 0.30/0.60  thf(fact_5234_diff__numeral__special_I12_J,axiom,
% 0.30/0.60      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60      = zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_numeral_special(12)
% 0.30/0.60  thf(fact_5235_diff__numeral__special_I12_J,axiom,
% 0.30/0.60      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60      = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_numeral_special(12)
% 0.30/0.60  thf(fact_5236_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ one_one_real )
% 0.30/0.60          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_eq_numeral_iff
% 0.30/0.60  thf(fact_5237_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_int @ one_one_int )
% 0.30/0.60          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_eq_numeral_iff
% 0.30/0.60  thf(fact_5238_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.30/0.60          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_eq_numeral_iff
% 0.30/0.60  thf(fact_5239_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.30/0.60          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_eq_numeral_iff
% 0.30/0.60  thf(fact_5240_neg__one__eq__numeral__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.30/0.60          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_eq_numeral_iff
% 0.30/0.60  thf(fact_5241_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.60          = ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_eq_neg_one_iff
% 0.30/0.60  thf(fact_5242_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.60          = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_eq_neg_one_iff
% 0.30/0.60  thf(fact_5243_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.60          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_eq_neg_one_iff
% 0.30/0.60  thf(fact_5244_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.60          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_eq_neg_one_iff
% 0.30/0.60  thf(fact_5245_numeral__eq__neg__one__iff,axiom,
% 0.30/0.60      ! [N: num] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.60          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60        = ( N = one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_eq_neg_one_iff
% 0.30/0.61  thf(fact_5246_minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.30/0.61        = one_one_real ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_mult_self
% 0.30/0.61  thf(fact_5247_minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.30/0.61        = one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_mult_self
% 0.30/0.61  thf(fact_5248_minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.30/0.61        = one_one_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_mult_self
% 0.30/0.61  thf(fact_5249_minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.30/0.61        = one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_mult_self
% 0.30/0.61  thf(fact_5250_minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.30/0.61        = one_one_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_mult_self
% 0.30/0.61  thf(fact_5251_left__minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat,A: real] :
% 0.30/0.61        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % left_minus_one_mult_self
% 0.30/0.61  thf(fact_5252_left__minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat,A: int] :
% 0.30/0.61        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % left_minus_one_mult_self
% 0.30/0.61  thf(fact_5253_left__minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat,A: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % left_minus_one_mult_self
% 0.30/0.61  thf(fact_5254_left__minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat,A: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % left_minus_one_mult_self
% 0.30/0.61  thf(fact_5255_left__minus__one__mult__self,axiom,
% 0.30/0.61      ! [N: nat,A: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % left_minus_one_mult_self
% 0.30/0.61  thf(fact_5256_mod__minus1__right,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61        = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus1_right
% 0.30/0.61  thf(fact_5257_mod__minus1__right,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.61        = zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus1_right
% 0.30/0.61  thf(fact_5258_max__number__of_I2_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61            = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(2)
% 0.30/0.61  thf(fact_5259_max__number__of_I2_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61            = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(2)
% 0.30/0.61  thf(fact_5260_max__number__of_I2_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(2)
% 0.30/0.61  thf(fact_5261_max__number__of_I2_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61            = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(2)
% 0.30/0.61  thf(fact_5262_max__number__of_I3_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.61         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.61            = ( numeral_numeral_real @ V ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.61         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.61            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(3)
% 0.30/0.61  thf(fact_5263_max__number__of_I3_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.61         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.61            = ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.61         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.30/0.61            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(3)
% 0.30/0.61  thf(fact_5264_max__number__of_I3_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.61         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.61            = ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.61         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.61            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(3)
% 0.30/0.61  thf(fact_5265_max__number__of_I3_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.61         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.61            = ( numeral_numeral_int @ V ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.61         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.61            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(3)
% 0.30/0.61  thf(fact_5266_max__number__of_I4_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(4)
% 0.30/0.61  thf(fact_5267_max__number__of_I4_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.30/0.61            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(4)
% 0.30/0.61  thf(fact_5268_max__number__of_I4_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(4)
% 0.30/0.61  thf(fact_5269_max__number__of_I4_J,axiom,
% 0.30/0.61      ! [U: num,V: num] :
% 0.30/0.61        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.30/0.61        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.61            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % max_number_of(4)
% 0.30/0.61  thf(fact_5270_semiring__norm_I168_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: real] :
% 0.30/0.61        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(168)
% 0.30/0.61  thf(fact_5271_semiring__norm_I168_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: int] :
% 0.30/0.61        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(168)
% 0.30/0.61  thf(fact_5272_semiring__norm_I168_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: complex] :
% 0.30/0.61        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(168)
% 0.30/0.61  thf(fact_5273_semiring__norm_I168_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.61        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(168)
% 0.30/0.61  thf(fact_5274_semiring__norm_I168_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: rat] :
% 0.30/0.61        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(168)
% 0.30/0.61  thf(fact_5275_diff__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.61        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(3)
% 0.30/0.61  thf(fact_5276_diff__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(3)
% 0.30/0.61  thf(fact_5277_diff__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(3)
% 0.30/0.61  thf(fact_5278_diff__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(3)
% 0.30/0.61  thf(fact_5279_diff__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.61        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(3)
% 0.30/0.61  thf(fact_5280_diff__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(2)
% 0.30/0.61  thf(fact_5281_diff__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(2)
% 0.30/0.61  thf(fact_5282_diff__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.61        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(2)
% 0.30/0.61  thf(fact_5283_diff__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.61        = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(2)
% 0.30/0.61  thf(fact_5284_diff__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_simps(2)
% 0.30/0.61  thf(fact_5285_semiring__norm_I172_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: real] :
% 0.30/0.61        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(172)
% 0.30/0.61  thf(fact_5286_semiring__norm_I172_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: int] :
% 0.30/0.61        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(172)
% 0.30/0.61  thf(fact_5287_semiring__norm_I172_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(172)
% 0.30/0.61  thf(fact_5288_semiring__norm_I172_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(172)
% 0.30/0.61  thf(fact_5289_semiring__norm_I172_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(172)
% 0.30/0.61  thf(fact_5290_semiring__norm_I171_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: real] :
% 0.30/0.61        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(171)
% 0.30/0.61  thf(fact_5291_semiring__norm_I171_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: int] :
% 0.30/0.61        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(171)
% 0.30/0.61  thf(fact_5292_semiring__norm_I171_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(171)
% 0.30/0.61  thf(fact_5293_semiring__norm_I171_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(171)
% 0.30/0.61  thf(fact_5294_semiring__norm_I171_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.30/0.61        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(171)
% 0.30/0.61  thf(fact_5295_semiring__norm_I170_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: real] :
% 0.30/0.61        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 0.30/0.61        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(170)
% 0.30/0.61  thf(fact_5296_semiring__norm_I170_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: int] :
% 0.30/0.61        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 0.30/0.61        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(170)
% 0.30/0.61  thf(fact_5297_semiring__norm_I170_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 0.30/0.61        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(170)
% 0.30/0.61  thf(fact_5298_semiring__norm_I170_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(170)
% 0.30/0.61  thf(fact_5299_semiring__norm_I170_J,axiom,
% 0.30/0.61      ! [V: num,W: num,Y3: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 0.30/0.61        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % semiring_norm(170)
% 0.30/0.61  thf(fact_5300_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.61        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(3)
% 0.30/0.61  thf(fact_5301_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(3)
% 0.30/0.61  thf(fact_5302_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(3)
% 0.30/0.61  thf(fact_5303_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(3)
% 0.30/0.61  thf(fact_5304_mult__neg__numeral__simps_I3_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.61        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(3)
% 0.30/0.61  thf(fact_5305_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.61        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(2)
% 0.30/0.61  thf(fact_5306_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(2)
% 0.30/0.61  thf(fact_5307_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(2)
% 0.30/0.61  thf(fact_5308_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(2)
% 0.30/0.61  thf(fact_5309_mult__neg__numeral__simps_I2_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.30/0.61        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(2)
% 0.30/0.61  thf(fact_5310_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(1)
% 0.30/0.61  thf(fact_5311_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(1)
% 0.30/0.61  thf(fact_5312_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.61        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(1)
% 0.30/0.61  thf(fact_5313_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.61        = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(1)
% 0.30/0.61  thf(fact_5314_mult__neg__numeral__simps_I1_J,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_neg_numeral_simps(1)
% 0.30/0.61  thf(fact_5315_neg__numeral__le__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.61        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_iff
% 0.30/0.61  thf(fact_5316_neg__numeral__le__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.61        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_iff
% 0.30/0.61  thf(fact_5317_neg__numeral__le__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.61        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_iff
% 0.30/0.61  thf(fact_5318_neg__numeral__le__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61        = ( ord_less_eq_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_iff
% 0.30/0.61  thf(fact_5319_neg__numeral__less__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.61        = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_iff
% 0.30/0.61  thf(fact_5320_neg__numeral__less__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61        = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_iff
% 0.30/0.61  thf(fact_5321_neg__numeral__less__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.61        = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_iff
% 0.30/0.61  thf(fact_5322_neg__numeral__less__iff,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.61        = ( ord_less_num @ N @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_iff
% 0.30/0.61  thf(fact_5323_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_le_neg_numeral_iff
% 0.30/0.61  thf(fact_5324_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_le_neg_numeral_iff
% 0.30/0.61  thf(fact_5325_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_le_neg_numeral_iff
% 0.30/0.61  thf(fact_5326_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_le_neg_numeral_iff
% 0.30/0.61  thf(fact_5327_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: real,W: num,A: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.30/0.61        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_le_eq_numeral1(2)
% 0.30/0.61  thf(fact_5328_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: rat,W: num,A: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.30/0.61        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_le_eq_numeral1(2)
% 0.30/0.61  thf(fact_5329_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: real,B2: real,W: num] :
% 0.30/0.61        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.61        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5330_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,W: num] :
% 0.30/0.61        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.61        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5331_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: real,W: num,A: real] :
% 0.30/0.61        ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61          = A )
% 0.30/0.61        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61             != zero_zero_real )
% 0.30/0.61           => ( B2
% 0.30/0.61              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.30/0.61          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61              = zero_zero_real )
% 0.30/0.61           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_eq_numeral1(2)
% 0.30/0.61  thf(fact_5332_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: complex,W: num,A: complex] :
% 0.30/0.61        ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.61          = A )
% 0.30/0.61        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61             != zero_zero_complex )
% 0.30/0.61           => ( B2
% 0.30/0.61              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.30/0.61          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61              = zero_zero_complex )
% 0.30/0.61           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_eq_numeral1(2)
% 0.30/0.61  thf(fact_5333_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: rat,W: num,A: rat] :
% 0.30/0.61        ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61          = A )
% 0.30/0.61        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61             != zero_zero_rat )
% 0.30/0.61           => ( B2
% 0.30/0.61              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.30/0.61          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61              = zero_zero_rat )
% 0.30/0.61           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_eq_numeral1(2)
% 0.30/0.61  thf(fact_5334_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: real,B2: real,W: num] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.61        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61             != zero_zero_real )
% 0.30/0.61           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61              = B2 ) )
% 0.30/0.61          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61              = zero_zero_real )
% 0.30/0.61           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5335_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: complex,B2: complex,W: num] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.30/0.61        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61             != zero_zero_complex )
% 0.30/0.61           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.61              = B2 ) )
% 0.30/0.61          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61              = zero_zero_complex )
% 0.30/0.61           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5336_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,W: num] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.61        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61             != zero_zero_rat )
% 0.30/0.61           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61              = B2 ) )
% 0.30/0.61          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61              = zero_zero_rat )
% 0.30/0.61           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5337_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_neg_one_iff
% 0.30/0.61  thf(fact_5338_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_neg_one_iff
% 0.30/0.61  thf(fact_5339_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_neg_one_iff
% 0.30/0.61  thf(fact_5340_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.61        = ( M != one ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_neg_one_iff
% 0.30/0.61  thf(fact_5341_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: real,W: num,A: real] :
% 0.30/0.61        ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.30/0.61        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_less_eq_numeral1(2)
% 0.30/0.61  thf(fact_5342_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [B2: rat,W: num,A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.30/0.61        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_less_eq_numeral1(2)
% 0.30/0.61  thf(fact_5343_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: real,B2: real,W: num] :
% 0.30/0.61        ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.61        = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5344_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,W: num] :
% 0.30/0.61        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.61        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_divide_eq_numeral1(2)
% 0.30/0.61  thf(fact_5345_power2__minus,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_minus
% 0.30/0.61  thf(fact_5346_power2__minus,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_minus
% 0.30/0.61  thf(fact_5347_power2__minus,axiom,
% 0.30/0.61      ! [A: complex] :
% 0.30/0.61        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_minus
% 0.30/0.61  thf(fact_5348_power2__minus,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_minus
% 0.30/0.61  thf(fact_5349_power2__minus,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_minus
% 0.30/0.61  thf(fact_5350_odd__of__bool__self,axiom,
% 0.30/0.61      ! [P4: $o] :
% 0.30/0.61        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % odd_of_bool_self
% 0.30/0.61  thf(fact_5351_odd__of__bool__self,axiom,
% 0.30/0.61      ! [P4: $o] :
% 0.30/0.61        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % odd_of_bool_self
% 0.30/0.61  thf(fact_5352_odd__of__bool__self,axiom,
% 0.30/0.61      ! [P4: $o] :
% 0.30/0.61        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % odd_of_bool_self
% 0.30/0.61  thf(fact_5353_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.61      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.61      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_neg_numeral_special(9)
% 0.30/0.61  thf(fact_5354_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.61      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_neg_numeral_special(9)
% 0.30/0.61  thf(fact_5355_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.61      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.61      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_neg_numeral_special(9)
% 0.30/0.61  thf(fact_5356_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.61      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.61      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_neg_numeral_special(9)
% 0.30/0.61  thf(fact_5357_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.61      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.61      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_neg_numeral_special(9)
% 0.30/0.61  thf(fact_5358_diff__numeral__special_I11_J,axiom,
% 0.30/0.61      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.61      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(11)
% 0.30/0.61  thf(fact_5359_diff__numeral__special_I11_J,axiom,
% 0.30/0.61      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(11)
% 0.30/0.61  thf(fact_5360_diff__numeral__special_I11_J,axiom,
% 0.30/0.61      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.61      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(11)
% 0.30/0.61  thf(fact_5361_diff__numeral__special_I11_J,axiom,
% 0.30/0.61      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.61      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(11)
% 0.30/0.61  thf(fact_5362_diff__numeral__special_I11_J,axiom,
% 0.30/0.61      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.61      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(11)
% 0.30/0.61  thf(fact_5363_diff__numeral__special_I10_J,axiom,
% 0.30/0.61      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.30/0.61      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(10)
% 0.30/0.61  thf(fact_5364_diff__numeral__special_I10_J,axiom,
% 0.30/0.61      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.30/0.61      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(10)
% 0.30/0.61  thf(fact_5365_diff__numeral__special_I10_J,axiom,
% 0.30/0.61      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.30/0.61      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(10)
% 0.30/0.61  thf(fact_5366_diff__numeral__special_I10_J,axiom,
% 0.30/0.61      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.30/0.61      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(10)
% 0.30/0.61  thf(fact_5367_diff__numeral__special_I10_J,axiom,
% 0.30/0.61      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.30/0.61      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(10)
% 0.30/0.61  thf(fact_5368_minus__1__div__2__eq,axiom,
% 0.30/0.61      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_1_div_2_eq
% 0.30/0.61  thf(fact_5369_minus__1__div__2__eq,axiom,
% 0.30/0.61      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.61      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_1_div_2_eq
% 0.30/0.61  thf(fact_5370_minus__1__mod__2__eq,axiom,
% 0.30/0.61      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61      = one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_1_mod_2_eq
% 0.30/0.61  thf(fact_5371_minus__1__mod__2__eq,axiom,
% 0.30/0.61      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.61      = one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_1_mod_2_eq
% 0.30/0.61  thf(fact_5372_bits__minus__1__mod__2__eq,axiom,
% 0.30/0.61      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61      = one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_minus_1_mod_2_eq
% 0.30/0.61  thf(fact_5373_bits__minus__1__mod__2__eq,axiom,
% 0.30/0.61      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.61      = one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_minus_1_mod_2_eq
% 0.30/0.61  thf(fact_5374_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [A: real,N: nat] :
% 0.30/0.61        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Power.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5375_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [A: int,N: nat] :
% 0.30/0.61        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Power.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5376_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [A: complex,N: nat] :
% 0.30/0.61        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Power.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5377_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [A: code_integer,N: nat] :
% 0.30/0.61        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Power.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5378_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [A: rat,N: nat] :
% 0.30/0.61        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Power.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5379_of__bool__half__eq__0,axiom,
% 0.30/0.61      ! [B2: $o] :
% 0.30/0.61        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61        = zero_zero_nat ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_half_eq_0
% 0.30/0.61  thf(fact_5380_of__bool__half__eq__0,axiom,
% 0.30/0.61      ! [B2: $o] :
% 0.30/0.61        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61        = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_half_eq_0
% 0.30/0.61  thf(fact_5381_of__bool__half__eq__0,axiom,
% 0.30/0.61      ! [B2: $o] :
% 0.30/0.61        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.61        = zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_half_eq_0
% 0.30/0.61  thf(fact_5382_power__minus__odd,axiom,
% 0.30/0.61      ! [N: nat,A: real] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_odd
% 0.30/0.61  thf(fact_5383_power__minus__odd,axiom,
% 0.30/0.61      ! [N: nat,A: int] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_odd
% 0.30/0.61  thf(fact_5384_power__minus__odd,axiom,
% 0.30/0.61      ! [N: nat,A: complex] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_odd
% 0.30/0.61  thf(fact_5385_power__minus__odd,axiom,
% 0.30/0.61      ! [N: nat,A: code_integer] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_odd
% 0.30/0.61  thf(fact_5386_power__minus__odd,axiom,
% 0.30/0.61      ! [N: nat,A: rat] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_odd
% 0.30/0.61  thf(fact_5387_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [N: nat,A: real] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61          = ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Parity.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5388_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [N: nat,A: int] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61          = ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Parity.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5389_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [N: nat,A: complex] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61          = ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Parity.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5390_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [N: nat,A: code_integer] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Parity.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5391_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.30/0.61      ! [N: nat,A: rat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61          = ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Parity.ring_1_class.power_minus_even
% 0.30/0.61  thf(fact_5392_diff__numeral__special_I3_J,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(3)
% 0.30/0.61  thf(fact_5393_diff__numeral__special_I3_J,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(3)
% 0.30/0.61  thf(fact_5394_diff__numeral__special_I3_J,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.30/0.61        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(3)
% 0.30/0.61  thf(fact_5395_diff__numeral__special_I3_J,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.61        = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(3)
% 0.30/0.61  thf(fact_5396_diff__numeral__special_I3_J,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.61        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(3)
% 0.30/0.61  thf(fact_5397_diff__numeral__special_I4_J,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.30/0.61        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(4)
% 0.30/0.61  thf(fact_5398_diff__numeral__special_I4_J,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.30/0.61        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(4)
% 0.30/0.61  thf(fact_5399_diff__numeral__special_I4_J,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(4)
% 0.30/0.61  thf(fact_5400_diff__numeral__special_I4_J,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(4)
% 0.30/0.61  thf(fact_5401_diff__numeral__special_I4_J,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.30/0.61        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_numeral_special(4)
% 0.30/0.61  thf(fact_5402_signed__take__bit__Suc__minus__bit0,axiom,
% 0.30/0.61      ! [N: nat,K: num] :
% 0.30/0.61        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.61        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_Suc_minus_bit0
% 0.30/0.61  thf(fact_5403_set__decode__0,axiom,
% 0.30/0.61      ! [X3: nat] :
% 0.30/0.61        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
% 0.30/0.61        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_decode_0
% 0.30/0.61  thf(fact_5404_dbl__simps_I4_J,axiom,
% 0.30/0.61      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.61      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_simps(4)
% 0.30/0.61  thf(fact_5405_dbl__simps_I4_J,axiom,
% 0.30/0.61      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_simps(4)
% 0.30/0.61  thf(fact_5406_dbl__simps_I4_J,axiom,
% 0.30/0.61      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.61      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_simps(4)
% 0.30/0.61  thf(fact_5407_dbl__simps_I4_J,axiom,
% 0.30/0.61      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.61      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_simps(4)
% 0.30/0.61  thf(fact_5408_dbl__simps_I4_J,axiom,
% 0.30/0.61      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.61      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_simps(4)
% 0.30/0.61  thf(fact_5409_power__minus1__even,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = one_one_real ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_even
% 0.30/0.61  thf(fact_5410_power__minus1__even,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_even
% 0.30/0.61  thf(fact_5411_power__minus1__even,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = one_one_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_even
% 0.30/0.61  thf(fact_5412_power__minus1__even,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_even
% 0.30/0.61  thf(fact_5413_power__minus1__even,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = one_one_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_even
% 0.30/0.61  thf(fact_5414_neg__one__odd__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.61          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_odd_power
% 0.30/0.61  thf(fact_5415_neg__one__odd__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.61          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_odd_power
% 0.30/0.61  thf(fact_5416_neg__one__odd__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.61          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_odd_power
% 0.30/0.61  thf(fact_5417_neg__one__odd__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.61          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_odd_power
% 0.30/0.61  thf(fact_5418_neg__one__odd__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.61          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_odd_power
% 0.30/0.61  thf(fact_5419_neg__one__even__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.61          = one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_even_power
% 0.30/0.61  thf(fact_5420_neg__one__even__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.61          = one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_even_power
% 0.30/0.61  thf(fact_5421_neg__one__even__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.61          = one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_even_power
% 0.30/0.61  thf(fact_5422_neg__one__even__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.61          = one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_even_power
% 0.30/0.61  thf(fact_5423_neg__one__even__power,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.61          = one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_even_power
% 0.30/0.61  thf(fact_5424_signed__take__bit__0,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_0
% 0.30/0.61  thf(fact_5425_signed__take__bit__0,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.30/0.61        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_0
% 0.30/0.61  thf(fact_5426_one__div__2__pow__eq,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_div_2_pow_eq
% 0.30/0.61  thf(fact_5427_one__div__2__pow__eq,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_div_2_pow_eq
% 0.30/0.61  thf(fact_5428_one__div__2__pow__eq,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_div_2_pow_eq
% 0.30/0.61  thf(fact_5429_bits__1__div__exp,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_1_div_exp
% 0.30/0.61  thf(fact_5430_bits__1__div__exp,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_1_div_exp
% 0.30/0.61  thf(fact_5431_bits__1__div__exp,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_1_div_exp
% 0.30/0.61  thf(fact_5432_one__mod__2__pow__eq,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_mod_2_pow_eq
% 0.30/0.61  thf(fact_5433_one__mod__2__pow__eq,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_mod_2_pow_eq
% 0.30/0.61  thf(fact_5434_one__mod__2__pow__eq,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_mod_2_pow_eq
% 0.30/0.61  thf(fact_5435_signed__take__bit__minus,axiom,
% 0.30/0.61      ! [N: nat,K: int] :
% 0.30/0.61        ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 0.30/0.61        = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_minus
% 0.30/0.61  thf(fact_5436_dvd__antisym,axiom,
% 0.30/0.61      ! [M: nat,N: nat] :
% 0.30/0.61        ( ( dvd_dvd_nat @ M @ N )
% 0.30/0.61       => ( ( dvd_dvd_nat @ N @ M )
% 0.30/0.61         => ( M = N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_antisym
% 0.30/0.61  thf(fact_5437_equation__minus__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_real @ B2 ) )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % equation_minus_iff
% 0.30/0.61  thf(fact_5438_equation__minus__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % equation_minus_iff
% 0.30/0.61  thf(fact_5439_equation__minus__iff,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % equation_minus_iff
% 0.30/0.61  thf(fact_5440_equation__minus__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % equation_minus_iff
% 0.30/0.61  thf(fact_5441_equation__minus__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % equation_minus_iff
% 0.30/0.61  thf(fact_5442_minus__equation__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ( uminus_uminus_real @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( uminus_uminus_real @ B2 )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_equation_iff
% 0.30/0.61  thf(fact_5443_minus__equation__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ( uminus_uminus_int @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( uminus_uminus_int @ B2 )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_equation_iff
% 0.30/0.61  thf(fact_5444_minus__equation__iff,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( uminus1482373934393186551omplex @ B2 )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_equation_iff
% 0.30/0.61  thf(fact_5445_minus__equation__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( uminus1351360451143612070nteger @ B2 )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_equation_iff
% 0.30/0.61  thf(fact_5446_minus__equation__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( uminus_uminus_rat @ B2 )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_equation_iff
% 0.30/0.61  thf(fact_5447_of__bool__eq__iff,axiom,
% 0.30/0.61      ! [P4: $o,Q2: $o] :
% 0.30/0.61        ( ( ( zero_n2687167440665602831ol_nat @ P4 )
% 0.30/0.61          = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 0.30/0.61        = ( P4 = Q2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_eq_iff
% 0.30/0.61  thf(fact_5448_of__bool__eq__iff,axiom,
% 0.30/0.61      ! [P4: $o,Q2: $o] :
% 0.30/0.61        ( ( ( zero_n2684676970156552555ol_int @ P4 )
% 0.30/0.61          = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 0.30/0.61        = ( P4 = Q2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_eq_iff
% 0.30/0.61  thf(fact_5449_of__bool__eq__iff,axiom,
% 0.30/0.61      ! [P4: $o,Q2: $o] :
% 0.30/0.61        ( ( ( zero_n356916108424825756nteger @ P4 )
% 0.30/0.61          = ( zero_n356916108424825756nteger @ Q2 ) )
% 0.30/0.61        = ( P4 = Q2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_eq_iff
% 0.30/0.61  thf(fact_5450_verit__negate__coefficient_I3_J,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( A = B2 )
% 0.30/0.61       => ( ( uminus_uminus_real @ A )
% 0.30/0.61          = ( uminus_uminus_real @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(3)
% 0.30/0.61  thf(fact_5451_verit__negate__coefficient_I3_J,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( A = B2 )
% 0.30/0.61       => ( ( uminus_uminus_int @ A )
% 0.30/0.61          = ( uminus_uminus_int @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(3)
% 0.30/0.61  thf(fact_5452_verit__negate__coefficient_I3_J,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( A = B2 )
% 0.30/0.61       => ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.61          = ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(3)
% 0.30/0.61  thf(fact_5453_verit__negate__coefficient_I3_J,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( A = B2 )
% 0.30/0.61       => ( ( uminus_uminus_rat @ A )
% 0.30/0.61          = ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(3)
% 0.30/0.61  thf(fact_5454_compl__mono,axiom,
% 0.30/0.61      ! [X3: set_int,Y3: set_int] :
% 0.30/0.61        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.61       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % compl_mono
% 0.30/0.61  thf(fact_5455_compl__le__swap1,axiom,
% 0.30/0.61      ! [Y3: set_int,X3: set_int] :
% 0.30/0.61        ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X3 ) )
% 0.30/0.61       => ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % compl_le_swap1
% 0.30/0.61  thf(fact_5456_compl__le__swap2,axiom,
% 0.30/0.61      ! [Y3: set_int,X3: set_int] :
% 0.30/0.61        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X3 )
% 0.30/0.61       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % compl_le_swap2
% 0.30/0.61  thf(fact_5457_of__bool__conj,axiom,
% 0.30/0.61      ! [P: $o,Q: $o] :
% 0.30/0.61        ( ( zero_n3304061248610475627l_real
% 0.30/0.61          @ ( P
% 0.30/0.61            & Q ) )
% 0.30/0.61        = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_conj
% 0.30/0.61  thf(fact_5458_of__bool__conj,axiom,
% 0.30/0.61      ! [P: $o,Q: $o] :
% 0.30/0.61        ( ( zero_n2052037380579107095ol_rat
% 0.30/0.61          @ ( P
% 0.30/0.61            & Q ) )
% 0.30/0.61        = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_conj
% 0.30/0.61  thf(fact_5459_of__bool__conj,axiom,
% 0.30/0.61      ! [P: $o,Q: $o] :
% 0.30/0.61        ( ( zero_n2687167440665602831ol_nat
% 0.30/0.61          @ ( P
% 0.30/0.61            & Q ) )
% 0.30/0.61        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_conj
% 0.30/0.61  thf(fact_5460_of__bool__conj,axiom,
% 0.30/0.61      ! [P: $o,Q: $o] :
% 0.30/0.61        ( ( zero_n2684676970156552555ol_int
% 0.30/0.61          @ ( P
% 0.30/0.61            & Q ) )
% 0.30/0.61        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_conj
% 0.30/0.61  thf(fact_5461_of__bool__conj,axiom,
% 0.30/0.61      ! [P: $o,Q: $o] :
% 0.30/0.61        ( ( zero_n356916108424825756nteger
% 0.30/0.61          @ ( P
% 0.30/0.61            & Q ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_conj
% 0.30/0.61  thf(fact_5462_le__imp__neg__le,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.61       => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_imp_neg_le
% 0.30/0.61  thf(fact_5463_le__imp__neg__le,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 0.30/0.61       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_imp_neg_le
% 0.30/0.61  thf(fact_5464_le__imp__neg__le,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.61       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_imp_neg_le
% 0.30/0.61  thf(fact_5465_le__imp__neg__le,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.61       => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_imp_neg_le
% 0.30/0.61  thf(fact_5466_minus__le__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.61        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_le_iff
% 0.30/0.61  thf(fact_5467_minus__le__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.61        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_le_iff
% 0.30/0.61  thf(fact_5468_minus__le__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.61        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_le_iff
% 0.30/0.61  thf(fact_5469_minus__le__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.61        = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_le_iff
% 0.30/0.61  thf(fact_5470_le__minus__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.61        = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_iff
% 0.30/0.61  thf(fact_5471_le__minus__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_iff
% 0.30/0.61  thf(fact_5472_le__minus__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61        = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_iff
% 0.30/0.61  thf(fact_5473_le__minus__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_iff
% 0.30/0.61  thf(fact_5474_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ A @ B2 )
% 0.30/0.61       => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(2)
% 0.30/0.61  thf(fact_5475_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ord_less_int @ A @ B2 )
% 0.30/0.61       => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(2)
% 0.30/0.61  thf(fact_5476_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 0.30/0.61       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(2)
% 0.30/0.61  thf(fact_5477_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ A @ B2 )
% 0.30/0.61       => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_negate_coefficient(2)
% 0.30/0.61  thf(fact_5478_minus__less__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.61        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_less_iff
% 0.30/0.61  thf(fact_5479_minus__less__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.61        = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_less_iff
% 0.30/0.61  thf(fact_5480_minus__less__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.61        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_less_iff
% 0.30/0.61  thf(fact_5481_minus__less__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.61        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_less_iff
% 0.30/0.61  thf(fact_5482_less__minus__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.61        = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_iff
% 0.30/0.61  thf(fact_5483_less__minus__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_iff
% 0.30/0.61  thf(fact_5484_less__minus__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_iff
% 0.30/0.61  thf(fact_5485_less__minus__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61        = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_iff
% 0.30/0.61  thf(fact_5486_neg__numeral__neq__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.30/0.61       != ( numeral_numeral_real @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_neq_numeral
% 0.30/0.61  thf(fact_5487_neg__numeral__neq__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.30/0.61       != ( numeral_numeral_int @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_neq_numeral
% 0.30/0.61  thf(fact_5488_neg__numeral__neq__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.30/0.61       != ( numera6690914467698888265omplex @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_neq_numeral
% 0.30/0.61  thf(fact_5489_neg__numeral__neq__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.30/0.61       != ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_neq_numeral
% 0.30/0.61  thf(fact_5490_neg__numeral__neq__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.30/0.61       != ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_neq_numeral
% 0.30/0.61  thf(fact_5491_numeral__neq__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( numeral_numeral_real @ M )
% 0.30/0.61       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_numeral
% 0.30/0.61  thf(fact_5492_numeral__neq__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( numeral_numeral_int @ M )
% 0.30/0.61       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_numeral
% 0.30/0.61  thf(fact_5493_numeral__neq__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( numera6690914467698888265omplex @ M )
% 0.30/0.61       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_numeral
% 0.30/0.61  thf(fact_5494_numeral__neq__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( numera6620942414471956472nteger @ M )
% 0.30/0.61       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_numeral
% 0.30/0.61  thf(fact_5495_numeral__neq__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ( ( numeral_numeral_rat @ M )
% 0.30/0.61       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_numeral
% 0.30/0.61  thf(fact_5496_square__eq__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ( times_times_real @ A @ A )
% 0.30/0.61          = ( times_times_real @ B2 @ B2 ) )
% 0.30/0.61        = ( ( A = B2 )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_iff
% 0.30/0.61  thf(fact_5497_square__eq__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ( times_times_int @ A @ A )
% 0.30/0.61          = ( times_times_int @ B2 @ B2 ) )
% 0.30/0.61        = ( ( A = B2 )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_iff
% 0.30/0.61  thf(fact_5498_square__eq__iff,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( ( times_times_complex @ A @ A )
% 0.30/0.61          = ( times_times_complex @ B2 @ B2 ) )
% 0.30/0.61        = ( ( A = B2 )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_iff
% 0.30/0.61  thf(fact_5499_square__eq__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.30/0.61          = ( times_3573771949741848930nteger @ B2 @ B2 ) )
% 0.30/0.61        = ( ( A = B2 )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_iff
% 0.30/0.61  thf(fact_5500_square__eq__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ( times_times_rat @ A @ A )
% 0.30/0.61          = ( times_times_rat @ B2 @ B2 ) )
% 0.30/0.61        = ( ( A = B2 )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_iff
% 0.30/0.61  thf(fact_5501_minus__mult__commute,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.61        = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_mult_commute
% 0.30/0.61  thf(fact_5502_minus__mult__commute,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.61        = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_mult_commute
% 0.30/0.61  thf(fact_5503_minus__mult__commute,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.30/0.61        = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_mult_commute
% 0.30/0.61  thf(fact_5504_minus__mult__commute,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.61        = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_mult_commute
% 0.30/0.61  thf(fact_5505_minus__mult__commute,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.61        = ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_mult_commute
% 0.30/0.61  thf(fact_5506_one__neq__neg__one,axiom,
% 0.30/0.61      ( one_one_real
% 0.30/0.61     != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_one
% 0.30/0.61  thf(fact_5507_one__neq__neg__one,axiom,
% 0.30/0.61      ( one_one_int
% 0.30/0.61     != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_one
% 0.30/0.61  thf(fact_5508_one__neq__neg__one,axiom,
% 0.30/0.61      ( one_one_complex
% 0.30/0.61     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_one
% 0.30/0.61  thf(fact_5509_one__neq__neg__one,axiom,
% 0.30/0.61      ( one_one_Code_integer
% 0.30/0.61     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_one
% 0.30/0.61  thf(fact_5510_one__neq__neg__one,axiom,
% 0.30/0.61      ( one_one_rat
% 0.30/0.61     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_one
% 0.30/0.61  thf(fact_5511_is__num__normalize_I8_J,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % is_num_normalize(8)
% 0.30/0.61  thf(fact_5512_is__num__normalize_I8_J,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % is_num_normalize(8)
% 0.30/0.61  thf(fact_5513_is__num__normalize_I8_J,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % is_num_normalize(8)
% 0.30/0.61  thf(fact_5514_is__num__normalize_I8_J,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.30/0.61        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % is_num_normalize(8)
% 0.30/0.61  thf(fact_5515_is__num__normalize_I8_J,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % is_num_normalize(8)
% 0.30/0.61  thf(fact_5516_add_Oinverse__distrib__swap,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_distrib_swap
% 0.30/0.61  thf(fact_5517_add_Oinverse__distrib__swap,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_distrib_swap
% 0.30/0.61  thf(fact_5518_add_Oinverse__distrib__swap,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_distrib_swap
% 0.30/0.61  thf(fact_5519_add_Oinverse__distrib__swap,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 0.30/0.61        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_distrib_swap
% 0.30/0.61  thf(fact_5520_add_Oinverse__distrib__swap,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 0.30/0.61        = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_distrib_swap
% 0.30/0.61  thf(fact_5521_group__cancel_Oneg1,axiom,
% 0.30/0.61      ! [A4: real,K: real,A: real] :
% 0.30/0.61        ( ( A4
% 0.30/0.61          = ( plus_plus_real @ K @ A ) )
% 0.30/0.61       => ( ( uminus_uminus_real @ A4 )
% 0.30/0.61          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.neg1
% 0.30/0.61  thf(fact_5522_group__cancel_Oneg1,axiom,
% 0.30/0.61      ! [A4: int,K: int,A: int] :
% 0.30/0.61        ( ( A4
% 0.30/0.61          = ( plus_plus_int @ K @ A ) )
% 0.30/0.61       => ( ( uminus_uminus_int @ A4 )
% 0.30/0.61          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.neg1
% 0.30/0.61  thf(fact_5523_group__cancel_Oneg1,axiom,
% 0.30/0.61      ! [A4: complex,K: complex,A: complex] :
% 0.30/0.61        ( ( A4
% 0.30/0.61          = ( plus_plus_complex @ K @ A ) )
% 0.30/0.61       => ( ( uminus1482373934393186551omplex @ A4 )
% 0.30/0.61          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.neg1
% 0.30/0.61  thf(fact_5524_group__cancel_Oneg1,axiom,
% 0.30/0.61      ! [A4: code_integer,K: code_integer,A: code_integer] :
% 0.30/0.61        ( ( A4
% 0.30/0.61          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.30/0.61       => ( ( uminus1351360451143612070nteger @ A4 )
% 0.30/0.61          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.neg1
% 0.30/0.61  thf(fact_5525_group__cancel_Oneg1,axiom,
% 0.30/0.61      ! [A4: rat,K: rat,A: rat] :
% 0.30/0.61        ( ( A4
% 0.30/0.61          = ( plus_plus_rat @ K @ A ) )
% 0.30/0.61       => ( ( uminus_uminus_rat @ A4 )
% 0.30/0.61          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.neg1
% 0.30/0.61  thf(fact_5526_minus__diff__minus,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.61        = ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_minus
% 0.30/0.61  thf(fact_5527_minus__diff__minus,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_minus
% 0.30/0.61  thf(fact_5528_minus__diff__minus,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_minus
% 0.30/0.61  thf(fact_5529_minus__diff__minus,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_minus
% 0.30/0.61  thf(fact_5530_minus__diff__minus,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61        = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_minus
% 0.30/0.61  thf(fact_5531_minus__diff__commute,axiom,
% 0.30/0.61      ! [B2: real,A: real] :
% 0.30/0.61        ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
% 0.30/0.61        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_commute
% 0.30/0.61  thf(fact_5532_minus__diff__commute,axiom,
% 0.30/0.61      ! [B2: int,A: int] :
% 0.30/0.61        ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
% 0.30/0.61        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_commute
% 0.30/0.61  thf(fact_5533_minus__diff__commute,axiom,
% 0.30/0.61      ! [B2: complex,A: complex] :
% 0.30/0.61        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A )
% 0.30/0.61        = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_commute
% 0.30/0.61  thf(fact_5534_minus__diff__commute,axiom,
% 0.30/0.61      ! [B2: code_integer,A: code_integer] :
% 0.30/0.61        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A )
% 0.30/0.61        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_commute
% 0.30/0.61  thf(fact_5535_minus__diff__commute,axiom,
% 0.30/0.61      ! [B2: rat,A: rat] :
% 0.30/0.61        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A )
% 0.30/0.61        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_diff_commute
% 0.30/0.61  thf(fact_5536_minus__divide__right,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.61        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_right
% 0.30/0.61  thf(fact_5537_minus__divide__right,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.61        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_right
% 0.30/0.61  thf(fact_5538_minus__divide__right,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.61        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_right
% 0.30/0.61  thf(fact_5539_minus__divide__divide,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.61        = ( divide_divide_real @ A @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_divide
% 0.30/0.61  thf(fact_5540_minus__divide__divide,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.61        = ( divide1717551699836669952omplex @ A @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_divide
% 0.30/0.61  thf(fact_5541_minus__divide__divide,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61        = ( divide_divide_rat @ A @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_divide
% 0.30/0.61  thf(fact_5542_minus__divide__left,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.61        = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_left
% 0.30/0.61  thf(fact_5543_minus__divide__left,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.61        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_left
% 0.30/0.61  thf(fact_5544_minus__divide__left,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.61        = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_left
% 0.30/0.61  thf(fact_5545_div__minus__right,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % div_minus_right
% 0.30/0.61  thf(fact_5546_div__minus__right,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % div_minus_right
% 0.30/0.61  thf(fact_5547_mod__minus__eq,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
% 0.30/0.61        = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus_eq
% 0.30/0.61  thf(fact_5548_mod__minus__eq,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ B2 )
% 0.30/0.61        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus_eq
% 0.30/0.61  thf(fact_5549_mod__minus__cong,axiom,
% 0.30/0.61      ! [A: int,B2: int,A2: int] :
% 0.30/0.61        ( ( ( modulo_modulo_int @ A @ B2 )
% 0.30/0.61          = ( modulo_modulo_int @ A2 @ B2 ) )
% 0.30/0.61       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.61          = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus_cong
% 0.30/0.61  thf(fact_5550_mod__minus__cong,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer,A2: code_integer] :
% 0.30/0.61        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 0.30/0.61          = ( modulo364778990260209775nteger @ A2 @ B2 ) )
% 0.30/0.61       => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.61          = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus_cong
% 0.30/0.61  thf(fact_5551_mod__minus__right,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus_right
% 0.30/0.61  thf(fact_5552_mod__minus__right,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mod_minus_right
% 0.30/0.61  thf(fact_5553_zero__less__eq__of__bool,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_eq_of_bool
% 0.30/0.61  thf(fact_5554_zero__less__eq__of__bool,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_eq_of_bool
% 0.30/0.61  thf(fact_5555_zero__less__eq__of__bool,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_eq_of_bool
% 0.30/0.61  thf(fact_5556_zero__less__eq__of__bool,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_eq_of_bool
% 0.30/0.61  thf(fact_5557_zero__less__eq__of__bool,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_eq_of_bool
% 0.30/0.61  thf(fact_5558_of__bool__less__eq__one,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_less_eq_one
% 0.30/0.61  thf(fact_5559_of__bool__less__eq__one,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_less_eq_one
% 0.30/0.61  thf(fact_5560_of__bool__less__eq__one,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_less_eq_one
% 0.30/0.61  thf(fact_5561_of__bool__less__eq__one,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_less_eq_one
% 0.30/0.61  thf(fact_5562_of__bool__less__eq__one,axiom,
% 0.30/0.61      ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_less_eq_one
% 0.30/0.61  thf(fact_5563_of__bool__def,axiom,
% 0.30/0.61      ( zero_n1201886186963655149omplex
% 0.30/0.61      = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_def
% 0.30/0.61  thf(fact_5564_of__bool__def,axiom,
% 0.30/0.61      ( zero_n3304061248610475627l_real
% 0.30/0.61      = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_def
% 0.30/0.61  thf(fact_5565_of__bool__def,axiom,
% 0.30/0.61      ( zero_n2052037380579107095ol_rat
% 0.30/0.61      = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_def
% 0.30/0.61  thf(fact_5566_of__bool__def,axiom,
% 0.30/0.61      ( zero_n2687167440665602831ol_nat
% 0.30/0.61      = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_def
% 0.30/0.61  thf(fact_5567_of__bool__def,axiom,
% 0.30/0.61      ( zero_n2684676970156552555ol_int
% 0.30/0.61      = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_def
% 0.30/0.61  thf(fact_5568_of__bool__def,axiom,
% 0.30/0.61      ( zero_n356916108424825756nteger
% 0.30/0.61      = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_def
% 0.30/0.61  thf(fact_5569_split__of__bool,axiom,
% 0.30/0.61      ! [P: complex > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.30/0.61        = ( ( P4
% 0.30/0.61           => ( P @ one_one_complex ) )
% 0.30/0.61          & ( ~ P4
% 0.30/0.61           => ( P @ zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool
% 0.30/0.61  thf(fact_5570_split__of__bool,axiom,
% 0.30/0.61      ! [P: real > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.30/0.61        = ( ( P4
% 0.30/0.61           => ( P @ one_one_real ) )
% 0.30/0.61          & ( ~ P4
% 0.30/0.61           => ( P @ zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool
% 0.30/0.61  thf(fact_5571_split__of__bool,axiom,
% 0.30/0.61      ! [P: rat > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.30/0.61        = ( ( P4
% 0.30/0.61           => ( P @ one_one_rat ) )
% 0.30/0.61          & ( ~ P4
% 0.30/0.61           => ( P @ zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool
% 0.30/0.61  thf(fact_5572_split__of__bool,axiom,
% 0.30/0.61      ! [P: nat > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.30/0.61        = ( ( P4
% 0.30/0.61           => ( P @ one_one_nat ) )
% 0.30/0.61          & ( ~ P4
% 0.30/0.61           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool
% 0.30/0.61  thf(fact_5573_split__of__bool,axiom,
% 0.30/0.61      ! [P: int > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.30/0.61        = ( ( P4
% 0.30/0.61           => ( P @ one_one_int ) )
% 0.30/0.61          & ( ~ P4
% 0.30/0.61           => ( P @ zero_zero_int ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool
% 0.30/0.61  thf(fact_5574_split__of__bool,axiom,
% 0.30/0.61      ! [P: code_integer > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 0.30/0.61        = ( ( P4
% 0.30/0.61           => ( P @ one_one_Code_integer ) )
% 0.30/0.61          & ( ~ P4
% 0.30/0.61           => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool
% 0.30/0.61  thf(fact_5575_split__of__bool__asm,axiom,
% 0.30/0.61      ! [P: complex > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.30/0.61        = ( ~ ( ( P4
% 0.30/0.61                & ~ ( P @ one_one_complex ) )
% 0.30/0.61              | ( ~ P4
% 0.30/0.61                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool_asm
% 0.30/0.61  thf(fact_5576_split__of__bool__asm,axiom,
% 0.30/0.61      ! [P: real > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.30/0.61        = ( ~ ( ( P4
% 0.30/0.61                & ~ ( P @ one_one_real ) )
% 0.30/0.61              | ( ~ P4
% 0.30/0.61                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool_asm
% 0.30/0.61  thf(fact_5577_split__of__bool__asm,axiom,
% 0.30/0.61      ! [P: rat > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.30/0.61        = ( ~ ( ( P4
% 0.30/0.61                & ~ ( P @ one_one_rat ) )
% 0.30/0.61              | ( ~ P4
% 0.30/0.61                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool_asm
% 0.30/0.61  thf(fact_5578_split__of__bool__asm,axiom,
% 0.30/0.61      ! [P: nat > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.30/0.61        = ( ~ ( ( P4
% 0.30/0.61                & ~ ( P @ one_one_nat ) )
% 0.30/0.61              | ( ~ P4
% 0.30/0.61                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool_asm
% 0.30/0.61  thf(fact_5579_split__of__bool__asm,axiom,
% 0.30/0.61      ! [P: int > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.30/0.61        = ( ~ ( ( P4
% 0.30/0.61                & ~ ( P @ one_one_int ) )
% 0.30/0.61              | ( ~ P4
% 0.30/0.61                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool_asm
% 0.30/0.61  thf(fact_5580_split__of__bool__asm,axiom,
% 0.30/0.61      ! [P: code_integer > $o,P4: $o] :
% 0.30/0.61        ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 0.30/0.61        = ( ~ ( ( P4
% 0.30/0.61                & ~ ( P @ one_one_Code_integer ) )
% 0.30/0.61              | ( ~ P4
% 0.30/0.61                & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_of_bool_asm
% 0.30/0.61  thf(fact_5581_not__numeral__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_numeral
% 0.30/0.61  thf(fact_5582_not__numeral__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_numeral
% 0.30/0.61  thf(fact_5583_not__numeral__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_numeral
% 0.30/0.61  thf(fact_5584_not__numeral__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_numeral
% 0.30/0.61  thf(fact_5585_neg__numeral__le__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_numeral
% 0.30/0.61  thf(fact_5586_neg__numeral__le__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_numeral
% 0.30/0.61  thf(fact_5587_neg__numeral__le__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_numeral
% 0.30/0.61  thf(fact_5588_neg__numeral__le__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_numeral
% 0.30/0.61  thf(fact_5589_zero__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( zero_zero_real
% 0.30/0.61       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_numeral
% 0.30/0.61  thf(fact_5590_zero__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( zero_zero_int
% 0.30/0.61       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_numeral
% 0.30/0.61  thf(fact_5591_zero__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( zero_zero_complex
% 0.30/0.61       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_numeral
% 0.30/0.61  thf(fact_5592_zero__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( zero_z3403309356797280102nteger
% 0.30/0.61       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_numeral
% 0.30/0.61  thf(fact_5593_zero__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( zero_zero_rat
% 0.30/0.61       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_numeral
% 0.30/0.61  thf(fact_5594_not__numeral__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_numeral
% 0.30/0.61  thf(fact_5595_not__numeral__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_numeral
% 0.30/0.61  thf(fact_5596_not__numeral__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_numeral
% 0.30/0.61  thf(fact_5597_not__numeral__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] :
% 0.30/0.61        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_numeral
% 0.30/0.61  thf(fact_5598_neg__numeral__less__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_numeral
% 0.30/0.61  thf(fact_5599_neg__numeral__less__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_numeral
% 0.30/0.61  thf(fact_5600_neg__numeral__less__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_numeral
% 0.30/0.61  thf(fact_5601_neg__numeral__less__numeral,axiom,
% 0.30/0.61      ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_numeral
% 0.30/0.61  thf(fact_5602_le__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(2)
% 0.30/0.61  thf(fact_5603_le__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(2)
% 0.30/0.61  thf(fact_5604_le__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(2)
% 0.30/0.61  thf(fact_5605_le__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(2)
% 0.30/0.61  thf(fact_5606_le__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(4)
% 0.30/0.61  thf(fact_5607_le__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(4)
% 0.30/0.61  thf(fact_5608_le__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(4)
% 0.30/0.61  thf(fact_5609_le__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(4)
% 0.30/0.61  thf(fact_5610_zero__neq__neg__one,axiom,
% 0.30/0.61      ( zero_zero_real
% 0.30/0.61     != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_one
% 0.30/0.61  thf(fact_5611_zero__neq__neg__one,axiom,
% 0.30/0.61      ( zero_zero_int
% 0.30/0.61     != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_one
% 0.30/0.61  thf(fact_5612_zero__neq__neg__one,axiom,
% 0.30/0.61      ( zero_zero_complex
% 0.30/0.61     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_one
% 0.30/0.61  thf(fact_5613_zero__neq__neg__one,axiom,
% 0.30/0.61      ( zero_z3403309356797280102nteger
% 0.30/0.61     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_one
% 0.30/0.61  thf(fact_5614_zero__neq__neg__one,axiom,
% 0.30/0.61      ( zero_zero_rat
% 0.30/0.61     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_neq_neg_one
% 0.30/0.61  thf(fact_5615_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ( uminus_uminus_real @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( plus_plus_real @ A @ B2 )
% 0.30/0.61          = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_eq_iff_add_eq_0
% 0.30/0.61  thf(fact_5616_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ( uminus_uminus_int @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( plus_plus_int @ A @ B2 )
% 0.30/0.61          = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_eq_iff_add_eq_0
% 0.30/0.61  thf(fact_5617_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( plus_plus_complex @ A @ B2 )
% 0.30/0.61          = zero_zero_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_eq_iff_add_eq_0
% 0.30/0.61  thf(fact_5618_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.30/0.61          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_eq_iff_add_eq_0
% 0.30/0.61  thf(fact_5619_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.61          = B2 )
% 0.30/0.61        = ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.61          = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_eq_iff_add_eq_0
% 0.30/0.61  thf(fact_5620_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_real @ B2 ) )
% 0.30/0.61        = ( ( plus_plus_real @ A @ B2 )
% 0.30/0.61          = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_neg_iff_add_eq_0
% 0.30/0.61  thf(fact_5621_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_int @ B2 ) )
% 0.30/0.61        = ( ( plus_plus_int @ A @ B2 )
% 0.30/0.61          = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_neg_iff_add_eq_0
% 0.30/0.61  thf(fact_5622_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.61        = ( ( plus_plus_complex @ A @ B2 )
% 0.30/0.61          = zero_zero_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_neg_iff_add_eq_0
% 0.30/0.61  thf(fact_5623_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61        = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.30/0.61          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_neg_iff_add_eq_0
% 0.30/0.61  thf(fact_5624_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61        = ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.61          = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_neg_iff_add_eq_0
% 0.30/0.61  thf(fact_5625_add_Oinverse__unique,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ( plus_plus_real @ A @ B2 )
% 0.30/0.61          = zero_zero_real )
% 0.30/0.61       => ( ( uminus_uminus_real @ A )
% 0.30/0.61          = B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_unique
% 0.30/0.61  thf(fact_5626_add_Oinverse__unique,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ( plus_plus_int @ A @ B2 )
% 0.30/0.61          = zero_zero_int )
% 0.30/0.61       => ( ( uminus_uminus_int @ A )
% 0.30/0.61          = B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_unique
% 0.30/0.61  thf(fact_5627_add_Oinverse__unique,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( ( plus_plus_complex @ A @ B2 )
% 0.30/0.61          = zero_zero_complex )
% 0.30/0.61       => ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.61          = B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_unique
% 0.30/0.61  thf(fact_5628_add_Oinverse__unique,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.30/0.61          = zero_z3403309356797280102nteger )
% 0.30/0.61       => ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.61          = B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_unique
% 0.30/0.61  thf(fact_5629_add_Oinverse__unique,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.61          = zero_zero_rat )
% 0.30/0.61       => ( ( uminus_uminus_rat @ A )
% 0.30/0.61          = B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add.inverse_unique
% 0.30/0.61  thf(fact_5630_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.61        = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_left_minus
% 0.30/0.61  thf(fact_5631_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.61        = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_left_minus
% 0.30/0.61  thf(fact_5632_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.61      ! [A: complex] :
% 0.30/0.61        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.30/0.61        = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_left_minus
% 0.30/0.61  thf(fact_5633_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.61        = zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_left_minus
% 0.30/0.61  thf(fact_5634_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.61        = zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_left_minus
% 0.30/0.61  thf(fact_5635_add__eq__0__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ( plus_plus_real @ A @ B2 )
% 0.30/0.61          = zero_zero_real )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_eq_0_iff
% 0.30/0.61  thf(fact_5636_add__eq__0__iff,axiom,
% 0.30/0.61      ! [A: int,B2: int] :
% 0.30/0.61        ( ( ( plus_plus_int @ A @ B2 )
% 0.30/0.61          = zero_zero_int )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_eq_0_iff
% 0.30/0.61  thf(fact_5637_add__eq__0__iff,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( ( plus_plus_complex @ A @ B2 )
% 0.30/0.61          = zero_zero_complex )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_eq_0_iff
% 0.30/0.61  thf(fact_5638_add__eq__0__iff,axiom,
% 0.30/0.61      ! [A: code_integer,B2: code_integer] :
% 0.30/0.61        ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 0.30/0.61          = zero_z3403309356797280102nteger )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_eq_0_iff
% 0.30/0.61  thf(fact_5639_add__eq__0__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ( plus_plus_rat @ A @ B2 )
% 0.30/0.61          = zero_zero_rat )
% 0.30/0.61        = ( B2
% 0.30/0.61          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_eq_0_iff
% 0.30/0.61  thf(fact_5640_less__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(2)
% 0.30/0.61  thf(fact_5641_less__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(2)
% 0.30/0.61  thf(fact_5642_less__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(2)
% 0.30/0.61  thf(fact_5643_less__minus__one__simps_I2_J,axiom,
% 0.30/0.61      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(2)
% 0.30/0.61  thf(fact_5644_less__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(4)
% 0.30/0.61  thf(fact_5645_less__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(4)
% 0.30/0.61  thf(fact_5646_less__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(4)
% 0.30/0.61  thf(fact_5647_less__minus__one__simps_I4_J,axiom,
% 0.30/0.61      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(4)
% 0.30/0.61  thf(fact_5648_numeral__times__minus__swap,axiom,
% 0.30/0.61      ! [W: num,X3: real] :
% 0.30/0.61        ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.61        = ( times_times_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_times_minus_swap
% 0.30/0.61  thf(fact_5649_numeral__times__minus__swap,axiom,
% 0.30/0.61      ! [W: num,X3: int] :
% 0.30/0.61        ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X3 ) )
% 0.30/0.61        = ( times_times_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_times_minus_swap
% 0.30/0.61  thf(fact_5650_numeral__times__minus__swap,axiom,
% 0.30/0.61      ! [W: num,X3: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X3 ) )
% 0.30/0.61        = ( times_times_complex @ X3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_times_minus_swap
% 0.30/0.61  thf(fact_5651_numeral__times__minus__swap,axiom,
% 0.30/0.61      ! [W: num,X3: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X3 ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ X3 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_times_minus_swap
% 0.30/0.61  thf(fact_5652_numeral__times__minus__swap,axiom,
% 0.30/0.61      ! [W: num,X3: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X3 ) )
% 0.30/0.61        = ( times_times_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_times_minus_swap
% 0.30/0.61  thf(fact_5653_nonzero__minus__divide__right,axiom,
% 0.30/0.61      ! [B2: real,A: real] :
% 0.30/0.61        ( ( B2 != zero_zero_real )
% 0.30/0.61       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.61          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_minus_divide_right
% 0.30/0.61  thf(fact_5654_nonzero__minus__divide__right,axiom,
% 0.30/0.61      ! [B2: complex,A: complex] :
% 0.30/0.61        ( ( B2 != zero_zero_complex )
% 0.30/0.61       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.61          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_minus_divide_right
% 0.30/0.61  thf(fact_5655_nonzero__minus__divide__right,axiom,
% 0.30/0.61      ! [B2: rat,A: rat] :
% 0.30/0.61        ( ( B2 != zero_zero_rat )
% 0.30/0.61       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.61          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_minus_divide_right
% 0.30/0.61  thf(fact_5656_nonzero__minus__divide__divide,axiom,
% 0.30/0.61      ! [B2: real,A: real] :
% 0.30/0.61        ( ( B2 != zero_zero_real )
% 0.30/0.61       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.61          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_minus_divide_divide
% 0.30/0.61  thf(fact_5657_nonzero__minus__divide__divide,axiom,
% 0.30/0.61      ! [B2: complex,A: complex] :
% 0.30/0.61        ( ( B2 != zero_zero_complex )
% 0.30/0.61       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.61          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_minus_divide_divide
% 0.30/0.61  thf(fact_5658_nonzero__minus__divide__divide,axiom,
% 0.30/0.61      ! [B2: rat,A: rat] :
% 0.30/0.61        ( ( B2 != zero_zero_rat )
% 0.30/0.61       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_minus_divide_divide
% 0.30/0.61  thf(fact_5659_one__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( one_one_real
% 0.30/0.61       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_numeral
% 0.30/0.61  thf(fact_5660_one__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( one_one_int
% 0.30/0.61       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_numeral
% 0.30/0.61  thf(fact_5661_one__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( one_one_complex
% 0.30/0.61       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_numeral
% 0.30/0.61  thf(fact_5662_one__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( one_one_Code_integer
% 0.30/0.61       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_numeral
% 0.30/0.61  thf(fact_5663_one__neq__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( one_one_rat
% 0.30/0.61       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_neq_neg_numeral
% 0.30/0.61  thf(fact_5664_numeral__neq__neg__one,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( numeral_numeral_real @ N )
% 0.30/0.61       != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_one
% 0.30/0.61  thf(fact_5665_numeral__neq__neg__one,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( numeral_numeral_int @ N )
% 0.30/0.61       != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_one
% 0.30/0.61  thf(fact_5666_numeral__neq__neg__one,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( numera6690914467698888265omplex @ N )
% 0.30/0.61       != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_one
% 0.30/0.61  thf(fact_5667_numeral__neq__neg__one,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( numera6620942414471956472nteger @ N )
% 0.30/0.61       != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_one
% 0.30/0.61  thf(fact_5668_numeral__neq__neg__one,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ( ( numeral_numeral_rat @ N )
% 0.30/0.61       != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_neq_neg_one
% 0.30/0.61  thf(fact_5669_square__eq__1__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ( times_times_real @ X3 @ X3 )
% 0.30/0.61          = one_one_real )
% 0.30/0.61        = ( ( X3 = one_one_real )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_1_iff
% 0.30/0.61  thf(fact_5670_square__eq__1__iff,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( ( times_times_int @ X3 @ X3 )
% 0.30/0.61          = one_one_int )
% 0.30/0.61        = ( ( X3 = one_one_int )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_1_iff
% 0.30/0.61  thf(fact_5671_square__eq__1__iff,axiom,
% 0.30/0.61      ! [X3: complex] :
% 0.30/0.61        ( ( ( times_times_complex @ X3 @ X3 )
% 0.30/0.61          = one_one_complex )
% 0.30/0.61        = ( ( X3 = one_one_complex )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_1_iff
% 0.30/0.61  thf(fact_5672_square__eq__1__iff,axiom,
% 0.30/0.61      ! [X3: code_integer] :
% 0.30/0.61        ( ( ( times_3573771949741848930nteger @ X3 @ X3 )
% 0.30/0.61          = one_one_Code_integer )
% 0.30/0.61        = ( ( X3 = one_one_Code_integer )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_1_iff
% 0.30/0.61  thf(fact_5673_square__eq__1__iff,axiom,
% 0.30/0.61      ! [X3: rat] :
% 0.30/0.61        ( ( ( times_times_rat @ X3 @ X3 )
% 0.30/0.61          = one_one_rat )
% 0.30/0.61        = ( ( X3 = one_one_rat )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_eq_1_iff
% 0.30/0.61  thf(fact_5674_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_real
% 0.30/0.61      = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.61  thf(fact_5675_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_int
% 0.30/0.61      = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.61  thf(fact_5676_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_complex
% 0.30/0.61      = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.61  thf(fact_5677_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_8373710615458151222nteger
% 0.30/0.61      = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.61  thf(fact_5678_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_rat
% 0.30/0.61      = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.61  thf(fact_5679_diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_real
% 0.30/0.61      = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_conv_add_uminus
% 0.30/0.61  thf(fact_5680_diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_int
% 0.30/0.61      = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_conv_add_uminus
% 0.30/0.61  thf(fact_5681_diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_complex
% 0.30/0.61      = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_conv_add_uminus
% 0.30/0.61  thf(fact_5682_diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_8373710615458151222nteger
% 0.30/0.61      = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_conv_add_uminus
% 0.30/0.61  thf(fact_5683_diff__conv__add__uminus,axiom,
% 0.30/0.61      ( minus_minus_rat
% 0.30/0.61      = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % diff_conv_add_uminus
% 0.30/0.61  thf(fact_5684_group__cancel_Osub2,axiom,
% 0.30/0.61      ! [B5: real,K: real,B2: real,A: real] :
% 0.30/0.61        ( ( B5
% 0.30/0.61          = ( plus_plus_real @ K @ B2 ) )
% 0.30/0.61       => ( ( minus_minus_real @ A @ B5 )
% 0.30/0.61          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.sub2
% 0.30/0.61  thf(fact_5685_group__cancel_Osub2,axiom,
% 0.30/0.61      ! [B5: int,K: int,B2: int,A: int] :
% 0.30/0.61        ( ( B5
% 0.30/0.61          = ( plus_plus_int @ K @ B2 ) )
% 0.30/0.61       => ( ( minus_minus_int @ A @ B5 )
% 0.30/0.61          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.sub2
% 0.30/0.61  thf(fact_5686_group__cancel_Osub2,axiom,
% 0.30/0.61      ! [B5: complex,K: complex,B2: complex,A: complex] :
% 0.30/0.61        ( ( B5
% 0.30/0.61          = ( plus_plus_complex @ K @ B2 ) )
% 0.30/0.61       => ( ( minus_minus_complex @ A @ B5 )
% 0.30/0.61          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.sub2
% 0.30/0.61  thf(fact_5687_group__cancel_Osub2,axiom,
% 0.30/0.61      ! [B5: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
% 0.30/0.61        ( ( B5
% 0.30/0.61          = ( plus_p5714425477246183910nteger @ K @ B2 ) )
% 0.30/0.61       => ( ( minus_8373710615458151222nteger @ A @ B5 )
% 0.30/0.61          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.sub2
% 0.30/0.61  thf(fact_5688_group__cancel_Osub2,axiom,
% 0.30/0.61      ! [B5: rat,K: rat,B2: rat,A: rat] :
% 0.30/0.61        ( ( B5
% 0.30/0.61          = ( plus_plus_rat @ K @ B2 ) )
% 0.30/0.61       => ( ( minus_minus_rat @ A @ B5 )
% 0.30/0.61          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % group_cancel.sub2
% 0.30/0.61  thf(fact_5689_replicate__length__same,axiom,
% 0.30/0.61      ! [Xs2: list_int,X3: int] :
% 0.30/0.61        ( ! [X5: int] :
% 0.30/0.61            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 0.30/0.61           => ( X5 = X3 ) )
% 0.30/0.61       => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X3 )
% 0.30/0.61          = Xs2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_length_same
% 0.30/0.61  thf(fact_5690_replicate__length__same,axiom,
% 0.30/0.61      ! [Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.30/0.61        ( ! [X5: vEBT_VEBT] :
% 0.30/0.61            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.61           => ( X5 = X3 ) )
% 0.30/0.61       => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X3 )
% 0.30/0.61          = Xs2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_length_same
% 0.30/0.61  thf(fact_5691_replicate__length__same,axiom,
% 0.30/0.61      ! [Xs2: list_o,X3: $o] :
% 0.30/0.61        ( ! [X5: $o] :
% 0.30/0.61            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 0.30/0.61           => ( X5 = X3 ) )
% 0.30/0.61       => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X3 )
% 0.30/0.61          = Xs2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_length_same
% 0.30/0.61  thf(fact_5692_replicate__length__same,axiom,
% 0.30/0.61      ! [Xs2: list_nat,X3: nat] :
% 0.30/0.61        ( ! [X5: nat] :
% 0.30/0.61            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.61           => ( X5 = X3 ) )
% 0.30/0.61       => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X3 )
% 0.30/0.61          = Xs2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_length_same
% 0.30/0.61  thf(fact_5693_replicate__eqI,axiom,
% 0.30/0.61      ! [Xs2: list_real,N: nat,X3: real] :
% 0.30/0.61        ( ( ( size_size_list_real @ Xs2 )
% 0.30/0.61          = N )
% 0.30/0.61       => ( ! [Y4: real] :
% 0.30/0.61              ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
% 0.30/0.61             => ( Y4 = X3 ) )
% 0.30/0.61         => ( Xs2
% 0.30/0.61            = ( replicate_real @ N @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_eqI
% 0.30/0.61  thf(fact_5694_replicate__eqI,axiom,
% 0.30/0.61      ! [Xs2: list_complex,N: nat,X3: complex] :
% 0.30/0.61        ( ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.30/0.61          = N )
% 0.30/0.61       => ( ! [Y4: complex] :
% 0.30/0.61              ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
% 0.30/0.61             => ( Y4 = X3 ) )
% 0.30/0.61         => ( Xs2
% 0.30/0.61            = ( replicate_complex @ N @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_eqI
% 0.30/0.61  thf(fact_5695_replicate__eqI,axiom,
% 0.30/0.61      ! [Xs2: list_int,N: nat,X3: int] :
% 0.30/0.61        ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.61          = N )
% 0.30/0.61       => ( ! [Y4: int] :
% 0.30/0.61              ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
% 0.30/0.61             => ( Y4 = X3 ) )
% 0.30/0.61         => ( Xs2
% 0.30/0.61            = ( replicate_int @ N @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_eqI
% 0.30/0.61  thf(fact_5696_replicate__eqI,axiom,
% 0.30/0.61      ! [Xs2: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
% 0.30/0.61        ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.61          = N )
% 0.30/0.61       => ( ! [Y4: vEBT_VEBT] :
% 0.30/0.61              ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.30/0.61             => ( Y4 = X3 ) )
% 0.30/0.61         => ( Xs2
% 0.30/0.61            = ( replicate_VEBT_VEBT @ N @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_eqI
% 0.30/0.61  thf(fact_5697_replicate__eqI,axiom,
% 0.30/0.61      ! [Xs2: list_o,N: nat,X3: $o] :
% 0.30/0.61        ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.61          = N )
% 0.30/0.61       => ( ! [Y4: $o] :
% 0.30/0.61              ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
% 0.30/0.61             => ( Y4 = X3 ) )
% 0.30/0.61         => ( Xs2
% 0.30/0.61            = ( replicate_o @ N @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_eqI
% 0.30/0.61  thf(fact_5698_replicate__eqI,axiom,
% 0.30/0.61      ! [Xs2: list_nat,N: nat,X3: nat] :
% 0.30/0.61        ( ( ( size_size_list_nat @ Xs2 )
% 0.30/0.61          = N )
% 0.30/0.61       => ( ! [Y4: nat] :
% 0.30/0.61              ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
% 0.30/0.61             => ( Y4 = X3 ) )
% 0.30/0.61         => ( Xs2
% 0.30/0.61            = ( replicate_nat @ N @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % replicate_eqI
% 0.30/0.61  thf(fact_5699_dvd__neg__div,axiom,
% 0.30/0.61      ! [B2: real,A: real] :
% 0.30/0.61        ( ( dvd_dvd_real @ B2 @ A )
% 0.30/0.61       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.61          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_neg_div
% 0.30/0.61  thf(fact_5700_dvd__neg__div,axiom,
% 0.30/0.61      ! [B2: int,A: int] :
% 0.30/0.61        ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.61       => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.61          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_neg_div
% 0.30/0.61  thf(fact_5701_dvd__neg__div,axiom,
% 0.30/0.61      ! [B2: complex,A: complex] :
% 0.30/0.61        ( ( dvd_dvd_complex @ B2 @ A )
% 0.30/0.61       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 0.30/0.61          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_neg_div
% 0.30/0.61  thf(fact_5702_dvd__neg__div,axiom,
% 0.30/0.61      ! [B2: code_integer,A: code_integer] :
% 0.30/0.61        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.61       => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.61          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_neg_div
% 0.30/0.61  thf(fact_5703_dvd__neg__div,axiom,
% 0.30/0.61      ! [B2: rat,A: rat] :
% 0.30/0.61        ( ( dvd_dvd_rat @ B2 @ A )
% 0.30/0.61       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.61          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_neg_div
% 0.30/0.61  thf(fact_5704_dvd__div__neg,axiom,
% 0.30/0.61      ! [B2: real,A: real] :
% 0.30/0.61        ( ( dvd_dvd_real @ B2 @ A )
% 0.30/0.61       => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
% 0.30/0.61          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_div_neg
% 0.30/0.61  thf(fact_5705_dvd__div__neg,axiom,
% 0.30/0.61      ! [B2: int,A: int] :
% 0.30/0.61        ( ( dvd_dvd_int @ B2 @ A )
% 0.30/0.61       => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 0.30/0.61          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_div_neg
% 0.30/0.61  thf(fact_5706_dvd__div__neg,axiom,
% 0.30/0.61      ! [B2: complex,A: complex] :
% 0.30/0.61        ( ( dvd_dvd_complex @ B2 @ A )
% 0.30/0.61       => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 0.30/0.61          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_div_neg
% 0.30/0.61  thf(fact_5707_dvd__div__neg,axiom,
% 0.30/0.61      ! [B2: code_integer,A: code_integer] :
% 0.30/0.61        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 0.30/0.61       => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 0.30/0.61          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_div_neg
% 0.30/0.61  thf(fact_5708_dvd__div__neg,axiom,
% 0.30/0.61      ! [B2: rat,A: rat] :
% 0.30/0.61        ( ( dvd_dvd_rat @ B2 @ A )
% 0.30/0.61       => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 0.30/0.61          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_div_neg
% 0.30/0.61  thf(fact_5709_subset__Compl__self__eq,axiom,
% 0.30/0.61      ! [A4: set_nat] :
% 0.30/0.61        ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
% 0.30/0.61        = ( A4 = bot_bot_set_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_Compl_self_eq
% 0.30/0.61  thf(fact_5710_subset__Compl__self__eq,axiom,
% 0.30/0.61      ! [A4: set_real] :
% 0.30/0.61        ( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ A4 ) )
% 0.30/0.61        = ( A4 = bot_bot_set_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_Compl_self_eq
% 0.30/0.61  thf(fact_5711_subset__Compl__self__eq,axiom,
% 0.30/0.61      ! [A4: set_int] :
% 0.30/0.61        ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
% 0.30/0.61        = ( A4 = bot_bot_set_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_Compl_self_eq
% 0.30/0.61  thf(fact_5712_real__minus__mult__self__le,axiom,
% 0.30/0.61      ! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X3 @ X3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % real_minus_mult_self_le
% 0.30/0.61  thf(fact_5713_pos__zmult__eq__1__iff__lemma,axiom,
% 0.30/0.61      ! [M: int,N: int] :
% 0.30/0.61        ( ( ( times_times_int @ M @ N )
% 0.30/0.61          = one_one_int )
% 0.30/0.61       => ( ( M = one_one_int )
% 0.30/0.61          | ( M
% 0.30/0.61            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_zmult_eq_1_iff_lemma
% 0.30/0.61  thf(fact_5714_zmult__eq__1__iff,axiom,
% 0.30/0.61      ! [M: int,N: int] :
% 0.30/0.61        ( ( ( times_times_int @ M @ N )
% 0.30/0.61          = one_one_int )
% 0.30/0.61        = ( ( ( M = one_one_int )
% 0.30/0.61            & ( N = one_one_int ) )
% 0.30/0.61          | ( ( M
% 0.30/0.61              = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.61            & ( N
% 0.30/0.61              = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zmult_eq_1_iff
% 0.30/0.61  thf(fact_5715_minus__real__def,axiom,
% 0.30/0.61      ( minus_minus_real
% 0.30/0.61      = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_real_def
% 0.30/0.61  thf(fact_5716_not__zero__le__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_le_neg_numeral
% 0.30/0.61  thf(fact_5717_not__zero__le__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_le_neg_numeral
% 0.30/0.61  thf(fact_5718_not__zero__le__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_le_neg_numeral
% 0.30/0.61  thf(fact_5719_not__zero__le__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_le_neg_numeral
% 0.30/0.61  thf(fact_5720_neg__numeral__le__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_zero
% 0.30/0.61  thf(fact_5721_neg__numeral__le__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_zero
% 0.30/0.61  thf(fact_5722_neg__numeral__le__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_zero
% 0.30/0.61  thf(fact_5723_neg__numeral__le__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_zero
% 0.30/0.61  thf(fact_5724_not__zero__less__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_less_neg_numeral
% 0.30/0.61  thf(fact_5725_not__zero__less__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_less_neg_numeral
% 0.30/0.61  thf(fact_5726_not__zero__less__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_less_neg_numeral
% 0.30/0.61  thf(fact_5727_not__zero__less__neg__numeral,axiom,
% 0.30/0.61      ! [N: num] :
% 0.30/0.61        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_zero_less_neg_numeral
% 0.30/0.61  thf(fact_5728_neg__numeral__less__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_zero
% 0.30/0.61  thf(fact_5729_neg__numeral__less__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_zero
% 0.30/0.61  thf(fact_5730_neg__numeral__less__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_zero
% 0.30/0.61  thf(fact_5731_neg__numeral__less__zero,axiom,
% 0.30/0.61      ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_zero
% 0.30/0.61  thf(fact_5732_le__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(3)
% 0.30/0.61  thf(fact_5733_le__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(3)
% 0.30/0.61  thf(fact_5734_le__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(3)
% 0.30/0.61  thf(fact_5735_le__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(3)
% 0.30/0.61  thf(fact_5736_le__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(1)
% 0.30/0.61  thf(fact_5737_le__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(1)
% 0.30/0.61  thf(fact_5738_le__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(1)
% 0.30/0.61  thf(fact_5739_le__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_one_simps(1)
% 0.30/0.61  thf(fact_5740_less__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(3)
% 0.30/0.61  thf(fact_5741_less__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(3)
% 0.30/0.61  thf(fact_5742_less__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(3)
% 0.30/0.61  thf(fact_5743_less__minus__one__simps_I3_J,axiom,
% 0.30/0.61      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(3)
% 0.30/0.61  thf(fact_5744_less__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(1)
% 0.30/0.61  thf(fact_5745_less__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(1)
% 0.30/0.61  thf(fact_5746_less__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(1)
% 0.30/0.61  thf(fact_5747_less__minus__one__simps_I1_J,axiom,
% 0.30/0.61      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_one_simps(1)
% 0.30/0.61  thf(fact_5748_not__one__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_le_neg_numeral
% 0.30/0.61  thf(fact_5749_not__one__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_le_neg_numeral
% 0.30/0.61  thf(fact_5750_not__one__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_le_neg_numeral
% 0.30/0.61  thf(fact_5751_not__one__le__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_le_neg_numeral
% 0.30/0.61  thf(fact_5752_not__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_one
% 0.30/0.61  thf(fact_5753_not__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_one
% 0.30/0.61  thf(fact_5754_not__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_one
% 0.30/0.61  thf(fact_5755_not__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_le_neg_one
% 0.30/0.61  thf(fact_5756_neg__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_neg_one
% 0.30/0.61  thf(fact_5757_neg__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_neg_one
% 0.30/0.61  thf(fact_5758_neg__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_neg_one
% 0.30/0.61  thf(fact_5759_neg__numeral__le__neg__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_neg_one
% 0.30/0.61  thf(fact_5760_neg__one__le__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_le_numeral
% 0.30/0.61  thf(fact_5761_neg__one__le__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_le_numeral
% 0.30/0.61  thf(fact_5762_neg__one__le__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_le_numeral
% 0.30/0.61  thf(fact_5763_neg__one__le__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_le_numeral
% 0.30/0.61  thf(fact_5764_neg__numeral__le__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_one
% 0.30/0.61  thf(fact_5765_neg__numeral__le__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_one
% 0.30/0.61  thf(fact_5766_neg__numeral__le__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_one
% 0.30/0.61  thf(fact_5767_neg__numeral__le__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_le_one
% 0.30/0.61  thf(fact_5768_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_less_neg_numeral
% 0.30/0.61  thf(fact_5769_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_less_neg_numeral
% 0.30/0.61  thf(fact_5770_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_less_neg_numeral
% 0.30/0.61  thf(fact_5771_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_neg_one_less_neg_numeral
% 0.30/0.61  thf(fact_5772_not__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_less_neg_numeral
% 0.30/0.61  thf(fact_5773_not__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_less_neg_numeral
% 0.30/0.61  thf(fact_5774_not__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_less_neg_numeral
% 0.30/0.61  thf(fact_5775_not__one__less__neg__numeral,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_one_less_neg_numeral
% 0.30/0.61  thf(fact_5776_not__numeral__less__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_one
% 0.30/0.61  thf(fact_5777_not__numeral__less__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_one
% 0.30/0.61  thf(fact_5778_not__numeral__less__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_one
% 0.30/0.61  thf(fact_5779_not__numeral__less__neg__one,axiom,
% 0.30/0.61      ! [M: num] :
% 0.30/0.61        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_numeral_less_neg_one
% 0.30/0.61  thf(fact_5780_neg__one__less__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_less_numeral
% 0.30/0.61  thf(fact_5781_neg__one__less__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_less_numeral
% 0.30/0.61  thf(fact_5782_neg__one__less__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_less_numeral
% 0.30/0.61  thf(fact_5783_neg__one__less__numeral,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_less_numeral
% 0.30/0.61  thf(fact_5784_neg__numeral__less__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_one
% 0.30/0.61  thf(fact_5785_neg__numeral__less__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_one
% 0.30/0.61  thf(fact_5786_neg__numeral__less__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_one
% 0.30/0.61  thf(fact_5787_neg__numeral__less__one,axiom,
% 0.30/0.61      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_numeral_less_one
% 0.30/0.61  thf(fact_5788_eq__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: real,B2: real,C: real] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( C != zero_zero_real )
% 0.30/0.61           => ( ( times_times_real @ A @ C )
% 0.30/0.61              = ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61          & ( ( C = zero_zero_real )
% 0.30/0.61           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_minus_divide_eq
% 0.30/0.61  thf(fact_5789_eq__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: complex,B2: complex,C: complex] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( C != zero_zero_complex )
% 0.30/0.61           => ( ( times_times_complex @ A @ C )
% 0.30/0.61              = ( uminus1482373934393186551omplex @ B2 ) ) )
% 0.30/0.61          & ( ( C = zero_zero_complex )
% 0.30/0.61           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_minus_divide_eq
% 0.30/0.61  thf(fact_5790_eq__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.61        ( ( A
% 0.30/0.61          = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( C != zero_zero_rat )
% 0.30/0.61           => ( ( times_times_rat @ A @ C )
% 0.30/0.61              = ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61          & ( ( C = zero_zero_rat )
% 0.30/0.61           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_minus_divide_eq
% 0.30/0.61  thf(fact_5791_minus__divide__eq__eq,axiom,
% 0.30/0.61      ! [B2: real,C: real,A: real] :
% 0.30/0.61        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.61          = A )
% 0.30/0.61        = ( ( ( C != zero_zero_real )
% 0.30/0.61           => ( ( uminus_uminus_real @ B2 )
% 0.30/0.61              = ( times_times_real @ A @ C ) ) )
% 0.30/0.61          & ( ( C = zero_zero_real )
% 0.30/0.61           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_eq_eq
% 0.30/0.61  thf(fact_5792_minus__divide__eq__eq,axiom,
% 0.30/0.61      ! [B2: complex,C: complex,A: complex] :
% 0.30/0.61        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.61          = A )
% 0.30/0.61        = ( ( ( C != zero_zero_complex )
% 0.30/0.61           => ( ( uminus1482373934393186551omplex @ B2 )
% 0.30/0.61              = ( times_times_complex @ A @ C ) ) )
% 0.30/0.61          & ( ( C = zero_zero_complex )
% 0.30/0.61           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_eq_eq
% 0.30/0.61  thf(fact_5793_minus__divide__eq__eq,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.61        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.61          = A )
% 0.30/0.61        = ( ( ( C != zero_zero_rat )
% 0.30/0.61           => ( ( uminus_uminus_rat @ B2 )
% 0.30/0.61              = ( times_times_rat @ A @ C ) ) )
% 0.30/0.61          & ( ( C = zero_zero_rat )
% 0.30/0.61           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_eq_eq
% 0.30/0.61  thf(fact_5794_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.61      ! [B2: real,A: real,C: real] :
% 0.30/0.61        ( ( B2 != zero_zero_real )
% 0.30/0.61       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.61            = C )
% 0.30/0.61          = ( ( uminus_uminus_real @ A )
% 0.30/0.61            = ( times_times_real @ C @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_neg_divide_eq_eq
% 0.30/0.61  thf(fact_5795_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.61      ! [B2: complex,A: complex,C: complex] :
% 0.30/0.61        ( ( B2 != zero_zero_complex )
% 0.30/0.61       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.61            = C )
% 0.30/0.61          = ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.61            = ( times_times_complex @ C @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_neg_divide_eq_eq
% 0.30/0.61  thf(fact_5796_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.61      ! [B2: rat,A: rat,C: rat] :
% 0.30/0.61        ( ( B2 != zero_zero_rat )
% 0.30/0.61       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.61            = C )
% 0.30/0.61          = ( ( uminus_uminus_rat @ A )
% 0.30/0.61            = ( times_times_rat @ C @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_neg_divide_eq_eq
% 0.30/0.61  thf(fact_5797_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.61      ! [B2: real,C: real,A: real] :
% 0.30/0.61        ( ( B2 != zero_zero_real )
% 0.30/0.61       => ( ( C
% 0.30/0.61            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
% 0.30/0.61          = ( ( times_times_real @ C @ B2 )
% 0.30/0.61            = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_neg_divide_eq_eq2
% 0.30/0.61  thf(fact_5798_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.61      ! [B2: complex,C: complex,A: complex] :
% 0.30/0.61        ( ( B2 != zero_zero_complex )
% 0.30/0.61       => ( ( C
% 0.30/0.61            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.61          = ( ( times_times_complex @ C @ B2 )
% 0.30/0.61            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_neg_divide_eq_eq2
% 0.30/0.61  thf(fact_5799_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.61        ( ( B2 != zero_zero_rat )
% 0.30/0.61       => ( ( C
% 0.30/0.61            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) )
% 0.30/0.61          = ( ( times_times_rat @ C @ B2 )
% 0.30/0.61            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % nonzero_neg_divide_eq_eq2
% 0.30/0.61  thf(fact_5800_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.61      ! [B2: real] :
% 0.30/0.61        ( ( times_times_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.30/0.61        = ( uminus_uminus_real @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(2)
% 0.30/0.61  thf(fact_5801_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.61      ! [B2: int] :
% 0.30/0.61        ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.30/0.61        = ( uminus_uminus_int @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(2)
% 0.30/0.61  thf(fact_5802_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.61      ! [B2: complex] :
% 0.30/0.61        ( ( times_times_complex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(2)
% 0.30/0.61  thf(fact_5803_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.61      ! [B2: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(2)
% 0.30/0.61  thf(fact_5804_mult__1s__ring__1_I2_J,axiom,
% 0.30/0.61      ! [B2: rat] :
% 0.30/0.61        ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.30/0.61        = ( uminus_uminus_rat @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(2)
% 0.30/0.61  thf(fact_5805_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.61      ! [B2: real] :
% 0.30/0.61        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B2 )
% 0.30/0.61        = ( uminus_uminus_real @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(1)
% 0.30/0.61  thf(fact_5806_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.61      ! [B2: int] :
% 0.30/0.61        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
% 0.30/0.61        = ( uminus_uminus_int @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(1)
% 0.30/0.61  thf(fact_5807_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.61      ! [B2: complex] :
% 0.30/0.61        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B2 )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(1)
% 0.30/0.61  thf(fact_5808_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.61      ! [B2: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(1)
% 0.30/0.61  thf(fact_5809_mult__1s__ring__1_I1_J,axiom,
% 0.30/0.61      ! [B2: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
% 0.30/0.61        = ( uminus_uminus_rat @ B2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mult_1s_ring_1(1)
% 0.30/0.61  thf(fact_5810_divide__eq__minus__1__iff,axiom,
% 0.30/0.61      ! [A: real,B2: real] :
% 0.30/0.61        ( ( ( divide_divide_real @ A @ B2 )
% 0.30/0.61          = ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.61        = ( ( B2 != zero_zero_real )
% 0.30/0.61          & ( A
% 0.30/0.61            = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_minus_1_iff
% 0.30/0.61  thf(fact_5811_divide__eq__minus__1__iff,axiom,
% 0.30/0.61      ! [A: complex,B2: complex] :
% 0.30/0.61        ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 0.30/0.61          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.61        = ( ( B2 != zero_zero_complex )
% 0.30/0.61          & ( A
% 0.30/0.61            = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_minus_1_iff
% 0.30/0.61  thf(fact_5812_divide__eq__minus__1__iff,axiom,
% 0.30/0.61      ! [A: rat,B2: rat] :
% 0.30/0.61        ( ( ( divide_divide_rat @ A @ B2 )
% 0.30/0.61          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.61        = ( ( B2 != zero_zero_rat )
% 0.30/0.61          & ( A
% 0.30/0.61            = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_minus_1_iff
% 0.30/0.61  thf(fact_5813_uminus__numeral__One,axiom,
% 0.30/0.61      ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.30/0.61      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_numeral_One
% 0.30/0.61  thf(fact_5814_uminus__numeral__One,axiom,
% 0.30/0.61      ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.30/0.61      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_numeral_One
% 0.30/0.61  thf(fact_5815_uminus__numeral__One,axiom,
% 0.30/0.61      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.61      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_numeral_One
% 0.30/0.61  thf(fact_5816_uminus__numeral__One,axiom,
% 0.30/0.61      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.61      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_numeral_One
% 0.30/0.61  thf(fact_5817_uminus__numeral__One,axiom,
% 0.30/0.61      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.30/0.61      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_numeral_One
% 0.30/0.61  thf(fact_5818_power__minus,axiom,
% 0.30/0.61      ! [A: real,N: nat] :
% 0.30/0.61        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus
% 0.30/0.61  thf(fact_5819_power__minus,axiom,
% 0.30/0.61      ! [A: int,N: nat] :
% 0.30/0.61        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus
% 0.30/0.61  thf(fact_5820_power__minus,axiom,
% 0.30/0.61      ! [A: complex,N: nat] :
% 0.30/0.61        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus
% 0.30/0.61  thf(fact_5821_power__minus,axiom,
% 0.30/0.61      ! [A: code_integer,N: nat] :
% 0.30/0.61        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus
% 0.30/0.61  thf(fact_5822_power__minus,axiom,
% 0.30/0.61      ! [A: rat,N: nat] :
% 0.30/0.61        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus
% 0.30/0.61  thf(fact_5823_power__minus__Bit0,axiom,
% 0.30/0.61      ! [X3: real,K: num] :
% 0.30/0.61        ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.61        = ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_Bit0
% 0.30/0.61  thf(fact_5824_power__minus__Bit0,axiom,
% 0.30/0.61      ! [X3: int,K: num] :
% 0.30/0.61        ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.61        = ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_Bit0
% 0.30/0.61  thf(fact_5825_power__minus__Bit0,axiom,
% 0.30/0.61      ! [X3: complex,K: num] :
% 0.30/0.61        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.61        = ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_Bit0
% 0.30/0.61  thf(fact_5826_power__minus__Bit0,axiom,
% 0.30/0.61      ! [X3: code_integer,K: num] :
% 0.30/0.61        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.61        = ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_Bit0
% 0.30/0.61  thf(fact_5827_power__minus__Bit0,axiom,
% 0.30/0.61      ! [X3: rat,K: num] :
% 0.30/0.61        ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.61        = ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus_Bit0
% 0.30/0.61  thf(fact_5828_real__add__less__0__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 0.30/0.61        = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % real_add_less_0_iff
% 0.30/0.61  thf(fact_5829_real__0__less__add__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.61        = ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % real_0_less_add_iff
% 0.30/0.61  thf(fact_5830_real__add__le__0__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 0.30/0.61        = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % real_add_le_0_iff
% 0.30/0.61  thf(fact_5831_real__0__le__add__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.61        = ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % real_0_le_add_iff
% 0.30/0.61  thf(fact_5832_pos__minus__divide__less__eq,axiom,
% 0.30/0.61      ! [C: real,B2: real,A: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_minus_divide_less_eq
% 0.30/0.61  thf(fact_5833_pos__minus__divide__less__eq,axiom,
% 0.30/0.61      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_minus_divide_less_eq
% 0.30/0.61  thf(fact_5834_pos__less__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: real,A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_less_minus_divide_eq
% 0.30/0.61  thf(fact_5835_pos__less__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_less_minus_divide_eq
% 0.30/0.61  thf(fact_5836_neg__minus__divide__less__eq,axiom,
% 0.30/0.61      ! [C: real,B2: real,A: real] :
% 0.30/0.61        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_minus_divide_less_eq
% 0.30/0.61  thf(fact_5837_neg__minus__divide__less__eq,axiom,
% 0.30/0.61      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_minus_divide_less_eq
% 0.30/0.61  thf(fact_5838_neg__less__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: real,A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_less_minus_divide_eq
% 0.30/0.61  thf(fact_5839_neg__less__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_less_minus_divide_eq
% 0.30/0.61  thf(fact_5840_minus__divide__less__eq,axiom,
% 0.30/0.61      ! [B2: real,C: real,A: real] :
% 0.30/0.61        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_less_eq
% 0.30/0.61  thf(fact_5841_minus__divide__less__eq,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_less_eq
% 0.30/0.61  thf(fact_5842_less__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: real,B2: real,C: real] :
% 0.30/0.61        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_divide_eq
% 0.30/0.61  thf(fact_5843_less__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.61        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_minus_divide_eq
% 0.30/0.61  thf(fact_5844_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: real,C: real,W: num] :
% 0.30/0.61        ( ( ( divide_divide_real @ B2 @ C )
% 0.30/0.61          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61        = ( ( ( C != zero_zero_real )
% 0.30/0.61           => ( B2
% 0.30/0.61              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ( C = zero_zero_real )
% 0.30/0.61           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61              = zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_eq_numeral(2)
% 0.30/0.61  thf(fact_5845_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: complex,C: complex,W: num] :
% 0.30/0.61        ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 0.30/0.61          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.61        = ( ( ( C != zero_zero_complex )
% 0.30/0.61           => ( B2
% 0.30/0.61              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ( C = zero_zero_complex )
% 0.30/0.61           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61              = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_eq_numeral(2)
% 0.30/0.61  thf(fact_5846_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,W: num] :
% 0.30/0.61        ( ( ( divide_divide_rat @ B2 @ C )
% 0.30/0.61          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61        = ( ( ( C != zero_zero_rat )
% 0.30/0.61           => ( B2
% 0.30/0.61              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ( C = zero_zero_rat )
% 0.30/0.61           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61              = zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_eq_eq_numeral(2)
% 0.30/0.61  thf(fact_5847_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: real,C: real] :
% 0.30/0.61        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61          = ( divide_divide_real @ B2 @ C ) )
% 0.30/0.61        = ( ( ( C != zero_zero_real )
% 0.30/0.61           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.30/0.61              = B2 ) )
% 0.30/0.61          & ( ( C = zero_zero_real )
% 0.30/0.61           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.61              = zero_zero_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5848_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: complex,C: complex] :
% 0.30/0.61        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 0.30/0.61        = ( ( ( C != zero_zero_complex )
% 0.30/0.61           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.30/0.61              = B2 ) )
% 0.30/0.61          & ( ( C = zero_zero_complex )
% 0.30/0.61           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.61              = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5849_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: rat,C: rat] :
% 0.30/0.61        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61          = ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.61        = ( ( ( C != zero_zero_rat )
% 0.30/0.61           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.30/0.61              = B2 ) )
% 0.30/0.61          & ( ( C = zero_zero_rat )
% 0.30/0.61           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.30/0.61              = zero_zero_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % eq_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5850_subset__decode__imp__le,axiom,
% 0.30/0.61      ! [M: nat,N: nat] :
% 0.30/0.61        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.30/0.61       => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_decode_imp_le
% 0.30/0.61  thf(fact_5851_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.61      ! [Z: real,A: real,B2: real] :
% 0.30/0.61        ( ( ( Z = zero_zero_real )
% 0.30/0.61         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.30/0.61            = B2 ) )
% 0.30/0.61        & ( ( Z != zero_zero_real )
% 0.30/0.61         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(3)
% 0.30/0.61  thf(fact_5852_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.61      ! [Z: complex,A: complex,B2: complex] :
% 0.30/0.61        ( ( ( Z = zero_zero_complex )
% 0.30/0.61         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.30/0.61            = B2 ) )
% 0.30/0.61        & ( ( Z != zero_zero_complex )
% 0.30/0.61         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(3)
% 0.30/0.61  thf(fact_5853_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.61      ! [Z: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ( Z = zero_zero_rat )
% 0.30/0.61         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.30/0.61            = B2 ) )
% 0.30/0.61        & ( ( Z != zero_zero_rat )
% 0.30/0.61         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(3)
% 0.30/0.61  thf(fact_5854_minus__divide__add__eq__iff,axiom,
% 0.30/0.61      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.61        ( ( Z != zero_zero_real )
% 0.30/0.61       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z ) ) @ Y3 )
% 0.30/0.61          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_add_eq_iff
% 0.30/0.61  thf(fact_5855_minus__divide__add__eq__iff,axiom,
% 0.30/0.61      ! [Z: complex,X3: complex,Y3: complex] :
% 0.30/0.61        ( ( Z != zero_zero_complex )
% 0.30/0.61       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z ) ) @ Y3 )
% 0.30/0.61          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_add_eq_iff
% 0.30/0.61  thf(fact_5856_minus__divide__add__eq__iff,axiom,
% 0.30/0.61      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.61        ( ( Z != zero_zero_rat )
% 0.30/0.61       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z ) ) @ Y3 )
% 0.30/0.61          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_add_eq_iff
% 0.30/0.61  thf(fact_5857_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.61      ! [Z: real,A: real,B2: real] :
% 0.30/0.61        ( ( ( Z = zero_zero_real )
% 0.30/0.61         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61        & ( ( Z != zero_zero_real )
% 0.30/0.61         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(6)
% 0.30/0.61  thf(fact_5858_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.61      ! [Z: complex,A: complex,B2: complex] :
% 0.30/0.61        ( ( ( Z = zero_zero_complex )
% 0.30/0.61         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( uminus1482373934393186551omplex @ B2 ) ) )
% 0.30/0.61        & ( ( Z != zero_zero_complex )
% 0.30/0.61         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(6)
% 0.30/0.61  thf(fact_5859_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.61      ! [Z: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ( Z = zero_zero_rat )
% 0.30/0.61         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61        & ( ( Z != zero_zero_rat )
% 0.30/0.61         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 0.30/0.61            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(6)
% 0.30/0.61  thf(fact_5860_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.61      ! [Z: real,A: real,B2: real] :
% 0.30/0.61        ( ( ( Z = zero_zero_real )
% 0.30/0.61         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.30/0.61            = ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61        & ( ( Z != zero_zero_real )
% 0.30/0.61         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 0.30/0.61            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(5)
% 0.30/0.61  thf(fact_5861_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.61      ! [Z: complex,A: complex,B2: complex] :
% 0.30/0.61        ( ( ( Z = zero_zero_complex )
% 0.30/0.61         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.30/0.61            = ( uminus1482373934393186551omplex @ B2 ) ) )
% 0.30/0.61        & ( ( Z != zero_zero_complex )
% 0.30/0.61         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 0.30/0.61            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(5)
% 0.30/0.61  thf(fact_5862_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.61      ! [Z: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ( Z = zero_zero_rat )
% 0.30/0.61         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.30/0.61            = ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61        & ( ( Z != zero_zero_rat )
% 0.30/0.61         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 0.30/0.61            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_divide_eq_if_simps(5)
% 0.30/0.61  thf(fact_5863_minus__divide__diff__eq__iff,axiom,
% 0.30/0.61      ! [Z: real,X3: real,Y3: real] :
% 0.30/0.61        ( ( Z != zero_zero_real )
% 0.30/0.61       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z ) ) @ Y3 )
% 0.30/0.61          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_diff_eq_iff
% 0.30/0.61  thf(fact_5864_minus__divide__diff__eq__iff,axiom,
% 0.30/0.61      ! [Z: complex,X3: complex,Y3: complex] :
% 0.30/0.61        ( ( Z != zero_zero_complex )
% 0.30/0.61       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z ) ) @ Y3 )
% 0.30/0.61          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_diff_eq_iff
% 0.30/0.61  thf(fact_5865_minus__divide__diff__eq__iff,axiom,
% 0.30/0.61      ! [Z: rat,X3: rat,Y3: rat] :
% 0.30/0.61        ( ( Z != zero_zero_rat )
% 0.30/0.61       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z ) ) @ Y3 )
% 0.30/0.61          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_diff_eq_iff
% 0.30/0.61  thf(fact_5866_even__minus,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.61        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % even_minus
% 0.30/0.61  thf(fact_5867_even__minus,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.61        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % even_minus
% 0.30/0.61  thf(fact_5868_power2__eq__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61        = ( ( X3 = Y3 )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_iff
% 0.30/0.61  thf(fact_5869_power2__eq__iff,axiom,
% 0.30/0.61      ! [X3: int,Y3: int] :
% 0.30/0.61        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61        = ( ( X3 = Y3 )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_iff
% 0.30/0.61  thf(fact_5870_power2__eq__iff,axiom,
% 0.30/0.61      ! [X3: complex,Y3: complex] :
% 0.30/0.61        ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61        = ( ( X3 = Y3 )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_iff
% 0.30/0.61  thf(fact_5871_power2__eq__iff,axiom,
% 0.30/0.61      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.61        ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61        = ( ( X3 = Y3 )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_iff
% 0.30/0.61  thf(fact_5872_power2__eq__iff,axiom,
% 0.30/0.61      ! [X3: rat,Y3: rat] :
% 0.30/0.61        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.61        = ( ( X3 = Y3 )
% 0.30/0.61          | ( X3
% 0.30/0.61            = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_iff
% 0.30/0.61  thf(fact_5873_uminus__power__if,axiom,
% 0.30/0.61      ! [N: nat,A: real] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61            = ( power_power_real @ A @ N ) ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.30/0.61            = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_power_if
% 0.30/0.61  thf(fact_5874_uminus__power__if,axiom,
% 0.30/0.61      ! [N: nat,A: int] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61            = ( power_power_int @ A @ N ) ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.30/0.61            = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_power_if
% 0.30/0.61  thf(fact_5875_uminus__power__if,axiom,
% 0.30/0.61      ! [N: nat,A: complex] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61            = ( power_power_complex @ A @ N ) ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.30/0.61            = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_power_if
% 0.30/0.61  thf(fact_5876_uminus__power__if,axiom,
% 0.30/0.61      ! [N: nat,A: code_integer] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61            = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.30/0.61            = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_power_if
% 0.30/0.61  thf(fact_5877_uminus__power__if,axiom,
% 0.30/0.61      ! [N: nat,A: rat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61            = ( power_power_rat @ A @ N ) ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.30/0.61            = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_power_if
% 0.30/0.61  thf(fact_5878_verit__less__mono__div__int2,axiom,
% 0.30/0.61      ! [A4: int,B5: int,N: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ A4 @ B5 )
% 0.30/0.61       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.30/0.61         => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % verit_less_mono_div_int2
% 0.30/0.61  thf(fact_5879_div__eq__minus1,axiom,
% 0.30/0.61      ! [B2: int] :
% 0.30/0.61        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.61       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 0.30/0.61          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % div_eq_minus1
% 0.30/0.61  thf(fact_5880_of__bool__odd__eq__mod__2,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( zero_n2687167440665602831ol_nat
% 0.30/0.61          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.61        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_odd_eq_mod_2
% 0.30/0.61  thf(fact_5881_of__bool__odd__eq__mod__2,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( zero_n2684676970156552555ol_int
% 0.30/0.61          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.61        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_odd_eq_mod_2
% 0.30/0.61  thf(fact_5882_of__bool__odd__eq__mod__2,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( zero_n356916108424825756nteger
% 0.30/0.61          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.61        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_bool_odd_eq_mod_2
% 0.30/0.61  thf(fact_5883_pos__minus__divide__le__eq,axiom,
% 0.30/0.61      ! [C: real,B2: real,A: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_minus_divide_le_eq
% 0.30/0.61  thf(fact_5884_pos__minus__divide__le__eq,axiom,
% 0.30/0.61      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_minus_divide_le_eq
% 0.30/0.61  thf(fact_5885_pos__le__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: real,A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_le_minus_divide_eq
% 0.30/0.61  thf(fact_5886_pos__le__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pos_le_minus_divide_eq
% 0.30/0.61  thf(fact_5887_neg__minus__divide__le__eq,axiom,
% 0.30/0.61      ! [C: real,B2: real,A: real] :
% 0.30/0.61        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_minus_divide_le_eq
% 0.30/0.61  thf(fact_5888_neg__minus__divide__le__eq,axiom,
% 0.30/0.61      ! [C: rat,B2: rat,A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.30/0.61          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_minus_divide_le_eq
% 0.30/0.61  thf(fact_5889_neg__le__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: real,A: real,B2: real] :
% 0.30/0.61        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_le_minus_divide_eq
% 0.30/0.61  thf(fact_5890_neg__le__minus__divide__eq,axiom,
% 0.30/0.61      ! [C: rat,A: rat,B2: rat] :
% 0.30/0.61        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_le_minus_divide_eq
% 0.30/0.61  thf(fact_5891_minus__divide__le__eq,axiom,
% 0.30/0.61      ! [B2: real,C: real,A: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_le_eq
% 0.30/0.61  thf(fact_5892_minus__divide__le__eq,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,A: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_divide_le_eq
% 0.30/0.61  thf(fact_5893_le__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: real,B2: real,C: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_divide_eq
% 0.30/0.61  thf(fact_5894_le__minus__divide__eq,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,C: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_minus_divide_eq
% 0.30/0.61  thf(fact_5895_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: real,C: real] :
% 0.30/0.61        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5896_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: rat,C: rat] :
% 0.30/0.61        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5897_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: real,C: real,W: num] :
% 0.30/0.61        ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_less_eq_numeral(2)
% 0.30/0.61  thf(fact_5898_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,W: num] :
% 0.30/0.61        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_less_eq_numeral(2)
% 0.30/0.61  thf(fact_5899_power2__eq__1__iff,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = one_one_real )
% 0.30/0.61        = ( ( A = one_one_real )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_1_iff
% 0.30/0.61  thf(fact_5900_power2__eq__1__iff,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = one_one_int )
% 0.30/0.61        = ( ( A = one_one_int )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_1_iff
% 0.30/0.61  thf(fact_5901_power2__eq__1__iff,axiom,
% 0.30/0.61      ! [A: complex] :
% 0.30/0.61        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = one_one_complex )
% 0.30/0.61        = ( ( A = one_one_complex )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_1_iff
% 0.30/0.61  thf(fact_5902_power2__eq__1__iff,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = one_one_Code_integer )
% 0.30/0.61        = ( ( A = one_one_Code_integer )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_1_iff
% 0.30/0.61  thf(fact_5903_power2__eq__1__iff,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61          = one_one_rat )
% 0.30/0.61        = ( ( A = one_one_rat )
% 0.30/0.61          | ( A
% 0.30/0.61            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power2_eq_1_iff
% 0.30/0.61  thf(fact_5904_minus__one__power__iff,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.61            = one_one_real ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.30/0.61            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_power_iff
% 0.30/0.61  thf(fact_5905_minus__one__power__iff,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.61            = one_one_int ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.30/0.61            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_power_iff
% 0.30/0.61  thf(fact_5906_minus__one__power__iff,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.61            = one_one_complex ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.30/0.61            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_power_iff
% 0.30/0.61  thf(fact_5907_minus__one__power__iff,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.61            = one_one_Code_integer ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.30/0.61            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_power_iff
% 0.30/0.61  thf(fact_5908_minus__one__power__iff,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.61            = one_one_rat ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.61         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.30/0.61            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_one_power_iff
% 0.30/0.61  thf(fact_5909_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61      ! [K: nat,N: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61  thf(fact_5910_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61      ! [K: nat,N: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61  thf(fact_5911_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61      ! [K: nat,N: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61  thf(fact_5912_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61      ! [K: nat,N: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61  thf(fact_5913_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.61      ! [K: nat,N: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.61       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.61          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.61  thf(fact_5914_realpow__square__minus__le,axiom,
% 0.30/0.61      ! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % realpow_square_minus_le
% 0.30/0.61  thf(fact_5915_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.30/0.61      ! [N: nat,K: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.30/0.61        = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_int_less_eq_self_iff
% 0.30/0.61  thf(fact_5916_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.30/0.61      ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_int_greater_eq_minus_exp
% 0.30/0.61  thf(fact_5917_signed__take__bit__int__greater__self__iff,axiom,
% 0.30/0.61      ! [K: int,N: nat] :
% 0.30/0.61        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.30/0.61        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_int_greater_self_iff
% 0.30/0.61  thf(fact_5918_minus__mod__int__eq,axiom,
% 0.30/0.61      ! [L: int,K: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.30/0.61       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.30/0.61          = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_mod_int_eq
% 0.30/0.61  thf(fact_5919_zmod__minus1,axiom,
% 0.30/0.61      ! [B2: int] :
% 0.30/0.61        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.61       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 0.30/0.61          = ( minus_minus_int @ B2 @ one_one_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zmod_minus1
% 0.30/0.61  thf(fact_5920_zminus1__lemma,axiom,
% 0.30/0.61      ! [A: int,B2: int,Q2: int,R: int] :
% 0.30/0.61        ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.61       => ( ( B2 != zero_zero_int )
% 0.30/0.61         => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zminus1_lemma
% 0.30/0.61  thf(fact_5921_bits__induct,axiom,
% 0.30/0.61      ! [P: nat > $o,A: nat] :
% 0.30/0.61        ( ! [A3: nat] :
% 0.30/0.61            ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61              = A3 )
% 0.30/0.61           => ( P @ A3 ) )
% 0.30/0.61       => ( ! [A3: nat,B3: $o] :
% 0.30/0.61              ( ( P @ A3 )
% 0.30/0.61             => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.61                  = A3 )
% 0.30/0.61               => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 0.30/0.61         => ( P @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_induct
% 0.30/0.61  thf(fact_5922_bits__induct,axiom,
% 0.30/0.61      ! [P: int > $o,A: int] :
% 0.30/0.61        ( ! [A3: int] :
% 0.30/0.61            ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61              = A3 )
% 0.30/0.61           => ( P @ A3 ) )
% 0.30/0.61       => ( ! [A3: int,B3: $o] :
% 0.30/0.61              ( ( P @ A3 )
% 0.30/0.61             => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61                  = A3 )
% 0.30/0.61               => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 0.30/0.61         => ( P @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_induct
% 0.30/0.61  thf(fact_5923_bits__induct,axiom,
% 0.30/0.61      ! [P: code_integer > $o,A: code_integer] :
% 0.30/0.61        ( ! [A3: code_integer] :
% 0.30/0.61            ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.61              = A3 )
% 0.30/0.61           => ( P @ A3 ) )
% 0.30/0.61       => ( ! [A3: code_integer,B3: $o] :
% 0.30/0.61              ( ( P @ A3 )
% 0.30/0.61             => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.61                  = A3 )
% 0.30/0.61               => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 0.30/0.61         => ( P @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bits_induct
% 0.30/0.61  thf(fact_5924_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: real,C: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5925_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [W: num,B2: rat,C: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % le_divide_eq_numeral(2)
% 0.30/0.61  thf(fact_5926_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: real,C: real,W: num] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.61        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.61           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.61               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_le_eq_numeral(2)
% 0.30/0.61  thf(fact_5927_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.61      ! [B2: rat,C: rat,W: num] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.61        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.30/0.61          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.61           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 0.30/0.61              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.61               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divide_le_eq_numeral(2)
% 0.30/0.61  thf(fact_5928_square__le__1,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.61       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.61         => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_le_1
% 0.30/0.61  thf(fact_5929_square__le__1,axiom,
% 0.30/0.61      ! [X3: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
% 0.30/0.61       => ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
% 0.30/0.61         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_le_1
% 0.30/0.61  thf(fact_5930_square__le__1,axiom,
% 0.30/0.61      ! [X3: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
% 0.30/0.61       => ( ( ord_less_eq_rat @ X3 @ one_one_rat )
% 0.30/0.61         => ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_le_1
% 0.30/0.61  thf(fact_5931_square__le__1,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
% 0.30/0.61       => ( ( ord_less_eq_int @ X3 @ one_one_int )
% 0.30/0.61         => ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % square_le_1
% 0.30/0.61  thf(fact_5932_minus__power__mult__self,axiom,
% 0.30/0.61      ! [A: real,N: nat] :
% 0.30/0.61        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.30/0.61        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_power_mult_self
% 0.30/0.61  thf(fact_5933_minus__power__mult__self,axiom,
% 0.30/0.61      ! [A: int,N: nat] :
% 0.30/0.61        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.30/0.61        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_power_mult_self
% 0.30/0.61  thf(fact_5934_minus__power__mult__self,axiom,
% 0.30/0.61      ! [A: complex,N: nat] :
% 0.30/0.61        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.30/0.61        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_power_mult_self
% 0.30/0.61  thf(fact_5935_minus__power__mult__self,axiom,
% 0.30/0.61      ! [A: code_integer,N: nat] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.30/0.61        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_power_mult_self
% 0.30/0.61  thf(fact_5936_minus__power__mult__self,axiom,
% 0.30/0.61      ! [A: rat,N: nat] :
% 0.30/0.61        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.30/0.61        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_power_mult_self
% 0.30/0.61  thf(fact_5937_signed__take__bit__int__eq__self,axiom,
% 0.30/0.61      ! [N: nat,K: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.30/0.61       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61         => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.30/0.61            = K ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_int_eq_self
% 0.30/0.61  thf(fact_5938_signed__take__bit__int__eq__self__iff,axiom,
% 0.30/0.61      ! [N: nat,K: int] :
% 0.30/0.61        ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.30/0.61          = K )
% 0.30/0.61        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.30/0.61          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % signed_take_bit_int_eq_self_iff
% 0.30/0.61  thf(fact_5939_minus__1__div__exp__eq__int,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_1_div_exp_eq_int
% 0.30/0.61  thf(fact_5940_div__pos__neg__trivial,axiom,
% 0.30/0.61      ! [K: int,L: int] :
% 0.30/0.61        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.61       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.30/0.61         => ( ( divide_divide_int @ K @ L )
% 0.30/0.61            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % div_pos_neg_trivial
% 0.30/0.61  thf(fact_5941_add__0__iff,axiom,
% 0.30/0.61      ! [B2: complex,A: complex] :
% 0.30/0.61        ( ( B2
% 0.30/0.61          = ( plus_plus_complex @ B2 @ A ) )
% 0.30/0.61        = ( A = zero_zero_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_0_iff
% 0.30/0.61  thf(fact_5942_add__0__iff,axiom,
% 0.30/0.61      ! [B2: real,A: real] :
% 0.30/0.61        ( ( B2
% 0.30/0.61          = ( plus_plus_real @ B2 @ A ) )
% 0.30/0.61        = ( A = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_0_iff
% 0.30/0.61  thf(fact_5943_add__0__iff,axiom,
% 0.30/0.61      ! [B2: rat,A: rat] :
% 0.30/0.61        ( ( B2
% 0.30/0.61          = ( plus_plus_rat @ B2 @ A ) )
% 0.30/0.61        = ( A = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_0_iff
% 0.30/0.61  thf(fact_5944_add__0__iff,axiom,
% 0.30/0.61      ! [B2: nat,A: nat] :
% 0.30/0.61        ( ( B2
% 0.30/0.61          = ( plus_plus_nat @ B2 @ A ) )
% 0.30/0.61        = ( A = zero_zero_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_0_iff
% 0.30/0.61  thf(fact_5945_add__0__iff,axiom,
% 0.30/0.61      ! [B2: int,A: int] :
% 0.30/0.61        ( ( B2
% 0.30/0.61          = ( plus_plus_int @ B2 @ A ) )
% 0.30/0.61        = ( A = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % add_0_iff
% 0.30/0.61  thf(fact_5946_exp__mod__exp,axiom,
% 0.30/0.61      ! [M: nat,N: nat] :
% 0.30/0.61        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % exp_mod_exp
% 0.30/0.61  thf(fact_5947_exp__mod__exp,axiom,
% 0.30/0.61      ! [M: nat,N: nat] :
% 0.30/0.61        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % exp_mod_exp
% 0.30/0.61  thf(fact_5948_exp__mod__exp,axiom,
% 0.30/0.61      ! [M: nat,N: nat] :
% 0.30/0.61        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % exp_mod_exp
% 0.30/0.61  thf(fact_5949_crossproduct__noteq,axiom,
% 0.30/0.61      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.61        ( ( ( A != B2 )
% 0.30/0.61          & ( C != D ) )
% 0.30/0.61        = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
% 0.30/0.61         != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_noteq
% 0.30/0.61  thf(fact_5950_crossproduct__noteq,axiom,
% 0.30/0.61      ! [A: rat,B2: rat,C: rat,D: rat] :
% 0.30/0.61        ( ( ( A != B2 )
% 0.30/0.61          & ( C != D ) )
% 0.30/0.61        = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) )
% 0.30/0.61         != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_noteq
% 0.30/0.61  thf(fact_5951_crossproduct__noteq,axiom,
% 0.30/0.61      ! [A: nat,B2: nat,C: nat,D: nat] :
% 0.30/0.61        ( ( ( A != B2 )
% 0.30/0.61          & ( C != D ) )
% 0.30/0.61        = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
% 0.30/0.61         != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_noteq
% 0.30/0.61  thf(fact_5952_crossproduct__noteq,axiom,
% 0.30/0.61      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.61        ( ( ( A != B2 )
% 0.30/0.61          & ( C != D ) )
% 0.30/0.61        = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
% 0.30/0.61         != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_noteq
% 0.30/0.61  thf(fact_5953_crossproduct__eq,axiom,
% 0.30/0.61      ! [W: real,Y3: real,X3: real,Z: real] :
% 0.30/0.61        ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X3 @ Z ) )
% 0.30/0.61          = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X3 @ Y3 ) ) )
% 0.30/0.61        = ( ( W = X3 )
% 0.30/0.61          | ( Y3 = Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_eq
% 0.30/0.61  thf(fact_5954_crossproduct__eq,axiom,
% 0.30/0.61      ! [W: rat,Y3: rat,X3: rat,Z: rat] :
% 0.30/0.61        ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X3 @ Z ) )
% 0.30/0.61          = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X3 @ Y3 ) ) )
% 0.30/0.61        = ( ( W = X3 )
% 0.30/0.61          | ( Y3 = Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_eq
% 0.30/0.61  thf(fact_5955_crossproduct__eq,axiom,
% 0.30/0.61      ! [W: nat,Y3: nat,X3: nat,Z: nat] :
% 0.30/0.61        ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X3 @ Z ) )
% 0.30/0.61          = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X3 @ Y3 ) ) )
% 0.30/0.61        = ( ( W = X3 )
% 0.30/0.61          | ( Y3 = Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_eq
% 0.30/0.61  thf(fact_5956_crossproduct__eq,axiom,
% 0.30/0.61      ! [W: int,Y3: int,X3: int,Z: int] :
% 0.30/0.61        ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X3 @ Z ) )
% 0.30/0.61          = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X3 @ Y3 ) ) )
% 0.30/0.61        = ( ( W = X3 )
% 0.30/0.61          | ( Y3 = Z ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % crossproduct_eq
% 0.30/0.61  thf(fact_5957_power__minus1__odd,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61        = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_odd
% 0.30/0.61  thf(fact_5958_power__minus1__odd,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_odd
% 0.30/0.61  thf(fact_5959_power__minus1__odd,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_odd
% 0.30/0.61  thf(fact_5960_power__minus1__odd,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.61        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.61        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.61  
% 0.30/0.61  % power_minus1_odd
% 0.30/0.61  thf(fact_5961_power__minus1__odd,axiom,
% 0.30/0.61      ! [N: nat] :
% 0.30/0.62        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.62        = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_minus1_odd
% 0.30/0.62  thf(fact_5962_int__bit__induct,axiom,
% 0.30/0.62      ! [P: int > $o,K: int] :
% 0.30/0.62        ( ( P @ zero_zero_int )
% 0.30/0.62       => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.62         => ( ! [K3: int] :
% 0.30/0.62                ( ( P @ K3 )
% 0.30/0.62               => ( ( K3 != zero_zero_int )
% 0.30/0.62                 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.62           => ( ! [K3: int] :
% 0.30/0.62                  ( ( P @ K3 )
% 0.30/0.62                 => ( ( K3
% 0.30/0.62                     != ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.62                   => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.30/0.62             => ( P @ K ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % int_bit_induct
% 0.30/0.62  thf(fact_5963_signed__take__bit__int__greater__eq,axiom,
% 0.30/0.62      ! [K: int,N: nat] :
% 0.30/0.62        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % signed_take_bit_int_greater_eq
% 0.30/0.62  thf(fact_5964_set__decode__def,axiom,
% 0.30/0.62      ( nat_set_decode
% 0.30/0.62      = ( ^ [X: nat] :
% 0.30/0.62            ( collect_nat
% 0.30/0.62            @ ^ [N2: nat] :
% 0.30/0.62                ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % set_decode_def
% 0.30/0.62  thf(fact_5965_exp__div__exp__eq,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.62        = ( times_times_nat
% 0.30/0.62          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.62            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.62               != zero_zero_nat )
% 0.30/0.62              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.30/0.62          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % exp_div_exp_eq
% 0.30/0.62  thf(fact_5966_exp__div__exp__eq,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.62        = ( times_times_int
% 0.30/0.62          @ ( zero_n2684676970156552555ol_int
% 0.30/0.62            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.30/0.62               != zero_zero_int )
% 0.30/0.62              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.30/0.62          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % exp_div_exp_eq
% 0.30/0.62  thf(fact_5967_exp__div__exp__eq,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.62        = ( times_3573771949741848930nteger
% 0.30/0.62          @ ( zero_n356916108424825756nteger
% 0.30/0.62            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.30/0.62               != zero_z3403309356797280102nteger )
% 0.30/0.62              & ( ord_less_eq_nat @ N @ M ) ) )
% 0.30/0.62          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % exp_div_exp_eq
% 0.30/0.62  thf(fact_5968_vebt__buildup_Osimps_I3_J,axiom,
% 0.30/0.62      ! [Va: nat] :
% 0.30/0.62        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.62         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.62            = ( 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 ) ) ) ) ) ) )
% 0.30/0.62        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.62         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.62            = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % vebt_buildup.simps(3)
% 0.30/0.62  thf(fact_5969_Divides_Oadjust__div__eq,axiom,
% 0.30/0.62      ! [Q2: int,R: int] :
% 0.30/0.62        ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.62        = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Divides.adjust_div_eq
% 0.30/0.62  thf(fact_5970_signed__take__bit__Suc__minus__bit1,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.62        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % signed_take_bit_Suc_minus_bit1
% 0.30/0.62  thf(fact_5971_vebt__buildup_Opelims,axiom,
% 0.30/0.62      ! [X3: nat,Y3: vEBT_VEBT] :
% 0.30/0.62        ( ( ( vEBT_vebt_buildup @ X3 )
% 0.30/0.62          = Y3 )
% 0.30/0.62       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
% 0.30/0.62         => ( ( ( X3 = zero_zero_nat )
% 0.30/0.62             => ( ( Y3
% 0.30/0.62                  = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.62               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.30/0.62           => ( ( ( X3
% 0.30/0.62                  = ( suc @ zero_zero_nat ) )
% 0.30/0.62               => ( ( Y3
% 0.30/0.62                    = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.62                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.62             => ~ ! [Va2: nat] :
% 0.30/0.62                    ( ( X3
% 0.30/0.62                      = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.62                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.62                         => ( Y3
% 0.30/0.62                            = ( 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 ) ) ) ) ) ) )
% 0.30/0.62                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.62                         => ( Y3
% 0.30/0.62                            = ( 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 ) ) ) ) ) ) ) ) )
% 0.30/0.62                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % vebt_buildup.pelims
% 0.30/0.62  thf(fact_5972_ln__one__minus__pos__lower__bound,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.62         => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_one_minus_pos_lower_bound
% 0.30/0.62  thf(fact_5973_artanh__def,axiom,
% 0.30/0.62      ( artanh_real
% 0.30/0.62      = ( ^ [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 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % artanh_def
% 0.30/0.62  thf(fact_5974_Sum__Icc__int,axiom,
% 0.30/0.62      ! [M: int,N: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ M @ N )
% 0.30/0.62       => ( ( groups4538972089207619220nt_int
% 0.30/0.62            @ ^ [X: int] : X
% 0.30/0.62            @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.30/0.62          = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Sum_Icc_int
% 0.30/0.62  thf(fact_5975_divmod__step__def,axiom,
% 0.30/0.62      ( unique5026877609467782581ep_nat
% 0.30/0.62      = ( ^ [L2: num] :
% 0.30/0.62            ( produc2626176000494625587at_nat
% 0.30/0.62            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_step_def
% 0.30/0.62  thf(fact_5976_divmod__step__def,axiom,
% 0.30/0.62      ( unique5024387138958732305ep_int
% 0.30/0.62      = ( ^ [L2: num] :
% 0.30/0.62            ( produc4245557441103728435nt_int
% 0.30/0.62            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_step_def
% 0.30/0.62  thf(fact_5977_divmod__step__def,axiom,
% 0.30/0.62      ( unique4921790084139445826nteger
% 0.30/0.62      = ( ^ [L2: num] :
% 0.30/0.62            ( produc6916734918728496179nteger
% 0.30/0.62            @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_step_def
% 0.30/0.62  thf(fact_5978_semiring__norm_I90_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ( bit1 @ M )
% 0.30/0.62          = ( bit1 @ N ) )
% 0.30/0.62        = ( M = N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(90)
% 0.30/0.62  thf(fact_5979_verit__eq__simplify_I9_J,axiom,
% 0.30/0.62      ! [X32: num,Y32: num] :
% 0.30/0.62        ( ( ( bit1 @ X32 )
% 0.30/0.62          = ( bit1 @ Y32 ) )
% 0.30/0.62        = ( X32 = Y32 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % verit_eq_simplify(9)
% 0.30/0.62  thf(fact_5980_semiring__norm_I88_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( bit0 @ M )
% 0.30/0.62       != ( bit1 @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(88)
% 0.30/0.62  thf(fact_5981_semiring__norm_I89_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( bit1 @ M )
% 0.30/0.62       != ( bit0 @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(89)
% 0.30/0.62  thf(fact_5982_semiring__norm_I84_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( one
% 0.30/0.62       != ( bit1 @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(84)
% 0.30/0.62  thf(fact_5983_semiring__norm_I86_J,axiom,
% 0.30/0.62      ! [M: num] :
% 0.30/0.62        ( ( bit1 @ M )
% 0.30/0.62       != one ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(86)
% 0.30/0.62  thf(fact_5984_semiring__norm_I73_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(73)
% 0.30/0.62  thf(fact_5985_semiring__norm_I80_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(80)
% 0.30/0.62  thf(fact_5986_case__prod__conv,axiom,
% 0.30/0.62      ! [F: nat > nat > nat,A: nat,B2: nat] :
% 0.30/0.62        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
% 0.30/0.62        = ( F @ A @ B2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_conv
% 0.30/0.62  thf(fact_5987_case__prod__conv,axiom,
% 0.30/0.62      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B2: nat] :
% 0.30/0.62        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
% 0.30/0.62        = ( F @ A @ B2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_conv
% 0.30/0.62  thf(fact_5988_case__prod__conv,axiom,
% 0.30/0.62      ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B2: nat] :
% 0.30/0.62        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
% 0.30/0.62        = ( F @ A @ B2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_conv
% 0.30/0.62  thf(fact_5989_case__prod__conv,axiom,
% 0.30/0.62      ! [F: int > int > product_prod_int_int,A: int,B2: int] :
% 0.30/0.62        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B2 ) )
% 0.30/0.62        = ( F @ A @ B2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_conv
% 0.30/0.62  thf(fact_5990_case__prod__conv,axiom,
% 0.30/0.62      ! [F: int > int > $o,A: int,B2: int] :
% 0.30/0.62        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) )
% 0.30/0.62        = ( F @ A @ B2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_conv
% 0.30/0.62  thf(fact_5991_ln__inj__iff,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62         => ( ( ( ln_ln_real @ X3 )
% 0.30/0.62              = ( ln_ln_real @ Y3 ) )
% 0.30/0.62            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_inj_iff
% 0.30/0.62  thf(fact_5992_ln__less__cancel__iff,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62         => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 0.30/0.62            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_less_cancel_iff
% 0.30/0.62  thf(fact_5993_semiring__norm_I9_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(9)
% 0.30/0.62  thf(fact_5994_semiring__norm_I7_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(7)
% 0.30/0.62  thf(fact_5995_semiring__norm_I15_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(15)
% 0.30/0.62  thf(fact_5996_semiring__norm_I14_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(14)
% 0.30/0.62  thf(fact_5997_semiring__norm_I72_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(72)
% 0.30/0.62  thf(fact_5998_semiring__norm_I81_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(81)
% 0.30/0.62  thf(fact_5999_semiring__norm_I70_J,axiom,
% 0.30/0.62      ! [M: num] :
% 0.30/0.62        ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(70)
% 0.30/0.62  thf(fact_6000_semiring__norm_I77_J,axiom,
% 0.30/0.62      ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(77)
% 0.30/0.62  thf(fact_6001_zdiv__numeral__Bit1,axiom,
% 0.30/0.62      ! [V: num,W: num] :
% 0.30/0.62        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.62        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zdiv_numeral_Bit1
% 0.30/0.62  thf(fact_6002_semiring__norm_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.30/0.62        = ( bit1 @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(3)
% 0.30/0.62  thf(fact_6003_semiring__norm_I4_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.30/0.62        = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(4)
% 0.30/0.62  thf(fact_6004_semiring__norm_I5_J,axiom,
% 0.30/0.62      ! [M: num] :
% 0.30/0.62        ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.30/0.62        = ( bit1 @ M ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(5)
% 0.30/0.62  thf(fact_6005_semiring__norm_I8_J,axiom,
% 0.30/0.62      ! [M: num] :
% 0.30/0.62        ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.30/0.62        = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(8)
% 0.30/0.62  thf(fact_6006_semiring__norm_I10_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(10)
% 0.30/0.62  thf(fact_6007_ln__le__cancel__iff,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62         => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 0.30/0.62            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_le_cancel_iff
% 0.30/0.62  thf(fact_6008_ln__less__zero__iff,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 0.30/0.62          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_less_zero_iff
% 0.30/0.62  thf(fact_6009_ln__gt__zero__iff,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.62          = ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_gt_zero_iff
% 0.30/0.62  thf(fact_6010_ln__eq__zero__iff,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ( ln_ln_real @ X3 )
% 0.30/0.62            = zero_zero_real )
% 0.30/0.62          = ( X3 = one_one_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_eq_zero_iff
% 0.30/0.62  thf(fact_6011_semiring__norm_I16_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(16)
% 0.30/0.62  thf(fact_6012_semiring__norm_I74_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(74)
% 0.30/0.62  thf(fact_6013_semiring__norm_I79_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.62        = ( ord_less_eq_num @ M @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % semiring_norm(79)
% 0.30/0.62  thf(fact_6014_ln__ge__zero__iff,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.62          = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_ge_zero_iff
% 0.30/0.62  thf(fact_6015_ln__le__zero__iff,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 0.30/0.62          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_le_zero_iff
% 0.30/0.62  thf(fact_6016_Suc__div__eq__add3__div__numeral,axiom,
% 0.30/0.62      ! [M: nat,V: num] :
% 0.30/0.62        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.62        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Suc_div_eq_add3_div_numeral
% 0.30/0.62  thf(fact_6017_div__Suc__eq__div__add3,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.30/0.62        = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % div_Suc_eq_div_add3
% 0.30/0.62  thf(fact_6018_mod__Suc__eq__mod__add3,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.30/0.62        = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mod_Suc_eq_mod_add3
% 0.30/0.62  thf(fact_6019_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.30/0.62      ! [M: nat,V: num] :
% 0.30/0.62        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.62        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Suc_mod_eq_add3_mod_numeral
% 0.30/0.62  thf(fact_6020_zmod__numeral__Bit1,axiom,
% 0.30/0.62      ! [V: num,W: num] :
% 0.30/0.62        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.62        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zmod_numeral_Bit1
% 0.30/0.62  thf(fact_6021_signed__take__bit__Suc__bit1,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.62        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % signed_take_bit_Suc_bit1
% 0.30/0.62  thf(fact_6022_verit__eq__simplify_I14_J,axiom,
% 0.30/0.62      ! [X2: num,X32: num] :
% 0.30/0.62        ( ( bit0 @ X2 )
% 0.30/0.62       != ( bit1 @ X32 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % verit_eq_simplify(14)
% 0.30/0.62  thf(fact_6023_verit__eq__simplify_I12_J,axiom,
% 0.30/0.62      ! [X32: num] :
% 0.30/0.62        ( one
% 0.30/0.62       != ( bit1 @ X32 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % verit_eq_simplify(12)
% 0.30/0.62  thf(fact_6024_old_Oprod_Ocase,axiom,
% 0.30/0.62      ! [F: nat > nat > nat,X1: nat,X2: nat] :
% 0.30/0.62        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
% 0.30/0.62        = ( F @ X1 @ X2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % old.prod.case
% 0.30/0.62  thf(fact_6025_old_Oprod_Ocase,axiom,
% 0.30/0.62      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
% 0.30/0.62        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
% 0.30/0.62        = ( F @ X1 @ X2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % old.prod.case
% 0.30/0.62  thf(fact_6026_old_Oprod_Ocase,axiom,
% 0.30/0.62      ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
% 0.30/0.62        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
% 0.30/0.62        = ( F @ X1 @ X2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % old.prod.case
% 0.30/0.62  thf(fact_6027_old_Oprod_Ocase,axiom,
% 0.30/0.62      ! [F: int > int > product_prod_int_int,X1: int,X2: int] :
% 0.30/0.62        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
% 0.30/0.62        = ( F @ X1 @ X2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % old.prod.case
% 0.30/0.62  thf(fact_6028_old_Oprod_Ocase,axiom,
% 0.30/0.62      ! [F: int > int > $o,X1: int,X2: int] :
% 0.30/0.62        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
% 0.30/0.62        = ( F @ X1 @ X2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % old.prod.case
% 0.30/0.62  thf(fact_6029_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_6030_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_6031_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_6032_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ! [I3: complex] :
% 0.30/0.62            ( ( member_complex @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_6033_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_6034_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_6035_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ! [I3: complex] :
% 0.30/0.62            ( ( member_complex @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_6036_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_6037_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_6038_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_complex,F: complex > int,G: complex > int] :
% 0.30/0.62        ( ! [I3: complex] :
% 0.30/0.62            ( ( member_complex @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_6039_sum__product,axiom,
% 0.30/0.62      ! [F: int > int,A4: set_int,G: int > int,B5: set_int] :
% 0.30/0.62        ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
% 0.30/0.62        = ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [I4: int] :
% 0.30/0.62              ( groups4538972089207619220nt_int
% 0.30/0.62              @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.62              @ B5 )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_product
% 0.30/0.62  thf(fact_6040_sum__product,axiom,
% 0.30/0.62      ! [F: complex > complex,A4: set_complex,G: complex > complex,B5: set_complex] :
% 0.30/0.62        ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
% 0.30/0.62        = ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [I4: complex] :
% 0.30/0.62              ( groups7754918857620584856omplex
% 0.30/0.62              @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.62              @ B5 )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_product
% 0.30/0.62  thf(fact_6041_sum__product,axiom,
% 0.30/0.62      ! [F: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat] :
% 0.30/0.62        ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
% 0.30/0.62        = ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [I4: nat] :
% 0.30/0.62              ( groups3542108847815614940at_nat
% 0.30/0.62              @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.62              @ B5 )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_product
% 0.30/0.62  thf(fact_6042_sum__product,axiom,
% 0.30/0.62      ! [F: nat > real,A4: set_nat,G: nat > real,B5: set_nat] :
% 0.30/0.62        ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I4: nat] :
% 0.30/0.62              ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.30/0.62              @ B5 )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_product
% 0.30/0.62  thf(fact_6043_sum__distrib__right,axiom,
% 0.30/0.62      ! [F: int > int,A4: set_int,R: int] :
% 0.30/0.62        ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ R )
% 0.30/0.62        = ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_right
% 0.30/0.62  thf(fact_6044_sum__distrib__right,axiom,
% 0.30/0.62      ! [F: complex > complex,A4: set_complex,R: complex] :
% 0.30/0.62        ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
% 0.30/0.62        = ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_right
% 0.30/0.62  thf(fact_6045_sum__distrib__right,axiom,
% 0.30/0.62      ! [F: nat > nat,A4: set_nat,R: nat] :
% 0.30/0.62        ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ R )
% 0.30/0.62        = ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_right
% 0.30/0.62  thf(fact_6046_sum__distrib__right,axiom,
% 0.30/0.62      ! [F: nat > real,A4: set_nat,R: real] :
% 0.30/0.62        ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_right
% 0.30/0.62  thf(fact_6047_sum__distrib__left,axiom,
% 0.30/0.62      ! [R: int,F: int > int,A4: set_int] :
% 0.30/0.62        ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A4 ) )
% 0.30/0.62        = ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [N2: int] : ( times_times_int @ R @ ( F @ N2 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_left
% 0.30/0.62  thf(fact_6048_sum__distrib__left,axiom,
% 0.30/0.62      ! [R: complex,F: complex > complex,A4: set_complex] :
% 0.30/0.62        ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A4 ) )
% 0.30/0.62        = ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_left
% 0.30/0.62  thf(fact_6049_sum__distrib__left,axiom,
% 0.30/0.62      ! [R: nat,F: nat > nat,A4: set_nat] :
% 0.30/0.62        ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 0.30/0.62        = ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_left
% 0.30/0.62  thf(fact_6050_sum__distrib__left,axiom,
% 0.30/0.62      ! [R: real,F: nat > real,A4: set_nat] :
% 0.30/0.62        ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A4 ) )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_distrib_left
% 0.30/0.62  thf(fact_6051_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: int > int,H2: int > int,A4: set_int] :
% 0.30/0.62        ( ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.62          @ A4 )
% 0.30/0.62        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_6052_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: complex > complex,H2: complex > complex,A4: set_complex] :
% 0.30/0.62        ( ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.62          @ A4 )
% 0.30/0.62        = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_6053_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: nat > nat,H2: nat > nat,A4: 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          @ A4 )
% 0.30/0.62        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_6054_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: nat > real,H2: nat > real,A4: 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          @ A4 )
% 0.30/0.62        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_6055_sum__subtractf,axiom,
% 0.30/0.62      ! [F: int > int,G: int > int,A4: set_int] :
% 0.30/0.62        ( ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62          @ A4 )
% 0.30/0.62        = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf
% 0.30/0.62  thf(fact_6056_sum__subtractf,axiom,
% 0.30/0.62      ! [F: complex > complex,G: complex > complex,A4: set_complex] :
% 0.30/0.62        ( ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62          @ A4 )
% 0.30/0.62        = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf
% 0.30/0.62  thf(fact_6057_sum__subtractf,axiom,
% 0.30/0.62      ! [F: nat > real,G: nat > real,A4: set_nat] :
% 0.30/0.62        ( ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62          @ A4 )
% 0.30/0.62        = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf
% 0.30/0.62  thf(fact_6058_sum__divide__distrib,axiom,
% 0.30/0.62      ! [F: complex > complex,A4: set_complex,R: complex] :
% 0.30/0.62        ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
% 0.30/0.62        = ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_divide_distrib
% 0.30/0.62  thf(fact_6059_sum__divide__distrib,axiom,
% 0.30/0.62      ! [F: nat > real,A4: set_nat,R: real] :
% 0.30/0.62        ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_divide_distrib
% 0.30/0.62  thf(fact_6060_mod__sum__eq,axiom,
% 0.30/0.62      ! [F: int > int,A: int,A4: set_int] :
% 0.30/0.62        ( ( modulo_modulo_int
% 0.30/0.62          @ ( groups4538972089207619220nt_int
% 0.30/0.62            @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.30/0.62            @ A4 )
% 0.30/0.62          @ A )
% 0.30/0.62        = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mod_sum_eq
% 0.30/0.62  thf(fact_6061_mod__sum__eq,axiom,
% 0.30/0.62      ! [F: nat > nat,A: nat,A4: set_nat] :
% 0.30/0.62        ( ( modulo_modulo_nat
% 0.30/0.62          @ ( groups3542108847815614940at_nat
% 0.30/0.62            @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 0.30/0.62            @ A4 )
% 0.30/0.62          @ A )
% 0.30/0.62        = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mod_sum_eq
% 0.30/0.62  thf(fact_6062_cond__case__prod__eta,axiom,
% 0.30/0.62      ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
% 0.30/0.62        ( ! [X5: nat,Y4: nat] :
% 0.30/0.62            ( ( F @ X5 @ Y4 )
% 0.30/0.62            = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 0.30/0.62       => ( ( produc6842872674320459806at_nat @ F )
% 0.30/0.62          = G ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cond_case_prod_eta
% 0.30/0.62  thf(fact_6063_cond__case__prod__eta,axiom,
% 0.30/0.62      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.30/0.62        ( ! [X5: nat,Y4: nat] :
% 0.30/0.62            ( ( F @ X5 @ Y4 )
% 0.30/0.62            = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 0.30/0.62       => ( ( produc27273713700761075at_nat @ F )
% 0.30/0.62          = G ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cond_case_prod_eta
% 0.30/0.62  thf(fact_6064_cond__case__prod__eta,axiom,
% 0.30/0.62      ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.62        ( ! [X5: nat,Y4: nat] :
% 0.30/0.62            ( ( F @ X5 @ Y4 )
% 0.30/0.62            = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 0.30/0.62       => ( ( produc8739625826339149834_nat_o @ F )
% 0.30/0.62          = G ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cond_case_prod_eta
% 0.30/0.62  thf(fact_6065_cond__case__prod__eta,axiom,
% 0.30/0.62      ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 0.30/0.62        ( ! [X5: int,Y4: int] :
% 0.30/0.62            ( ( F @ X5 @ Y4 )
% 0.30/0.62            = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 0.30/0.62       => ( ( produc4245557441103728435nt_int @ F )
% 0.30/0.62          = G ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cond_case_prod_eta
% 0.30/0.62  thf(fact_6066_cond__case__prod__eta,axiom,
% 0.30/0.62      ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 0.30/0.62        ( ! [X5: int,Y4: int] :
% 0.30/0.62            ( ( F @ X5 @ Y4 )
% 0.30/0.62            = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 0.30/0.62       => ( ( produc4947309494688390418_int_o @ F )
% 0.30/0.62          = G ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cond_case_prod_eta
% 0.30/0.62  thf(fact_6067_case__prod__eta,axiom,
% 0.30/0.62      ! [F: product_prod_nat_nat > nat] :
% 0.30/0.62        ( ( produc6842872674320459806at_nat
% 0.30/0.62          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.62        = F ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_eta
% 0.30/0.62  thf(fact_6068_case__prod__eta,axiom,
% 0.30/0.62      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.30/0.62        ( ( produc27273713700761075at_nat
% 0.30/0.62          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.62        = F ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_eta
% 0.30/0.62  thf(fact_6069_case__prod__eta,axiom,
% 0.30/0.62      ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.62        ( ( produc8739625826339149834_nat_o
% 0.30/0.62          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.62        = F ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_eta
% 0.30/0.62  thf(fact_6070_case__prod__eta,axiom,
% 0.30/0.62      ! [F: product_prod_int_int > product_prod_int_int] :
% 0.30/0.62        ( ( produc4245557441103728435nt_int
% 0.30/0.62          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 0.30/0.62        = F ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_eta
% 0.30/0.62  thf(fact_6071_case__prod__eta,axiom,
% 0.30/0.62      ! [F: product_prod_int_int > $o] :
% 0.30/0.62        ( ( produc4947309494688390418_int_o
% 0.30/0.62          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 0.30/0.62        = F ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prod_eta
% 0.30/0.62  thf(fact_6072_case__prodE2,axiom,
% 0.30/0.62      ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
% 0.30/0.62        ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
% 0.30/0.62       => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.62              ( ( Z
% 0.30/0.62                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.62             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE2
% 0.30/0.62  thf(fact_6073_case__prodE2,axiom,
% 0.30/0.62      ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
% 0.30/0.62        ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
% 0.30/0.62       => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.62              ( ( Z
% 0.30/0.62                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.62             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE2
% 0.30/0.62  thf(fact_6074_case__prodE2,axiom,
% 0.30/0.62      ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
% 0.30/0.62        ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
% 0.30/0.62       => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.62              ( ( Z
% 0.30/0.62                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.62             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE2
% 0.30/0.62  thf(fact_6075_case__prodE2,axiom,
% 0.30/0.62      ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
% 0.30/0.62        ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
% 0.30/0.62       => ~ ! [X5: int,Y4: int] :
% 0.30/0.62              ( ( Z
% 0.30/0.62                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.62             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE2
% 0.30/0.62  thf(fact_6076_case__prodE2,axiom,
% 0.30/0.62      ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
% 0.30/0.62        ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
% 0.30/0.62       => ~ ! [X5: int,Y4: int] :
% 0.30/0.62              ( ( Z
% 0.30/0.62                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.62             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE2
% 0.30/0.62  thf(fact_6077_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > real] :
% 0.30/0.62        ( ! [X5: real] :
% 0.30/0.62            ( ( member_real @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6078_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > real] :
% 0.30/0.62        ( ! [X5: int] :
% 0.30/0.62            ( ( member_int @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6079_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > real] :
% 0.30/0.62        ( ! [X5: complex] :
% 0.30/0.62            ( ( member_complex @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6080_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_nat,F: nat > rat] :
% 0.30/0.62        ( ! [X5: nat] :
% 0.30/0.62            ( ( member_nat @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6081_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > rat] :
% 0.30/0.62        ( ! [X5: real] :
% 0.30/0.62            ( ( member_real @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6082_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > rat] :
% 0.30/0.62        ( ! [X5: int] :
% 0.30/0.62            ( ( member_int @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6083_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > rat] :
% 0.30/0.62        ( ! [X5: complex] :
% 0.30/0.62            ( ( member_complex @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6084_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > nat] :
% 0.30/0.62        ( ! [X5: real] :
% 0.30/0.62            ( ( member_real @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6085_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > nat] :
% 0.30/0.62        ( ! [X5: int] :
% 0.30/0.62            ( ( member_int @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6086_sum__nonneg,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > nat] :
% 0.30/0.62        ( ! [X5: complex] :
% 0.30/0.62            ( ( member_complex @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_6087_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > real] :
% 0.30/0.62        ( ! [X5: real] :
% 0.30/0.62            ( ( member_real @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.30/0.62       => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6088_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > real] :
% 0.30/0.62        ( ! [X5: int] :
% 0.30/0.62            ( ( member_int @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.30/0.62       => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6089_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > real] :
% 0.30/0.62        ( ! [X5: complex] :
% 0.30/0.62            ( ( member_complex @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.30/0.62       => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6090_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_nat,F: nat > rat] :
% 0.30/0.62        ( ! [X5: nat] :
% 0.30/0.62            ( ( member_nat @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6091_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > rat] :
% 0.30/0.62        ( ! [X5: real] :
% 0.30/0.62            ( ( member_real @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6092_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > rat] :
% 0.30/0.62        ( ! [X5: int] :
% 0.30/0.62            ( ( member_int @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6093_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > rat] :
% 0.30/0.62        ( ! [X5: complex] :
% 0.30/0.62            ( ( member_complex @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6094_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > nat] :
% 0.30/0.62        ( ! [X5: real] :
% 0.30/0.62            ( ( member_real @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6095_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > nat] :
% 0.30/0.62        ( ! [X5: int] :
% 0.30/0.62            ( ( member_int @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6096_sum__nonpos,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > nat] :
% 0.30/0.62        ( ! [X5: complex] :
% 0.30/0.62            ( ( member_complex @ X5 @ A4 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_6097_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_6098_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_6099_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_6100_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_6101_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_6102_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_6103_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_6104_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_6105_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_6106_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
% 0.30/0.62        ( ( ( groups5690904116761175830ex_int @ F @ I5 )
% 0.30/0.62          = ( groups5690904116761175830ex_int @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_int @ ( 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_6107_ln__less__self,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_less_self
% 0.30/0.62  thf(fact_6108_xor__num_Ocases,axiom,
% 0.30/0.62      ! [X3: product_prod_num_num] :
% 0.30/0.62        ( ( X3
% 0.30/0.62         != ( product_Pair_num_num @ one @ one ) )
% 0.30/0.62       => ( ! [N3: num] :
% 0.30/0.62              ( X3
% 0.30/0.62             != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
% 0.30/0.62         => ( ! [N3: num] :
% 0.30/0.62                ( X3
% 0.30/0.62               != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
% 0.30/0.62           => ( ! [M4: num] :
% 0.30/0.62                  ( X3
% 0.30/0.62                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 0.30/0.62             => ( ! [M4: num,N3: num] :
% 0.30/0.62                    ( X3
% 0.30/0.62                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
% 0.30/0.62               => ( ! [M4: num,N3: num] :
% 0.30/0.62                      ( X3
% 0.30/0.62                     != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
% 0.30/0.62                 => ( ! [M4: num] :
% 0.30/0.62                        ( X3
% 0.30/0.62                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 0.30/0.62                   => ( ! [M4: num,N3: num] :
% 0.30/0.62                          ( X3
% 0.30/0.62                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
% 0.30/0.62                     => ~ ! [M4: num,N3: num] :
% 0.30/0.62                            ( X3
% 0.30/0.62                           != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % xor_num.cases
% 0.30/0.62  thf(fact_6109_num_Oexhaust,axiom,
% 0.30/0.62      ! [Y3: num] :
% 0.30/0.62        ( ( Y3 != one )
% 0.30/0.62       => ( ! [X22: num] :
% 0.30/0.62              ( Y3
% 0.30/0.62             != ( bit0 @ X22 ) )
% 0.30/0.62         => ~ ! [X33: num] :
% 0.30/0.62                ( Y3
% 0.30/0.62               != ( bit1 @ X33 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % num.exhaust
% 0.30/0.62  thf(fact_6110_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         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( 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_6111_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         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( 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_6112_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         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( 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_6113_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         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( 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_6114_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         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( 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_6115_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         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( 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_6116_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         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( 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_6117_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         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( 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_6118_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         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( 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_6119_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         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ 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                        = X5 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( 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_6120_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A4 )
% 0.30/0.62       => ( ! [X5: real] :
% 0.30/0.62              ( ( member_real @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups8097168146408367636l_real @ F @ A4 )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62            = ( ! [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A4 )
% 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_6121_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups8778361861064173332t_real @ F @ A4 )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62            = ( ! [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A4 )
% 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_6122_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups5808333547571424918x_real @ F @ A4 )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62            = ( ! [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A4 )
% 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_6123_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A4 )
% 0.30/0.62       => ( ! [X5: real] :
% 0.30/0.62              ( ( member_real @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups1300246762558778688al_rat @ F @ A4 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A4 )
% 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_6124_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ! [X5: nat] :
% 0.30/0.62              ( ( member_nat @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: nat] :
% 0.30/0.62                  ( ( member_nat @ X @ A4 )
% 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_6125_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A4 )
% 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_6126_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A4 )
% 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_6127_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A4 )
% 0.30/0.62       => ( ! [X5: real] :
% 0.30/0.62              ( ( member_real @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups1935376822645274424al_nat @ F @ A4 )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62            = ( ! [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A4 )
% 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_6128_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62            = ( ! [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A4 )
% 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_6129_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62         => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62            = ( ! [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A4 )
% 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_6130_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > real,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: int] :
% 0.30/0.62                ( ( member_int @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6131_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: complex] :
% 0.30/0.62                ( ( member_complex @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6132_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ! [X5: nat] :
% 0.30/0.62              ( ( member_nat @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: nat] :
% 0.30/0.62                ( ( member_nat @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6133_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > rat,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: int] :
% 0.30/0.62                ( ( member_int @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6134_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: complex] :
% 0.30/0.62                ( ( member_complex @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6135_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > nat,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: int] :
% 0.30/0.62                ( ( member_int @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6136_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: complex] :
% 0.30/0.62                ( ( member_complex @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6137_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_nat,F: nat > int,G: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ! [X5: nat] :
% 0.30/0.62              ( ( member_nat @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: nat] :
% 0.30/0.62                ( ( member_nat @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6138_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > int,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ! [X5: complex] :
% 0.30/0.62              ( ( member_complex @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: complex] :
% 0.30/0.62                ( ( member_complex @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6139_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > int,G: int > int] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ! [X5: int] :
% 0.30/0.62              ( ( member_int @ X5 @ A4 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62         => ( ? [X6: int] :
% 0.30/0.62                ( ( member_int @ X6 @ A4 )
% 0.30/0.62                & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 0.30/0.62           => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_6140_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.62        ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 0.30/0.62       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_nat @ S2 )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6141_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 0.30/0.62        ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 0.30/0.62       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S2 )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6142_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 0.30/0.62        ( ( R2 @ zero_zero_real @ zero_zero_real )
% 0.30/0.62       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S2 )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6143_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.62        ( ( R2 @ zero_zero_real @ zero_zero_real )
% 0.30/0.62       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite3207457112153483333omplex @ S2 )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6144_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.62       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_nat @ S2 )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6145_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 0.30/0.62        ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.62       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S2 )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6146_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.62       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite3207457112153483333omplex @ S2 )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6147_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 0.30/0.62        ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 0.30/0.62       => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S2 )
% 0.30/0.62           => ( ! [X5: int] :
% 0.30/0.62                  ( ( member_int @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6148_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 0.30/0.62       => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite3207457112153483333omplex @ S2 )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6149_sum_Orelated,axiom,
% 0.30/0.62      ! [R2: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
% 0.30/0.62        ( ( R2 @ zero_zero_int @ zero_zero_int )
% 0.30/0.62       => ( ! [X15: int,Y15: int,X22: int,Y22: int] :
% 0.30/0.62              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.62                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.62             => ( R2 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
% 0.30/0.62         => ( ( finite_finite_nat @ S2 )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62             => ( R2 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_6150_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6151_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > real,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_int )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6152_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > real,G: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_real )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6153_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6154_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_nat )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6155_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > rat,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_int )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6156_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > rat,G: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_real )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6157_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6158_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_int,F: int > nat,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_int )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6159_sum__strict__mono,axiom,
% 0.30/0.62      ! [A4: set_real,F: real > nat,G: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A4 )
% 0.30/0.62       => ( ( A4 != bot_bot_set_real )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ A4 )
% 0.30/0.62               => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_6160_ln__bound,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_bound
% 0.30/0.62  thf(fact_6161_ln__gt__zero__imp__gt__one,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62         => ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_gt_zero_imp_gt_one
% 0.30/0.62  thf(fact_6162_ln__less__zero,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.62         => ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_less_zero
% 0.30/0.62  thf(fact_6163_ln__gt__zero,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.62       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_gt_zero
% 0.30/0.62  thf(fact_6164_ln__ge__zero,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.62       => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_ge_zero
% 0.30/0.62  thf(fact_6165_numeral__Bit1,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1
% 0.30/0.62  thf(fact_6166_numeral__Bit1,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1
% 0.30/0.62  thf(fact_6167_numeral__Bit1,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1
% 0.30/0.62  thf(fact_6168_numeral__Bit1,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1
% 0.30/0.62  thf(fact_6169_numeral__Bit1,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1
% 0.30/0.62  thf(fact_6170_eval__nat__numeral_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.30/0.62        = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % eval_nat_numeral(3)
% 0.30/0.62  thf(fact_6171_cong__exp__iff__simps_I10_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(10)
% 0.30/0.62  thf(fact_6172_cong__exp__iff__simps_I10_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(10)
% 0.30/0.62  thf(fact_6173_cong__exp__iff__simps_I10_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(10)
% 0.30/0.62  thf(fact_6174_cong__exp__iff__simps_I12_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(12)
% 0.30/0.62  thf(fact_6175_cong__exp__iff__simps_I12_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(12)
% 0.30/0.62  thf(fact_6176_cong__exp__iff__simps_I12_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(12)
% 0.30/0.62  thf(fact_6177_cong__exp__iff__simps_I13_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.62          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(13)
% 0.30/0.62  thf(fact_6178_cong__exp__iff__simps_I13_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.62          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(13)
% 0.30/0.62  thf(fact_6179_cong__exp__iff__simps_I13_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num,N: num] :
% 0.30/0.62        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.62          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(13)
% 0.30/0.62  thf(fact_6180_power__minus__Bit1,axiom,
% 0.30/0.62      ! [X3: real,K: num] :
% 0.30/0.62        ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.62        = ( uminus_uminus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_minus_Bit1
% 0.30/0.62  thf(fact_6181_power__minus__Bit1,axiom,
% 0.30/0.62      ! [X3: int,K: num] :
% 0.30/0.62        ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.62        = ( uminus_uminus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_minus_Bit1
% 0.30/0.62  thf(fact_6182_power__minus__Bit1,axiom,
% 0.30/0.62      ! [X3: complex,K: num] :
% 0.30/0.62        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.62        = ( uminus1482373934393186551omplex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_minus_Bit1
% 0.30/0.62  thf(fact_6183_power__minus__Bit1,axiom,
% 0.30/0.62      ! [X3: code_integer,K: num] :
% 0.30/0.62        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.62        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_minus_Bit1
% 0.30/0.62  thf(fact_6184_power__minus__Bit1,axiom,
% 0.30/0.62      ! [X3: rat,K: num] :
% 0.30/0.62        ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.62        = ( uminus_uminus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_minus_Bit1
% 0.30/0.62  thf(fact_6185_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_real,F: real > real,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6186_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_int,F: int > real,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6187_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > real,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6188_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_real,F: real > rat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6189_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_nat,F: nat > rat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_nat @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6190_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_int,F: int > rat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6191_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > rat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6192_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_real,F: real > nat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6193_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_int,F: int > nat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6194_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > nat,B5: 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              = B5 )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_6195_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_6196_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_6197_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_6198_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_6199_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_6200_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_6201_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_6202_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_6203_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_6204_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_6205_numeral__code_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_code(3)
% 0.30/0.62  thf(fact_6206_numeral__code_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_code(3)
% 0.30/0.62  thf(fact_6207_numeral__code_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_code(3)
% 0.30/0.62  thf(fact_6208_numeral__code_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_code(3)
% 0.30/0.62  thf(fact_6209_numeral__code_I3_J,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.30/0.62        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_code(3)
% 0.30/0.62  thf(fact_6210_power__numeral__odd,axiom,
% 0.30/0.62      ! [Z: complex,W: num] :
% 0.30/0.62        ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.62        = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_numeral_odd
% 0.30/0.62  thf(fact_6211_power__numeral__odd,axiom,
% 0.30/0.62      ! [Z: real,W: num] :
% 0.30/0.62        ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.62        = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_numeral_odd
% 0.30/0.62  thf(fact_6212_power__numeral__odd,axiom,
% 0.30/0.62      ! [Z: rat,W: num] :
% 0.30/0.62        ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.62        = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_numeral_odd
% 0.30/0.62  thf(fact_6213_power__numeral__odd,axiom,
% 0.30/0.62      ! [Z: nat,W: num] :
% 0.30/0.62        ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.62        = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_numeral_odd
% 0.30/0.62  thf(fact_6214_power__numeral__odd,axiom,
% 0.30/0.62      ! [Z: int,W: num] :
% 0.30/0.62        ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.30/0.62        = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power_numeral_odd
% 0.30/0.62  thf(fact_6215_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_6216_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_6217_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_6218_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_6219_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_6220_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_6221_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_6222_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_6223_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_6224_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_6225_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_6226_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_6227_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_6228_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_6229_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_6230_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_6231_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_6232_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_6233_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_6234_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_6235_ln__ge__zero__imp__ge__one,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62         => ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_ge_zero_imp_ge_one
% 0.30/0.62  thf(fact_6236_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: 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 @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_complex ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_complex ) )
% 0.30/0.62               => ( ( ( groups5754745047067104278omplex @ G @ A4 )
% 0.30/0.62                    = ( groups5754745047067104278omplex @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.30/0.62                    = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6237_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: nat] :
% 0.30/0.62                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_complex ) )
% 0.30/0.62             => ( ! [B3: nat] :
% 0.30/0.62                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_complex ) )
% 0.30/0.62               => ( ( ( groups2073611262835488442omplex @ G @ A4 )
% 0.30/0.62                    = ( groups2073611262835488442omplex @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.30/0.62                    = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6238_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: 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 @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_real ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_real ) )
% 0.30/0.62               => ( ( ( groups8097168146408367636l_real @ G @ A4 )
% 0.30/0.62                    = ( groups8097168146408367636l_real @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.30/0.62                    = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6239_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: complex] :
% 0.30/0.62                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_real ) )
% 0.30/0.62             => ( ! [B3: complex] :
% 0.30/0.62                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_real ) )
% 0.30/0.62               => ( ( ( groups5808333547571424918x_real @ G @ A4 )
% 0.30/0.62                    = ( groups5808333547571424918x_real @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.30/0.62                    = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6240_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: 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 @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_rat ) )
% 0.30/0.62               => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
% 0.30/0.62                    = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.30/0.62                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6241_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: nat] :
% 0.30/0.62                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B3: nat] :
% 0.30/0.62                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_rat ) )
% 0.30/0.62               => ( ( ( groups2906978787729119204at_rat @ G @ A4 )
% 0.30/0.62                    = ( groups2906978787729119204at_rat @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups2906978787729119204at_rat @ G @ C4 )
% 0.30/0.62                    = ( groups2906978787729119204at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6242_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: complex] :
% 0.30/0.62                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B3: complex] :
% 0.30/0.62                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_rat ) )
% 0.30/0.62               => ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
% 0.30/0.62                    = ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.30/0.62                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6243_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_nat ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_nat ) )
% 0.30/0.62               => ( ( ( groups1935376822645274424al_nat @ G @ A4 )
% 0.30/0.62                    = ( groups1935376822645274424al_nat @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.30/0.62                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6244_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: complex] :
% 0.30/0.62                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_nat ) )
% 0.30/0.62             => ( ! [B3: complex] :
% 0.30/0.62                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_nat ) )
% 0.30/0.62               => ( ( ( groups5693394587270226106ex_nat @ G @ A4 )
% 0.30/0.62                    = ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.30/0.62                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6245_sum_Osame__carrier,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_int ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_int ) )
% 0.30/0.62               => ( ( ( groups1932886352136224148al_int @ G @ A4 )
% 0.30/0.62                    = ( groups1932886352136224148al_int @ H2 @ B5 ) )
% 0.30/0.62                  = ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.30/0.62                    = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrier
% 0.30/0.62  thf(fact_6246_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: 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 @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_complex ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 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 @ A4 )
% 0.30/0.62                    = ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6247_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: nat] :
% 0.30/0.62                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_complex ) )
% 0.30/0.62             => ( ! [B3: nat] :
% 0.30/0.62                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_complex ) )
% 0.30/0.62               => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.30/0.62                    = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups2073611262835488442omplex @ G @ A4 )
% 0.30/0.62                    = ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6248_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: 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 @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_real ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 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 @ A4 )
% 0.30/0.62                    = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6249_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: complex] :
% 0.30/0.62                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_real ) )
% 0.30/0.62             => ( ! [B3: complex] :
% 0.30/0.62                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 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 @ A4 )
% 0.30/0.62                    = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6250_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: 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 @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_rat ) )
% 0.30/0.62               => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.30/0.62                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups1300246762558778688al_rat @ G @ A4 )
% 0.30/0.62                    = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6251_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: nat] :
% 0.30/0.62                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B3: nat] :
% 0.30/0.62                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_rat ) )
% 0.30/0.62               => ( ( ( groups2906978787729119204at_rat @ G @ C4 )
% 0.30/0.62                    = ( groups2906978787729119204at_rat @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups2906978787729119204at_rat @ G @ A4 )
% 0.30/0.62                    = ( groups2906978787729119204at_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6252_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: complex] :
% 0.30/0.62                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B3: complex] :
% 0.30/0.62                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_rat ) )
% 0.30/0.62               => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.30/0.62                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups5058264527183730370ex_rat @ G @ A4 )
% 0.30/0.62                    = ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6253_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_nat ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_nat ) )
% 0.30/0.62               => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.30/0.62                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups1935376822645274424al_nat @ G @ A4 )
% 0.30/0.62                    = ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6254_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: complex] :
% 0.30/0.62                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_nat ) )
% 0.30/0.62             => ( ! [B3: complex] :
% 0.30/0.62                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_nat ) )
% 0.30/0.62               => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.30/0.62                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups5693394587270226106ex_nat @ G @ A4 )
% 0.30/0.62                    = ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6255_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.62           => ( ! [A3: real] :
% 0.30/0.62                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.62                 => ( ( G @ A3 )
% 0.30/0.62                    = zero_zero_int ) )
% 0.30/0.62             => ( ! [B3: real] :
% 0.30/0.62                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.62                   => ( ( H2 @ B3 )
% 0.30/0.62                      = zero_zero_int ) )
% 0.30/0.62               => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.30/0.62                    = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups1932886352136224148al_int @ G @ A4 )
% 0.30/0.62                    = ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_6256_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups2073611262835488442omplex @ G @ S2 )
% 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_6257_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups5808333547571424918x_real @ G @ S2 )
% 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_6258_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 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_6259_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 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_6260_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 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_6261_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups3539618377306564664at_int @ G @ S2 )
% 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_6262_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 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_6263_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups3049146728041665814omplex @ G @ S2 )
% 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_6264_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_int,S2: set_int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups8778361861064173332t_real @ G @ S2 )
% 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_6265_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_int,S2: set_int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat @ G @ S2 )
% 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_6266_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.30/0.62              = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6267_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.30/0.62              = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6268_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.30/0.62              = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6269_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.30/0.62              = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6270_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.30/0.62              = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6271_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups3539618377306564664at_int @ G @ T3 )
% 0.30/0.62              = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6272_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.30/0.62              = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6273_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.30/0.62              = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6274_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_int,S2: set_int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.30/0.62              = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6275_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_int,S2: set_int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: int] :
% 0.30/0.62                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.30/0.62              = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_6276_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5754745047067104278omplex @ G @ S2 )
% 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_6277_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups2073611262835488442omplex @ G @ S2 )
% 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_6278_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups8097168146408367636l_real @ G @ S2 )
% 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_6279_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5808333547571424918x_real @ G @ S2 )
% 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_6280_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups1300246762558778688al_rat @ G @ S2 )
% 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_6281_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 0.30/0.62                = ( groups2906978787729119204at_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_6282_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 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_6283_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups1935376822645274424al_nat @ G @ S2 )
% 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_6284_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 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_6285_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( H2 @ X5 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups1932886352136224148al_int @ G @ S2 )
% 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_6286_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5754745047067104278omplex @ G @ T3 )
% 0.30/0.62                = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6287_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.30/0.62                = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6288_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups8097168146408367636l_real @ G @ T3 )
% 0.30/0.62                = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6289_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.30/0.62                = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6290_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 0.30/0.62                = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6291_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S2: set_nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: nat] :
% 0.30/0.62                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X5: nat] :
% 0.30/0.62                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.30/0.62                = ( groups2906978787729119204at_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6292_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.30/0.62                = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6293_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 0.30/0.62                = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6294_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: complex] :
% 0.30/0.62                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X5: complex] :
% 0.30/0.62                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.30/0.62                = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6295_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S2: 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 @ S2 @ T3 )
% 0.30/0.62         => ( ! [X5: real] :
% 0.30/0.62                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.62               => ( ( G @ X5 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ! [X5: real] :
% 0.30/0.62                  ( ( member_real @ X5 @ S2 )
% 0.30/0.62                 => ( ( G @ X5 )
% 0.30/0.62                    = ( H2 @ X5 ) ) )
% 0.30/0.62             => ( ( groups1932886352136224148al_int @ G @ T3 )
% 0.30/0.62                = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_6296_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,G: complex > real] :
% 0.30/0.62        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62         => ( ( groups5808333547571424918x_real @ G @ A4 )
% 0.30/0.62            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6297_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_nat,A4: set_nat,G: nat > rat] :
% 0.30/0.62        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.62       => ( ( finite_finite_nat @ A4 )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ G @ A4 )
% 0.30/0.62            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6298_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,G: complex > rat] :
% 0.30/0.62        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62         => ( ( groups5058264527183730370ex_rat @ G @ A4 )
% 0.30/0.62            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6299_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,G: complex > nat] :
% 0.30/0.62        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62         => ( ( groups5693394587270226106ex_nat @ G @ A4 )
% 0.30/0.62            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6300_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_nat,A4: set_nat,G: nat > int] :
% 0.30/0.62        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.62       => ( ( finite_finite_nat @ A4 )
% 0.30/0.62         => ( ( groups3539618377306564664at_int @ G @ A4 )
% 0.30/0.62            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6301_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,G: complex > int] :
% 0.30/0.62        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62         => ( ( groups5690904116761175830ex_int @ G @ A4 )
% 0.30/0.62            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6302_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_int,A4: set_int,G: int > real] :
% 0.30/0.62        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62       => ( ( finite_finite_int @ A4 )
% 0.30/0.62         => ( ( groups8778361861064173332t_real @ G @ A4 )
% 0.30/0.62            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6303_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_int,A4: set_int,G: int > rat] :
% 0.30/0.62        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62       => ( ( finite_finite_int @ A4 )
% 0.30/0.62         => ( ( groups3906332499630173760nt_rat @ G @ A4 )
% 0.30/0.62            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6304_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_int,A4: set_int,G: int > nat] :
% 0.30/0.62        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62       => ( ( finite_finite_int @ A4 )
% 0.30/0.62         => ( ( groups4541462559716669496nt_nat @ G @ A4 )
% 0.30/0.62            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6305_sum_Osubset__diff,axiom,
% 0.30/0.62      ! [B5: set_int,A4: set_int,G: int > int] :
% 0.30/0.62        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62       => ( ( finite_finite_int @ A4 )
% 0.30/0.62         => ( ( groups4538972089207619220nt_int @ G @ A4 )
% 0.30/0.62            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.subset_diff
% 0.30/0.62  thf(fact_6306_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_nat,B5: set_nat,F: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.62         => ( ( groups2073611262835488442omplex @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ A4 ) @ ( groups2073611262835488442omplex @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6307_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_complex,B5: set_complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6308_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_nat,B5: set_nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6309_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_complex,B5: set_complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6310_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_nat,B5: set_nat,F: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.62         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6311_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_complex,B5: set_complex,F: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.62         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6312_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_int,B5: set_int,F: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62         => ( ( groups3049146728041665814omplex @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_complex @ ( groups3049146728041665814omplex @ F @ A4 ) @ ( groups3049146728041665814omplex @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6313_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_int,B5: set_int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6314_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_int,B5: set_int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6315_sum__diff,axiom,
% 0.30/0.62      ! [A4: set_int,B5: set_int,F: int > int] :
% 0.30/0.62        ( ( finite_finite_int @ A4 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.62         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 0.30/0.62            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_diff
% 0.30/0.62  thf(fact_6316_ln__add__one__self__le__self,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_add_one_self_le_self
% 0.30/0.62  thf(fact_6317_ln__add__one__self__le__self2,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.62       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_add_one_self_le_self2
% 0.30/0.62  thf(fact_6318_ln__mult,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62         => ( ( ln_ln_real @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.62            = ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_mult
% 0.30/0.62  thf(fact_6319_ln__eq__minus__one,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ( ln_ln_real @ X3 )
% 0.30/0.62            = ( minus_minus_real @ X3 @ one_one_real ) )
% 0.30/0.62         => ( X3 = one_one_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_eq_minus_one
% 0.30/0.62  thf(fact_6320_ln__div,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62         => ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.62            = ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_div
% 0.30/0.62  thf(fact_6321_numeral__Bit1__div__2,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62        = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1_div_2
% 0.30/0.62  thf(fact_6322_numeral__Bit1__div__2,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.62        = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_Bit1_div_2
% 0.30/0.62  thf(fact_6323_odd__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % odd_numeral
% 0.30/0.62  thf(fact_6324_odd__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % odd_numeral
% 0.30/0.62  thf(fact_6325_odd__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % odd_numeral
% 0.30/0.62  thf(fact_6326_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.62      ! [N: num,Q2: num] :
% 0.30/0.62        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != zero_zero_nat ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(3)
% 0.30/0.62  thf(fact_6327_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.62      ! [N: num,Q2: num] :
% 0.30/0.62        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != zero_zero_int ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(3)
% 0.30/0.62  thf(fact_6328_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.62      ! [N: num,Q2: num] :
% 0.30/0.62        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.62       != zero_z3403309356797280102nteger ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(3)
% 0.30/0.62  thf(fact_6329_power3__eq__cube,axiom,
% 0.30/0.62      ! [A: complex] :
% 0.30/0.62        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.62        = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power3_eq_cube
% 0.30/0.62  thf(fact_6330_power3__eq__cube,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.62        = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power3_eq_cube
% 0.30/0.62  thf(fact_6331_power3__eq__cube,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.62        = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power3_eq_cube
% 0.30/0.62  thf(fact_6332_power3__eq__cube,axiom,
% 0.30/0.62      ! [A: nat] :
% 0.30/0.62        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.62        = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power3_eq_cube
% 0.30/0.62  thf(fact_6333_power3__eq__cube,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.62        = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power3_eq_cube
% 0.30/0.62  thf(fact_6334_numeral__3__eq__3,axiom,
% 0.30/0.62      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.30/0.62      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_3_eq_3
% 0.30/0.62  thf(fact_6335_Suc3__eq__add__3,axiom,
% 0.30/0.62      ! [N: nat] :
% 0.30/0.62        ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.30/0.62        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Suc3_eq_add_3
% 0.30/0.62  thf(fact_6336_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,F: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: real] :
% 0.30/0.62                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6337_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: complex] :
% 0.30/0.62                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6338_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,F: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: real] :
% 0.30/0.62                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6339_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_nat,A4: set_nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: nat] :
% 0.30/0.62                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6340_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: complex] :
% 0.30/0.62                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6341_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,F: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: real] :
% 0.30/0.62                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6342_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: complex] :
% 0.30/0.62                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6343_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,F: real > int] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: real] :
% 0.30/0.62                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6344_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_nat,A4: set_nat,F: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: nat] :
% 0.30/0.62                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6345_sum__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,F: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ! [B3: complex] :
% 0.30/0.62                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.30/0.62           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono2
% 0.30/0.62  thf(fact_6346_num_Osize_I6_J,axiom,
% 0.30/0.62      ! [X32: num] :
% 0.30/0.62        ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.30/0.62        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % num.size(6)
% 0.30/0.62  thf(fact_6347_ln__le__minus__one,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_le_minus_one
% 0.30/0.62  thf(fact_6348_ln__one__minus__pos__upper__bound,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.62         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_one_minus_pos_upper_bound
% 0.30/0.62  thf(fact_6349_ln__diff__le,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_diff_le
% 0.30/0.62  thf(fact_6350_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.62      ! [Q2: num,N: num] :
% 0.30/0.62        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.62          = zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(7)
% 0.30/0.62  thf(fact_6351_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.62      ! [Q2: num,N: num] :
% 0.30/0.62        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.62          = zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(7)
% 0.30/0.62  thf(fact_6352_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.62      ! [Q2: num,N: num] :
% 0.30/0.62        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(7)
% 0.30/0.62  thf(fact_6353_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num] :
% 0.30/0.62        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.62          = zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(11)
% 0.30/0.62  thf(fact_6354_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num] :
% 0.30/0.62        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.62          = zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(11)
% 0.30/0.62  thf(fact_6355_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.62      ! [M: num,Q2: num] :
% 0.30/0.62        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.62          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.62        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.62          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62  
% 0.30/0.62  % cong_exp_iff_simps(11)
% 0.30/0.62  thf(fact_6356_Suc__div__eq__add3__div,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.30/0.62        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Suc_div_eq_add3_div
% 0.30/0.62  thf(fact_6357_Suc__mod__eq__add3__mod,axiom,
% 0.30/0.62      ! [M: nat,N: nat] :
% 0.30/0.62        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.30/0.62        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Suc_mod_eq_add3_mod
% 0.30/0.62  thf(fact_6358_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,B2: real,F: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: real] :
% 0.30/0.62                    ( ( member_real @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6359_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: complex] :
% 0.30/0.62                    ( ( member_complex @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6360_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,B2: real,F: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: real] :
% 0.30/0.62                    ( ( member_real @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6361_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_nat,A4: set_nat,B2: nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.62         => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: nat] :
% 0.30/0.62                    ( ( member_nat @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6362_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: complex] :
% 0.30/0.62                    ( ( member_complex @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6363_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,B2: real,F: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: real] :
% 0.30/0.62                    ( ( member_real @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6364_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: complex] :
% 0.30/0.62                    ( ( member_complex @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6365_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_real,A4: set_real,B2: real,F: real > int] :
% 0.30/0.62        ( ( finite_finite_real @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.62         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: real] :
% 0.30/0.62                    ( ( member_real @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6366_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_nat,A4: set_nat,B2: nat,F: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ B5 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.62         => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: nat] :
% 0.30/0.62                    ( ( member_nat @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6367_sum__strict__mono2,axiom,
% 0.30/0.62      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.62         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.62           => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
% 0.30/0.62             => ( ! [X5: complex] :
% 0.30/0.62                    ( ( member_complex @ X5 @ B5 )
% 0.30/0.62                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.62               => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono2
% 0.30/0.62  thf(fact_6368_divmod__step__nat__def,axiom,
% 0.30/0.62      ( unique5026877609467782581ep_nat
% 0.30/0.62      = ( ^ [L2: num] :
% 0.30/0.62            ( produc2626176000494625587at_nat
% 0.30/0.62            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_step_nat_def
% 0.30/0.62  thf(fact_6369_ln__one__plus__pos__lower__bound,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.62         => ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % ln_one_plus_pos_lower_bound
% 0.30/0.62  thf(fact_6370_divmod__step__int__def,axiom,
% 0.30/0.62      ( unique5024387138958732305ep_int
% 0.30/0.62      = ( ^ [L2: num] :
% 0.30/0.62            ( produc4245557441103728435nt_int
% 0.30/0.62            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_step_int_def
% 0.30/0.62  thf(fact_6371_odd__mod__4__div__2,axiom,
% 0.30/0.62      ! [N: nat] :
% 0.30/0.62        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.62          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.62       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % odd_mod_4_div_2
% 0.30/0.62  thf(fact_6372_mod__exhaust__less__4,axiom,
% 0.30/0.62      ! [M: nat] :
% 0.30/0.62        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.62          = zero_zero_nat )
% 0.30/0.62        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.62          = one_one_nat )
% 0.30/0.62        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.62          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.62          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mod_exhaust_less_4
% 0.30/0.62  thf(fact_6373_ln__2__less__1,axiom,
% 0.30/0.62      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.30/0.62  
% 0.30/0.62  % ln_2_less_1
% 0.30/0.62  thf(fact_6374_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( produc4245557441103728435nt_int
% 0.30/0.62          @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 0.30/0.62          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_algorithm_code(6)
% 0.30/0.62  thf(fact_6375_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( produc2626176000494625587at_nat
% 0.30/0.62          @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 0.30/0.62          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_algorithm_code(6)
% 0.30/0.62  thf(fact_6376_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.62        = ( produc6916734918728496179nteger
% 0.30/0.62          @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 0.30/0.62          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_algorithm_code(6)
% 0.30/0.62  thf(fact_6377_signed__take__bit__numeral__minus__bit1,axiom,
% 0.30/0.62      ! [L: num,K: num] :
% 0.30/0.62        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.62        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % signed_take_bit_numeral_minus_bit1
% 0.30/0.62  thf(fact_6378_dbl__dec__simps_I4_J,axiom,
% 0.30/0.62      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.62      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(4)
% 0.30/0.62  thf(fact_6379_dbl__dec__simps_I4_J,axiom,
% 0.30/0.62      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.62      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(4)
% 0.30/0.62  thf(fact_6380_dbl__dec__simps_I4_J,axiom,
% 0.30/0.62      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.62      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(4)
% 0.30/0.62  thf(fact_6381_dbl__dec__simps_I4_J,axiom,
% 0.30/0.62      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.62      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(4)
% 0.30/0.62  thf(fact_6382_dbl__dec__simps_I4_J,axiom,
% 0.30/0.62      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.62      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(4)
% 0.30/0.62  thf(fact_6383_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.62       => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.62         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.30/0.62  thf(fact_6384_abs__idempotent,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/0.62        = ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_idempotent
% 0.30/0.62  thf(fact_6385_abs__idempotent,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.30/0.62        = ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_idempotent
% 0.30/0.62  thf(fact_6386_abs__idempotent,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_idempotent
% 0.30/0.62  thf(fact_6387_abs__idempotent,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.62        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_idempotent
% 0.30/0.62  thf(fact_6388_abs__abs,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/0.62        = ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_abs
% 0.30/0.62  thf(fact_6389_abs__abs,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.30/0.62        = ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_abs
% 0.30/0.62  thf(fact_6390_abs__abs,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_abs
% 0.30/0.62  thf(fact_6391_abs__abs,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.62        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_abs
% 0.30/0.62  thf(fact_6392_case__prodI,axiom,
% 0.30/0.62      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( F @ A @ B2 )
% 0.30/0.62       => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI
% 0.30/0.62  thf(fact_6393_case__prodI,axiom,
% 0.30/0.62      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( F @ A @ B2 )
% 0.30/0.62       => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI
% 0.30/0.62  thf(fact_6394_case__prodI,axiom,
% 0.30/0.62      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
% 0.30/0.62        ( ( F @ A @ B2 )
% 0.30/0.62       => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI
% 0.30/0.62  thf(fact_6395_case__prodI,axiom,
% 0.30/0.62      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 0.30/0.62        ( ( F @ A @ B2 )
% 0.30/0.62       => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI
% 0.30/0.62  thf(fact_6396_case__prodI,axiom,
% 0.30/0.62      ! [F: int > int > $o,A: int,B2: int] :
% 0.30/0.62        ( ( F @ A @ B2 )
% 0.30/0.62       => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI
% 0.30/0.62  thf(fact_6397_case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 0.30/0.62        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.30/0.62           => ( C @ A3 @ B3 ) )
% 0.30/0.62       => ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI2
% 0.30/0.62  thf(fact_6398_case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 0.30/0.62        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 0.30/0.62           => ( C @ A3 @ B3 ) )
% 0.30/0.62       => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI2
% 0.30/0.62  thf(fact_6399_case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 0.30/0.62        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 0.30/0.62           => ( C @ A3 @ B3 ) )
% 0.30/0.62       => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI2
% 0.30/0.62  thf(fact_6400_case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 0.30/0.62        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.30/0.62           => ( C @ A3 @ B3 ) )
% 0.30/0.62       => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI2
% 0.30/0.62  thf(fact_6401_case__prodI2,axiom,
% 0.30/0.62      ! [P4: product_prod_int_int,C: int > int > $o] :
% 0.30/0.62        ( ! [A3: int,B3: int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.30/0.62           => ( C @ A3 @ B3 ) )
% 0.30/0.62       => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI2
% 0.30/0.62  thf(fact_6402_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: nat,C: int > int > set_nat,A: int,B2: int] :
% 0.30/0.62        ( ( member_nat @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6403_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: real,C: int > int > set_real,A: int,B2: int] :
% 0.30/0.62        ( ( member_real @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6404_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: int,C: int > int > set_int,A: int,B2: int] :
% 0.30/0.62        ( ( member_int @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6405_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: complex,C: int > int > set_complex,A: int,B2: int] :
% 0.30/0.62        ( ( member_complex @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6406_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( member_nat @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6407_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( member_real @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6408_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( member_int @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6409_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( member_complex @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6410_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 0.30/0.62        ( ( member_nat @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6411_mem__case__prodI,axiom,
% 0.30/0.62      ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 0.30/0.62        ( ( member_real @ Z @ ( C @ A @ B2 ) )
% 0.30/0.62       => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI
% 0.30/0.62  thf(fact_6412_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: product_prod_int_int,Z: nat,C: int > int > set_nat] :
% 0.30/0.62        ( ! [A3: int,B3: int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.30/0.62           => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6413_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: product_prod_int_int,Z: real,C: int > int > set_real] :
% 0.30/0.62        ( ! [A3: int,B3: int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.30/0.62           => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6414_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: product_prod_int_int,Z: int,C: int > int > set_int] :
% 0.30/0.62        ( ! [A3: int,B3: int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.30/0.62           => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6415_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: product_prod_int_int,Z: complex,C: int > int > set_complex] :
% 0.30/0.62        ( ! [A3: int,B3: int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( product_Pair_int_int @ A3 @ B3 ) )
% 0.30/0.62           => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6416_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc8763457246119570046nteger,Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
% 0.30/0.62        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.30/0.62           => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6417_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc8763457246119570046nteger,Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
% 0.30/0.62        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.30/0.62           => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6418_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc8763457246119570046nteger,Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
% 0.30/0.62        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.30/0.62           => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6419_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc8763457246119570046nteger,Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
% 0.30/0.62        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 0.30/0.62           => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6420_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
% 0.30/0.62        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.30/0.62           => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6421_mem__case__prodI2,axiom,
% 0.30/0.62      ! [P4: produc7773217078559923341nt_int,Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
% 0.30/0.62        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 0.30/0.62            ( ( P4
% 0.30/0.62              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 0.30/0.62           => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
% 0.30/0.62       => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodI2
% 0.30/0.62  thf(fact_6422_case__prodI2_H,axiom,
% 0.30/0.62      ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
% 0.30/0.62        ( ! [A3: nat,B3: nat] :
% 0.30/0.62            ( ( ( product_Pair_nat_nat @ A3 @ B3 )
% 0.30/0.62              = P4 )
% 0.30/0.62           => ( C @ A3 @ B3 @ X3 ) )
% 0.30/0.62       => ( produc8739625826339149834_nat_o @ C @ P4 @ X3 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodI2'
% 0.30/0.62  thf(fact_6423_abs__0__eq,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( zero_z3403309356797280102nteger
% 0.30/0.62          = ( abs_abs_Code_integer @ A ) )
% 0.30/0.62        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0_eq
% 0.30/0.62  thf(fact_6424_abs__0__eq,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( zero_zero_real
% 0.30/0.62          = ( abs_abs_real @ A ) )
% 0.30/0.62        = ( A = zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0_eq
% 0.30/0.62  thf(fact_6425_abs__0__eq,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( zero_zero_rat
% 0.30/0.62          = ( abs_abs_rat @ A ) )
% 0.30/0.62        = ( A = zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0_eq
% 0.30/0.62  thf(fact_6426_abs__0__eq,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( zero_zero_int
% 0.30/0.62          = ( abs_abs_int @ A ) )
% 0.30/0.62        = ( A = zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0_eq
% 0.30/0.62  thf(fact_6427_abs__eq__0,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
% 0.30/0.62  thf(fact_6428_abs__eq__0,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
% 0.30/0.62  thf(fact_6429_abs__eq__0,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
% 0.30/0.62  thf(fact_6430_abs__eq__0,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
% 0.30/0.62  thf(fact_6431_abs__zero,axiom,
% 0.30/0.62      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.30/0.62      = zero_z3403309356797280102nteger ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_zero
% 0.30/0.62  thf(fact_6432_abs__zero,axiom,
% 0.30/0.62      ( ( abs_abs_real @ zero_zero_real )
% 0.30/0.62      = zero_zero_real ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_zero
% 0.30/0.62  thf(fact_6433_abs__zero,axiom,
% 0.30/0.62      ( ( abs_abs_rat @ zero_zero_rat )
% 0.30/0.62      = zero_zero_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_zero
% 0.30/0.62  thf(fact_6434_abs__zero,axiom,
% 0.30/0.62      ( ( abs_abs_int @ zero_zero_int )
% 0.30/0.62      = zero_zero_int ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_zero
% 0.30/0.62  thf(fact_6435_abs__0,axiom,
% 0.30/0.62      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.30/0.62      = zero_z3403309356797280102nteger ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0
% 0.30/0.62  thf(fact_6436_abs__0,axiom,
% 0.30/0.62      ( ( abs_abs_complex @ zero_zero_complex )
% 0.30/0.62      = zero_zero_complex ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0
% 0.30/0.62  thf(fact_6437_abs__0,axiom,
% 0.30/0.62      ( ( abs_abs_real @ zero_zero_real )
% 0.30/0.62      = zero_zero_real ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0
% 0.30/0.62  thf(fact_6438_abs__0,axiom,
% 0.30/0.62      ( ( abs_abs_rat @ zero_zero_rat )
% 0.30/0.62      = zero_zero_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0
% 0.30/0.62  thf(fact_6439_abs__0,axiom,
% 0.30/0.62      ( ( abs_abs_int @ zero_zero_int )
% 0.30/0.62      = zero_zero_int ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_0
% 0.30/0.62  thf(fact_6440_abs__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.62        = ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_numeral
% 0.30/0.62  thf(fact_6441_abs__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.30/0.62        = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_numeral
% 0.30/0.62  thf(fact_6442_abs__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.30/0.62        = ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_numeral
% 0.30/0.62  thf(fact_6443_abs__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.30/0.62        = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_numeral
% 0.30/0.62  thf(fact_6444_abs__mult__self__eq,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62        = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_self_eq
% 0.30/0.62  thf(fact_6445_abs__mult__self__eq,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.30/0.62        = ( times_times_real @ A @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_self_eq
% 0.30/0.62  thf(fact_6446_abs__mult__self__eq,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.30/0.62        = ( times_times_rat @ A @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_self_eq
% 0.30/0.62  thf(fact_6447_abs__mult__self__eq,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.30/0.62        = ( times_times_int @ A @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_self_eq
% 0.30/0.62  thf(fact_6448_abs__1,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_1
% 0.30/0.62  thf(fact_6449_abs__1,axiom,
% 0.30/0.62      ( ( abs_abs_complex @ one_one_complex )
% 0.30/0.62      = one_one_complex ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_1
% 0.30/0.62  thf(fact_6450_abs__1,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_1
% 0.30/0.62  thf(fact_6451_abs__1,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_1
% 0.30/0.62  thf(fact_6452_abs__1,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_1
% 0.30/0.62  thf(fact_6453_abs__add__abs,axiom,
% 0.30/0.62      ! [A: code_integer,B2: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) )
% 0.30/0.62        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_abs
% 0.30/0.62  thf(fact_6454_abs__add__abs,axiom,
% 0.30/0.62      ! [A: real,B2: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
% 0.30/0.62        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_abs
% 0.30/0.62  thf(fact_6455_abs__add__abs,axiom,
% 0.30/0.62      ! [A: rat,B2: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) )
% 0.30/0.62        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_abs
% 0.30/0.62  thf(fact_6456_abs__add__abs,axiom,
% 0.30/0.62      ! [A: int,B2: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
% 0.30/0.62        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_abs
% 0.30/0.62  thf(fact_6457_abs__divide,axiom,
% 0.30/0.62      ! [A: complex,B2: complex] :
% 0.30/0.62        ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.62        = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_divide
% 0.30/0.62  thf(fact_6458_abs__divide,axiom,
% 0.30/0.62      ! [A: real,B2: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.62        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_divide
% 0.30/0.62  thf(fact_6459_abs__divide,axiom,
% 0.30/0.62      ! [A: rat,B2: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.62        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_divide
% 0.30/0.62  thf(fact_6460_abs__minus__cancel,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.62        = ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_cancel
% 0.30/0.62  thf(fact_6461_abs__minus__cancel,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.62        = ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_cancel
% 0.30/0.62  thf(fact_6462_abs__minus__cancel,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.62        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_cancel
% 0.30/0.62  thf(fact_6463_abs__minus__cancel,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.62        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_cancel
% 0.30/0.62  thf(fact_6464_abs__minus,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.62        = ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus
% 0.30/0.62  thf(fact_6465_abs__minus,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.62        = ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus
% 0.30/0.62  thf(fact_6466_abs__minus,axiom,
% 0.30/0.62      ! [A: complex] :
% 0.30/0.62        ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.62        = ( abs_abs_complex @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus
% 0.30/0.62  thf(fact_6467_abs__minus,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.62        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus
% 0.30/0.62  thf(fact_6468_abs__minus,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.62        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus
% 0.30/0.62  thf(fact_6469_abs__dvd__iff,axiom,
% 0.30/0.62      ! [M: real,K: real] :
% 0.30/0.62        ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.30/0.62        = ( dvd_dvd_real @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_dvd_iff
% 0.30/0.62  thf(fact_6470_abs__dvd__iff,axiom,
% 0.30/0.62      ! [M: int,K: int] :
% 0.30/0.62        ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.30/0.62        = ( dvd_dvd_int @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_dvd_iff
% 0.30/0.62  thf(fact_6471_abs__dvd__iff,axiom,
% 0.30/0.62      ! [M: code_integer,K: code_integer] :
% 0.30/0.62        ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.30/0.62        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_dvd_iff
% 0.30/0.62  thf(fact_6472_abs__dvd__iff,axiom,
% 0.30/0.62      ! [M: rat,K: rat] :
% 0.30/0.62        ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 0.30/0.62        = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_dvd_iff
% 0.30/0.62  thf(fact_6473_dvd__abs__iff,axiom,
% 0.30/0.62      ! [M: real,K: real] :
% 0.30/0.62        ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.30/0.62        = ( dvd_dvd_real @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_abs_iff
% 0.30/0.62  thf(fact_6474_dvd__abs__iff,axiom,
% 0.30/0.62      ! [M: int,K: int] :
% 0.30/0.62        ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.30/0.62        = ( dvd_dvd_int @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_abs_iff
% 0.30/0.62  thf(fact_6475_dvd__abs__iff,axiom,
% 0.30/0.62      ! [M: code_integer,K: code_integer] :
% 0.30/0.62        ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.30/0.62        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_abs_iff
% 0.30/0.62  thf(fact_6476_dvd__abs__iff,axiom,
% 0.30/0.62      ! [M: rat,K: rat] :
% 0.30/0.62        ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 0.30/0.62        = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_abs_iff
% 0.30/0.62  thf(fact_6477_abs__bool__eq,axiom,
% 0.30/0.62      ! [P: $o] :
% 0.30/0.62        ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.30/0.62        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_bool_eq
% 0.30/0.62  thf(fact_6478_abs__bool__eq,axiom,
% 0.30/0.62      ! [P: $o] :
% 0.30/0.62        ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.30/0.62        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_bool_eq
% 0.30/0.62  thf(fact_6479_abs__bool__eq,axiom,
% 0.30/0.62      ! [P: $o] :
% 0.30/0.62        ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.62        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_bool_eq
% 0.30/0.62  thf(fact_6480_abs__bool__eq,axiom,
% 0.30/0.62      ! [P: $o] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.30/0.62        = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_bool_eq
% 0.30/0.62  thf(fact_6481_dbl__dec__simps_I3_J,axiom,
% 0.30/0.62      ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.30/0.62      = one_one_complex ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(3)
% 0.30/0.62  thf(fact_6482_dbl__dec__simps_I3_J,axiom,
% 0.30/0.62      ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.30/0.62      = one_one_real ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(3)
% 0.30/0.62  thf(fact_6483_dbl__dec__simps_I3_J,axiom,
% 0.30/0.62      ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.30/0.62      = one_one_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(3)
% 0.30/0.62  thf(fact_6484_dbl__dec__simps_I3_J,axiom,
% 0.30/0.62      ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.30/0.62      = one_one_int ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(3)
% 0.30/0.62  thf(fact_6485_abs__le__zero__iff,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.62        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_zero_iff
% 0.30/0.62  thf(fact_6486_abs__le__zero__iff,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.30/0.62        = ( A = zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_zero_iff
% 0.30/0.62  thf(fact_6487_abs__le__zero__iff,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.30/0.62        = ( A = zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_zero_iff
% 0.30/0.62  thf(fact_6488_abs__le__zero__iff,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.30/0.62        = ( A = zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_zero_iff
% 0.30/0.62  thf(fact_6489_abs__le__self__iff,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.30/0.62        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_self_iff
% 0.30/0.62  thf(fact_6490_abs__le__self__iff,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.30/0.62        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_self_iff
% 0.30/0.62  thf(fact_6491_abs__le__self__iff,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.30/0.62        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_self_iff
% 0.30/0.62  thf(fact_6492_abs__le__self__iff,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.30/0.62        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_self_iff
% 0.30/0.62  thf(fact_6493_abs__of__nonneg,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ 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_nonneg
% 0.30/0.62  thf(fact_6494_abs__of__nonneg,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_eq_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_nonneg
% 0.30/0.62  thf(fact_6495_abs__of__nonneg,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_eq_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_nonneg
% 0.30/0.62  thf(fact_6496_abs__of__nonneg,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_eq_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_nonneg
% 0.30/0.62  thf(fact_6497_zero__less__abs__iff,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.62        = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_abs_iff
% 0.30/0.62  thf(fact_6498_zero__less__abs__iff,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.30/0.62        = ( A != zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_abs_iff
% 0.30/0.62  thf(fact_6499_zero__less__abs__iff,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.62        = ( A != zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_abs_iff
% 0.30/0.62  thf(fact_6500_zero__less__abs__iff,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.30/0.62        = ( A != zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_abs_iff
% 0.30/0.62  thf(fact_6501_abs__neg__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.62        = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_numeral
% 0.30/0.62  thf(fact_6502_abs__neg__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.62        = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_numeral
% 0.30/0.62  thf(fact_6503_abs__neg__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.30/0.62        = ( numera6620942414471956472nteger @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_numeral
% 0.30/0.62  thf(fact_6504_abs__neg__numeral,axiom,
% 0.30/0.62      ! [N: num] :
% 0.30/0.62        ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.30/0.62        = ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_numeral
% 0.30/0.62  thf(fact_6505_abs__neg__one,axiom,
% 0.30/0.62      ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.62      = one_one_real ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_one
% 0.30/0.62  thf(fact_6506_abs__neg__one,axiom,
% 0.30/0.62      ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.62      = one_one_int ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_one
% 0.30/0.62  thf(fact_6507_abs__neg__one,axiom,
% 0.30/0.62      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.62      = one_one_Code_integer ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_one
% 0.30/0.62  thf(fact_6508_abs__neg__one,axiom,
% 0.30/0.62      ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.62      = one_one_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_neg_one
% 0.30/0.62  thf(fact_6509_abs__power__minus,axiom,
% 0.30/0.62      ! [A: real,N: nat] :
% 0.30/0.62        ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.30/0.62        = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power_minus
% 0.30/0.62  thf(fact_6510_abs__power__minus,axiom,
% 0.30/0.62      ! [A: int,N: nat] :
% 0.30/0.62        ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.30/0.62        = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power_minus
% 0.30/0.62  thf(fact_6511_abs__power__minus,axiom,
% 0.30/0.62      ! [A: code_integer,N: nat] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.30/0.62        = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power_minus
% 0.30/0.62  thf(fact_6512_abs__power__minus,axiom,
% 0.30/0.62      ! [A: rat,N: nat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.30/0.62        = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power_minus
% 0.30/0.62  thf(fact_6513_pred__numeral__simps_I1_J,axiom,
% 0.30/0.62      ( ( pred_numeral @ one )
% 0.30/0.62      = zero_zero_nat ) ).
% 0.30/0.62  
% 0.30/0.62  % pred_numeral_simps(1)
% 0.30/0.62  thf(fact_6514_eq__numeral__Suc,axiom,
% 0.30/0.62      ! [K: num,N: nat] :
% 0.30/0.62        ( ( ( numeral_numeral_nat @ K )
% 0.30/0.62          = ( suc @ N ) )
% 0.30/0.62        = ( ( pred_numeral @ K )
% 0.30/0.62          = N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % eq_numeral_Suc
% 0.30/0.62  thf(fact_6515_Suc__eq__numeral,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( ( suc @ N )
% 0.30/0.62          = ( numeral_numeral_nat @ K ) )
% 0.30/0.62        = ( N
% 0.30/0.62          = ( pred_numeral @ K ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Suc_eq_numeral
% 0.30/0.62  thf(fact_6516_sum__abs,axiom,
% 0.30/0.62      ! [F: int > int,A4: set_int] :
% 0.30/0.62        ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
% 0.30/0.62        @ ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_abs
% 0.30/0.62  thf(fact_6517_sum__abs,axiom,
% 0.30/0.62      ! [F: nat > real,A4: set_nat] :
% 0.30/0.62        ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
% 0.30/0.62        @ ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_abs
% 0.30/0.62  thf(fact_6518_zero__le__divide__abs__iff,axiom,
% 0.30/0.62      ! [A: real,B2: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
% 0.30/0.62        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62          | ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_divide_abs_iff
% 0.30/0.62  thf(fact_6519_zero__le__divide__abs__iff,axiom,
% 0.30/0.62      ! [A: rat,B2: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) )
% 0.30/0.62        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62          | ( B2 = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_divide_abs_iff
% 0.30/0.62  thf(fact_6520_divide__le__0__abs__iff,axiom,
% 0.30/0.62      ! [A: real,B2: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
% 0.30/0.62        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.62          | ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divide_le_0_abs_iff
% 0.30/0.62  thf(fact_6521_divide__le__0__abs__iff,axiom,
% 0.30/0.62      ! [A: rat,B2: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
% 0.30/0.62        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.62          | ( B2 = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divide_le_0_abs_iff
% 0.30/0.62  thf(fact_6522_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_6523_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_6524_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_6525_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_6526_less__numeral__Suc,axiom,
% 0.30/0.62      ! [K: num,N: nat] :
% 0.30/0.62        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.62        = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % less_numeral_Suc
% 0.30/0.62  thf(fact_6527_less__Suc__numeral,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.62        = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % less_Suc_numeral
% 0.30/0.62  thf(fact_6528_artanh__minus__real,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.62       => ( ( artanh_real @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.62          = ( uminus_uminus_real @ ( artanh_real @ X3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % artanh_minus_real
% 0.30/0.62  thf(fact_6529_pred__numeral__simps_I3_J,axiom,
% 0.30/0.62      ! [K: num] :
% 0.30/0.62        ( ( pred_numeral @ ( bit1 @ K ) )
% 0.30/0.62        = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % pred_numeral_simps(3)
% 0.30/0.62  thf(fact_6530_le__numeral__Suc,axiom,
% 0.30/0.62      ! [K: num,N: nat] :
% 0.30/0.62        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.62        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % le_numeral_Suc
% 0.30/0.62  thf(fact_6531_le__Suc__numeral,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.62        = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % le_Suc_numeral
% 0.30/0.62  thf(fact_6532_diff__Suc__numeral,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.62        = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % diff_Suc_numeral
% 0.30/0.62  thf(fact_6533_diff__numeral__Suc,axiom,
% 0.30/0.62      ! [K: num,N: nat] :
% 0.30/0.62        ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.62        = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.30/0.62  
% 0.30/0.62  % diff_numeral_Suc
% 0.30/0.62  thf(fact_6534_max__Suc__numeral,axiom,
% 0.30/0.62      ! [N: nat,K: num] :
% 0.30/0.62        ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.62        = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % max_Suc_numeral
% 0.30/0.62  thf(fact_6535_max__numeral__Suc,axiom,
% 0.30/0.62      ! [K: num,N: nat] :
% 0.30/0.62        ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.62        = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % max_numeral_Suc
% 0.30/0.62  thf(fact_6536_sum__abs__ge__zero,axiom,
% 0.30/0.62      ! [F: int > int,A4: set_int] :
% 0.30/0.62        ( ord_less_eq_int @ zero_zero_int
% 0.30/0.62        @ ( groups4538972089207619220nt_int
% 0.30/0.62          @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_abs_ge_zero
% 0.30/0.62  thf(fact_6537_sum__abs__ge__zero,axiom,
% 0.30/0.62      ! [F: nat > real,A4: set_nat] :
% 0.30/0.62        ( ord_less_eq_real @ zero_zero_real
% 0.30/0.62        @ ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.30/0.62          @ A4 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_abs_ge_zero
% 0.30/0.62  thf(fact_6538_dbl__dec__simps_I2_J,axiom,
% 0.30/0.62      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.30/0.62      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(2)
% 0.30/0.62  thf(fact_6539_dbl__dec__simps_I2_J,axiom,
% 0.30/0.62      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.30/0.62      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(2)
% 0.30/0.62  thf(fact_6540_dbl__dec__simps_I2_J,axiom,
% 0.30/0.62      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.30/0.62      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(2)
% 0.30/0.62  thf(fact_6541_dbl__dec__simps_I2_J,axiom,
% 0.30/0.62      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.30/0.62      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(2)
% 0.30/0.62  thf(fact_6542_dbl__dec__simps_I2_J,axiom,
% 0.30/0.62      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.30/0.62      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dbl_dec_simps(2)
% 0.30/0.62  thf(fact_6543_numeral__div__minus__numeral,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.62        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % numeral_div_minus_numeral
% 0.30/0.62  thf(fact_6544_minus__numeral__div__numeral,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.62        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % minus_numeral_div_numeral
% 0.30/0.62  thf(fact_6545_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: code_integer,N: nat] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.30/0.62        = ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6546_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: real,N: nat] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.30/0.62        = ( ( A != zero_zero_real )
% 0.30/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6547_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: rat,N: nat] :
% 0.30/0.62        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.30/0.62        = ( ( A != zero_zero_rat )
% 0.30/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6548_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: int,N: nat] :
% 0.30/0.62        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.30/0.62        = ( ( A != zero_zero_int )
% 0.30/0.62          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6549_abs__power2,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power2
% 0.30/0.62  thf(fact_6550_abs__power2,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power2
% 0.30/0.62  thf(fact_6551_abs__power2,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power2
% 0.30/0.62  thf(fact_6552_abs__power2,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.62        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_power2
% 0.30/0.62  thf(fact_6553_power2__abs,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power2_abs
% 0.30/0.62  thf(fact_6554_power2__abs,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power2_abs
% 0.30/0.62  thf(fact_6555_power2__abs,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power2_abs
% 0.30/0.62  thf(fact_6556_power2__abs,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.62        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % power2_abs
% 0.30/0.62  thf(fact_6557_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N: nat,M: nat,G: nat > complex] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = zero_zero_complex ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6558_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N: nat,M: nat,G: nat > rat] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = zero_zero_rat ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6559_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N: nat,M: nat,G: nat > int] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = zero_zero_int ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6560_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N: nat,M: nat,G: nat > nat] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = zero_zero_nat ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6561_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N: nat,M: nat,G: nat > real] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = zero_zero_real ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.62         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.62            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6562_dvd__numeral__simp,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.62        = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_numeral_simp
% 0.30/0.62  thf(fact_6563_dvd__numeral__simp,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.62        = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_numeral_simp
% 0.30/0.62  thf(fact_6564_dvd__numeral__simp,axiom,
% 0.30/0.62      ! [M: num,N: num] :
% 0.30/0.62        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.30/0.62        = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_numeral_simp
% 0.30/0.62  thf(fact_6565_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.62      ! [M: num] :
% 0.30/0.62        ( ( unique5052692396658037445od_int @ M @ one )
% 0.30/0.62        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_algorithm_code(2)
% 0.30/0.62  thf(fact_6566_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.62      ! [M: num] :
% 0.30/0.62        ( ( unique5055182867167087721od_nat @ M @ one )
% 0.30/0.62        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divmod_algorithm_code(2)
% 0.30/0.62  thf(fact_6567_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.63      ! [M: num] :
% 0.30/0.63        ( ( unique3479559517661332726nteger @ M @ one )
% 0.30/0.63        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(2)
% 0.30/0.63  thf(fact_6568_sum__zero__power,axiom,
% 0.30/0.63      ! [A4: set_nat,C: nat > complex] :
% 0.30/0.63        ( ( ( ( finite_finite_nat @ A4 )
% 0.30/0.63            & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = ( C @ zero_zero_nat ) ) )
% 0.30/0.63        & ( ~ ( ( finite_finite_nat @ A4 )
% 0.30/0.63              & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = zero_zero_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_zero_power
% 0.30/0.63  thf(fact_6569_sum__zero__power,axiom,
% 0.30/0.63      ! [A4: set_nat,C: nat > rat] :
% 0.30/0.63        ( ( ( ( finite_finite_nat @ A4 )
% 0.30/0.63            & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = ( C @ zero_zero_nat ) ) )
% 0.30/0.63        & ( ~ ( ( finite_finite_nat @ A4 )
% 0.30/0.63              & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = zero_zero_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_zero_power
% 0.30/0.63  thf(fact_6570_sum__zero__power,axiom,
% 0.30/0.63      ! [A4: set_nat,C: nat > real] :
% 0.30/0.63        ( ( ( ( finite_finite_nat @ A4 )
% 0.30/0.63            & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = ( C @ zero_zero_nat ) ) )
% 0.30/0.63        & ( ~ ( ( finite_finite_nat @ A4 )
% 0.30/0.63              & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = zero_zero_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_zero_power
% 0.30/0.63  thf(fact_6571_power__even__abs__numeral,axiom,
% 0.30/0.63      ! [W: num,A: code_integer] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63          = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs_numeral
% 0.30/0.63  thf(fact_6572_power__even__abs__numeral,axiom,
% 0.30/0.63      ! [W: num,A: rat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63          = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs_numeral
% 0.30/0.63  thf(fact_6573_power__even__abs__numeral,axiom,
% 0.30/0.63      ! [W: num,A: real] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63       => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63          = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs_numeral
% 0.30/0.63  thf(fact_6574_power__even__abs__numeral,axiom,
% 0.30/0.63      ! [W: num,A: int] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63       => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.30/0.63          = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs_numeral
% 0.30/0.63  thf(fact_6575_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.30/0.63        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(3)
% 0.30/0.63  thf(fact_6576_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.30/0.63        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(3)
% 0.30/0.63  thf(fact_6577_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.30/0.63        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(3)
% 0.30/0.63  thf(fact_6578_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.30/0.63        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(4)
% 0.30/0.63  thf(fact_6579_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.30/0.63        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(4)
% 0.30/0.63  thf(fact_6580_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.30/0.63        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(4)
% 0.30/0.63  thf(fact_6581_sum__zero__power_H,axiom,
% 0.30/0.63      ! [A4: set_nat,C: nat > complex,D: nat > complex] :
% 0.30/0.63        ( ( ( ( finite_finite_nat @ A4 )
% 0.30/0.63            & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.63        & ( ~ ( ( finite_finite_nat @ A4 )
% 0.30/0.63              & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = zero_zero_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_zero_power'
% 0.30/0.63  thf(fact_6582_sum__zero__power_H,axiom,
% 0.30/0.63      ! [A4: set_nat,C: nat > rat,D: nat > rat] :
% 0.30/0.63        ( ( ( ( finite_finite_nat @ A4 )
% 0.30/0.63            & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.63        & ( ~ ( ( finite_finite_nat @ A4 )
% 0.30/0.63              & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = zero_zero_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_zero_power'
% 0.30/0.63  thf(fact_6583_sum__zero__power_H,axiom,
% 0.30/0.63      ! [A4: set_nat,C: nat > real,D: nat > real] :
% 0.30/0.63        ( ( ( ( finite_finite_nat @ A4 )
% 0.30/0.63            & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.63        & ( ~ ( ( finite_finite_nat @ A4 )
% 0.30/0.63              & ( member_nat @ zero_zero_nat @ A4 ) )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.30/0.63              @ A4 )
% 0.30/0.63            = zero_zero_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_zero_power'
% 0.30/0.63  thf(fact_6584_one__div__minus__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.63        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_div_minus_numeral
% 0.30/0.63  thf(fact_6585_minus__one__div__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.63        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % minus_one_div_numeral
% 0.30/0.63  thf(fact_6586_signed__take__bit__numeral__bit0,axiom,
% 0.30/0.63      ! [L: num,K: num] :
% 0.30/0.63        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.30/0.63        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % signed_take_bit_numeral_bit0
% 0.30/0.63  thf(fact_6587_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.63        = ( produc4245557441103728435nt_int
% 0.30/0.63          @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.63          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(5)
% 0.30/0.63  thf(fact_6588_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.63        = ( produc2626176000494625587at_nat
% 0.30/0.63          @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.63          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(5)
% 0.30/0.63  thf(fact_6589_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.63        = ( produc6916734918728496179nteger
% 0.30/0.63          @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.63          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(5)
% 0.30/0.63  thf(fact_6590_signed__take__bit__numeral__minus__bit0,axiom,
% 0.30/0.63      ! [L: num,K: num] :
% 0.30/0.63        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.63        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % signed_take_bit_numeral_minus_bit0
% 0.30/0.63  thf(fact_6591_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( ( ord_less_num @ M @ N )
% 0.30/0.63         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_num @ M @ N )
% 0.30/0.63         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(8)
% 0.30/0.63  thf(fact_6592_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( ( ord_less_num @ M @ N )
% 0.30/0.63         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_num @ M @ N )
% 0.30/0.63         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(8)
% 0.30/0.63  thf(fact_6593_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( ( ord_less_num @ M @ N )
% 0.30/0.63         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_num @ M @ N )
% 0.30/0.63         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(8)
% 0.30/0.63  thf(fact_6594_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( ( ord_less_eq_num @ M @ N )
% 0.30/0.63         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.30/0.63         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(7)
% 0.30/0.63  thf(fact_6595_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( ( ord_less_eq_num @ M @ N )
% 0.30/0.63         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.30/0.63         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(7)
% 0.30/0.63  thf(fact_6596_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.63      ! [M: num,N: num] :
% 0.30/0.63        ( ( ( ord_less_eq_num @ M @ N )
% 0.30/0.63         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_num @ M @ N )
% 0.30/0.63         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.63            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_algorithm_code(7)
% 0.30/0.63  thf(fact_6597_signed__take__bit__numeral__bit1,axiom,
% 0.30/0.63      ! [L: num,K: num] :
% 0.30/0.63        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.63        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % signed_take_bit_numeral_bit1
% 0.30/0.63  thf(fact_6598_abs__ge__self,axiom,
% 0.30/0.63      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_self
% 0.30/0.63  thf(fact_6599_abs__ge__self,axiom,
% 0.30/0.63      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_self
% 0.30/0.63  thf(fact_6600_abs__ge__self,axiom,
% 0.30/0.63      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_self
% 0.30/0.63  thf(fact_6601_abs__ge__self,axiom,
% 0.30/0.63      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_self
% 0.30/0.63  thf(fact_6602_abs__le__D1,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 0.30/0.63       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D1
% 0.30/0.63  thf(fact_6603_abs__le__D1,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.30/0.63       => ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D1
% 0.30/0.63  thf(fact_6604_abs__le__D1,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.30/0.63       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D1
% 0.30/0.63  thf(fact_6605_abs__le__D1,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 0.30/0.63       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D1
% 0.30/0.63  thf(fact_6606_abs__eq__0__iff,axiom,
% 0.30/0.63      ! [A: code_integer] :
% 0.30/0.63        ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.63          = zero_z3403309356797280102nteger )
% 0.30/0.63        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_0_iff
% 0.30/0.63  thf(fact_6607_abs__eq__0__iff,axiom,
% 0.30/0.63      ! [A: complex] :
% 0.30/0.63        ( ( ( abs_abs_complex @ A )
% 0.30/0.63          = zero_zero_complex )
% 0.30/0.63        = ( A = zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_0_iff
% 0.30/0.63  thf(fact_6608_abs__eq__0__iff,axiom,
% 0.30/0.63      ! [A: real] :
% 0.30/0.63        ( ( ( abs_abs_real @ A )
% 0.30/0.63          = zero_zero_real )
% 0.30/0.63        = ( A = zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_0_iff
% 0.30/0.63  thf(fact_6609_abs__eq__0__iff,axiom,
% 0.30/0.63      ! [A: rat] :
% 0.30/0.63        ( ( ( abs_abs_rat @ A )
% 0.30/0.63          = zero_zero_rat )
% 0.30/0.63        = ( A = zero_zero_rat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_0_iff
% 0.30/0.63  thf(fact_6610_abs__eq__0__iff,axiom,
% 0.30/0.63      ! [A: int] :
% 0.30/0.63        ( ( ( abs_abs_int @ A )
% 0.30/0.63          = zero_zero_int )
% 0.30/0.63        = ( A = zero_zero_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_0_iff
% 0.30/0.63  thf(fact_6611_abs__mult,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.30/0.63        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult
% 0.30/0.63  thf(fact_6612_abs__mult,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.63        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult
% 0.30/0.63  thf(fact_6613_abs__mult,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.63        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult
% 0.30/0.63  thf(fact_6614_abs__mult,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 0.30/0.63        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult
% 0.30/0.63  thf(fact_6615_abs__one,axiom,
% 0.30/0.63      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.30/0.63      = one_one_Code_integer ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_one
% 0.30/0.63  thf(fact_6616_abs__one,axiom,
% 0.30/0.63      ( ( abs_abs_real @ one_one_real )
% 0.30/0.63      = one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_one
% 0.30/0.63  thf(fact_6617_abs__one,axiom,
% 0.30/0.63      ( ( abs_abs_rat @ one_one_rat )
% 0.30/0.63      = one_one_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_one
% 0.30/0.63  thf(fact_6618_abs__one,axiom,
% 0.30/0.63      ( ( abs_abs_int @ one_one_int )
% 0.30/0.63      = one_one_int ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_one
% 0.30/0.63  thf(fact_6619_abs__minus__commute,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 0.30/0.63        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_commute
% 0.30/0.63  thf(fact_6620_abs__minus__commute,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.63        = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_commute
% 0.30/0.63  thf(fact_6621_abs__minus__commute,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) )
% 0.30/0.63        = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_commute
% 0.30/0.63  thf(fact_6622_abs__minus__commute,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
% 0.30/0.63        = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_commute
% 0.30/0.63  thf(fact_6623_abs__eq__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ( abs_abs_real @ X3 )
% 0.30/0.63          = ( abs_abs_real @ Y3 ) )
% 0.30/0.63        = ( ( X3 = Y3 )
% 0.30/0.63          | ( X3
% 0.30/0.63            = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff
% 0.30/0.63  thf(fact_6624_abs__eq__iff,axiom,
% 0.30/0.63      ! [X3: int,Y3: int] :
% 0.30/0.63        ( ( ( abs_abs_int @ X3 )
% 0.30/0.63          = ( abs_abs_int @ Y3 ) )
% 0.30/0.63        = ( ( X3 = Y3 )
% 0.30/0.63          | ( X3
% 0.30/0.63            = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff
% 0.30/0.63  thf(fact_6625_abs__eq__iff,axiom,
% 0.30/0.63      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.63        ( ( ( abs_abs_Code_integer @ X3 )
% 0.30/0.63          = ( abs_abs_Code_integer @ Y3 ) )
% 0.30/0.63        = ( ( X3 = Y3 )
% 0.30/0.63          | ( X3
% 0.30/0.63            = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff
% 0.30/0.63  thf(fact_6626_abs__eq__iff,axiom,
% 0.30/0.63      ! [X3: rat,Y3: rat] :
% 0.30/0.63        ( ( ( abs_abs_rat @ X3 )
% 0.30/0.63          = ( abs_abs_rat @ Y3 ) )
% 0.30/0.63        = ( ( X3 = Y3 )
% 0.30/0.63          | ( X3
% 0.30/0.63            = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff
% 0.30/0.63  thf(fact_6627_power__abs,axiom,
% 0.30/0.63      ! [A: code_integer,N: nat] :
% 0.30/0.63        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.30/0.63        = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_abs
% 0.30/0.63  thf(fact_6628_power__abs,axiom,
% 0.30/0.63      ! [A: rat,N: nat] :
% 0.30/0.63        ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 0.30/0.63        = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_abs
% 0.30/0.63  thf(fact_6629_power__abs,axiom,
% 0.30/0.63      ! [A: real,N: nat] :
% 0.30/0.63        ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.30/0.63        = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_abs
% 0.30/0.63  thf(fact_6630_power__abs,axiom,
% 0.30/0.63      ! [A: int,N: nat] :
% 0.30/0.63        ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.30/0.63        = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_abs
% 0.30/0.63  thf(fact_6631_dvd__if__abs__eq,axiom,
% 0.30/0.63      ! [L: real,K: real] :
% 0.30/0.63        ( ( ( abs_abs_real @ L )
% 0.30/0.63          = ( abs_abs_real @ K ) )
% 0.30/0.63       => ( dvd_dvd_real @ L @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dvd_if_abs_eq
% 0.30/0.63  thf(fact_6632_dvd__if__abs__eq,axiom,
% 0.30/0.63      ! [L: int,K: int] :
% 0.30/0.63        ( ( ( abs_abs_int @ L )
% 0.30/0.63          = ( abs_abs_int @ K ) )
% 0.30/0.63       => ( dvd_dvd_int @ L @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dvd_if_abs_eq
% 0.30/0.63  thf(fact_6633_dvd__if__abs__eq,axiom,
% 0.30/0.63      ! [L: code_integer,K: code_integer] :
% 0.30/0.63        ( ( ( abs_abs_Code_integer @ L )
% 0.30/0.63          = ( abs_abs_Code_integer @ K ) )
% 0.30/0.63       => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dvd_if_abs_eq
% 0.30/0.63  thf(fact_6634_dvd__if__abs__eq,axiom,
% 0.30/0.63      ! [L: rat,K: rat] :
% 0.30/0.63        ( ( ( abs_abs_rat @ L )
% 0.30/0.63          = ( abs_abs_rat @ K ) )
% 0.30/0.63       => ( dvd_dvd_rat @ L @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dvd_if_abs_eq
% 0.30/0.63  thf(fact_6635_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: nat,C: int > int > set_nat,P4: product_prod_int_int] :
% 0.30/0.63        ( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: int,Y4: int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6636_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: real,C: int > int > set_real,P4: product_prod_int_int] :
% 0.30/0.63        ( ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: int,Y4: int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6637_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: int,C: int > int > set_int,P4: product_prod_int_int] :
% 0.30/0.63        ( ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: int,Y4: int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6638_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: complex,C: int > int > set_complex,P4: product_prod_int_int] :
% 0.30/0.63        ( ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: int,Y4: int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6639_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
% 0.30/0.63        ( ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6640_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
% 0.30/0.63        ( ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6641_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
% 0.30/0.63        ( ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6642_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P4: produc8763457246119570046nteger] :
% 0.30/0.63        ( ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6643_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
% 0.30/0.63        ( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6644_mem__case__prodE,axiom,
% 0.30/0.63      ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
% 0.30/0.63        ( ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) )
% 0.30/0.63       => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mem_case_prodE
% 0.30/0.63  thf(fact_6645_case__prodD,axiom,
% 0.30/0.63      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.63        ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B2 ) )
% 0.30/0.63       => ( F @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodD
% 0.30/0.63  thf(fact_6646_case__prodD,axiom,
% 0.30/0.63      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 0.30/0.63        ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B2 ) )
% 0.30/0.63       => ( F @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodD
% 0.30/0.63  thf(fact_6647_case__prodD,axiom,
% 0.30/0.63      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
% 0.30/0.63        ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B2 ) )
% 0.30/0.63       => ( F @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodD
% 0.30/0.63  thf(fact_6648_case__prodD,axiom,
% 0.30/0.63      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 0.30/0.63        ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B2 ) )
% 0.30/0.63       => ( F @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodD
% 0.30/0.63  thf(fact_6649_case__prodD,axiom,
% 0.30/0.63      ! [F: int > int > $o,A: int,B2: int] :
% 0.30/0.63        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) )
% 0.30/0.63       => ( F @ A @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodD
% 0.30/0.63  thf(fact_6650_case__prodE,axiom,
% 0.30/0.63      ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
% 0.30/0.63        ( ( produc127349428274296955eger_o @ C @ P4 )
% 0.30/0.63       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodE
% 0.30/0.63  thf(fact_6651_case__prodE,axiom,
% 0.30/0.63      ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
% 0.30/0.63        ( ( produc6253627499356882019eger_o @ C @ P4 )
% 0.30/0.63       => ~ ! [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc8603105652947943368nteger @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodE
% 0.30/0.63  thf(fact_6652_case__prodE,axiom,
% 0.30/0.63      ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
% 0.30/0.63        ( ( produc1573362020775583542_int_o @ C @ P4 )
% 0.30/0.63       => ~ ! [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc5700946648718959541nt_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodE
% 0.30/0.63  thf(fact_6653_case__prodE,axiom,
% 0.30/0.63      ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
% 0.30/0.63        ( ( produc2558449545302689196_int_o @ C @ P4 )
% 0.30/0.63       => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodE
% 0.30/0.63  thf(fact_6654_case__prodE,axiom,
% 0.30/0.63      ! [C: int > int > $o,P4: product_prod_int_int] :
% 0.30/0.63        ( ( produc4947309494688390418_int_o @ C @ P4 )
% 0.30/0.63       => ~ ! [X5: int,Y4: int] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodE
% 0.30/0.63  thf(fact_6655_case__prodD_H,axiom,
% 0.30/0.63      ! [R2: nat > nat > product_prod_nat_nat > $o,A: nat,B2: nat,C: product_prod_nat_nat] :
% 0.30/0.63        ( ( produc8739625826339149834_nat_o @ R2 @ ( product_Pair_nat_nat @ A @ B2 ) @ C )
% 0.30/0.63       => ( R2 @ A @ B2 @ C ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodD'
% 0.30/0.63  thf(fact_6656_case__prodE_H,axiom,
% 0.30/0.63      ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
% 0.30/0.63        ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
% 0.30/0.63       => ~ ! [X5: nat,Y4: nat] :
% 0.30/0.63              ( ( P4
% 0.30/0.63                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 0.30/0.63             => ~ ( C @ X5 @ Y4 @ Z ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % case_prodE'
% 0.30/0.63  thf(fact_6657_abs__ge__zero,axiom,
% 0.30/0.63      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_zero
% 0.30/0.63  thf(fact_6658_abs__ge__zero,axiom,
% 0.30/0.63      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_zero
% 0.30/0.63  thf(fact_6659_abs__ge__zero,axiom,
% 0.30/0.63      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_zero
% 0.30/0.63  thf(fact_6660_abs__ge__zero,axiom,
% 0.30/0.63      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_zero
% 0.30/0.63  thf(fact_6661_abs__not__less__zero,axiom,
% 0.30/0.63      ! [A: code_integer] :
% 0.30/0.63        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_not_less_zero
% 0.30/0.63  thf(fact_6662_abs__not__less__zero,axiom,
% 0.30/0.63      ! [A: real] :
% 0.30/0.63        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_not_less_zero
% 0.30/0.63  thf(fact_6663_abs__not__less__zero,axiom,
% 0.30/0.63      ! [A: rat] :
% 0.30/0.63        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_not_less_zero
% 0.30/0.63  thf(fact_6664_abs__not__less__zero,axiom,
% 0.30/0.63      ! [A: int] :
% 0.30/0.63        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_not_less_zero
% 0.30/0.63  thf(fact_6665_abs__of__pos,axiom,
% 0.30/0.63      ! [A: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.63       => ( ( abs_abs_Code_integer @ A )
% 0.30/0.63          = A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_pos
% 0.30/0.63  thf(fact_6666_abs__of__pos,axiom,
% 0.30/0.63      ! [A: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63       => ( ( abs_abs_real @ A )
% 0.30/0.63          = A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_pos
% 0.30/0.63  thf(fact_6667_abs__of__pos,axiom,
% 0.30/0.63      ! [A: rat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.63       => ( ( abs_abs_rat @ A )
% 0.30/0.63          = A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_pos
% 0.30/0.63  thf(fact_6668_abs__of__pos,axiom,
% 0.30/0.63      ! [A: int] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.63       => ( ( abs_abs_int @ A )
% 0.30/0.63          = A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_pos
% 0.30/0.63  thf(fact_6669_abs__triangle__ineq,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq
% 0.30/0.63  thf(fact_6670_abs__triangle__ineq,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq
% 0.30/0.63  thf(fact_6671_abs__triangle__ineq,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq
% 0.30/0.63  thf(fact_6672_abs__triangle__ineq,axiom,
% 0.30/0.63      ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq
% 0.30/0.63  thf(fact_6673_abs__mult__less,axiom,
% 0.30/0.63      ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.30/0.63       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
% 0.30/0.63         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_less
% 0.30/0.63  thf(fact_6674_abs__mult__less,axiom,
% 0.30/0.63      ! [A: real,C: real,B2: real,D: real] :
% 0.30/0.63        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.30/0.63       => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
% 0.30/0.63         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_less
% 0.30/0.63  thf(fact_6675_abs__mult__less,axiom,
% 0.30/0.63      ! [A: rat,C: rat,B2: rat,D: rat] :
% 0.30/0.63        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.30/0.63       => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
% 0.30/0.63         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_less
% 0.30/0.63  thf(fact_6676_abs__mult__less,axiom,
% 0.30/0.63      ! [A: int,C: int,B2: int,D: int] :
% 0.30/0.63        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.30/0.63       => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
% 0.30/0.63         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_less
% 0.30/0.63  thf(fact_6677_abs__triangle__ineq2,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2
% 0.30/0.63  thf(fact_6678_abs__triangle__ineq2,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2
% 0.30/0.63  thf(fact_6679_abs__triangle__ineq2,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2
% 0.30/0.63  thf(fact_6680_abs__triangle__ineq2,axiom,
% 0.30/0.63      ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2
% 0.30/0.63  thf(fact_6681_abs__triangle__ineq3,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq3
% 0.30/0.63  thf(fact_6682_abs__triangle__ineq3,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq3
% 0.30/0.63  thf(fact_6683_abs__triangle__ineq3,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq3
% 0.30/0.63  thf(fact_6684_abs__triangle__ineq3,axiom,
% 0.30/0.63      ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq3
% 0.30/0.63  thf(fact_6685_abs__triangle__ineq2__sym,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2_sym
% 0.30/0.63  thf(fact_6686_abs__triangle__ineq2__sym,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2_sym
% 0.30/0.63  thf(fact_6687_abs__triangle__ineq2__sym,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2_sym
% 0.30/0.63  thf(fact_6688_abs__triangle__ineq2__sym,axiom,
% 0.30/0.63      ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq2_sym
% 0.30/0.63  thf(fact_6689_nonzero__abs__divide,axiom,
% 0.30/0.63      ! [B2: real,A: real] :
% 0.30/0.63        ( ( B2 != zero_zero_real )
% 0.30/0.63       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.63          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonzero_abs_divide
% 0.30/0.63  thf(fact_6690_nonzero__abs__divide,axiom,
% 0.30/0.63      ! [B2: rat,A: rat] :
% 0.30/0.63        ( ( B2 != zero_zero_rat )
% 0.30/0.63       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.63          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonzero_abs_divide
% 0.30/0.63  thf(fact_6691_abs__leI,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 0.30/0.63         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_leI
% 0.30/0.63  thf(fact_6692_abs__leI,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 0.30/0.63       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 0.30/0.63         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_leI
% 0.30/0.63  thf(fact_6693_abs__leI,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 0.30/0.63         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_leI
% 0.30/0.63  thf(fact_6694_abs__leI,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 0.30/0.63         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_leI
% 0.30/0.63  thf(fact_6695_abs__le__D2,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 0.30/0.63       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D2
% 0.30/0.63  thf(fact_6696_abs__le__D2,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.30/0.63       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D2
% 0.30/0.63  thf(fact_6697_abs__le__D2,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.30/0.63       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D2
% 0.30/0.63  thf(fact_6698_abs__le__D2,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 0.30/0.63       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_D2
% 0.30/0.63  thf(fact_6699_abs__le__iff,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.63          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_iff
% 0.30/0.63  thf(fact_6700_abs__le__iff,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 0.30/0.63          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_iff
% 0.30/0.63  thf(fact_6701_abs__le__iff,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_less_eq_rat @ A @ B2 )
% 0.30/0.63          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_iff
% 0.30/0.63  thf(fact_6702_abs__le__iff,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_less_eq_int @ A @ B2 )
% 0.30/0.63          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_iff
% 0.30/0.63  thf(fact_6703_abs__ge__minus__self,axiom,
% 0.30/0.63      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_minus_self
% 0.30/0.63  thf(fact_6704_abs__ge__minus__self,axiom,
% 0.30/0.63      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_minus_self
% 0.30/0.63  thf(fact_6705_abs__ge__minus__self,axiom,
% 0.30/0.63      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_minus_self
% 0.30/0.63  thf(fact_6706_abs__ge__minus__self,axiom,
% 0.30/0.63      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ge_minus_self
% 0.30/0.63  thf(fact_6707_abs__less__iff,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_less_real @ A @ B2 )
% 0.30/0.63          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_less_iff
% 0.30/0.63  thf(fact_6708_abs__less__iff,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_less_int @ A @ B2 )
% 0.30/0.63          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_less_iff
% 0.30/0.63  thf(fact_6709_abs__less__iff,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 0.30/0.63          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_less_iff
% 0.30/0.63  thf(fact_6710_abs__less__iff,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B2 )
% 0.30/0.63        = ( ( ord_less_rat @ A @ B2 )
% 0.30/0.63          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_less_iff
% 0.30/0.63  thf(fact_6711_sin__bound__lemma,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,U: real,V: real] :
% 0.30/0.63        ( ( X3 = Y3 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.30/0.63         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X3 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sin_bound_lemma
% 0.30/0.63  thf(fact_6712_numeral__eq__Suc,axiom,
% 0.30/0.63      ( numeral_numeral_nat
% 0.30/0.63      = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_eq_Suc
% 0.30/0.63  thf(fact_6713_sum__subtractf__nat,axiom,
% 0.30/0.63      ! [A4: set_real,G: real > nat,F: real > nat] :
% 0.30/0.63        ( ! [X5: real] :
% 0.30/0.63            ( ( member_real @ X5 @ A4 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.63       => ( ( groups1935376822645274424al_nat
% 0.30/0.63            @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.63            @ A4 )
% 0.30/0.63          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_subtractf_nat
% 0.30/0.63  thf(fact_6714_sum__subtractf__nat,axiom,
% 0.30/0.63      ! [A4: set_int,G: int > nat,F: int > nat] :
% 0.30/0.63        ( ! [X5: int] :
% 0.30/0.63            ( ( member_int @ X5 @ A4 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.63       => ( ( groups4541462559716669496nt_nat
% 0.30/0.63            @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.63            @ A4 )
% 0.30/0.63          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_subtractf_nat
% 0.30/0.63  thf(fact_6715_sum__subtractf__nat,axiom,
% 0.30/0.63      ! [A4: set_complex,G: complex > nat,F: complex > nat] :
% 0.30/0.63        ( ! [X5: complex] :
% 0.30/0.63            ( ( member_complex @ X5 @ A4 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.63       => ( ( groups5693394587270226106ex_nat
% 0.30/0.63            @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.63            @ A4 )
% 0.30/0.63          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_subtractf_nat
% 0.30/0.63  thf(fact_6716_sum__subtractf__nat,axiom,
% 0.30/0.63      ! [A4: set_nat,G: nat > nat,F: nat > nat] :
% 0.30/0.63        ( ! [X5: nat] :
% 0.30/0.63            ( ( member_nat @ X5 @ A4 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat
% 0.30/0.63            @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.63            @ A4 )
% 0.30/0.63          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_subtractf_nat
% 0.30/0.63  thf(fact_6717_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.63      ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.shift_bounds_cl_nat_ivl
% 0.30/0.63  thf(fact_6718_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.63      ! [G: nat > real,M: nat,K: nat,N: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.shift_bounds_cl_nat_ivl
% 0.30/0.63  thf(fact_6719_dense__eq0__I,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ! [E2: real] :
% 0.30/0.63            ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E2 ) )
% 0.30/0.63       => ( X3 = zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dense_eq0_I
% 0.30/0.63  thf(fact_6720_dense__eq0__I,axiom,
% 0.30/0.63      ! [X3: rat] :
% 0.30/0.63        ( ! [E2: rat] :
% 0.30/0.63            ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.63           => ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E2 ) )
% 0.30/0.63       => ( X3 = zero_zero_rat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dense_eq0_I
% 0.30/0.63  thf(fact_6721_abs__mult__pos,axiom,
% 0.30/0.63      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.30/0.63       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X3 )
% 0.30/0.63          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_pos
% 0.30/0.63  thf(fact_6722_abs__mult__pos,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X3 )
% 0.30/0.63          = ( abs_abs_real @ ( times_times_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_pos
% 0.30/0.63  thf(fact_6723_abs__mult__pos,axiom,
% 0.30/0.63      ! [X3: rat,Y3: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.63       => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X3 )
% 0.30/0.63          = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_pos
% 0.30/0.63  thf(fact_6724_abs__mult__pos,axiom,
% 0.30/0.63      ! [X3: int,Y3: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.63       => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X3 )
% 0.30/0.63          = ( abs_abs_int @ ( times_times_int @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mult_pos
% 0.30/0.63  thf(fact_6725_abs__eq__mult,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.63            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.30/0.63          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.63            | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
% 0.30/0.63       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 0.30/0.63          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_mult
% 0.30/0.63  thf(fact_6726_abs__eq__mult,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.63            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.63          & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.63            | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 0.30/0.63       => ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.63          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_mult
% 0.30/0.63  thf(fact_6727_abs__eq__mult,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.63            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.63          & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.63            | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 0.30/0.63       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.63          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_mult
% 0.30/0.63  thf(fact_6728_abs__eq__mult,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.63            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.63          & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.63            | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 0.30/0.63       => ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 0.30/0.63          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_mult
% 0.30/0.63  thf(fact_6729_abs__eq__iff_H,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( ( abs_abs_real @ A )
% 0.30/0.63          = B2 )
% 0.30/0.63        = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 0.30/0.63          & ( ( A = B2 )
% 0.30/0.63            | ( A
% 0.30/0.63              = ( uminus_uminus_real @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff'
% 0.30/0.63  thf(fact_6730_abs__eq__iff_H,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.63          = B2 )
% 0.30/0.63        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 0.30/0.63          & ( ( A = B2 )
% 0.30/0.63            | ( A
% 0.30/0.63              = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff'
% 0.30/0.63  thf(fact_6731_abs__eq__iff_H,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( ( abs_abs_rat @ A )
% 0.30/0.63          = B2 )
% 0.30/0.63        = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 0.30/0.63          & ( ( A = B2 )
% 0.30/0.63            | ( A
% 0.30/0.63              = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff'
% 0.30/0.63  thf(fact_6732_abs__eq__iff_H,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( ( abs_abs_int @ A )
% 0.30/0.63          = B2 )
% 0.30/0.63        = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.63          & ( ( A = B2 )
% 0.30/0.63            | ( A
% 0.30/0.63              = ( uminus_uminus_int @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_eq_iff'
% 0.30/0.63  thf(fact_6733_eq__abs__iff_H,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( A
% 0.30/0.63          = ( abs_abs_real @ B2 ) )
% 0.30/0.63        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.63          & ( ( B2 = A )
% 0.30/0.63            | ( B2
% 0.30/0.63              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % eq_abs_iff'
% 0.30/0.63  thf(fact_6734_eq__abs__iff_H,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( A
% 0.30/0.63          = ( abs_abs_Code_integer @ B2 ) )
% 0.30/0.63        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.63          & ( ( B2 = A )
% 0.30/0.63            | ( B2
% 0.30/0.63              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % eq_abs_iff'
% 0.30/0.63  thf(fact_6735_eq__abs__iff_H,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] :
% 0.30/0.63        ( ( A
% 0.30/0.63          = ( abs_abs_rat @ B2 ) )
% 0.30/0.63        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.63          & ( ( B2 = A )
% 0.30/0.63            | ( B2
% 0.30/0.63              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % eq_abs_iff'
% 0.30/0.63  thf(fact_6736_eq__abs__iff_H,axiom,
% 0.30/0.63      ! [A: int,B2: int] :
% 0.30/0.63        ( ( A
% 0.30/0.63          = ( abs_abs_int @ B2 ) )
% 0.30/0.63        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.63          & ( ( B2 = A )
% 0.30/0.63            | ( B2
% 0.30/0.63              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % eq_abs_iff'
% 0.30/0.63  thf(fact_6737_abs__minus__le__zero,axiom,
% 0.30/0.63      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_le_zero
% 0.30/0.63  thf(fact_6738_abs__minus__le__zero,axiom,
% 0.30/0.63      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_le_zero
% 0.30/0.63  thf(fact_6739_abs__minus__le__zero,axiom,
% 0.30/0.63      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_le_zero
% 0.30/0.63  thf(fact_6740_abs__minus__le__zero,axiom,
% 0.30/0.63      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_minus_le_zero
% 0.30/0.63  thf(fact_6741_abs__div__pos,axiom,
% 0.30/0.63      ! [Y3: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63       => ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y3 )
% 0.30/0.63          = ( abs_abs_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_div_pos
% 0.30/0.63  thf(fact_6742_abs__div__pos,axiom,
% 0.30/0.63      ! [Y3: rat,X3: rat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.63       => ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y3 )
% 0.30/0.63          = ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_div_pos
% 0.30/0.63  thf(fact_6743_zero__le__power__abs,axiom,
% 0.30/0.63      ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_power_abs
% 0.30/0.63  thf(fact_6744_zero__le__power__abs,axiom,
% 0.30/0.63      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_power_abs
% 0.30/0.63  thf(fact_6745_zero__le__power__abs,axiom,
% 0.30/0.63      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_power_abs
% 0.30/0.63  thf(fact_6746_zero__le__power__abs,axiom,
% 0.30/0.63      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_power_abs
% 0.30/0.63  thf(fact_6747_abs__if,axiom,
% 0.30/0.63      ( abs_abs_real
% 0.30/0.63      = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if
% 0.30/0.63  thf(fact_6748_abs__if,axiom,
% 0.30/0.63      ( abs_abs_int
% 0.30/0.63      = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if
% 0.30/0.63  thf(fact_6749_abs__if,axiom,
% 0.30/0.63      ( abs_abs_Code_integer
% 0.30/0.63      = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if
% 0.30/0.63  thf(fact_6750_abs__if,axiom,
% 0.30/0.63      ( abs_abs_rat
% 0.30/0.63      = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if
% 0.30/0.63  thf(fact_6751_abs__if__raw,axiom,
% 0.30/0.63      ( abs_abs_real
% 0.30/0.63      = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if_raw
% 0.30/0.63  thf(fact_6752_abs__if__raw,axiom,
% 0.30/0.63      ( abs_abs_int
% 0.30/0.63      = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if_raw
% 0.30/0.63  thf(fact_6753_abs__if__raw,axiom,
% 0.30/0.63      ( abs_abs_Code_integer
% 0.30/0.63      = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if_raw
% 0.30/0.63  thf(fact_6754_abs__if__raw,axiom,
% 0.30/0.63      ( abs_abs_rat
% 0.30/0.63      = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_if_raw
% 0.30/0.63  thf(fact_6755_abs__of__neg,axiom,
% 0.30/0.63      ! [A: real] :
% 0.30/0.63        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.63       => ( ( abs_abs_real @ A )
% 0.30/0.63          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_neg
% 0.30/0.63  thf(fact_6756_abs__of__neg,axiom,
% 0.30/0.63      ! [A: int] :
% 0.30/0.63        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.63       => ( ( abs_abs_int @ A )
% 0.30/0.63          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_neg
% 0.30/0.63  thf(fact_6757_abs__of__neg,axiom,
% 0.30/0.63      ! [A: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.63       => ( ( abs_abs_Code_integer @ A )
% 0.30/0.63          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_neg
% 0.30/0.63  thf(fact_6758_abs__of__neg,axiom,
% 0.30/0.63      ! [A: rat] :
% 0.30/0.63        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.63       => ( ( abs_abs_rat @ A )
% 0.30/0.63          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_neg
% 0.30/0.63  thf(fact_6759_abs__diff__le__iff,axiom,
% 0.30/0.63      ! [X3: code_integer,A: code_integer,R: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_le_iff
% 0.30/0.63  thf(fact_6760_abs__diff__le__iff,axiom,
% 0.30/0.63      ! [X3: real,A: real,R: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_le_iff
% 0.30/0.63  thf(fact_6761_abs__diff__le__iff,axiom,
% 0.30/0.63      ! [X3: rat,A: rat,R: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_le_iff
% 0.30/0.63  thf(fact_6762_abs__diff__le__iff,axiom,
% 0.30/0.63      ! [X3: int,A: int,R: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_le_iff
% 0.30/0.63  thf(fact_6763_abs__triangle__ineq4,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq4
% 0.30/0.63  thf(fact_6764_abs__triangle__ineq4,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq4
% 0.30/0.63  thf(fact_6765_abs__triangle__ineq4,axiom,
% 0.30/0.63      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq4
% 0.30/0.63  thf(fact_6766_abs__triangle__ineq4,axiom,
% 0.30/0.63      ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_triangle_ineq4
% 0.30/0.63  thf(fact_6767_abs__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_triangle_ineq
% 0.30/0.63  thf(fact_6768_abs__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_triangle_ineq
% 0.30/0.63  thf(fact_6769_abs__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_triangle_ineq
% 0.30/0.63  thf(fact_6770_abs__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_triangle_ineq
% 0.30/0.63  thf(fact_6771_abs__diff__less__iff,axiom,
% 0.30/0.63      ! [X3: code_integer,A: code_integer,R: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_less_iff
% 0.30/0.63  thf(fact_6772_abs__diff__less__iff,axiom,
% 0.30/0.63      ! [X3: real,A: real,R: real] :
% 0.30/0.63        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_less_iff
% 0.30/0.63  thf(fact_6773_abs__diff__less__iff,axiom,
% 0.30/0.63      ! [X3: rat,A: rat,R: rat] :
% 0.30/0.63        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_less_iff
% 0.30/0.63  thf(fact_6774_abs__diff__less__iff,axiom,
% 0.30/0.63      ! [X3: int,A: int,R: int] :
% 0.30/0.63        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R )
% 0.30/0.63        = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X3 )
% 0.30/0.63          & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_diff_less_iff
% 0.30/0.63  thf(fact_6775_sum__SucD,axiom,
% 0.30/0.63      ! [F: nat > nat,A4: set_nat,N: nat] :
% 0.30/0.63        ( ( ( groups3542108847815614940at_nat @ F @ A4 )
% 0.30/0.63          = ( suc @ N ) )
% 0.30/0.63       => ? [X5: nat] :
% 0.30/0.63            ( ( member_nat @ X5 @ A4 )
% 0.30/0.63            & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_SucD
% 0.30/0.63  thf(fact_6776_abs__real__def,axiom,
% 0.30/0.63      ( abs_abs_real
% 0.30/0.63      = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_real_def
% 0.30/0.63  thf(fact_6777_lemma__interval__lt,axiom,
% 0.30/0.63      ! [A: real,X3: real,B2: real] :
% 0.30/0.63        ( ( ord_less_real @ A @ X3 )
% 0.30/0.63       => ( ( ord_less_real @ X3 @ B2 )
% 0.30/0.63         => ? [D3: real] :
% 0.30/0.63              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.63              & ! [Y5: real] :
% 0.30/0.63                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D3 )
% 0.30/0.63                 => ( ( ord_less_real @ A @ Y5 )
% 0.30/0.63                    & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_interval_lt
% 0.30/0.63  thf(fact_6778_sum__power__add,axiom,
% 0.30/0.63      ! [X3: complex,M: nat,I5: set_nat] :
% 0.30/0.63        ( ( groups2073611262835488442omplex
% 0.30/0.63          @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.63          @ I5 )
% 0.30/0.63        = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_power_add
% 0.30/0.63  thf(fact_6779_sum__power__add,axiom,
% 0.30/0.63      ! [X3: rat,M: nat,I5: set_nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.63          @ I5 )
% 0.30/0.63        = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_power_add
% 0.30/0.63  thf(fact_6780_sum__power__add,axiom,
% 0.30/0.63      ! [X3: int,M: nat,I5: set_nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [I4: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.63          @ I5 )
% 0.30/0.63        = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_power_add
% 0.30/0.63  thf(fact_6781_sum__power__add,axiom,
% 0.30/0.63      ! [X3: real,M: nat,I5: set_nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I4: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 0.30/0.63          @ I5 )
% 0.30/0.63        = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_power_add
% 0.30/0.63  thf(fact_6782_sum_OatLeastAtMost__rev,axiom,
% 0.30/0.63      ! [G: nat > nat,N: nat,M: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeastAtMost_rev
% 0.30/0.63  thf(fact_6783_sum_OatLeastAtMost__rev,axiom,
% 0.30/0.63      ! [G: nat > real,N: nat,M: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeastAtMost_rev
% 0.30/0.63  thf(fact_6784_pred__numeral__def,axiom,
% 0.30/0.63      ( pred_numeral
% 0.30/0.63      = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % pred_numeral_def
% 0.30/0.63  thf(fact_6785_sum__nth__roots,axiom,
% 0.30/0.63      ! [N: nat,C: complex] :
% 0.30/0.63        ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.63       => ( ( groups7754918857620584856omplex
% 0.30/0.63            @ ^ [X: complex] : X
% 0.30/0.63            @ ( collect_complex
% 0.30/0.63              @ ^ [Z2: complex] :
% 0.30/0.63                  ( ( power_power_complex @ Z2 @ N )
% 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_6786_sum__roots__unity,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ one_one_nat @ N )
% 0.30/0.63       => ( ( groups7754918857620584856omplex
% 0.30/0.63            @ ^ [X: complex] : X
% 0.30/0.63            @ ( collect_complex
% 0.30/0.63              @ ^ [Z2: complex] :
% 0.30/0.63                  ( ( power_power_complex @ Z2 @ N )
% 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_6787_abs__add__one__gt__zero,axiom,
% 0.30/0.63      ! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_add_one_gt_zero
% 0.30/0.63  thf(fact_6788_abs__add__one__gt__zero,axiom,
% 0.30/0.63      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_add_one_gt_zero
% 0.30/0.63  thf(fact_6789_abs__add__one__gt__zero,axiom,
% 0.30/0.63      ! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_add_one_gt_zero
% 0.30/0.63  thf(fact_6790_abs__add__one__gt__zero,axiom,
% 0.30/0.63      ! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_add_one_gt_zero
% 0.30/0.63  thf(fact_6791_sum__diff__nat,axiom,
% 0.30/0.63      ! [B5: set_complex,A4: set_complex,F: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.63         => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_6792_sum__diff__nat,axiom,
% 0.30/0.63      ! [B5: set_int,A4: set_int,F: int > nat] :
% 0.30/0.63        ( ( finite_finite_int @ B5 )
% 0.30/0.63       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.63         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_6793_sum__diff__nat,axiom,
% 0.30/0.63      ! [B5: set_nat,A4: set_nat,F: nat > nat] :
% 0.30/0.63        ( ( finite_finite_nat @ B5 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.63         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_6794_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > rat,N: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.63        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_6795_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > int,N: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.63        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_6796_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > nat,N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.63        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_6797_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > real,N: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.63        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_6798_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_6799_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_6800_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_6801_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_6802_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_6803_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_6804_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_6805_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_6806_lemma__interval,axiom,
% 0.30/0.63      ! [A: real,X3: real,B2: real] :
% 0.30/0.63        ( ( ord_less_real @ A @ X3 )
% 0.30/0.63       => ( ( ord_less_real @ X3 @ B2 )
% 0.30/0.63         => ? [D3: real] :
% 0.30/0.63              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.63              & ! [Y5: real] :
% 0.30/0.63                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D3 )
% 0.30/0.63                 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.30/0.63                    & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_interval
% 0.30/0.63  thf(fact_6807_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ M )
% 0.30/0.63            @ ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_6808_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ M )
% 0.30/0.63            @ ( groups3539618377306564664at_int
% 0.30/0.63              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_6809_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ M )
% 0.30/0.63            @ ( groups3542108847815614940at_nat
% 0.30/0.63              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_6810_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ M )
% 0.30/0.63            @ ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_6811_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > complex] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_complex @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_6812_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_6813_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_6814_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.63       => ( ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_6815_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_6816_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_6817_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_6818_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_6819_abs__square__eq__1,axiom,
% 0.30/0.63      ! [X3: code_integer] :
% 0.30/0.63        ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          = one_one_Code_integer )
% 0.30/0.63        = ( ( abs_abs_Code_integer @ X3 )
% 0.30/0.63          = one_one_Code_integer ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_eq_1
% 0.30/0.63  thf(fact_6820_abs__square__eq__1,axiom,
% 0.30/0.63      ! [X3: rat] :
% 0.30/0.63        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          = one_one_rat )
% 0.30/0.63        = ( ( abs_abs_rat @ X3 )
% 0.30/0.63          = one_one_rat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_eq_1
% 0.30/0.63  thf(fact_6821_abs__square__eq__1,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          = one_one_real )
% 0.30/0.63        = ( ( abs_abs_real @ X3 )
% 0.30/0.63          = one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_eq_1
% 0.30/0.63  thf(fact_6822_abs__square__eq__1,axiom,
% 0.30/0.63      ! [X3: int] :
% 0.30/0.63        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          = one_one_int )
% 0.30/0.63        = ( ( abs_abs_int @ X3 )
% 0.30/0.63          = one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_eq_1
% 0.30/0.63  thf(fact_6823_power__even__abs,axiom,
% 0.30/0.63      ! [N: nat,A: code_integer] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.30/0.63          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs
% 0.30/0.63  thf(fact_6824_power__even__abs,axiom,
% 0.30/0.63      ! [N: nat,A: rat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 0.30/0.63          = ( power_power_rat @ A @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs
% 0.30/0.63  thf(fact_6825_power__even__abs,axiom,
% 0.30/0.63      ! [N: nat,A: real] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.30/0.63          = ( power_power_real @ A @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs
% 0.30/0.63  thf(fact_6826_power__even__abs,axiom,
% 0.30/0.63      ! [N: nat,A: int] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.30/0.63          = ( power_power_int @ A @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_even_abs
% 0.30/0.63  thf(fact_6827_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > rat,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_6828_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > int,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_6829_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > nat,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_6830_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N: nat,G: nat > real,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_6831_divmod__int__def,axiom,
% 0.30/0.63      ( unique5052692396658037445od_int
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_int_def
% 0.30/0.63  thf(fact_6832_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_6833_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_6834_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_6835_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_6836_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: code_integer > code_integer > $o,X3: code_integer] :
% 0.30/0.63        ( ! [X5: code_integer] :
% 0.30/0.63            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
% 0.30/0.63           => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( 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_6837_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: real > real > $o,X3: real] :
% 0.30/0.63        ( ! [X5: real] :
% 0.30/0.63            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.63           => ( P @ X5 @ ( power_power_real @ X5 @ ( 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_6838_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: rat > rat > $o,X3: rat] :
% 0.30/0.63        ( ! [X5: rat] :
% 0.30/0.63            ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
% 0.30/0.63           => ( P @ X5 @ ( power_power_rat @ X5 @ ( 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_6839_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: int > int > $o,X3: int] :
% 0.30/0.63        ( ! [X5: int] :
% 0.30/0.63            ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.30/0.63           => ( P @ X5 @ ( power_power_int @ X5 @ ( 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_6840_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_6841_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_6842_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_6843_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_6844_divmod__def,axiom,
% 0.30/0.63      ( unique5052692396658037445od_int
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_def
% 0.30/0.63  thf(fact_6845_divmod__def,axiom,
% 0.30/0.63      ( unique5055182867167087721od_nat
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_def
% 0.30/0.63  thf(fact_6846_divmod__def,axiom,
% 0.30/0.63      ( unique3479559517661332726nteger
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_def
% 0.30/0.63  thf(fact_6847_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_6848_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_6849_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_6850_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_6851_power__mono__even,axiom,
% 0.30/0.63      ! [N: nat,A: code_integer,B2: code_integer] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) )
% 0.30/0.63         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_6852_power__mono__even,axiom,
% 0.30/0.63      ! [N: nat,A: real,B2: real] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
% 0.30/0.63         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_6853_power__mono__even,axiom,
% 0.30/0.63      ! [N: nat,A: rat,B2: rat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) )
% 0.30/0.63         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_6854_power__mono__even,axiom,
% 0.30/0.63      ! [N: nat,A: int,B2: int] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
% 0.30/0.63         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_6855_divmod_H__nat__def,axiom,
% 0.30/0.63      ( unique5055182867167087721od_nat
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod'_nat_def
% 0.30/0.63  thf(fact_6856_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B2: code_integer,Delta: code_integer] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( member_nat @ I3 @ I5 )
% 0.30/0.63           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups7501900531339628137nteger @ X3 @ I5 )
% 0.30/0.63            = one_one_Code_integer )
% 0.30/0.63         => ( ! [I3: nat] :
% 0.30/0.63                ( ( member_nat @ I3 @ I5 )
% 0.30/0.63               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_le3102999989581377725nteger
% 0.30/0.63              @ ( abs_abs_Code_integer
% 0.30/0.63                @ ( minus_8373710615458151222nteger
% 0.30/0.63                  @ ( groups7501900531339628137nteger
% 0.30/0.63                    @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6857_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B2: code_integer,Delta: code_integer] :
% 0.30/0.63        ( ! [I3: real] :
% 0.30/0.63            ( ( member_real @ I3 @ I5 )
% 0.30/0.63           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups7713935264441627589nteger @ X3 @ I5 )
% 0.30/0.63            = one_one_Code_integer )
% 0.30/0.63         => ( ! [I3: real] :
% 0.30/0.63                ( ( member_real @ I3 @ I5 )
% 0.30/0.63               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_le3102999989581377725nteger
% 0.30/0.63              @ ( abs_abs_Code_integer
% 0.30/0.63                @ ( minus_8373710615458151222nteger
% 0.30/0.63                  @ ( groups7713935264441627589nteger
% 0.30/0.63                    @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6858_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B2: code_integer,Delta: code_integer] :
% 0.30/0.63        ( ! [I3: int] :
% 0.30/0.63            ( ( member_int @ I3 @ I5 )
% 0.30/0.63           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups7873554091576472773nteger @ X3 @ I5 )
% 0.30/0.63            = one_one_Code_integer )
% 0.30/0.63         => ( ! [I3: int] :
% 0.30/0.63                ( ( member_int @ I3 @ I5 )
% 0.30/0.63               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_le3102999989581377725nteger
% 0.30/0.63              @ ( abs_abs_Code_integer
% 0.30/0.63                @ ( minus_8373710615458151222nteger
% 0.30/0.63                  @ ( groups7873554091576472773nteger
% 0.30/0.63                    @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6859_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_complex,X3: complex > code_integer,A: complex > code_integer,B2: code_integer,Delta: code_integer] :
% 0.30/0.63        ( ! [I3: complex] :
% 0.30/0.63            ( ( member_complex @ I3 @ I5 )
% 0.30/0.63           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups6621422865394947399nteger @ X3 @ I5 )
% 0.30/0.63            = one_one_Code_integer )
% 0.30/0.63         => ( ! [I3: complex] :
% 0.30/0.63                ( ( member_complex @ I3 @ I5 )
% 0.30/0.63               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_le3102999989581377725nteger
% 0.30/0.63              @ ( abs_abs_Code_integer
% 0.30/0.63                @ ( minus_8373710615458151222nteger
% 0.30/0.63                  @ ( groups6621422865394947399nteger
% 0.30/0.63                    @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6860_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_real,X3: real > real,A: real > real,B2: real,Delta: real] :
% 0.30/0.63        ( ! [I3: real] :
% 0.30/0.63            ( ( member_real @ I3 @ I5 )
% 0.30/0.63           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups8097168146408367636l_real @ X3 @ I5 )
% 0.30/0.63            = one_one_real )
% 0.30/0.63         => ( ! [I3: real] :
% 0.30/0.63                ( ( member_real @ I3 @ I5 )
% 0.30/0.63               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_less_eq_real
% 0.30/0.63              @ ( abs_abs_real
% 0.30/0.63                @ ( minus_minus_real
% 0.30/0.63                  @ ( groups8097168146408367636l_real
% 0.30/0.63                    @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6861_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_int,X3: int > real,A: int > real,B2: real,Delta: real] :
% 0.30/0.63        ( ! [I3: int] :
% 0.30/0.63            ( ( member_int @ I3 @ I5 )
% 0.30/0.63           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups8778361861064173332t_real @ X3 @ I5 )
% 0.30/0.63            = one_one_real )
% 0.30/0.63         => ( ! [I3: int] :
% 0.30/0.63                ( ( member_int @ I3 @ I5 )
% 0.30/0.63               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_less_eq_real
% 0.30/0.63              @ ( abs_abs_real
% 0.30/0.63                @ ( minus_minus_real
% 0.30/0.63                  @ ( groups8778361861064173332t_real
% 0.30/0.63                    @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6862_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_complex,X3: complex > real,A: complex > real,B2: real,Delta: real] :
% 0.30/0.63        ( ! [I3: complex] :
% 0.30/0.63            ( ( member_complex @ I3 @ I5 )
% 0.30/0.63           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups5808333547571424918x_real @ X3 @ I5 )
% 0.30/0.63            = one_one_real )
% 0.30/0.63         => ( ! [I3: complex] :
% 0.30/0.63                ( ( member_complex @ I3 @ I5 )
% 0.30/0.63               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_less_eq_real
% 0.30/0.63              @ ( abs_abs_real
% 0.30/0.63                @ ( minus_minus_real
% 0.30/0.63                  @ ( groups5808333547571424918x_real
% 0.30/0.63                    @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6863_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_nat,X3: nat > rat,A: nat > rat,B2: rat,Delta: rat] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( member_nat @ I3 @ I5 )
% 0.30/0.63           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups2906978787729119204at_rat @ X3 @ I5 )
% 0.30/0.63            = one_one_rat )
% 0.30/0.63         => ( ! [I3: nat] :
% 0.30/0.63                ( ( member_nat @ I3 @ I5 )
% 0.30/0.63               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_less_eq_rat
% 0.30/0.63              @ ( abs_abs_rat
% 0.30/0.63                @ ( minus_minus_rat
% 0.30/0.63                  @ ( groups2906978787729119204at_rat
% 0.30/0.63                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6864_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_real,X3: real > rat,A: real > rat,B2: rat,Delta: rat] :
% 0.30/0.63        ( ! [I3: real] :
% 0.30/0.63            ( ( member_real @ I3 @ I5 )
% 0.30/0.63           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups1300246762558778688al_rat @ X3 @ I5 )
% 0.30/0.63            = one_one_rat )
% 0.30/0.63         => ( ! [I3: real] :
% 0.30/0.63                ( ( member_real @ I3 @ I5 )
% 0.30/0.63               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_less_eq_rat
% 0.30/0.63              @ ( abs_abs_rat
% 0.30/0.63                @ ( minus_minus_rat
% 0.30/0.63                  @ ( groups1300246762558778688al_rat
% 0.30/0.63                    @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6865_convex__sum__bound__le,axiom,
% 0.30/0.63      ! [I5: set_int,X3: int > rat,A: int > rat,B2: rat,Delta: rat] :
% 0.30/0.63        ( ! [I3: int] :
% 0.30/0.63            ( ( member_int @ I3 @ I5 )
% 0.30/0.63           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 0.30/0.63       => ( ( ( groups3906332499630173760nt_rat @ X3 @ I5 )
% 0.30/0.63            = one_one_rat )
% 0.30/0.63         => ( ! [I3: int] :
% 0.30/0.63                ( ( member_int @ I3 @ I5 )
% 0.30/0.63               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 0.30/0.63           => ( ord_less_eq_rat
% 0.30/0.63              @ ( abs_abs_rat
% 0.30/0.63                @ ( minus_minus_rat
% 0.30/0.63                  @ ( groups3906332499630173760nt_rat
% 0.30/0.63                    @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 0.30/0.63                    @ I5 )
% 0.30/0.63                  @ B2 ) )
% 0.30/0.63              @ Delta ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % convex_sum_bound_le
% 0.30/0.63  thf(fact_6866_dbl__dec__def,axiom,
% 0.30/0.63      ( neg_nu6511756317524482435omplex
% 0.30/0.63      = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_def
% 0.30/0.63  thf(fact_6867_dbl__dec__def,axiom,
% 0.30/0.63      ( neg_nu6075765906172075777c_real
% 0.30/0.63      = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_def
% 0.30/0.63  thf(fact_6868_dbl__dec__def,axiom,
% 0.30/0.63      ( neg_nu3179335615603231917ec_rat
% 0.30/0.63      = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_def
% 0.30/0.63  thf(fact_6869_dbl__dec__def,axiom,
% 0.30/0.63      ( neg_nu3811975205180677377ec_int
% 0.30/0.63      = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_def
% 0.30/0.63  thf(fact_6870_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > complex] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_6871_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > rat] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_6872_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > int] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups3539618377306564664at_int
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups3539618377306564664at_int
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_6873_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > real] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_6874_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > complex] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.63          = ( minus_minus_complex @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_6875_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.63          = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_6876_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.63          = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_6877_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N: nat,F: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.63          = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_6878_mask__eq__sum__exp,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ 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            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mask_eq_sum_exp
% 0.30/0.63  thf(fact_6879_mask__eq__sum__exp,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ 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            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mask_eq_sum_exp
% 0.30/0.63  thf(fact_6880_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N: nat,X3: complex] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.63          = ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_6881_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N: nat,X3: rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.63          = ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_6882_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N: nat,X3: int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.63          = ( minus_minus_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_6883_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N: nat,X3: real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.63       => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.30/0.63          = ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_6884_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > rat,M: nat,N: 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 ) ) @ N ) ) ) )
% 0.30/0.63        = ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_6885_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > int,M: nat,N: 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 ) ) @ N ) ) ) )
% 0.30/0.63        = ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_6886_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > nat,M: nat,N: 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 ) ) @ N ) ) ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_6887_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > real,M: nat,N: 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 ) ) @ N ) ) ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_6888_mask__eq__sum__exp__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( 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            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mask_eq_sum_exp_nat
% 0.30/0.63  thf(fact_6889_gauss__sum__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [X: nat] : X
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum_nat
% 0.30/0.63  thf(fact_6890_arith__series__nat,axiom,
% 0.30/0.63      ! [A: nat,D: nat,N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arith_series_nat
% 0.30/0.63  thf(fact_6891_Sum__Icc__nat,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [X: nat] : X
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % Sum_Icc_nat
% 0.30/0.63  thf(fact_6892_divmod__divmod__step,axiom,
% 0.30/0.63      ( unique5055182867167087721od_nat
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_divmod_step
% 0.30/0.63  thf(fact_6893_divmod__divmod__step,axiom,
% 0.30/0.63      ( unique5052692396658037445od_int
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_divmod_step
% 0.30/0.63  thf(fact_6894_divmod__divmod__step,axiom,
% 0.30/0.63      ( unique3479559517661332726nteger
% 0.30/0.63      = ( ^ [M3: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_divmod_step
% 0.30/0.63  thf(fact_6895_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.63         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.30/0.63  thf(fact_6896_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.63       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_ln_one_plus_x_minus_x_bound
% 0.30/0.63  thf(fact_6897_divmod__nat__if,axiom,
% 0.30/0.63      ( divmod_nat
% 0.30/0.63      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.63            ( if_Pro6206227464963214023at_nat
% 0.30/0.63            @ ( ( N2 = zero_zero_nat )
% 0.30/0.63              | ( ord_less_nat @ M3 @ N2 ) )
% 0.30/0.63            @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 0.30/0.63            @ ( produc2626176000494625587at_nat
% 0.30/0.63              @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 0.30/0.63              @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_nat_if
% 0.30/0.63  thf(fact_6898_tanh__ln__real,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( tanh_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.63          = ( 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 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_ln_real
% 0.30/0.63  thf(fact_6899_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_6900_dbl__inc__simps_I3_J,axiom,
% 0.30/0.63      ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.30/0.63      = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(3)
% 0.30/0.63  thf(fact_6901_dbl__inc__simps_I3_J,axiom,
% 0.30/0.63      ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.30/0.63      = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(3)
% 0.30/0.63  thf(fact_6902_dbl__inc__simps_I3_J,axiom,
% 0.30/0.63      ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.30/0.63      = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(3)
% 0.30/0.63  thf(fact_6903_dbl__inc__simps_I3_J,axiom,
% 0.30/0.63      ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.30/0.63      = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(3)
% 0.30/0.63  thf(fact_6904_sum__gp,axiom,
% 0.30/0.63      ! [N: nat,M: nat,X3: complex] :
% 0.30/0.63        ( ( ( ord_less_nat @ N @ M )
% 0.30/0.63         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_complex ) )
% 0.30/0.63        & ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.63         => ( ( ( X3 = one_one_complex )
% 0.30/0.63             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63                = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ 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 @ N ) )
% 0.30/0.63                = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp
% 0.30/0.63  thf(fact_6905_sum__gp,axiom,
% 0.30/0.63      ! [N: nat,M: nat,X3: rat] :
% 0.30/0.63        ( ( ( ord_less_nat @ N @ M )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_rat ) )
% 0.30/0.63        & ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.63         => ( ( ( X3 = one_one_rat )
% 0.30/0.63             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63                = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ 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 @ N ) )
% 0.30/0.63                = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp
% 0.30/0.63  thf(fact_6906_sum__gp,axiom,
% 0.30/0.63      ! [N: nat,M: nat,X3: real] :
% 0.30/0.63        ( ( ( ord_less_nat @ N @ M )
% 0.30/0.63         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63            = zero_zero_real ) )
% 0.30/0.63        & ( ~ ( ord_less_nat @ N @ M )
% 0.30/0.63         => ( ( ( X3 = one_one_real )
% 0.30/0.63             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.63                = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ 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 @ N ) )
% 0.30/0.63                = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp
% 0.30/0.63  thf(fact_6907_of__nat__eq__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( M = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_iff
% 0.30/0.63  thf(fact_6908_of__nat__eq__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.63          = ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( M = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_iff
% 0.30/0.63  thf(fact_6909_of__nat__eq__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ M )
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( M = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_iff
% 0.30/0.63  thf(fact_6910_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_6911_abs__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.30/0.63        = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_6912_abs__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63        = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_6913_abs__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_6914_abs__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_6915_negative__zle,axiom,
% 0.30/0.63      ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zle
% 0.30/0.63  thf(fact_6916_tanh__real__less__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
% 0.30/0.63        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_less_iff
% 0.30/0.63  thf(fact_6917_tanh__real__le__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_le_iff
% 0.30/0.63  thf(fact_6918_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_6919_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_6920_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_6921_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_6922_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_6923_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( zero_zero_complex
% 0.30/0.63          = ( semiri8010041392384452111omplex @ N ) )
% 0.30/0.63        = ( zero_zero_nat = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_6924_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( zero_zero_rat
% 0.30/0.63          = ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63        = ( zero_zero_nat = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_6925_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( zero_zero_int
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( zero_zero_nat = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_6926_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( zero_zero_real
% 0.30/0.63          = ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( zero_zero_nat = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_6927_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( zero_zero_nat
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( zero_zero_nat = N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_6928_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_6929_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_6930_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_6931_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_6932_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_6933_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_6934_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_6935_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_6936_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_6937_of__nat__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63        = ( numera6690914467698888265omplex @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_numeral
% 0.30/0.63  thf(fact_6938_of__nat__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63        = ( numeral_numeral_rat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_numeral
% 0.30/0.63  thf(fact_6939_of__nat__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63        = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_numeral
% 0.30/0.63  thf(fact_6940_of__nat__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63        = ( numeral_numeral_real @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_numeral
% 0.30/0.63  thf(fact_6941_of__nat__numeral,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63        = ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_numeral
% 0.30/0.63  thf(fact_6942_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_6943_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_6944_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_6945_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_6946_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.63        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_6947_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_6948_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.63        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_6949_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.63        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_6950_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.30/0.63        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_6951_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_6952_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.30/0.63        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_6953_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.30/0.63        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_6954_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ( semiri8010041392384452111omplex @ N )
% 0.30/0.63          = one_one_complex )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_6955_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ( semiri681578069525770553at_rat @ N )
% 0.30/0.63          = one_one_rat )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_6956_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ N )
% 0.30/0.63          = one_one_int )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_6957_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ N )
% 0.30/0.63          = one_one_real )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_6958_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ N )
% 0.30/0.63          = one_one_nat )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_6959_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( one_one_complex
% 0.30/0.63          = ( semiri8010041392384452111omplex @ N ) )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_6960_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( one_one_rat
% 0.30/0.63          = ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_6961_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( one_one_int
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_6962_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( one_one_real
% 0.30/0.63          = ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_6963_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( one_one_nat
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( N = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_6964_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_6965_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_6966_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_6967_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_6968_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_6969_of__nat__power,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.30/0.63        = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power
% 0.30/0.63  thf(fact_6970_of__nat__power,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.30/0.63        = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power
% 0.30/0.63  thf(fact_6971_of__nat__power,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.30/0.63        = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power
% 0.30/0.63  thf(fact_6972_of__nat__power,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.30/0.63        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power
% 0.30/0.63  thf(fact_6973_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
% 0.30/0.63          = ( semiri8010041392384452111omplex @ X3 ) )
% 0.30/0.63        = ( ( power_power_nat @ B2 @ W )
% 0.30/0.63          = X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_6974_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
% 0.30/0.63          = ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ( power_power_nat @ B2 @ W )
% 0.30/0.63          = X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_6975_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
% 0.30/0.63          = ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ( power_power_nat @ B2 @ W )
% 0.30/0.63          = X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_6976_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ( power_power_nat @ B2 @ W )
% 0.30/0.63          = X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_6977_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ( semiri8010041392384452111omplex @ X3 )
% 0.30/0.63          = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
% 0.30/0.63        = ( X3
% 0.30/0.63          = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_6978_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ X3 )
% 0.30/0.63          = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 0.30/0.63        = ( X3
% 0.30/0.63          = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_6979_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ X3 )
% 0.30/0.63          = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 0.30/0.63        = ( X3
% 0.30/0.63          = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_6980_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ X3 )
% 0.30/0.63          = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 0.30/0.63        = ( X3
% 0.30/0.63          = ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_6981_negative__zless,axiom,
% 0.30/0.63      ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zless
% 0.30/0.63  thf(fact_6982_zabs__less__one__iff,axiom,
% 0.30/0.63      ! [Z: int] :
% 0.30/0.63        ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.30/0.63        = ( Z = zero_zero_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zabs_less_one_iff
% 0.30/0.63  thf(fact_6983_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_6984_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_6985_zero__le__arctan__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_arctan_iff
% 0.30/0.63  thf(fact_6986_arctan__le__zero__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( arctan @ X3 ) @ zero_zero_real )
% 0.30/0.63        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_le_zero_iff
% 0.30/0.63  thf(fact_6987_tanh__real__pos__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
% 0.30/0.63        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_pos_iff
% 0.30/0.63  thf(fact_6988_tanh__real__neg__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
% 0.30/0.63        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_neg_iff
% 0.30/0.63  thf(fact_6989_tanh__real__nonpos__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ zero_zero_real )
% 0.30/0.63        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_nonpos_iff
% 0.30/0.63  thf(fact_6990_tanh__real__nonneg__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_nonneg_iff
% 0.30/0.63  thf(fact_6991_dbl__inc__simps_I2_J,axiom,
% 0.30/0.63      ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.30/0.63      = one_one_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(2)
% 0.30/0.63  thf(fact_6992_dbl__inc__simps_I2_J,axiom,
% 0.30/0.63      ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.30/0.63      = one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(2)
% 0.30/0.63  thf(fact_6993_dbl__inc__simps_I2_J,axiom,
% 0.30/0.63      ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.30/0.63      = one_one_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(2)
% 0.30/0.63  thf(fact_6994_dbl__inc__simps_I2_J,axiom,
% 0.30/0.63      ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.30/0.63      = one_one_int ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(2)
% 0.30/0.63  thf(fact_6995_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_6996_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_6997_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_6998_of__nat__of__bool,axiom,
% 0.30/0.63      ! [P: $o] :
% 0.30/0.63        ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63        = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_of_bool
% 0.30/0.63  thf(fact_6999_dbl__inc__simps_I4_J,axiom,
% 0.30/0.63      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.63      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(4)
% 0.30/0.63  thf(fact_7000_dbl__inc__simps_I4_J,axiom,
% 0.30/0.63      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.63      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(4)
% 0.30/0.63  thf(fact_7001_dbl__inc__simps_I4_J,axiom,
% 0.30/0.63      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.63      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(4)
% 0.30/0.63  thf(fact_7002_dbl__inc__simps_I4_J,axiom,
% 0.30/0.63      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.63      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(4)
% 0.30/0.63  thf(fact_7003_dbl__inc__simps_I4_J,axiom,
% 0.30/0.63      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.63      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(4)
% 0.30/0.63  thf(fact_7004_dbl__inc__simps_I5_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.30/0.63        = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(5)
% 0.30/0.63  thf(fact_7005_dbl__inc__simps_I5_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.30/0.63        = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(5)
% 0.30/0.63  thf(fact_7006_dbl__inc__simps_I5_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.63        = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(5)
% 0.30/0.63  thf(fact_7007_dbl__inc__simps_I5_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.63        = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(5)
% 0.30/0.63  thf(fact_7008_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_7009_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_7010_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_7011_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_7012_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_7013_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_7014_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_7015_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_7016_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_7017_numeral__less__real__of__nat__iff,axiom,
% 0.30/0.63      ! [W: num,N: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_less_real_of_nat_iff
% 0.30/0.63  thf(fact_7018_real__of__nat__less__numeral__iff,axiom,
% 0.30/0.63      ! [N: nat,W: num] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.30/0.63        = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_less_numeral_iff
% 0.30/0.63  thf(fact_7019_numeral__le__real__of__nat__iff,axiom,
% 0.30/0.63      ! [N: num,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_le_real_of_nat_iff
% 0.30/0.63  thf(fact_7020_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7021_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7022_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7023_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7024_dbl__inc__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.30/0.63        = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(1)
% 0.30/0.63  thf(fact_7025_dbl__inc__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.63        = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(1)
% 0.30/0.63  thf(fact_7026_dbl__inc__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.30/0.63        = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(1)
% 0.30/0.63  thf(fact_7027_dbl__inc__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.30/0.63        = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(1)
% 0.30/0.63  thf(fact_7028_dbl__inc__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.63        = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_simps(1)
% 0.30/0.63  thf(fact_7029_dbl__dec__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.30/0.63        = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_simps(1)
% 0.30/0.63  thf(fact_7030_dbl__dec__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.63        = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_simps(1)
% 0.30/0.63  thf(fact_7031_dbl__dec__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.30/0.63        = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_simps(1)
% 0.30/0.63  thf(fact_7032_dbl__dec__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.30/0.63        = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_simps(1)
% 0.30/0.63  thf(fact_7033_dbl__dec__simps_I1_J,axiom,
% 0.30/0.63      ! [K: num] :
% 0.30/0.63        ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.63        = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_dec_simps(1)
% 0.30/0.63  thf(fact_7034_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7035_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7036_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7037_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7038_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7039_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7040_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7041_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7042_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [Y3: nat,X3: num,N: nat] :
% 0.30/0.63        ( ( ( semiri8010041392384452111omplex @ Y3 )
% 0.30/0.63          = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N ) )
% 0.30/0.63        = ( Y3
% 0.30/0.63          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7043_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [Y3: nat,X3: num,N: nat] :
% 0.30/0.63        ( ( ( semiri681578069525770553at_rat @ Y3 )
% 0.30/0.63          = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
% 0.30/0.63        = ( Y3
% 0.30/0.63          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7044_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [Y3: nat,X3: num,N: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ Y3 )
% 0.30/0.63          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
% 0.30/0.63        = ( Y3
% 0.30/0.63          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7045_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [Y3: nat,X3: num,N: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ Y3 )
% 0.30/0.63          = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
% 0.30/0.63        = ( Y3
% 0.30/0.63          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7046_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [Y3: nat,X3: num,N: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 0.30/0.63          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 0.30/0.63        = ( Y3
% 0.30/0.63          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_eq_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7047_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: num,N: nat,Y3: nat] :
% 0.30/0.63        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N )
% 0.30/0.63          = ( semiri8010041392384452111omplex @ Y3 ) )
% 0.30/0.63        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.63          = Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7048_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: num,N: nat,Y3: nat] :
% 0.30/0.63        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N )
% 0.30/0.63          = ( semiri681578069525770553at_rat @ Y3 ) )
% 0.30/0.63        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.63          = Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7049_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: num,N: nat,Y3: nat] :
% 0.30/0.63        ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
% 0.30/0.63          = ( semiri1314217659103216013at_int @ Y3 ) )
% 0.30/0.63        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.63          = Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7050_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: num,N: nat,Y3: nat] :
% 0.30/0.63        ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N )
% 0.30/0.63          = ( semiri5074537144036343181t_real @ Y3 ) )
% 0.30/0.63        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.63          = Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7051_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: num,N: nat,Y3: nat] :
% 0.30/0.63        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ Y3 ) )
% 0.30/0.63        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.63          = Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_eq_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7052_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7053_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7054_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7055_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B2: nat,W: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7056_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7057_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7058_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7059_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B2: nat,W: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7060_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7061_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7062_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7063_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7064_even__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.30/0.63        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % even_of_nat
% 0.30/0.63  thf(fact_7065_even__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % even_of_nat
% 0.30/0.63  thf(fact_7066_even__of__nat,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % even_of_nat
% 0.30/0.63  thf(fact_7067_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7068_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7069_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7070_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7071_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7072_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7073_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7074_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7075_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7076_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7077_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7078_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7079_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7080_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7081_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7082_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7083_Collect__case__prod__mono,axiom,
% 0.30/0.63      ! [A4: int > int > $o,B5: int > int > $o] :
% 0.30/0.63        ( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
% 0.30/0.63       => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % Collect_case_prod_mono
% 0.30/0.63  thf(fact_7084_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_7085_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_7086_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_7087_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_7088_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_7089_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_7090_arctan__monotone_H,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.63       => ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_monotone'
% 0.30/0.63  thf(fact_7091_arctan__le__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_le_iff
% 0.30/0.63  thf(fact_7092_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7093_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7094_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7095_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7096_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_7097_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_7098_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_7099_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_7100_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 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_7101_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 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_7102_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 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_7103_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 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_7104_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 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_7105_div__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: int,M: nat,N: nat] :
% 0.30/0.63        ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.63        = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % div_mult2_eq'
% 0.30/0.63  thf(fact_7106_div__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: nat,M: nat,N: nat] :
% 0.30/0.63        ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.30/0.63        = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % div_mult2_eq'
% 0.30/0.63  thf(fact_7107_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N )
% 0.30/0.63       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7108_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N )
% 0.30/0.63       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7109_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N )
% 0.30/0.63       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7110_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N )
% 0.30/0.63       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7111_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7112_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7113_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7114_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7115_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_7116_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_7117_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_7118_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_7119_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.30/0.63  thf(fact_7120_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.30/0.63  thf(fact_7121_of__nat__dvd__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.30/0.63        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_dvd_iff
% 0.30/0.63  thf(fact_7122_of__nat__dvd__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_dvd_iff
% 0.30/0.63  thf(fact_7123_of__nat__dvd__iff,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.30/0.63        = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_dvd_iff
% 0.30/0.63  thf(fact_7124_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_7125_int__ops_I3_J,axiom,
% 0.30/0.63      ! [N: num] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.30/0.63        = ( numeral_numeral_int @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(3)
% 0.30/0.63  thf(fact_7126_abs__zmult__eq__1,axiom,
% 0.30/0.63      ! [M: int,N: int] :
% 0.30/0.63        ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.30/0.63          = one_one_int )
% 0.30/0.63       => ( ( abs_abs_int @ M )
% 0.30/0.63          = one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_zmult_eq_1
% 0.30/0.63  thf(fact_7127_nat__int__comparison_I2_J,axiom,
% 0.30/0.63      ( ord_less_nat
% 0.30/0.63      = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_int_comparison(2)
% 0.30/0.63  thf(fact_7128_zle__int,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zle_int
% 0.30/0.63  thf(fact_7129_nat__int__comparison_I3_J,axiom,
% 0.30/0.63      ( ord_less_eq_nat
% 0.30/0.63      = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_int_comparison(3)
% 0.30/0.63  thf(fact_7130_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_7131_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_7132_of__nat__mod,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.63        = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mod
% 0.30/0.63  thf(fact_7133_of__nat__mod,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.63        = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mod
% 0.30/0.63  thf(fact_7134_of__nat__mod,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.30/0.63        = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mod
% 0.30/0.63  thf(fact_7135_zadd__int__left,axiom,
% 0.30/0.63      ! [M: nat,N: nat,Z: int] :
% 0.30/0.63        ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zadd_int_left
% 0.30/0.63  thf(fact_7136_int__plus,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_plus
% 0.30/0.63  thf(fact_7137_int__ops_I5_J,axiom,
% 0.30/0.63      ! [A: nat,B2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(5)
% 0.30/0.63  thf(fact_7138_int__ops_I2_J,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  % int_ops(2)
% 0.30/0.63  thf(fact_7139_int__ops_I7_J,axiom,
% 0.30/0.63      ! [A: nat,B2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(7)
% 0.30/0.63  thf(fact_7140_zle__iff__zadd,axiom,
% 0.30/0.63      ( ord_less_eq_int
% 0.30/0.63      = ( ^ [W3: int,Z2: int] :
% 0.30/0.63          ? [N2: nat] :
% 0.30/0.63            ( Z2
% 0.30/0.63            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zle_iff_zadd
% 0.30/0.63  thf(fact_7141_not__int__zless__negative,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % not_int_zless_negative
% 0.30/0.63  thf(fact_7142_zdiv__int,axiom,
% 0.30/0.63      ! [A: nat,B2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
% 0.30/0.63        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zdiv_int
% 0.30/0.63  thf(fact_7143_of__nat__max,axiom,
% 0.30/0.63      ! [X3: nat,Y3: nat] :
% 0.30/0.63        ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X3 @ Y3 ) )
% 0.30/0.63        = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X3 ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_max
% 0.30/0.63  thf(fact_7144_of__nat__max,axiom,
% 0.30/0.63      ! [X3: nat,Y3: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X3 @ Y3 ) )
% 0.30/0.63        = ( ord_max_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_max
% 0.30/0.63  thf(fact_7145_of__nat__max,axiom,
% 0.30/0.63      ! [X3: nat,Y3: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X3 @ Y3 ) )
% 0.30/0.63        = ( ord_max_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_max
% 0.30/0.63  thf(fact_7146_of__nat__max,axiom,
% 0.30/0.63      ! [X3: nat,Y3: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X3 @ Y3 ) )
% 0.30/0.63        = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_max
% 0.30/0.63  thf(fact_7147_tanh__real__lt__1,axiom,
% 0.30/0.63      ! [X3: real] : ( ord_less_real @ ( tanh_real @ X3 ) @ one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_lt_1
% 0.30/0.63  thf(fact_7148_nat__less__as__int,axiom,
% 0.30/0.63      ( ord_less_nat
% 0.30/0.63      = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_less_as_int
% 0.30/0.63  thf(fact_7149_nat__leq__as__int,axiom,
% 0.30/0.63      ( ord_less_eq_nat
% 0.30/0.63      = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_leq_as_int
% 0.30/0.63  thf(fact_7150_of__nat__diff,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7151_of__nat__diff,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7152_of__nat__diff,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7153_of__nat__diff,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7154_of__nat__diff,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.30/0.63          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7155_zabs__def,axiom,
% 0.30/0.63      ( abs_abs_int
% 0.30/0.63      = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zabs_def
% 0.30/0.63  thf(fact_7156_reals__Archimedean3,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ! [Y5: real] :
% 0.30/0.63          ? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % reals_Archimedean3
% 0.30/0.63  thf(fact_7157_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_7158_dvd__imp__le__int,axiom,
% 0.30/0.63      ! [I: int,D: int] :
% 0.30/0.63        ( ( I != zero_zero_int )
% 0.30/0.63       => ( ( dvd_dvd_int @ D @ I )
% 0.30/0.63         => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dvd_imp_le_int
% 0.30/0.63  thf(fact_7159_real__of__nat__div4,axiom,
% 0.30/0.63      ! [N: nat,X3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_div4
% 0.30/0.63  thf(fact_7160_int__zle__neg,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.63        = ( ( N = 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_7161_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_7162_int__Suc,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_Suc
% 0.30/0.63  thf(fact_7163_zless__iff__Suc__zadd,axiom,
% 0.30/0.63      ( ord_less_int
% 0.30/0.63      = ( ^ [W3: int,Z2: int] :
% 0.30/0.63          ? [N2: nat] :
% 0.30/0.63            ( Z2
% 0.30/0.63            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zless_iff_Suc_zadd
% 0.30/0.63  thf(fact_7164_abs__mod__less,axiom,
% 0.30/0.63      ! [L: int,K: int] :
% 0.30/0.63        ( ( L != zero_zero_int )
% 0.30/0.63       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_mod_less
% 0.30/0.63  thf(fact_7165_negative__zle__0,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zle_0
% 0.30/0.63  thf(fact_7166_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_7167_real__of__nat__div,axiom,
% 0.30/0.63      ! [D: nat,N: nat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ D @ N )
% 0.30/0.63       => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.30/0.63          = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_div
% 0.30/0.63  thf(fact_7168_tanh__real__gt__neg1,axiom,
% 0.30/0.63      ! [X3: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % tanh_real_gt_neg1
% 0.30/0.63  thf(fact_7169_mod__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: code_integer,M: nat,N: nat] :
% 0.30/0.63        ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.30/0.63        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mod_mult2_eq'
% 0.30/0.63  thf(fact_7170_mod__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: int,M: nat,N: nat] :
% 0.30/0.63        ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 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 @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mod_mult2_eq'
% 0.30/0.63  thf(fact_7171_mod__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: nat,M: nat,N: nat] :
% 0.30/0.63        ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 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 @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mod_mult2_eq'
% 0.30/0.63  thf(fact_7172_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % field_char_0_class.of_nat_div
% 0.30/0.63  thf(fact_7173_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63        = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % field_char_0_class.of_nat_div
% 0.30/0.63  thf(fact_7174_field__char__0__class_Oof__nat__div,axiom,
% 0.30/0.63      ! [M: nat,N: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.30/0.63        = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % field_char_0_class.of_nat_div
% 0.30/0.63  thf(fact_7175_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_7176_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_7177_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_7178_nat__less__real__le,axiom,
% 0.30/0.63      ( ord_less_nat
% 0.30/0.63      = ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_less_real_le
% 0.30/0.63  thf(fact_7179_nat__le__real__less,axiom,
% 0.30/0.63      ( ord_less_eq_nat
% 0.30/0.63      = ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_le_real_less
% 0.30/0.63  thf(fact_7180_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_7181_zdvd__mult__cancel1,axiom,
% 0.30/0.63      ! [M: int,N: int] :
% 0.30/0.63        ( ( M != zero_zero_int )
% 0.30/0.63       => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.30/0.63          = ( ( abs_abs_int @ N )
% 0.30/0.63            = one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zdvd_mult_cancel1
% 0.30/0.63  thf(fact_7182_not__zle__0__negative,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % not_zle_0_negative
% 0.30/0.63  thf(fact_7183_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_7184_negative__zless__0,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zless_0
% 0.30/0.63  thf(fact_7185_dbl__inc__def,axiom,
% 0.30/0.63      ( neg_nu8557863876264182079omplex
% 0.30/0.63      = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_def
% 0.30/0.63  thf(fact_7186_dbl__inc__def,axiom,
% 0.30/0.63      ( neg_nu8295874005876285629c_real
% 0.30/0.63      = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_def
% 0.30/0.63  thf(fact_7187_dbl__inc__def,axiom,
% 0.30/0.63      ( neg_nu5219082963157363817nc_rat
% 0.30/0.63      = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_def
% 0.30/0.63  thf(fact_7188_dbl__inc__def,axiom,
% 0.30/0.63      ( neg_nu5851722552734809277nc_int
% 0.30/0.63      = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % dbl_inc_def
% 0.30/0.63  thf(fact_7189_int__ops_I6_J,axiom,
% 0.30/0.63      ! [A: nat,B2: nat] :
% 0.30/0.63        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 0.30/0.63         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.63            = zero_zero_int ) )
% 0.30/0.63        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 0.30/0.63         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 0.30/0.63            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(6)
% 0.30/0.63  thf(fact_7190_real__of__nat__div__aux,axiom,
% 0.30/0.63      ! [X3: nat,D: nat] :
% 0.30/0.63        ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.30/0.63        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X3 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X3 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_div_aux
% 0.30/0.63  thf(fact_7191_of__nat__less__two__power,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_two_power
% 0.30/0.63  thf(fact_7192_of__nat__less__two__power,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_two_power
% 0.30/0.63  thf(fact_7193_of__nat__less__two__power,axiom,
% 0.30/0.63      ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_two_power
% 0.30/0.63  thf(fact_7194_inverse__of__nat__le,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( N != 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 @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % inverse_of_nat_le
% 0.30/0.63  thf(fact_7195_inverse__of__nat__le,axiom,
% 0.30/0.63      ! [N: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.63       => ( ( N != 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 @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % inverse_of_nat_le
% 0.30/0.63  thf(fact_7196_even__abs__add__iff,axiom,
% 0.30/0.63      ! [K: int,L: int] :
% 0.30/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 0.30/0.63        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % even_abs_add_iff
% 0.30/0.63  thf(fact_7197_even__add__abs__iff,axiom,
% 0.30/0.63      ! [K: int,L: int] :
% 0.30/0.63        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 0.30/0.63        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % even_add_abs_iff
% 0.30/0.63  thf(fact_7198_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         => ( ! [M4: nat] :
% 0.30/0.63                ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.30/0.63               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ 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_7199_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_7200_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_7201_real__of__nat__div2,axiom,
% 0.30/0.63      ! [N: nat,X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_div2
% 0.30/0.63  thf(fact_7202_real__of__nat__div3,axiom,
% 0.30/0.63      ! [N: nat,X3: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) @ one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_div3
% 0.30/0.63  thf(fact_7203_ln__realpow,axiom,
% 0.30/0.63      ! [X3: real,N: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( ln_ln_real @ ( power_power_real @ X3 @ N ) )
% 0.30/0.63          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ln_realpow
% 0.30/0.63  thf(fact_7204_nat__intermed__int__val,axiom,
% 0.30/0.63      ! [M: nat,N: 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 @ N ) )
% 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 @ N )
% 0.30/0.63         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.30/0.63           => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 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 @ N )
% 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_7205_incr__lemma,axiom,
% 0.30/0.63      ! [D: int,Z: int,X3: int] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.63       => ( ord_less_int @ Z @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % incr_lemma
% 0.30/0.63  thf(fact_7206_decr__lemma,axiom,
% 0.30/0.63      ! [D: int,X3: int,Z: 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 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.30/0.63  
% 0.30/0.63  % decr_lemma
% 0.30/0.63  thf(fact_7207_linear__plus__1__le__power,axiom,
% 0.30/0.63      ! [X3: real,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X3 @ one_one_real ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % linear_plus_1_le_power
% 0.30/0.63  thf(fact_7208_Bernoulli__inequality,axiom,
% 0.30/0.63      ! [X3: real,N: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.63       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % Bernoulli_inequality
% 0.30/0.63  thf(fact_7209_divmod__nat__def,axiom,
% 0.30/0.63      ( divmod_nat
% 0.30/0.63      = ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_nat_def
% 0.30/0.63  thf(fact_7210_nat__ivt__aux,axiom,
% 0.30/0.63      ! [N: nat,F: nat > int,K: int] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( ord_less_nat @ I3 @ N )
% 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 @ N ) )
% 0.30/0.63           => ? [I3: nat] :
% 0.30/0.63                ( ( ord_less_eq_nat @ I3 @ N )
% 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_7211_nat0__intermed__int__val,axiom,
% 0.30/0.63      ! [N: nat,F: nat > int,K: int] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( ord_less_nat @ I3 @ N )
% 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 @ N ) )
% 0.30/0.63           => ? [I3: nat] :
% 0.30/0.63                ( ( ord_less_eq_nat @ I3 @ N )
% 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_7212_double__arith__series,axiom,
% 0.30/0.63      ! [A: complex,D: complex,N: nat] :
% 0.30/0.63        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7213_double__arith__series,axiom,
% 0.30/0.63      ! [A: rat,D: rat,N: nat] :
% 0.30/0.63        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7214_double__arith__series,axiom,
% 0.30/0.63      ! [A: int,D: int,N: nat] :
% 0.30/0.63        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7215_double__arith__series,axiom,
% 0.30/0.63      ! [A: nat,D: nat,N: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups3542108847815614940at_nat
% 0.30/0.63            @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7216_double__arith__series,axiom,
% 0.30/0.63      ! [A: real,D: real,N: nat] :
% 0.30/0.63        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7217_double__gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7218_double__gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7219_double__gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7220_double__gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7221_double__gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.63        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7222_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_7223_arith__series,axiom,
% 0.30/0.63      ! [A: int,D: int,N: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63        = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arith_series
% 0.30/0.63  thf(fact_7224_arith__series,axiom,
% 0.30/0.63      ! [A: nat,D: nat,N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arith_series
% 0.30/0.63  thf(fact_7225_gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum
% 0.30/0.63  thf(fact_7226_gauss__sum,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum
% 0.30/0.63  thf(fact_7227_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7228_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7229_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7230_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7231_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.30/0.63        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7232_Bernoulli__inequality__even,axiom,
% 0.30/0.63      ! [N: nat,X3: real] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.63       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % Bernoulli_inequality_even
% 0.30/0.63  thf(fact_7233_sum__gp__offset,axiom,
% 0.30/0.63      ! [X3: complex,M: nat,N: 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 @ N ) ) )
% 0.30/0.63            = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ 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 @ N ) ) )
% 0.30/0.63            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_offset
% 0.30/0.63  thf(fact_7234_sum__gp__offset,axiom,
% 0.30/0.63      ! [X3: rat,M: nat,N: 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 @ N ) ) )
% 0.30/0.63            = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ 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 @ N ) ) )
% 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 @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_offset
% 0.30/0.63  thf(fact_7235_sum__gp__offset,axiom,
% 0.30/0.63      ! [X3: real,M: nat,N: 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 @ N ) ) )
% 0.30/0.63            = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ 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 @ N ) ) )
% 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 @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_offset
% 0.30/0.63  thf(fact_7236_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.63        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ 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_7237_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ 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_7238_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri8010041392384452111omplex
% 0.30/0.63      = ( ^ [N2: nat] :
% 0.30/0.63            ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
% 0.30/0.63            @ ( produc1917071388513777916omplex
% 0.30/0.63              @ ^ [M3: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
% 0.30/0.63              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7239_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri681578069525770553at_rat
% 0.30/0.63      = ( ^ [N2: nat] :
% 0.30/0.63            ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
% 0.30/0.63            @ ( produc6207742614233964070at_rat
% 0.30/0.63              @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
% 0.30/0.63              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7240_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri1314217659103216013at_int
% 0.30/0.63      = ( ^ [N2: nat] :
% 0.30/0.63            ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
% 0.30/0.63            @ ( produc6840382203811409530at_int
% 0.30/0.63              @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
% 0.30/0.63              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7241_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri5074537144036343181t_real
% 0.30/0.63      = ( ^ [N2: nat] :
% 0.30/0.63            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.30/0.63            @ ( produc1703576794950452218t_real
% 0.30/0.63              @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
% 0.30/0.63              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7242_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri1316708129612266289at_nat
% 0.30/0.63      = ( ^ [N2: nat] :
% 0.30/0.63            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.30/0.63            @ ( produc6842872674320459806at_nat
% 0.30/0.63              @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
% 0.30/0.63              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7243_nat__approx__posE,axiom,
% 0.30/0.63      ! [E: rat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_approx_posE
% 0.30/0.63  thf(fact_7244_nat__approx__posE,axiom,
% 0.30/0.63      ! [E: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_approx_posE
% 0.30/0.63  thf(fact_7245_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          @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( 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_7246_lemma__termdiff3,axiom,
% 0.30/0.63      ! [H2: real,Z: real,K5: real,N: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_real )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 0.30/0.63         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ 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 @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( 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_7247_lemma__termdiff3,axiom,
% 0.30/0.63      ! [H2: complex,Z: complex,K5: real,N: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_complex )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 0.30/0.63         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ 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 @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff3
% 0.30/0.63  thf(fact_7248_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_7249_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_7250_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              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ln_series
% 0.30/0.63  thf(fact_7251_powser__zero,axiom,
% 0.30/0.63      ! [F: nat > complex] :
% 0.30/0.63        ( ( suminf_complex
% 0.30/0.63          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 0.30/0.63        = ( F @ zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % powser_zero
% 0.30/0.63  thf(fact_7252_powser__zero,axiom,
% 0.30/0.63      ! [F: nat > real] :
% 0.30/0.63        ( ( suminf_real
% 0.30/0.63          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 0.30/0.63        = ( F @ zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % powser_zero
% 0.30/0.63  thf(fact_7253_int__if,axiom,
% 0.30/0.63      ! [P: $o,A: nat,B2: nat] :
% 0.30/0.63        ( ( P
% 0.30/0.63         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 0.30/0.63            = ( semiri1314217659103216013at_int @ A ) ) )
% 0.30/0.63        & ( ~ P
% 0.30/0.63         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 0.30/0.63            = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_if
% 0.30/0.63  thf(fact_7254_nat__int__comparison_I1_J,axiom,
% 0.30/0.63      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.63      = ( ^ [A5: nat,B4: nat] :
% 0.30/0.63            ( ( semiri1314217659103216013at_int @ A5 )
% 0.30/0.63            = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_int_comparison(1)
% 0.30/0.63  thf(fact_7255_complex__mod__minus__le__complex__mod,axiom,
% 0.30/0.63      ! [X3: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % complex_mod_minus_le_complex_mod
% 0.30/0.63  thf(fact_7256_complex__mod__triangle__ineq2,axiom,
% 0.30/0.63      ! [B2: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % complex_mod_triangle_ineq2
% 0.30/0.63  thf(fact_7257_monoseq__realpow,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.63         => ( topolo6980174941875973593q_real @ ( power_power_real @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % monoseq_realpow
% 0.30/0.63  thf(fact_7258_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_7259_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_7260_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_7261_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_7262_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            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_series
% 0.30/0.63  thf(fact_7263_norm__divide__numeral,axiom,
% 0.30/0.63      ! [A: real,W: num] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.63        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_divide_numeral
% 0.30/0.63  thf(fact_7264_norm__divide__numeral,axiom,
% 0.30/0.63      ! [A: complex,W: num] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.63        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_divide_numeral
% 0.30/0.63  thf(fact_7265_norm__mult__numeral1,axiom,
% 0.30/0.63      ! [W: num,A: real] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.30/0.63        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_numeral1
% 0.30/0.63  thf(fact_7266_norm__mult__numeral1,axiom,
% 0.30/0.63      ! [W: num,A: complex] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.30/0.63        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_numeral1
% 0.30/0.63  thf(fact_7267_norm__mult__numeral2,axiom,
% 0.30/0.63      ! [A: real,W: num] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.63        = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_numeral2
% 0.30/0.63  thf(fact_7268_norm__mult__numeral2,axiom,
% 0.30/0.63      ! [A: complex,W: num] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.63        = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_numeral2
% 0.30/0.63  thf(fact_7269_norm__neg__numeral,axiom,
% 0.30/0.63      ! [W: num] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.63        = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_neg_numeral
% 0.30/0.63  thf(fact_7270_norm__neg__numeral,axiom,
% 0.30/0.63      ! [W: num] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.63        = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_neg_numeral
% 0.30/0.63  thf(fact_7271_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_7272_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_7273_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_7274_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_7275_norm__numeral,axiom,
% 0.30/0.63      ! [W: num] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.30/0.63        = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_numeral
% 0.30/0.63  thf(fact_7276_norm__numeral,axiom,
% 0.30/0.63      ! [W: num] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.63        = ( numeral_numeral_real @ W ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_numeral
% 0.30/0.63  thf(fact_7277_norm__minus__commute,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
% 0.30/0.63        = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_minus_commute
% 0.30/0.63  thf(fact_7278_norm__minus__commute,axiom,
% 0.30/0.63      ! [A: complex,B2: complex] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
% 0.30/0.63        = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_minus_commute
% 0.30/0.63  thf(fact_7279_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_7280_norm__ge__zero,axiom,
% 0.30/0.63      ! [X3: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_ge_zero
% 0.30/0.63  thf(fact_7281_norm__mult,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.63        = ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult
% 0.30/0.63  thf(fact_7282_norm__mult,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) )
% 0.30/0.63        = ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult
% 0.30/0.63  thf(fact_7283_sum__norm__le,axiom,
% 0.30/0.63      ! [S2: set_real,F: real > complex,G: real > real] :
% 0.30/0.63        ( ! [X5: real] :
% 0.30/0.63            ( ( member_real @ X5 @ S2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_norm_le
% 0.30/0.63  thf(fact_7284_sum__norm__le,axiom,
% 0.30/0.63      ! [S2: set_int,F: int > complex,G: int > real] :
% 0.30/0.63        ( ! [X5: int] :
% 0.30/0.63            ( ( member_int @ X5 @ S2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_norm_le
% 0.30/0.63  thf(fact_7285_sum__norm__le,axiom,
% 0.30/0.63      ! [S2: set_nat,F: nat > complex,G: nat > real] :
% 0.30/0.63        ( ! [X5: nat] :
% 0.30/0.63            ( ( member_nat @ X5 @ S2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_norm_le
% 0.30/0.63  thf(fact_7286_sum__norm__le,axiom,
% 0.30/0.63      ! [S2: set_complex,F: complex > complex,G: complex > real] :
% 0.30/0.63        ( ! [X5: complex] :
% 0.30/0.63            ( ( member_complex @ X5 @ S2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_norm_le
% 0.30/0.63  thf(fact_7287_sum__norm__le,axiom,
% 0.30/0.63      ! [S2: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.63        ( ! [X5: nat] :
% 0.30/0.63            ( ( member_nat @ X5 @ S2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_norm_le
% 0.30/0.63  thf(fact_7288_norm__divide,axiom,
% 0.30/0.63      ! [A: real,B2: real] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.63        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_divide
% 0.30/0.63  thf(fact_7289_norm__divide,axiom,
% 0.30/0.63      ! [A: complex,B2: complex] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.63        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_divide
% 0.30/0.63  thf(fact_7290_norm__power,axiom,
% 0.30/0.63      ! [X3: real,N: nat] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) )
% 0.30/0.63        = ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_power
% 0.30/0.63  thf(fact_7291_norm__power,axiom,
% 0.30/0.63      ! [X3: complex,N: nat] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) )
% 0.30/0.63        = ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_power
% 0.30/0.63  thf(fact_7292_norm__sum,axiom,
% 0.30/0.63      ! [F: nat > complex,A4: set_nat] :
% 0.30/0.63        ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A4 ) )
% 0.30/0.63        @ ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.30/0.63          @ A4 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_sum
% 0.30/0.63  thf(fact_7293_norm__sum,axiom,
% 0.30/0.63      ! [F: complex > complex,A4: set_complex] :
% 0.30/0.63        ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A4 ) )
% 0.30/0.63        @ ( groups5808333547571424918x_real
% 0.30/0.63          @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.30/0.63          @ A4 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_sum
% 0.30/0.63  thf(fact_7294_norm__sum,axiom,
% 0.30/0.63      ! [F: nat > real,A4: set_nat] :
% 0.30/0.63        ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
% 0.30/0.63        @ ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 0.30/0.63          @ A4 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_sum
% 0.30/0.63  thf(fact_7295_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_7296_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_7297_nonzero__norm__divide,axiom,
% 0.30/0.63      ! [B2: real,A: real] :
% 0.30/0.63        ( ( B2 != zero_zero_real )
% 0.30/0.63       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.63          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonzero_norm_divide
% 0.30/0.63  thf(fact_7298_nonzero__norm__divide,axiom,
% 0.30/0.63      ! [B2: complex,A: complex] :
% 0.30/0.63        ( ( B2 != zero_zero_complex )
% 0.30/0.63       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.63          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonzero_norm_divide
% 0.30/0.63  thf(fact_7299_power__eq__imp__eq__norm,axiom,
% 0.30/0.63      ! [W: real,N: nat,Z: real] :
% 0.30/0.63        ( ( ( power_power_real @ W @ N )
% 0.30/0.63          = ( power_power_real @ Z @ N ) )
% 0.30/0.63       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.63         => ( ( real_V7735802525324610683m_real @ W )
% 0.30/0.63            = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_imp_eq_norm
% 0.30/0.63  thf(fact_7300_power__eq__imp__eq__norm,axiom,
% 0.30/0.63      ! [W: complex,N: nat,Z: complex] :
% 0.30/0.63        ( ( ( power_power_complex @ W @ N )
% 0.30/0.63          = ( power_power_complex @ Z @ N ) )
% 0.30/0.63       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.63         => ( ( real_V1022390504157884413omplex @ W )
% 0.30/0.63            = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_imp_eq_norm
% 0.30/0.63  thf(fact_7301_norm__mult__less,axiom,
% 0.30/0.63      ! [X3: real,R: real,Y3: real,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R )
% 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 @ R @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_less
% 0.30/0.63  thf(fact_7302_norm__mult__less,axiom,
% 0.30/0.63      ! [X3: complex,R: real,Y3: complex,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R )
% 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 @ R @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_less
% 0.30/0.63  thf(fact_7303_norm__mult__ineq,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_ineq
% 0.30/0.63  thf(fact_7304_norm__mult__ineq,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_ineq
% 0.30/0.63  thf(fact_7305_norm__triangle__lt,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E: real] :
% 0.30/0.63        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.30/0.63       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_lt
% 0.30/0.63  thf(fact_7306_norm__triangle__lt,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E: real] :
% 0.30/0.63        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.30/0.63       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_lt
% 0.30/0.63  thf(fact_7307_norm__add__less,axiom,
% 0.30/0.63      ! [X3: real,R: real,Y3: real,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R )
% 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 @ R @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_less
% 0.30/0.63  thf(fact_7308_norm__add__less,axiom,
% 0.30/0.63      ! [X3: complex,R: real,Y3: complex,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R )
% 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 @ R @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_less
% 0.30/0.63  thf(fact_7309_norm__power__ineq,axiom,
% 0.30/0.63      ! [X3: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_power_ineq
% 0.30/0.63  thf(fact_7310_norm__power__ineq,axiom,
% 0.30/0.63      ! [X3: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_power_ineq
% 0.30/0.63  thf(fact_7311_norm__triangle__mono,axiom,
% 0.30/0.63      ! [A: real,R: real,B2: real,S: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
% 0.30/0.63         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_mono
% 0.30/0.63  thf(fact_7312_norm__triangle__mono,axiom,
% 0.30/0.63      ! [A: complex,R: real,B2: complex,S: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
% 0.30/0.63         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_mono
% 0.30/0.63  thf(fact_7313_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_7314_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_7315_norm__triangle__le,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_le
% 0.30/0.63  thf(fact_7316_norm__triangle__le,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_le
% 0.30/0.63  thf(fact_7317_norm__add__leD,axiom,
% 0.30/0.63      ! [A: real,B2: real,C: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( 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_7318_norm__add__leD,axiom,
% 0.30/0.63      ! [A: complex,B2: complex,C: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_leD
% 0.30/0.63  thf(fact_7319_norm__diff__triangle__less,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E1: real,Z: 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 @ Z ) ) @ E22 )
% 0.30/0.63         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_less
% 0.30/0.63  thf(fact_7320_norm__diff__triangle__less,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E1: real,Z: 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 @ Z ) ) @ E22 )
% 0.30/0.63         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_less
% 0.30/0.63  thf(fact_7321_norm__triangle__sub,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_sub
% 0.30/0.63  thf(fact_7322_norm__triangle__sub,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_sub
% 0.30/0.63  thf(fact_7323_norm__triangle__ineq4,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq4
% 0.30/0.63  thf(fact_7324_norm__triangle__ineq4,axiom,
% 0.30/0.63      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq4
% 0.30/0.63  thf(fact_7325_norm__diff__triangle__le,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.30/0.63         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_le
% 0.30/0.63  thf(fact_7326_norm__diff__triangle__le,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.30/0.63         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_le
% 0.30/0.63  thf(fact_7327_norm__triangle__le__diff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_le_diff
% 0.30/0.63  thf(fact_7328_norm__triangle__le__diff,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_le_diff
% 0.30/0.63  thf(fact_7329_norm__diff__ineq,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_ineq
% 0.30/0.63  thf(fact_7330_norm__diff__ineq,axiom,
% 0.30/0.63      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_ineq
% 0.30/0.63  thf(fact_7331_norm__triangle__ineq2,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq2
% 0.30/0.63  thf(fact_7332_norm__triangle__ineq2,axiom,
% 0.30/0.63      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq2
% 0.30/0.63  thf(fact_7333_power__eq__1__iff,axiom,
% 0.30/0.63      ! [W: real,N: nat] :
% 0.30/0.63        ( ( ( power_power_real @ W @ N )
% 0.30/0.63          = one_one_real )
% 0.30/0.63       => ( ( ( real_V7735802525324610683m_real @ W )
% 0.30/0.63            = one_one_real )
% 0.30/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_1_iff
% 0.30/0.63  thf(fact_7334_power__eq__1__iff,axiom,
% 0.30/0.63      ! [W: complex,N: nat] :
% 0.30/0.63        ( ( ( power_power_complex @ W @ N )
% 0.30/0.63          = one_one_complex )
% 0.30/0.63       => ( ( ( real_V1022390504157884413omplex @ W )
% 0.30/0.63            = one_one_real )
% 0.30/0.63          | ( N = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_1_iff
% 0.30/0.63  thf(fact_7335_norm__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_ineq
% 0.30/0.63  thf(fact_7336_norm__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_ineq
% 0.30/0.63  thf(fact_7337_norm__triangle__ineq3,axiom,
% 0.30/0.63      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq3
% 0.30/0.63  thf(fact_7338_norm__triangle__ineq3,axiom,
% 0.30/0.63      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq3
% 0.30/0.63  thf(fact_7339_square__norm__one,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          = one_one_real )
% 0.30/0.63       => ( ( real_V7735802525324610683m_real @ X3 )
% 0.30/0.63          = one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % square_norm_one
% 0.30/0.63  thf(fact_7340_square__norm__one,axiom,
% 0.30/0.63      ! [X3: complex] :
% 0.30/0.63        ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64          = one_one_complex )
% 0.30/0.64       => ( ( real_V1022390504157884413omplex @ X3 )
% 0.30/0.64          = one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % square_norm_one
% 0.30/0.64  thf(fact_7341_norm__power__diff,axiom,
% 0.30/0.64      ! [Z: real,W: real,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.30/0.64       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.30/0.64         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_power_diff
% 0.30/0.64  thf(fact_7342_norm__power__diff,axiom,
% 0.30/0.64      ! [Z: complex,W: complex,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.30/0.64       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.30/0.64         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_power_diff
% 0.30/0.64  thf(fact_7343_suminf__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       => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.30/0.64          = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_geometric
% 0.30/0.64  thf(fact_7344_suminf__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       => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.30/0.64          = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_geometric
% 0.30/0.64  thf(fact_7345_pi__series,axiom,
% 0.30/0.64      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.64      = ( suminf_real
% 0.30/0.64        @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_series
% 0.30/0.64  thf(fact_7346_lemma__termdiff2,axiom,
% 0.30/0.64      ! [H2: complex,Z: complex,N: nat] :
% 0.30/0.64        ( ( H2 != zero_zero_complex )
% 0.30/0.64       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.64          = ( times_times_complex @ H2
% 0.30/0.64            @ ( groups2073611262835488442omplex
% 0.30/0.64              @ ^ [P5: nat] :
% 0.30/0.64                  ( groups2073611262835488442omplex
% 0.30/0.64                  @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.30/0.64                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff2
% 0.30/0.64  thf(fact_7347_lemma__termdiff2,axiom,
% 0.30/0.64      ! [H2: rat,Z: rat,N: nat] :
% 0.30/0.64        ( ( H2 != zero_zero_rat )
% 0.30/0.64       => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.64          = ( times_times_rat @ H2
% 0.30/0.64            @ ( groups2906978787729119204at_rat
% 0.30/0.64              @ ^ [P5: nat] :
% 0.30/0.64                  ( groups2906978787729119204at_rat
% 0.30/0.64                  @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.30/0.64                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff2
% 0.30/0.64  thf(fact_7348_lemma__termdiff2,axiom,
% 0.30/0.64      ! [H2: real,Z: real,N: nat] :
% 0.30/0.64        ( ( H2 != zero_zero_real )
% 0.30/0.64       => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.64          = ( times_times_real @ H2
% 0.30/0.64            @ ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [P5: nat] :
% 0.30/0.64                  ( groups6591440286371151544t_real
% 0.30/0.64                  @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.30/0.64                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff2
% 0.30/0.64  thf(fact_7349_summable__arctan__series,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_arctan_series
% 0.30/0.64  thf(fact_7350_pred__subset__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.30/0.64        ( ( ord_le6741204236512500942_int_o
% 0.30/0.64          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 0.30/0.64          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) )
% 0.30/0.64        = ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq2
% 0.30/0.64  thf(fact_7351_pred__subset__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 0.30/0.64        ( ( ord_le3636971675376928563eger_o
% 0.30/0.64          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R2 )
% 0.30/0.64          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S2 ) )
% 0.30/0.64        = ( ord_le3216752416896350996nteger @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq2
% 0.30/0.64  thf(fact_7352_pred__subset__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 0.30/0.64        ( ( ord_le4340812435750786203eger_o
% 0.30/0.64          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 )
% 0.30/0.64          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) )
% 0.30/0.64        = ( ord_le653643898420964396nteger @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq2
% 0.30/0.64  thf(fact_7353_pred__subset__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 0.30/0.64        ( ( ord_le5643404153117327598_int_o
% 0.30/0.64          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 )
% 0.30/0.64          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) )
% 0.30/0.64        = ( ord_le8725513860283290265nt_int @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq2
% 0.30/0.64  thf(fact_7354_pred__subset__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 0.30/0.64        ( ( ord_le2124322318746777828_int_o
% 0.30/0.64          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 )
% 0.30/0.64          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) )
% 0.30/0.64        = ( ord_le135402666524580259nt_int @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq2
% 0.30/0.64  thf(fact_7355_infinite__int__iff__unbounded,axiom,
% 0.30/0.64      ! [S2: set_int] :
% 0.30/0.64        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.30/0.64        = ( ! [M3: int] :
% 0.30/0.64            ? [N2: int] :
% 0.30/0.64              ( ( ord_less_int @ M3 @ ( abs_abs_int @ N2 ) )
% 0.30/0.64              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % infinite_int_iff_unbounded
% 0.30/0.64  thf(fact_7356_lessThan__iff,axiom,
% 0.30/0.64      ! [I: rat,K: rat] :
% 0.30/0.64        ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 0.30/0.64        = ( ord_less_rat @ I @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_iff
% 0.30/0.64  thf(fact_7357_lessThan__iff,axiom,
% 0.30/0.64      ! [I: num,K: num] :
% 0.30/0.64        ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 0.30/0.64        = ( ord_less_num @ I @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_iff
% 0.30/0.64  thf(fact_7358_lessThan__iff,axiom,
% 0.30/0.64      ! [I: nat,K: nat] :
% 0.30/0.64        ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.30/0.64        = ( ord_less_nat @ I @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_iff
% 0.30/0.64  thf(fact_7359_lessThan__iff,axiom,
% 0.30/0.64      ! [I: int,K: int] :
% 0.30/0.64        ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 0.30/0.64        = ( ord_less_int @ I @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_iff
% 0.30/0.64  thf(fact_7360_lessThan__iff,axiom,
% 0.30/0.64      ! [I: real,K: real] :
% 0.30/0.64        ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 0.30/0.64        = ( ord_less_real @ I @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_iff
% 0.30/0.64  thf(fact_7361_summable__iff__shift,axiom,
% 0.30/0.64      ! [F: nat > real,K: nat] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.30/0.64        = ( summable_real @ F ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_iff_shift
% 0.30/0.64  thf(fact_7362_lessThan__subset__iff,axiom,
% 0.30/0.64      ! [X3: rat,Y3: rat] :
% 0.30/0.64        ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_subset_iff
% 0.30/0.64  thf(fact_7363_lessThan__subset__iff,axiom,
% 0.30/0.64      ! [X3: num,Y3: num] :
% 0.30/0.64        ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_subset_iff
% 0.30/0.64  thf(fact_7364_lessThan__subset__iff,axiom,
% 0.30/0.64      ! [X3: nat,Y3: nat] :
% 0.30/0.64        ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_subset_iff
% 0.30/0.64  thf(fact_7365_lessThan__subset__iff,axiom,
% 0.30/0.64      ! [X3: int,Y3: int] :
% 0.30/0.64        ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_subset_iff
% 0.30/0.64  thf(fact_7366_lessThan__subset__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_subset_iff
% 0.30/0.64  thf(fact_7367_summable__cmult__iff,axiom,
% 0.30/0.64      ! [C: complex,F: nat > complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.30/0.64        = ( ( C = zero_zero_complex )
% 0.30/0.64          | ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_cmult_iff
% 0.30/0.64  thf(fact_7368_summable__cmult__iff,axiom,
% 0.30/0.64      ! [C: real,F: nat > real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.30/0.64        = ( ( C = zero_zero_real )
% 0.30/0.64          | ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_cmult_iff
% 0.30/0.64  thf(fact_7369_summable__divide__iff,axiom,
% 0.30/0.64      ! [F: nat > complex,C: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.30/0.64        = ( ( C = zero_zero_complex )
% 0.30/0.64          | ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_divide_iff
% 0.30/0.64  thf(fact_7370_summable__divide__iff,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.30/0.64        = ( ( C = zero_zero_real )
% 0.30/0.64          | ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_divide_iff
% 0.30/0.64  thf(fact_7371_sum_OlessThan__Suc,axiom,
% 0.30/0.64      ! [G: nat > rat,N: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc
% 0.30/0.64  thf(fact_7372_sum_OlessThan__Suc,axiom,
% 0.30/0.64      ! [G: nat > int,N: nat] :
% 0.30/0.64        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc
% 0.30/0.64  thf(fact_7373_sum_OlessThan__Suc,axiom,
% 0.30/0.64      ! [G: nat > nat,N: nat] :
% 0.30/0.64        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc
% 0.30/0.64  thf(fact_7374_sum_OlessThan__Suc,axiom,
% 0.30/0.64      ! [G: nat > real,N: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc
% 0.30/0.64  thf(fact_7375_summable__geometric__iff,axiom,
% 0.30/0.64      ! [C: real] :
% 0.30/0.64        ( ( summable_real @ ( power_power_real @ C ) )
% 0.30/0.64        = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_geometric_iff
% 0.30/0.64  thf(fact_7376_summable__geometric__iff,axiom,
% 0.30/0.64      ! [C: complex] :
% 0.30/0.64        ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.30/0.64        = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_geometric_iff
% 0.30/0.64  thf(fact_7377_summable__comparison__test_H,axiom,
% 0.30/0.64      ! [G: nat > real,N4: nat,F: nat > real] :
% 0.30/0.64        ( ( summable_real @ G )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N4 @ N3 )
% 0.30/0.64             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64         => ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_comparison_test'
% 0.30/0.64  thf(fact_7378_summable__comparison__test_H,axiom,
% 0.30/0.64      ! [G: nat > real,N4: nat,F: nat > complex] :
% 0.30/0.64        ( ( summable_real @ G )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N4 @ N3 )
% 0.30/0.64             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64         => ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_comparison_test'
% 0.30/0.64  thf(fact_7379_summable__comparison__test,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real] :
% 0.30/0.64        ( ? [N7: nat] :
% 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 @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64       => ( ( summable_real @ G )
% 0.30/0.64         => ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_comparison_test
% 0.30/0.64  thf(fact_7380_summable__comparison__test,axiom,
% 0.30/0.64      ! [F: nat > complex,G: nat > real] :
% 0.30/0.64        ( ? [N7: nat] :
% 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 @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64       => ( ( summable_real @ G )
% 0.30/0.64         => ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_comparison_test
% 0.30/0.64  thf(fact_7381_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          @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_ignore_initial_segment
% 0.30/0.64  thf(fact_7382_summable__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         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_add
% 0.30/0.64  thf(fact_7383_summable__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         => ( summable_nat
% 0.30/0.64            @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_add
% 0.30/0.64  thf(fact_7384_summable__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         => ( summable_int
% 0.30/0.64            @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_add
% 0.30/0.64  thf(fact_7385_summable__diff,axiom,
% 0.30/0.64      ! [F: nat > complex,G: nat > complex] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( summable_complex @ G )
% 0.30/0.64         => ( summable_complex
% 0.30/0.64            @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_diff
% 0.30/0.64  thf(fact_7386_summable__diff,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         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_diff
% 0.30/0.64  thf(fact_7387_summable__divide,axiom,
% 0.30/0.64      ! [F: nat > complex,C: complex] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_divide
% 0.30/0.64  thf(fact_7388_summable__divide,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_divide
% 0.30/0.64  thf(fact_7389_summable__mult2,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_mult2
% 0.30/0.64  thf(fact_7390_summable__mult,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_mult
% 0.30/0.64  thf(fact_7391_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_7392_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_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_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_7400_summable__mult__D,axiom,
% 0.30/0.64      ! [C: complex,F: nat > complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 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_7401_summable__mult__D,axiom,
% 0.30/0.64      ! [C: real,F: nat > real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 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_7402_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_7403_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_7404_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_7405_lessThan__def,axiom,
% 0.30/0.64      ( set_ord_lessThan_rat
% 0.30/0.64      = ( ^ [U2: rat] :
% 0.30/0.64            ( collect_rat
% 0.30/0.64            @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_def
% 0.30/0.64  thf(fact_7406_lessThan__def,axiom,
% 0.30/0.64      ( set_ord_lessThan_num
% 0.30/0.64      = ( ^ [U2: num] :
% 0.30/0.64            ( collect_num
% 0.30/0.64            @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_def
% 0.30/0.64  thf(fact_7407_lessThan__def,axiom,
% 0.30/0.64      ( set_ord_lessThan_nat
% 0.30/0.64      = ( ^ [U2: nat] :
% 0.30/0.64            ( collect_nat
% 0.30/0.64            @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_def
% 0.30/0.64  thf(fact_7408_lessThan__def,axiom,
% 0.30/0.64      ( set_ord_lessThan_int
% 0.30/0.64      = ( ^ [U2: int] :
% 0.30/0.64            ( collect_int
% 0.30/0.64            @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_def
% 0.30/0.64  thf(fact_7409_lessThan__def,axiom,
% 0.30/0.64      ( set_or5984915006950818249n_real
% 0.30/0.64      = ( ^ [U2: real] :
% 0.30/0.64            ( collect_real
% 0.30/0.64            @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_def
% 0.30/0.64  thf(fact_7410_suminf__mult,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.30/0.64          = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_mult
% 0.30/0.64  thf(fact_7411_suminf__mult2,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 0.30/0.64          = ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_mult2
% 0.30/0.64  thf(fact_7412_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              @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_add
% 0.30/0.64  thf(fact_7413_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              @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_add
% 0.30/0.64  thf(fact_7414_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              @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_add
% 0.30/0.64  thf(fact_7415_suminf__diff,axiom,
% 0.30/0.64      ! [F: nat > complex,G: nat > complex] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( summable_complex @ G )
% 0.30/0.64         => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
% 0.30/0.64            = ( suminf_complex
% 0.30/0.64              @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_diff
% 0.30/0.64  thf(fact_7416_suminf__diff,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         => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.30/0.64            = ( suminf_real
% 0.30/0.64              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_diff
% 0.30/0.64  thf(fact_7417_suminf__divide,axiom,
% 0.30/0.64      ! [F: nat > complex,C: complex] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( suminf_complex
% 0.30/0.64            @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.30/0.64          = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_divide
% 0.30/0.64  thf(fact_7418_suminf__divide,axiom,
% 0.30/0.64      ! [F: nat > real,C: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.30/0.64          = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_divide
% 0.30/0.64  thf(fact_7419_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              @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ 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_7420_suminf__minus__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > complex,K: nat] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( suminf_complex
% 0.30/0.64            @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.30/0.64          = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_minus_initial_segment
% 0.30/0.64  thf(fact_7421_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            @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ 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_7422_finite__nat__bounded,axiom,
% 0.30/0.64      ! [S2: set_nat] :
% 0.30/0.64        ( ( finite_finite_nat @ S2 )
% 0.30/0.64       => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % finite_nat_bounded
% 0.30/0.64  thf(fact_7423_finite__nat__iff__bounded,axiom,
% 0.30/0.64      ( finite_finite_nat
% 0.30/0.64      = ( ^ [S4: set_nat] :
% 0.30/0.64          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % finite_nat_iff_bounded
% 0.30/0.64  thf(fact_7424_sum__less__suminf,axiom,
% 0.30/0.64      ! [F: nat > int,N: nat] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [M4: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.64             => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.30/0.64         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf
% 0.30/0.64  thf(fact_7425_sum__less__suminf,axiom,
% 0.30/0.64      ! [F: nat > nat,N: nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [M4: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.64             => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.30/0.64         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf
% 0.30/0.64  thf(fact_7426_sum__less__suminf,axiom,
% 0.30/0.64      ! [F: nat > real,N: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [M4: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.64             => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.30/0.64         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf
% 0.30/0.64  thf(fact_7427_powser__insidea,axiom,
% 0.30/0.64      ! [F: nat > real,X3: real,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_insidea
% 0.30/0.64  thf(fact_7428_powser__insidea,axiom,
% 0.30/0.64      ! [F: nat > complex,X3: complex,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_insidea
% 0.30/0.64  thf(fact_7429_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            = ( ! [N2: nat] :
% 0.30/0.64                  ( ( F @ N2 )
% 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_7430_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            = ( ! [N2: nat] :
% 0.30/0.64                  ( ( F @ N2 )
% 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_7431_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            = ( ! [N2: nat] :
% 0.30/0.64                  ( ( F @ N2 )
% 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_7432_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_7433_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_7434_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_7435_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_7436_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_7437_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_7438_lessThan__strict__subset__iff,axiom,
% 0.30/0.64      ! [M: rat,N: rat] :
% 0.30/0.64        ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 0.30/0.64        = ( ord_less_rat @ M @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_strict_subset_iff
% 0.30/0.64  thf(fact_7439_lessThan__strict__subset__iff,axiom,
% 0.30/0.64      ! [M: num,N: num] :
% 0.30/0.64        ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 0.30/0.64        = ( ord_less_num @ M @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_strict_subset_iff
% 0.30/0.64  thf(fact_7440_lessThan__strict__subset__iff,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64        = ( ord_less_nat @ M @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_strict_subset_iff
% 0.30/0.64  thf(fact_7441_lessThan__strict__subset__iff,axiom,
% 0.30/0.64      ! [M: int,N: int] :
% 0.30/0.64        ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 0.30/0.64        = ( ord_less_int @ M @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_strict_subset_iff
% 0.30/0.64  thf(fact_7442_lessThan__strict__subset__iff,axiom,
% 0.30/0.64      ! [M: real,N: real] :
% 0.30/0.64        ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 0.30/0.64        = ( ord_less_real @ M @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lessThan_strict_subset_iff
% 0.30/0.64  thf(fact_7443_summable__0__powser,axiom,
% 0.30/0.64      ! [F: nat > complex] :
% 0.30/0.64        ( summable_complex
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_0_powser
% 0.30/0.64  thf(fact_7444_summable__0__powser,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( summable_real
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_0_powser
% 0.30/0.64  thf(fact_7445_summable__zero__power_H,axiom,
% 0.30/0.64      ! [F: nat > complex] :
% 0.30/0.64        ( summable_complex
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power'
% 0.30/0.64  thf(fact_7446_summable__zero__power_H,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( summable_real
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power'
% 0.30/0.64  thf(fact_7447_summable__zero__power_H,axiom,
% 0.30/0.64      ! [F: nat > int] :
% 0.30/0.64        ( summable_int
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power'
% 0.30/0.64  thf(fact_7448_powser__split__head_I3_J,axiom,
% 0.30/0.64      ! [F: nat > complex,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64       => ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(3)
% 0.30/0.64  thf(fact_7449_powser__split__head_I3_J,axiom,
% 0.30/0.64      ! [F: nat > real,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(3)
% 0.30/0.64  thf(fact_7450_summable__powser__split__head,axiom,
% 0.30/0.64      ! [F: nat > complex,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64        = ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_split_head
% 0.30/0.64  thf(fact_7451_summable__powser__split__head,axiom,
% 0.30/0.64      ! [F: nat > real,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64        = ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_split_head
% 0.30/0.64  thf(fact_7452_summable__powser__ignore__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > complex,M: nat,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64        = ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_ignore_initial_segment
% 0.30/0.64  thf(fact_7453_summable__powser__ignore__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,M: nat,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64        = ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_ignore_initial_segment
% 0.30/0.64  thf(fact_7454_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_7455_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_7456_pi__ge__zero,axiom,
% 0.30/0.64      ord_less_eq_real @ zero_zero_real @ pi ).
% 0.30/0.64  
% 0.30/0.64  % pi_ge_zero
% 0.30/0.64  thf(fact_7457_sum__less__suminf2,axiom,
% 0.30/0.64      ! [F: nat > int,N: nat,I: nat] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [M4: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.64             => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ N @ 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 @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf2
% 0.30/0.64  thf(fact_7458_sum__less__suminf2,axiom,
% 0.30/0.64      ! [F: nat > nat,N: nat,I: nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [M4: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.64             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ N @ 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 @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf2
% 0.30/0.64  thf(fact_7459_sum__less__suminf2,axiom,
% 0.30/0.64      ! [F: nat > real,N: nat,I: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [M4: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N @ M4 )
% 0.30/0.64             => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ N @ 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 @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf2
% 0.30/0.64  thf(fact_7460_summable__norm__comparison__test,axiom,
% 0.30/0.64      ! [F: nat > complex,G: nat > real] :
% 0.30/0.64        ( ? [N7: nat] :
% 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 @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64       => ( ( summable_real @ G )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_norm_comparison_test
% 0.30/0.64  thf(fact_7461_summable__rabs__comparison__test,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real] :
% 0.30/0.64        ( ? [N7: nat] :
% 0.30/0.64          ! [N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ N7 @ 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            @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_rabs_comparison_test
% 0.30/0.64  thf(fact_7462_summable__rabs,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.30/0.64          @ ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_rabs
% 0.30/0.64  thf(fact_7463_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_7464_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_7465_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_7466_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            = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos_iff
% 0.30/0.64  thf(fact_7467_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            = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos_iff
% 0.30/0.64  thf(fact_7468_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            = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos_iff
% 0.30/0.64  thf(fact_7469_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_7470_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_7471_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_7472_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_7473_suminf__split__head,axiom,
% 0.30/0.64      ! [F: nat > complex] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( suminf_complex
% 0.30/0.64            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 0.30/0.64          = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_split_head
% 0.30/0.64  thf(fact_7474_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            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 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_7475_summable__norm,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 0.30/0.64          @ ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_norm
% 0.30/0.64  thf(fact_7476_summable__norm,axiom,
% 0.30/0.64      ! [F: nat > complex] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 0.30/0.64          @ ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_norm
% 0.30/0.64  thf(fact_7477_sum_Onat__diff__reindex,axiom,
% 0.30/0.64      ! [G: nat > nat,N: nat] :
% 0.30/0.64        ( ( groups3542108847815614940at_nat
% 0.30/0.64          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64        = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.nat_diff_reindex
% 0.30/0.64  thf(fact_7478_sum_Onat__diff__reindex,axiom,
% 0.30/0.64      ! [G: nat > real,N: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64        = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.nat_diff_reindex
% 0.30/0.64  thf(fact_7479_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_7480_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_7481_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_7482_sum__diff__distrib,axiom,
% 0.30/0.64      ! [Q: int > nat,P: int > nat,N: int] :
% 0.30/0.64        ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.30/0.64       => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 0.30/0.64          = ( groups4541462559716669496nt_nat
% 0.30/0.64            @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.64            @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_diff_distrib
% 0.30/0.64  thf(fact_7483_sum__diff__distrib,axiom,
% 0.30/0.64      ! [Q: real > nat,P: real > nat,N: real] :
% 0.30/0.64        ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.30/0.64       => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 0.30/0.64          = ( groups1935376822645274424al_nat
% 0.30/0.64            @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.64            @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_diff_distrib
% 0.30/0.64  thf(fact_7484_sum__diff__distrib,axiom,
% 0.30/0.64      ! [Q: nat > nat,P: nat > nat,N: nat] :
% 0.30/0.64        ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.30/0.64       => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 0.30/0.64          = ( groups3542108847815614940at_nat
% 0.30/0.64            @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_diff_distrib
% 0.30/0.64  thf(fact_7485_powser__inside,axiom,
% 0.30/0.64      ! [F: nat > real,X3: real,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_inside
% 0.30/0.64  thf(fact_7486_powser__inside,axiom,
% 0.30/0.64      ! [F: nat > complex,X3: complex,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 0.30/0.64         => ( summable_complex
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_inside
% 0.30/0.64  thf(fact_7487_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_7488_pi__ge__two,axiom,
% 0.30/0.64      ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.30/0.64  
% 0.30/0.64  % pi_ge_two
% 0.30/0.64  thf(fact_7489_pi__half__neq__two,axiom,
% 0.30/0.64      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64     != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_half_neq_two
% 0.30/0.64  thf(fact_7490_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       => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ 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_7491_powser__split__head_I1_J,axiom,
% 0.30/0.64      ! [F: nat > complex,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64       => ( ( suminf_complex
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 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                @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64              @ Z ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(1)
% 0.30/0.64  thf(fact_7492_powser__split__head_I1_J,axiom,
% 0.30/0.64      ! [F: nat > real,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64       => ( ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 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                @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64              @ Z ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(1)
% 0.30/0.64  thf(fact_7493_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.64      ! [G: nat > rat,N: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.30/0.64          @ ( groups2906978787729119204at_rat
% 0.30/0.64            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc_shift
% 0.30/0.64  thf(fact_7494_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.64      ! [G: nat > int,N: nat] :
% 0.30/0.64        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.30/0.64          @ ( groups3539618377306564664at_int
% 0.30/0.64            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc_shift
% 0.30/0.64  thf(fact_7495_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.64      ! [G: nat > nat,N: nat] :
% 0.30/0.64        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.30/0.64          @ ( groups3542108847815614940at_nat
% 0.30/0.64            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc_shift
% 0.30/0.64  thf(fact_7496_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.64      ! [G: nat > real,N: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum.lessThan_Suc_shift
% 0.30/0.64  thf(fact_7497_powser__split__head_I2_J,axiom,
% 0.30/0.64      ! [F: nat > complex,Z: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64       => ( ( times_times_complex
% 0.30/0.64            @ ( suminf_complex
% 0.30/0.64              @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.30/0.64            @ Z )
% 0.30/0.64          = ( minus_minus_complex
% 0.30/0.64            @ ( suminf_complex
% 0.30/0.64              @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 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_7498_powser__split__head_I2_J,axiom,
% 0.30/0.64      ! [F: nat > real,Z: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64       => ( ( times_times_real
% 0.30/0.64            @ ( suminf_real
% 0.30/0.64              @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.30/0.64            @ Z )
% 0.30/0.64          = ( minus_minus_real
% 0.30/0.64            @ ( suminf_real
% 0.30/0.64              @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 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_7499_sumr__diff__mult__const2,axiom,
% 0.30/0.64      ! [F: nat > complex,N: nat,R: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R ) )
% 0.30/0.64        = ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ R )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sumr_diff_mult_const2
% 0.30/0.64  thf(fact_7500_sumr__diff__mult__const2,axiom,
% 0.30/0.64      ! [F: nat > rat,N: nat,R: rat] :
% 0.30/0.64        ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R ) )
% 0.30/0.64        = ( groups2906978787729119204at_rat
% 0.30/0.64          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sumr_diff_mult_const2
% 0.30/0.64  thf(fact_7501_sumr__diff__mult__const2,axiom,
% 0.30/0.64      ! [F: nat > int,N: nat,R: int] :
% 0.30/0.64        ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
% 0.30/0.64        = ( groups3539618377306564664at_int
% 0.30/0.64          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sumr_diff_mult_const2
% 0.30/0.64  thf(fact_7502_sumr__diff__mult__const2,axiom,
% 0.30/0.64      ! [F: nat > real,N: nat,R: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
% 0.30/0.64        = ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sumr_diff_mult_const2
% 0.30/0.64  thf(fact_7503_sum__lessThan__telescope,axiom,
% 0.30/0.64      ! [F: nat > complex,M: nat] :
% 0.30/0.64        ( ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_complex @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope
% 0.30/0.64  thf(fact_7504_sum__lessThan__telescope,axiom,
% 0.30/0.64      ! [F: nat > rat,M: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope
% 0.30/0.64  thf(fact_7505_sum__lessThan__telescope,axiom,
% 0.30/0.64      ! [F: nat > int,M: nat] :
% 0.30/0.64        ( ( groups3539618377306564664at_int
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope
% 0.30/0.64  thf(fact_7506_sum__lessThan__telescope,axiom,
% 0.30/0.64      ! [F: nat > real,M: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope
% 0.30/0.64  thf(fact_7507_sum__lessThan__telescope_H,axiom,
% 0.30/0.64      ! [F: nat > complex,M: nat] :
% 0.30/0.64        ( ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope'
% 0.30/0.64  thf(fact_7508_sum__lessThan__telescope_H,axiom,
% 0.30/0.64      ! [F: nat > rat,M: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope'
% 0.30/0.64  thf(fact_7509_sum__lessThan__telescope_H,axiom,
% 0.30/0.64      ! [F: nat > int,M: nat] :
% 0.30/0.64        ( ( groups3539618377306564664at_int
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope'
% 0.30/0.64  thf(fact_7510_sum__lessThan__telescope_H,axiom,
% 0.30/0.64      ! [F: nat > real,M: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_lessThan_telescope'
% 0.30/0.64  thf(fact_7511_summable__partial__sum__bound,axiom,
% 0.30/0.64      ! [F: nat > complex,E: real] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.64         => ~ ! [N8: nat] :
% 0.30/0.64                ~ ! [M2: nat] :
% 0.30/0.64                    ( ( ord_less_eq_nat @ N8 @ M2 )
% 0.30/0.64                   => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_partial_sum_bound
% 0.30/0.64  thf(fact_7512_summable__partial__sum__bound,axiom,
% 0.30/0.64      ! [F: nat > real,E: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.64         => ~ ! [N8: nat] :
% 0.30/0.64                ~ ! [M2: nat] :
% 0.30/0.64                    ( ( ord_less_eq_nat @ N8 @ M2 )
% 0.30/0.64                   => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_partial_sum_bound
% 0.30/0.64  thf(fact_7513_suminf__exist__split,axiom,
% 0.30/0.64      ! [R: real,F: nat > real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ R )
% 0.30/0.64       => ( ( summable_real @ F )
% 0.30/0.64         => ? [N8: nat] :
% 0.30/0.64            ! [N9: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N8 @ N9 )
% 0.30/0.64             => ( ord_less_real
% 0.30/0.64                @ ( real_V7735802525324610683m_real
% 0.30/0.64                  @ ( suminf_real
% 0.30/0.64                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 0.30/0.64                @ R ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_exist_split
% 0.30/0.64  thf(fact_7514_suminf__exist__split,axiom,
% 0.30/0.64      ! [R: real,F: nat > complex] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ R )
% 0.30/0.64       => ( ( summable_complex @ F )
% 0.30/0.64         => ? [N8: nat] :
% 0.30/0.64            ! [N9: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N8 @ N9 )
% 0.30/0.64             => ( ord_less_real
% 0.30/0.64                @ ( real_V1022390504157884413omplex
% 0.30/0.64                  @ ( suminf_complex
% 0.30/0.64                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 0.30/0.64                @ R ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_exist_split
% 0.30/0.64  thf(fact_7515_summable__power__series,axiom,
% 0.30/0.64      ! [F: nat > real,Z: 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 @ Z )
% 0.30/0.64           => ( ( ord_less_real @ Z @ one_one_real )
% 0.30/0.64             => ( summable_real
% 0.30/0.64                @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_power_series
% 0.30/0.64  thf(fact_7516_Abel__lemma,axiom,
% 0.30/0.64      ! [R: real,R0: real,A: nat > complex,M7: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.30/0.64       => ( ( ord_less_real @ R @ 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              @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Abel_lemma
% 0.30/0.64  thf(fact_7517_bot__empty__eq2,axiom,
% 0.30/0.64      ( bot_bot_int_int_o
% 0.30/0.64      = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % bot_empty_eq2
% 0.30/0.64  thf(fact_7518_bot__empty__eq2,axiom,
% 0.30/0.64      ( bot_bo5358457235160185703eger_o
% 0.30/0.64      = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % bot_empty_eq2
% 0.30/0.64  thf(fact_7519_bot__empty__eq2,axiom,
% 0.30/0.64      ( bot_bo3000040243691356879eger_o
% 0.30/0.64      = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % bot_empty_eq2
% 0.30/0.64  thf(fact_7520_bot__empty__eq2,axiom,
% 0.30/0.64      ( bot_bo8662317086119403298_int_o
% 0.30/0.64      = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % bot_empty_eq2
% 0.30/0.64  thf(fact_7521_bot__empty__eq2,axiom,
% 0.30/0.64      ( bot_bo1403522918969695512_int_o
% 0.30/0.64      = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % bot_empty_eq2
% 0.30/0.64  thf(fact_7522_pred__equals__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.30/0.64        ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) )
% 0.30/0.64          = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) ) )
% 0.30/0.64        = ( R2 = S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_equals_eq2
% 0.30/0.64  thf(fact_7523_pred__equals__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 0.30/0.64        ( ( ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R2 ) )
% 0.30/0.64          = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S2 ) ) )
% 0.30/0.64        = ( R2 = S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_equals_eq2
% 0.30/0.64  thf(fact_7524_pred__equals__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 0.30/0.64        ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 ) )
% 0.30/0.64          = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) ) )
% 0.30/0.64        = ( R2 = S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_equals_eq2
% 0.30/0.64  thf(fact_7525_pred__equals__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 0.30/0.64        ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 ) )
% 0.30/0.64          = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) ) )
% 0.30/0.64        = ( R2 = S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_equals_eq2
% 0.30/0.64  thf(fact_7526_pred__equals__eq2,axiom,
% 0.30/0.64      ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 0.30/0.64        ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 ) )
% 0.30/0.64          = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) ) )
% 0.30/0.64        = ( R2 = S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_equals_eq2
% 0.30/0.64  thf(fact_7527_summable__ratio__test,axiom,
% 0.30/0.64      ! [C: real,N4: 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 @ N4 @ 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_7528_summable__ratio__test,axiom,
% 0.30/0.64      ! [C: real,N4: 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 @ N4 @ 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_7529_pi__half__neq__zero,axiom,
% 0.30/0.64      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64     != zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_half_neq_zero
% 0.30/0.64  thf(fact_7530_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_7531_pi__half__le__two,axiom,
% 0.30/0.64      ord_less_eq_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_le_two
% 0.30/0.64  thf(fact_7532_power__diff__1__eq,axiom,
% 0.30/0.64      ! [X3: complex,N: nat] :
% 0.30/0.64        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex )
% 0.30/0.64        = ( times_times_complex @ ( minus_minus_complex @ X3 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_1_eq
% 0.30/0.64  thf(fact_7533_power__diff__1__eq,axiom,
% 0.30/0.64      ! [X3: rat,N: nat] :
% 0.30/0.64        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat )
% 0.30/0.64        = ( times_times_rat @ ( minus_minus_rat @ X3 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_1_eq
% 0.30/0.64  thf(fact_7534_power__diff__1__eq,axiom,
% 0.30/0.64      ! [X3: int,N: nat] :
% 0.30/0.64        ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ one_one_int )
% 0.30/0.64        = ( times_times_int @ ( minus_minus_int @ X3 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_1_eq
% 0.30/0.64  thf(fact_7535_power__diff__1__eq,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real )
% 0.30/0.64        = ( times_times_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_1_eq
% 0.30/0.64  thf(fact_7536_one__diff__power__eq,axiom,
% 0.30/0.64      ! [X3: complex,N: nat] :
% 0.30/0.64        ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
% 0.30/0.64        = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq
% 0.30/0.64  thf(fact_7537_one__diff__power__eq,axiom,
% 0.30/0.64      ! [X3: rat,N: nat] :
% 0.30/0.64        ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
% 0.30/0.64        = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq
% 0.30/0.64  thf(fact_7538_one__diff__power__eq,axiom,
% 0.30/0.64      ! [X3: int,N: nat] :
% 0.30/0.64        ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
% 0.30/0.64        = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq
% 0.30/0.64  thf(fact_7539_one__diff__power__eq,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
% 0.30/0.64        = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq
% 0.30/0.64  thf(fact_7540_geometric__sum,axiom,
% 0.30/0.64      ! [X3: complex,N: nat] :
% 0.30/0.64        ( ( X3 != one_one_complex )
% 0.30/0.64       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_sum
% 0.30/0.64  thf(fact_7541_geometric__sum,axiom,
% 0.30/0.64      ! [X3: rat,N: nat] :
% 0.30/0.64        ( ( X3 != one_one_rat )
% 0.30/0.64       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64          = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_sum
% 0.30/0.64  thf(fact_7542_geometric__sum,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( X3 != one_one_real )
% 0.30/0.64       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_sum
% 0.30/0.64  thf(fact_7543_sum__gp__strict,axiom,
% 0.30/0.64      ! [X3: complex,N: nat] :
% 0.30/0.64        ( ( ( X3 = one_one_complex )
% 0.30/0.64         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            = ( semiri8010041392384452111omplex @ N ) ) )
% 0.30/0.64        & ( ( X3 != one_one_complex )
% 0.30/0.64         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_gp_strict
% 0.30/0.64  thf(fact_7544_sum__gp__strict,axiom,
% 0.30/0.64      ! [X3: rat,N: nat] :
% 0.30/0.64        ( ( ( X3 = one_one_rat )
% 0.30/0.64         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            = ( semiri681578069525770553at_rat @ N ) ) )
% 0.30/0.64        & ( ( X3 != one_one_rat )
% 0.30/0.64         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_gp_strict
% 0.30/0.64  thf(fact_7545_sum__gp__strict,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( ( X3 = one_one_real )
% 0.30/0.64         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            = ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64        & ( ( X3 != one_one_real )
% 0.30/0.64         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_gp_strict
% 0.30/0.64  thf(fact_7546_lemma__termdiff1,axiom,
% 0.30/0.64      ! [Z: complex,H2: complex,M: nat] :
% 0.30/0.64        ( ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff1
% 0.30/0.64  thf(fact_7547_lemma__termdiff1,axiom,
% 0.30/0.64      ! [Z: rat,H2: rat,M: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat
% 0.30/0.64          @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( groups2906978787729119204at_rat
% 0.30/0.64          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff1
% 0.30/0.64  thf(fact_7548_lemma__termdiff1,axiom,
% 0.30/0.64      ! [Z: int,H2: int,M: nat] :
% 0.30/0.64        ( ( groups3539618377306564664at_int
% 0.30/0.64          @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( groups3539618377306564664at_int
% 0.30/0.64          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff1
% 0.30/0.64  thf(fact_7549_lemma__termdiff1,axiom,
% 0.30/0.64      ! [Z: real,H2: real,M: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.64        = ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_termdiff1
% 0.30/0.64  thf(fact_7550_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_7551_power__diff__sumr2,axiom,
% 0.30/0.64      ! [X3: complex,N: nat,Y3: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.30/0.64        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 0.30/0.64          @ ( groups2073611262835488442omplex
% 0.30/0.64            @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_sumr2
% 0.30/0.64  thf(fact_7552_power__diff__sumr2,axiom,
% 0.30/0.64      ! [X3: rat,N: nat,Y3: rat] :
% 0.30/0.64        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.30/0.64        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 0.30/0.64          @ ( groups2906978787729119204at_rat
% 0.30/0.64            @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X3 @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_sumr2
% 0.30/0.64  thf(fact_7553_power__diff__sumr2,axiom,
% 0.30/0.64      ! [X3: int,N: nat,Y3: int] :
% 0.30/0.64        ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.30/0.64        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 0.30/0.64          @ ( groups3539618377306564664at_int
% 0.30/0.64            @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X3 @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_sumr2
% 0.30/0.64  thf(fact_7554_power__diff__sumr2,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Y3: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.30/0.64        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X3 @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % power_diff_sumr2
% 0.30/0.64  thf(fact_7555_diff__power__eq__sum,axiom,
% 0.30/0.64      ! [X3: complex,N: nat,Y3: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 0.30/0.64        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 0.30/0.64          @ ( groups2073611262835488442omplex
% 0.30/0.64            @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P5 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diff_power_eq_sum
% 0.30/0.64  thf(fact_7556_diff__power__eq__sum,axiom,
% 0.30/0.64      ! [X3: rat,N: nat,Y3: rat] :
% 0.30/0.64        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 0.30/0.64        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 0.30/0.64          @ ( groups2906978787729119204at_rat
% 0.30/0.64            @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P5 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diff_power_eq_sum
% 0.30/0.64  thf(fact_7557_diff__power__eq__sum,axiom,
% 0.30/0.64      ! [X3: int,N: nat,Y3: int] :
% 0.30/0.64        ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 0.30/0.64        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 0.30/0.64          @ ( groups3539618377306564664at_int
% 0.30/0.64            @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X3 @ P5 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diff_power_eq_sum
% 0.30/0.64  thf(fact_7558_diff__power__eq__sum,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Y3: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 0.30/0.64        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X3 @ P5 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diff_power_eq_sum
% 0.30/0.64  thf(fact_7559_pi__half__ge__zero,axiom,
% 0.30/0.64      ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_half_ge_zero
% 0.30/0.64  thf(fact_7560_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_7561_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_7562_arctan__one,axiom,
% 0.30/0.64      ( ( arctan @ one_one_real )
% 0.30/0.64      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan_one
% 0.30/0.64  thf(fact_7563_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.64      ! [N: nat,F: nat > rat,K5: rat,K: nat] :
% 0.30/0.64        ( ! [P7: nat] :
% 0.30/0.64            ( ( ord_less_nat @ P7 @ N )
% 0.30/0.64           => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 0.30/0.64       => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 0.30/0.64         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sum_nat_ivl_bounded2
% 0.30/0.64  thf(fact_7564_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.64      ! [N: nat,F: nat > int,K5: int,K: nat] :
% 0.30/0.64        ( ! [P7: nat] :
% 0.30/0.64            ( ( ord_less_nat @ P7 @ N )
% 0.30/0.64           => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 0.30/0.64       => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 0.30/0.64         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sum_nat_ivl_bounded2
% 0.30/0.64  thf(fact_7565_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.64      ! [N: nat,F: nat > nat,K5: nat,K: nat] :
% 0.30/0.64        ( ! [P7: nat] :
% 0.30/0.64            ( ( ord_less_nat @ P7 @ N )
% 0.30/0.64           => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 0.30/0.64       => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 0.30/0.64         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sum_nat_ivl_bounded2
% 0.30/0.64  thf(fact_7566_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.64      ! [N: nat,F: nat > real,K5: real,K: nat] :
% 0.30/0.64        ( ! [P7: nat] :
% 0.30/0.64            ( ( ord_less_nat @ P7 @ N )
% 0.30/0.64           => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 0.30/0.64       => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 0.30/0.64         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sum_nat_ivl_bounded2
% 0.30/0.64  thf(fact_7567_one__diff__power__eq_H,axiom,
% 0.30/0.64      ! [X3: complex,N: nat] :
% 0.30/0.64        ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
% 0.30/0.64        = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
% 0.30/0.64          @ ( groups2073611262835488442omplex
% 0.30/0.64            @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq'
% 0.30/0.64  thf(fact_7568_one__diff__power__eq_H,axiom,
% 0.30/0.64      ! [X3: rat,N: nat] :
% 0.30/0.64        ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
% 0.30/0.64        = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
% 0.30/0.64          @ ( groups2906978787729119204at_rat
% 0.30/0.64            @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq'
% 0.30/0.64  thf(fact_7569_one__diff__power__eq_H,axiom,
% 0.30/0.64      ! [X3: int,N: nat] :
% 0.30/0.64        ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
% 0.30/0.64        = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
% 0.30/0.64          @ ( groups3539618377306564664at_int
% 0.30/0.64            @ ^ [I4: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq'
% 0.30/0.64  thf(fact_7570_one__diff__power__eq_H,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
% 0.30/0.64        = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [I4: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_diff_power_eq'
% 0.30/0.64  thf(fact_7571_subrelI,axiom,
% 0.30/0.64      ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.30/0.64        ( ! [X5: int,Y4: int] :
% 0.30/0.64            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R )
% 0.30/0.64           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S ) )
% 0.30/0.64       => ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % subrelI
% 0.30/0.64  thf(fact_7572_subrelI,axiom,
% 0.30/0.64      ! [R: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
% 0.30/0.64        ( ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.64            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y4 ) @ R )
% 0.30/0.64           => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y4 ) @ S ) )
% 0.30/0.64       => ( ord_le3216752416896350996nteger @ R @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % subrelI
% 0.30/0.64  thf(fact_7573_subrelI,axiom,
% 0.30/0.64      ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
% 0.30/0.64        ( ! [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 0.30/0.64            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y4 ) @ R )
% 0.30/0.64           => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y4 ) @ S ) )
% 0.30/0.64       => ( ord_le653643898420964396nteger @ R @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % subrelI
% 0.30/0.64  thf(fact_7574_subrelI,axiom,
% 0.30/0.64      ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
% 0.30/0.64        ( ! [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.64            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y4 ) @ R )
% 0.30/0.64           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y4 ) @ S ) )
% 0.30/0.64       => ( ord_le8725513860283290265nt_int @ R @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % subrelI
% 0.30/0.64  thf(fact_7575_subrelI,axiom,
% 0.30/0.64      ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
% 0.30/0.64        ( ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 0.30/0.64            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y4 ) @ R )
% 0.30/0.64           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y4 ) @ S ) )
% 0.30/0.64       => ( ord_le135402666524580259nt_int @ R @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % subrelI
% 0.30/0.64  thf(fact_7576_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_7577_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_7578_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_7579_unbounded__k__infinite,axiom,
% 0.30/0.64      ! [K: nat,S2: set_nat] :
% 0.30/0.64        ( ! [M4: nat] :
% 0.30/0.64            ( ( ord_less_nat @ K @ M4 )
% 0.30/0.64           => ? [N9: nat] :
% 0.30/0.64                ( ( ord_less_nat @ M4 @ N9 )
% 0.30/0.64                & ( member_nat @ N9 @ S2 ) ) )
% 0.30/0.64       => ~ ( finite_finite_nat @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % unbounded_k_infinite
% 0.30/0.64  thf(fact_7580_infinite__nat__iff__unbounded,axiom,
% 0.30/0.64      ! [S2: set_nat] :
% 0.30/0.64        ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.30/0.64        = ( ! [M3: nat] :
% 0.30/0.64            ? [N2: nat] :
% 0.30/0.64              ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.64              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % infinite_nat_iff_unbounded
% 0.30/0.64  thf(fact_7581_infinite__nat__iff__unbounded__le,axiom,
% 0.30/0.64      ! [S2: set_nat] :
% 0.30/0.64        ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.30/0.64        = ( ! [M3: nat] :
% 0.30/0.64            ? [N2: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % infinite_nat_iff_unbounded_le
% 0.30/0.64  thf(fact_7582_sum__split__even__odd,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real,N: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.64        = ( plus_plus_real
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_split_even_odd
% 0.30/0.64  thf(fact_7583_pred__subset__eq,axiom,
% 0.30/0.64      ! [R2: set_nat,S2: set_nat] :
% 0.30/0.64        ( ( ord_less_eq_nat_o
% 0.30/0.64          @ ^ [X: nat] : ( member_nat @ X @ R2 )
% 0.30/0.64          @ ^ [X: nat] : ( member_nat @ X @ S2 ) )
% 0.30/0.64        = ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq
% 0.30/0.64  thf(fact_7584_pred__subset__eq,axiom,
% 0.30/0.64      ! [R2: set_real,S2: set_real] :
% 0.30/0.64        ( ( ord_less_eq_real_o
% 0.30/0.64          @ ^ [X: real] : ( member_real @ X @ R2 )
% 0.30/0.64          @ ^ [X: real] : ( member_real @ X @ S2 ) )
% 0.30/0.64        = ( ord_less_eq_set_real @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq
% 0.30/0.64  thf(fact_7585_pred__subset__eq,axiom,
% 0.30/0.64      ! [R2: set_complex,S2: set_complex] :
% 0.30/0.64        ( ( ord_le4573692005234683329plex_o
% 0.30/0.64          @ ^ [X: complex] : ( member_complex @ X @ R2 )
% 0.30/0.64          @ ^ [X: complex] : ( member_complex @ X @ S2 ) )
% 0.30/0.64        = ( ord_le211207098394363844omplex @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq
% 0.30/0.64  thf(fact_7586_pred__subset__eq,axiom,
% 0.30/0.64      ! [R2: set_int,S2: set_int] :
% 0.30/0.64        ( ( ord_less_eq_int_o
% 0.30/0.64          @ ^ [X: int] : ( member_int @ X @ R2 )
% 0.30/0.64          @ ^ [X: int] : ( member_int @ X @ S2 ) )
% 0.30/0.64        = ( ord_less_eq_set_int @ R2 @ S2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pred_subset_eq
% 0.30/0.64  thf(fact_7587_machin__Euler,axiom,
% 0.30/0.64      ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.64      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % machin_Euler
% 0.30/0.64  thf(fact_7588_machin,axiom,
% 0.30/0.64      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.64      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % machin
% 0.30/0.64  thf(fact_7589_infinite__int__iff__unbounded__le,axiom,
% 0.30/0.64      ! [S2: set_int] :
% 0.30/0.64        ( ( ~ ( finite_finite_int @ S2 ) )
% 0.30/0.64        = ( ! [M3: int] :
% 0.30/0.64            ? [N2: int] :
% 0.30/0.64              ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
% 0.30/0.64              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % infinite_int_iff_unbounded_le
% 0.30/0.64  thf(fact_7590_sin__cos__npi,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_npi
% 0.30/0.64  thf(fact_7591_sumr__cos__zero__one,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 0.30/0.64          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sumr_cos_zero_one
% 0.30/0.64  thf(fact_7592_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_7593_accp__subset,axiom,
% 0.30/0.64      ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
% 0.30/0.64        ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
% 0.30/0.64       => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset
% 0.30/0.64  thf(fact_7594_accp__subset,axiom,
% 0.30/0.64      ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.64        ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
% 0.30/0.64       => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset
% 0.30/0.64  thf(fact_7595_accp__subset,axiom,
% 0.30/0.64      ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
% 0.30/0.64        ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
% 0.30/0.64       => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset
% 0.30/0.64  thf(fact_7596_accp__subset,axiom,
% 0.30/0.64      ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
% 0.30/0.64        ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
% 0.30/0.64       => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset
% 0.30/0.64  thf(fact_7597_accp__subset,axiom,
% 0.30/0.64      ! [R1: nat > nat > $o,R22: nat > nat > $o] :
% 0.30/0.64        ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
% 0.30/0.64       => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset
% 0.30/0.64  thf(fact_7598_geometric__deriv__sums,axiom,
% 0.30/0.64      ! [Z: real] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
% 0.30/0.64          @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_deriv_sums
% 0.30/0.64  thf(fact_7599_geometric__deriv__sums,axiom,
% 0.30/0.64      ! [Z: complex] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
% 0.30/0.64          @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_deriv_sums
% 0.30/0.64  thf(fact_7600_monoseq__def,axiom,
% 0.30/0.64      ( topolo6980174941875973593q_real
% 0.30/0.64      = ( ^ [X7: nat > real] :
% 0.30/0.64            ( ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 0.30/0.64            | ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_real @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_7601_monoseq__def,axiom,
% 0.30/0.64      ( topolo3100542954746470799et_int
% 0.30/0.64      = ( ^ [X7: nat > set_int] :
% 0.30/0.64            ( ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_set_int @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 0.30/0.64            | ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_set_int @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_7602_monoseq__def,axiom,
% 0.30/0.64      ( topolo4267028734544971653eq_rat
% 0.30/0.64      = ( ^ [X7: nat > rat] :
% 0.30/0.64            ( ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_rat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 0.30/0.64            | ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_7603_monoseq__def,axiom,
% 0.30/0.64      ( topolo1459490580787246023eq_num
% 0.30/0.64      = ( ^ [X7: nat > num] :
% 0.30/0.64            ( ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_num @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 0.30/0.64            | ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_num @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_7604_monoseq__def,axiom,
% 0.30/0.64      ( topolo4902158794631467389eq_nat
% 0.30/0.64      = ( ^ [X7: nat > nat] :
% 0.30/0.64            ( ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 0.30/0.64            | ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_nat @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_7605_monoseq__def,axiom,
% 0.30/0.64      ( topolo4899668324122417113eq_int
% 0.30/0.64      = ( ^ [X7: nat > int] :
% 0.30/0.64            ( ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 0.30/0.64            | ! [M3: nat,N2: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.30/0.64               => ( ord_less_eq_int @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_7606_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_7607_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_7608_sin__npi2,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_npi2
% 0.30/0.64  thf(fact_7609_sin__npi,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_npi
% 0.30/0.64  thf(fact_7610_powser__sums__zero__iff,axiom,
% 0.30/0.64      ! [A: nat > complex,X3: complex] :
% 0.30/0.64        ( ( sums_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.30/0.64          @ X3 )
% 0.30/0.64        = ( ( A @ zero_zero_nat )
% 0.30/0.64          = X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_zero_iff
% 0.30/0.64  thf(fact_7611_powser__sums__zero__iff,axiom,
% 0.30/0.64      ! [A: nat > real,X3: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.30/0.64          @ X3 )
% 0.30/0.64        = ( ( A @ zero_zero_nat )
% 0.30/0.64          = X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_zero_iff
% 0.30/0.64  thf(fact_7612_cos__pi__half,axiom,
% 0.30/0.64      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_pi_half
% 0.30/0.64  thf(fact_7613_sin__two__pi,axiom,
% 0.30/0.64      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_two_pi
% 0.30/0.64  thf(fact_7614_sin__pi__half,axiom,
% 0.30/0.64      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_pi_half
% 0.30/0.64  thf(fact_7615_cos__two__pi,axiom,
% 0.30/0.64      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_two_pi
% 0.30/0.64  thf(fact_7616_cos__periodic,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( cos_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.64        = ( cos_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_periodic
% 0.30/0.64  thf(fact_7617_sin__periodic,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( sin_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.64        = ( sin_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_periodic
% 0.30/0.64  thf(fact_7618_cos__2pi__minus,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X3 ) )
% 0.30/0.64        = ( cos_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_2pi_minus
% 0.30/0.64  thf(fact_7619_cos__npi2,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_npi2
% 0.30/0.64  thf(fact_7620_cos__npi,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.64        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_npi
% 0.30/0.64  thf(fact_7621_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_7622_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_7623_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_7624_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_7625_sin__2npi,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_2npi
% 0.30/0.64  thf(fact_7626_cos__2npi,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_2npi
% 0.30/0.64  thf(fact_7627_sin__2pi__minus,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X3 ) )
% 0.30/0.64        = ( uminus_uminus_real @ ( sin_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_2pi_minus
% 0.30/0.64  thf(fact_7628_cos__3over2__pi,axiom,
% 0.30/0.64      ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_3over2_pi
% 0.30/0.64  thf(fact_7629_sin__3over2__pi,axiom,
% 0.30/0.64      ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.30/0.64      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_3over2_pi
% 0.30/0.64  thf(fact_7630_sums__le,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real,S: real,T: real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64       => ( ( sums_real @ F @ S )
% 0.30/0.64         => ( ( sums_real @ G @ T )
% 0.30/0.64           => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_le
% 0.30/0.64  thf(fact_7631_sums__le,axiom,
% 0.30/0.64      ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64       => ( ( sums_nat @ F @ S )
% 0.30/0.64         => ( ( sums_nat @ G @ T )
% 0.30/0.64           => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_le
% 0.30/0.64  thf(fact_7632_sums__le,axiom,
% 0.30/0.64      ! [F: nat > int,G: nat > int,S: int,T: int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64       => ( ( sums_int @ F @ S )
% 0.30/0.64         => ( ( sums_int @ G @ T )
% 0.30/0.64           => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_le
% 0.30/0.64  thf(fact_7633_polar__Ex,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64      ? [R3: real,A3: real] :
% 0.30/0.64        ( ( X3
% 0.30/0.64          = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
% 0.30/0.64        & ( Y3
% 0.30/0.64          = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % polar_Ex
% 0.30/0.64  thf(fact_7634_sin__diff,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( sin_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 0.30/0.64        = ( minus_minus_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_diff
% 0.30/0.64  thf(fact_7635_sin__diff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( sin_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( minus_minus_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_diff
% 0.30/0.64  thf(fact_7636_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_7637_sums__mult,axiom,
% 0.30/0.64      ! [F: nat > real,A: real,C: real] :
% 0.30/0.64        ( ( sums_real @ F @ A )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.30/0.64          @ ( times_times_real @ C @ A ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult
% 0.30/0.64  thf(fact_7638_sums__mult2,axiom,
% 0.30/0.64      ! [F: nat > real,A: real,C: real] :
% 0.30/0.64        ( ( sums_real @ F @ A )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 0.30/0.64          @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult2
% 0.30/0.64  thf(fact_7639_sums__add,axiom,
% 0.30/0.64      ! [F: nat > real,A: real,G: nat > real,B2: real] :
% 0.30/0.64        ( ( sums_real @ F @ A )
% 0.30/0.64       => ( ( sums_real @ G @ B2 )
% 0.30/0.64         => ( sums_real
% 0.30/0.64            @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64            @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_add
% 0.30/0.64  thf(fact_7640_sums__add,axiom,
% 0.30/0.64      ! [F: nat > nat,A: nat,G: nat > nat,B2: nat] :
% 0.30/0.64        ( ( sums_nat @ F @ A )
% 0.30/0.64       => ( ( sums_nat @ G @ B2 )
% 0.30/0.64         => ( sums_nat
% 0.30/0.64            @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64            @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_add
% 0.30/0.64  thf(fact_7641_sums__add,axiom,
% 0.30/0.64      ! [F: nat > int,A: int,G: nat > int,B2: int] :
% 0.30/0.64        ( ( sums_int @ F @ A )
% 0.30/0.64       => ( ( sums_int @ G @ B2 )
% 0.30/0.64         => ( sums_int
% 0.30/0.64            @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64            @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_add
% 0.30/0.64  thf(fact_7642_sums__diff,axiom,
% 0.30/0.64      ! [F: nat > complex,A: complex,G: nat > complex,B2: complex] :
% 0.30/0.64        ( ( sums_complex @ F @ A )
% 0.30/0.64       => ( ( sums_complex @ G @ B2 )
% 0.30/0.64         => ( sums_complex
% 0.30/0.64            @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64            @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_diff
% 0.30/0.64  thf(fact_7643_sums__diff,axiom,
% 0.30/0.64      ! [F: nat > real,A: real,G: nat > real,B2: real] :
% 0.30/0.64        ( ( sums_real @ F @ A )
% 0.30/0.64       => ( ( sums_real @ G @ B2 )
% 0.30/0.64         => ( sums_real
% 0.30/0.64            @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64            @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_diff
% 0.30/0.64  thf(fact_7644_sums__divide,axiom,
% 0.30/0.64      ! [F: nat > complex,A: complex,C: complex] :
% 0.30/0.64        ( ( sums_complex @ F @ A )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
% 0.30/0.64          @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_divide
% 0.30/0.64  thf(fact_7645_sums__divide,axiom,
% 0.30/0.64      ! [F: nat > real,A: real,C: real] :
% 0.30/0.64        ( ( sums_real @ F @ A )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
% 0.30/0.64          @ ( divide_divide_real @ A @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_divide
% 0.30/0.64  thf(fact_7646_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_7647_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_7648_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_7649_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_7650_sin__double,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X3 ) ) @ ( cos_complex @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_double
% 0.30/0.64  thf(fact_7651_sin__double,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X3 ) ) @ ( cos_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_double
% 0.30/0.64  thf(fact_7652_sincos__principal__value,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64      ? [Y4: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.30/0.64        & ( ord_less_eq_real @ Y4 @ pi )
% 0.30/0.64        & ( ( sin_real @ Y4 )
% 0.30/0.64          = ( sin_real @ X3 ) )
% 0.30/0.64        & ( ( cos_real @ Y4 )
% 0.30/0.64          = ( cos_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_principal_value
% 0.30/0.64  thf(fact_7653_sin__x__le__x,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_eq_real @ ( sin_real @ X3 ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_x_le_x
% 0.30/0.64  thf(fact_7654_sin__le__one,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( sin_real @ X3 ) @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_le_one
% 0.30/0.64  thf(fact_7655_cos__le__one,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( cos_real @ X3 ) @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_le_one
% 0.30/0.64  thf(fact_7656_abs__sin__x__le__abs__x,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ ( abs_abs_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % abs_sin_x_le_abs_x
% 0.30/0.64  thf(fact_7657_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            @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 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_7658_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            @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 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_7659_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            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ 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_7660_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            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ 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_7661_sin__cos__le1,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_le1
% 0.30/0.64  thf(fact_7662_cos__squared__eq,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_squared_eq
% 0.30/0.64  thf(fact_7663_cos__squared__eq,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_squared_eq
% 0.30/0.64  thf(fact_7664_sin__squared__eq,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_squared_eq
% 0.30/0.64  thf(fact_7665_sin__squared__eq,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_squared_eq
% 0.30/0.64  thf(fact_7666_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          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 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_7667_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          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 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_7668_sums__Suc__iff,axiom,
% 0.30/0.64      ! [F: nat > real,S: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 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_7669_sums__Suc,axiom,
% 0.30/0.64      ! [F: nat > real,L: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 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_7670_sums__Suc,axiom,
% 0.30/0.64      ! [F: nat > nat,L: nat] :
% 0.30/0.64        ( ( sums_nat
% 0.30/0.64          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 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_7671_sums__Suc,axiom,
% 0.30/0.64      ! [F: nat > int,L: int] :
% 0.30/0.64        ( ( sums_int
% 0.30/0.64          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 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_7672_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_7673_sin__x__ge__neg__x,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_eq_real @ ( uminus_uminus_real @ X3 ) @ ( sin_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_x_ge_neg_x
% 0.30/0.64  thf(fact_7674_sin__ge__zero,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_eq_real @ X3 @ pi )
% 0.30/0.64         => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_ge_zero
% 0.30/0.64  thf(fact_7675_sums__zero__iff__shift,axiom,
% 0.30/0.64      ! [N: nat,F: nat > complex,S: complex] :
% 0.30/0.64        ( ! [I3: nat] :
% 0.30/0.64            ( ( ord_less_nat @ I3 @ N )
% 0.30/0.64           => ( ( F @ I3 )
% 0.30/0.64              = zero_zero_complex ) )
% 0.30/0.64       => ( ( sums_complex
% 0.30/0.64            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 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_7676_sums__zero__iff__shift,axiom,
% 0.30/0.64      ! [N: nat,F: nat > real,S: real] :
% 0.30/0.64        ( ! [I3: nat] :
% 0.30/0.64            ( ( ord_less_nat @ I3 @ N )
% 0.30/0.64           => ( ( F @ I3 )
% 0.30/0.64              = zero_zero_real ) )
% 0.30/0.64       => ( ( sums_real
% 0.30/0.64            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 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_7677_sin__ge__minus__one,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_ge_minus_one
% 0.30/0.64  thf(fact_7678_cos__monotone__0__pi__le,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_eq_real @ Y3 @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ pi )
% 0.30/0.64           => ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_monotone_0_pi_le
% 0.30/0.64  thf(fact_7679_cos__mono__le__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_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
% 0.30/0.64                = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_mono_le_eq
% 0.30/0.64  thf(fact_7680_cos__inj__pi,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             => ( ( ( cos_real @ X3 )
% 0.30/0.64                  = ( cos_real @ Y3 ) )
% 0.30/0.64               => ( X3 = Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_inj_pi
% 0.30/0.64  thf(fact_7681_cos__ge__minus__one,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_ge_minus_one
% 0.30/0.64  thf(fact_7682_abs__sin__le__one,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % abs_sin_le_one
% 0.30/0.64  thf(fact_7683_abs__cos__le__one,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X3 ) ) @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % abs_cos_le_one
% 0.30/0.64  thf(fact_7684_sin__times__sin,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_sin
% 0.30/0.64  thf(fact_7685_sin__times__sin,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64        = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_sin
% 0.30/0.64  thf(fact_7686_sin__times__cos,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_cos
% 0.30/0.64  thf(fact_7687_sin__times__cos,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64        = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_cos
% 0.30/0.64  thf(fact_7688_cos__times__sin,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_sin
% 0.30/0.64  thf(fact_7689_cos__times__sin,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64        = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_sin
% 0.30/0.64  thf(fact_7690_sin__plus__sin,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_plus_sin
% 0.30/0.64  thf(fact_7691_sin__plus__sin,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_plus_sin
% 0.30/0.64  thf(fact_7692_sin__diff__sin,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_diff_sin
% 0.30/0.64  thf(fact_7693_sin__diff__sin,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_diff_sin
% 0.30/0.64  thf(fact_7694_cos__diff__cos,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_diff_cos
% 0.30/0.64  thf(fact_7695_cos__diff__cos,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_diff_cos
% 0.30/0.64  thf(fact_7696_cos__double,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64        = ( minus_minus_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  
% 0.30/0.64  % cos_double
% 0.30/0.64  thf(fact_7697_cos__double,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64        = ( minus_minus_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  
% 0.30/0.64  % cos_double
% 0.30/0.64  thf(fact_7698_cos__double__sin,axiom,
% 0.30/0.64      ! [W: complex] :
% 0.30/0.64        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64        = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_double_sin
% 0.30/0.64  thf(fact_7699_cos__double__sin,axiom,
% 0.30/0.64      ! [W: real] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64        = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_double_sin
% 0.30/0.64  thf(fact_7700_powser__sums__if,axiom,
% 0.30/0.64      ! [M: nat,Z: complex] :
% 0.30/0.64        ( sums_complex
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
% 0.30/0.64        @ ( power_power_complex @ Z @ M ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_if
% 0.30/0.64  thf(fact_7701_powser__sums__if,axiom,
% 0.30/0.64      ! [M: nat,Z: real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
% 0.30/0.64        @ ( power_power_real @ Z @ M ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_if
% 0.30/0.64  thf(fact_7702_powser__sums__if,axiom,
% 0.30/0.64      ! [M: nat,Z: int] :
% 0.30/0.64        ( sums_int
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
% 0.30/0.64        @ ( power_power_int @ Z @ M ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_if
% 0.30/0.64  thf(fact_7703_powser__sums__zero,axiom,
% 0.30/0.64      ! [A: nat > complex] :
% 0.30/0.64        ( sums_complex
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 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_7704_powser__sums__zero,axiom,
% 0.30/0.64      ! [A: nat > real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 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_7705_cos__two__neq__zero,axiom,
% 0.30/0.64      ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64     != zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_two_neq_zero
% 0.30/0.64  thf(fact_7706_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_7707_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_7708_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_7709_sums__iff__shift,axiom,
% 0.30/0.64      ! [F: nat > real,N: nat,S: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64          @ S )
% 0.30/0.64        = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_iff_shift
% 0.30/0.64  thf(fact_7710_sums__split__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > complex,S: complex,N: nat] :
% 0.30/0.64        ( ( sums_complex @ F @ S )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64          @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_split_initial_segment
% 0.30/0.64  thf(fact_7711_sums__split__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,S: real,N: nat] :
% 0.30/0.64        ( ( sums_real @ F @ S )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_split_initial_segment
% 0.30/0.64  thf(fact_7712_sums__iff__shift_H,axiom,
% 0.30/0.64      ! [F: nat > complex,N: nat,S: complex] :
% 0.30/0.64        ( ( sums_complex
% 0.30/0.64          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64          @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.30/0.64        = ( sums_complex @ F @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_iff_shift'
% 0.30/0.64  thf(fact_7713_sums__iff__shift_H,axiom,
% 0.30/0.64      ! [F: nat > real,N: nat,S: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.30/0.64          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 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_7714_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_7715_cos__monotone__minus__pi__0_H,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_eq_real @ Y3 @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.64           => ( ord_less_eq_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_7716_sums__If__finite__set_H,axiom,
% 0.30/0.64      ! [G: nat > complex,S2: complex,A4: set_nat,S5: complex,F: nat > complex] :
% 0.30/0.64        ( ( sums_complex @ G @ S2 )
% 0.30/0.64       => ( ( finite_finite_nat @ A4 )
% 0.30/0.64         => ( ( S5
% 0.30/0.64              = ( plus_plus_complex @ S2
% 0.30/0.64                @ ( groups2073611262835488442omplex
% 0.30/0.64                  @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64                  @ A4 ) ) )
% 0.30/0.64           => ( sums_complex
% 0.30/0.64              @ ^ [N2: nat] : ( if_complex @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64              @ S5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set'
% 0.30/0.64  thf(fact_7717_sums__If__finite__set_H,axiom,
% 0.30/0.64      ! [G: nat > real,S2: real,A4: set_nat,S5: real,F: nat > real] :
% 0.30/0.64        ( ( sums_real @ G @ S2 )
% 0.30/0.64       => ( ( finite_finite_nat @ A4 )
% 0.30/0.64         => ( ( S5
% 0.30/0.64              = ( plus_plus_real @ S2
% 0.30/0.64                @ ( groups6591440286371151544t_real
% 0.30/0.64                  @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64                  @ A4 ) ) )
% 0.30/0.64           => ( sums_real
% 0.30/0.64              @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.64              @ S5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set'
% 0.30/0.64  thf(fact_7718_sincos__total__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       => ( ( ( 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         => ? [T4: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.64              & ( ord_less_eq_real @ T4 @ pi )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( cos_real @ T4 ) )
% 0.30/0.64              & ( Y3
% 0.30/0.64                = ( sin_real @ T4 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_total_pi
% 0.30/0.64  thf(fact_7719_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_7720_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_7721_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_7722_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_7723_cos__is__zero,axiom,
% 0.30/0.64      ? [X5: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64        & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64        & ( ( cos_real @ X5 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64        & ! [Y5: real] :
% 0.30/0.64            ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.30/0.64              & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64              & ( ( cos_real @ Y5 )
% 0.30/0.64                = zero_zero_real ) )
% 0.30/0.64           => ( Y5 = X5 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_is_zero
% 0.30/0.64  thf(fact_7724_cos__two__le__zero,axiom,
% 0.30/0.64      ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.30/0.64  
% 0.30/0.64  % cos_two_le_zero
% 0.30/0.64  thf(fact_7725_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_7726_cos__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64         => ? [X5: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64              & ( ord_less_eq_real @ X5 @ pi )
% 0.30/0.64              & ( ( cos_real @ X5 )
% 0.30/0.64                = Y3 )
% 0.30/0.64              & ! [Y5: real] :
% 0.30/0.64                  ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.30/0.64                    & ( ord_less_eq_real @ Y5 @ pi )
% 0.30/0.64                    & ( ( cos_real @ Y5 )
% 0.30/0.64                      = Y3 ) )
% 0.30/0.64                 => ( Y5 = X5 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_total
% 0.30/0.64  thf(fact_7727_sincos__total__pi__half,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         => ( ( ( 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           => ? [T4: real] :
% 0.30/0.64                ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.64                & ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64                & ( X3
% 0.30/0.64                  = ( cos_real @ T4 ) )
% 0.30/0.64                & ( Y3
% 0.30/0.64                  = ( sin_real @ T4 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_total_pi_half
% 0.30/0.64  thf(fact_7728_sincos__total__2pi__le,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       => ? [T4: real] :
% 0.30/0.64            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.64            & ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64            & ( X3
% 0.30/0.64              = ( cos_real @ T4 ) )
% 0.30/0.64            & ( Y3
% 0.30/0.64              = ( sin_real @ T4 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_total_2pi_le
% 0.30/0.64  thf(fact_7729_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       => ~ ! [T4: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.64             => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64               => ( ( X3
% 0.30/0.64                    = ( cos_real @ T4 ) )
% 0.30/0.64                 => ( Y3
% 0.30/0.64                   != ( sin_real @ T4 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_total_2pi
% 0.30/0.64  thf(fact_7730_sin__pi__divide__n__ge__0,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( N != zero_zero_nat )
% 0.30/0.64       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_pi_divide_n_ge_0
% 0.30/0.64  thf(fact_7731_cos__times__cos,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_cos
% 0.30/0.64  thf(fact_7732_cos__times__cos,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64        = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_cos
% 0.30/0.64  thf(fact_7733_cos__plus__cos,axiom,
% 0.30/0.64      ! [W: complex,Z: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_plus_cos
% 0.30/0.64  thf(fact_7734_cos__plus__cos,axiom,
% 0.30/0.64      ! [W: real,Z: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_plus_cos
% 0.30/0.64  thf(fact_7735_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_7736_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_7737_power__half__series,axiom,
% 0.30/0.64      ( sums_real
% 0.30/0.64      @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
% 0.30/0.64      @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % power_half_series
% 0.30/0.64  thf(fact_7738_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_7739_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_7740_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_7741_sin__30,axiom,
% 0.30/0.64      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.30/0.64      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_30
% 0.30/0.64  thf(fact_7742_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_7743_sin__monotone__2pi__le,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_eq_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_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_monotone_2pi_le
% 0.30/0.64  thf(fact_7744_sin__mono__le__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_eq_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
% 0.30/0.64                = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_mono_le_eq
% 0.30/0.64  thf(fact_7745_sin__inj__pi,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             => ( ( ( sin_real @ X3 )
% 0.30/0.64                  = ( sin_real @ Y3 ) )
% 0.30/0.64               => ( X3 = Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_inj_pi
% 0.30/0.64  thf(fact_7746_cos__60,axiom,
% 0.30/0.64      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.30/0.64      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_60
% 0.30/0.64  thf(fact_7747_cos__double__cos,axiom,
% 0.30/0.64      ! [W: complex] :
% 0.30/0.64        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_double_cos
% 0.30/0.64  thf(fact_7748_cos__double__cos,axiom,
% 0.30/0.64      ! [W: real] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.30/0.64        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_double_cos
% 0.30/0.64  thf(fact_7749_cos__treble__cos,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X3 ) )
% 0.30/0.64        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_treble_cos
% 0.30/0.64  thf(fact_7750_cos__treble__cos,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X3 ) )
% 0.30/0.64        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_treble_cos
% 0.30/0.64  thf(fact_7751_sums__if_H,axiom,
% 0.30/0.64      ! [G: nat > real,X3: real] :
% 0.30/0.64        ( ( sums_real @ G @ X3 )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64          @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_if'
% 0.30/0.64  thf(fact_7752_sums__if,axiom,
% 0.30/0.64      ! [G: nat > real,X3: real,F: nat > real,Y3: real] :
% 0.30/0.64        ( ( sums_real @ G @ X3 )
% 0.30/0.64       => ( ( sums_real @ F @ Y3 )
% 0.30/0.64         => ( sums_real
% 0.30/0.64            @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64            @ ( plus_plus_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_if
% 0.30/0.64  thf(fact_7753_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_7754_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_7755_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_7756_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_7757_sin__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.64         => ? [X5: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.30/0.64              & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64              & ( ( sin_real @ X5 )
% 0.30/0.64                = Y3 )
% 0.30/0.64              & ! [Y5: real] :
% 0.30/0.64                  ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.30/0.64                    & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64                    & ( ( sin_real @ Y5 )
% 0.30/0.64                      = Y3 ) )
% 0.30/0.64                 => ( Y5 = X5 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_total
% 0.30/0.64  thf(fact_7758_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_7759_cos__ge__zero,axiom,
% 0.30/0.64      ! [X3: 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 @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_ge_zero
% 0.30/0.64  thf(fact_7760_cos__one__2pi,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        = ( ? [X: nat] :
% 0.30/0.64              ( X3
% 0.30/0.64              = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.30/0.64          | ? [X: nat] :
% 0.30/0.64              ( X3
% 0.30/0.64              = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_one_2pi
% 0.30/0.64  thf(fact_7761_accp__subset__induct,axiom,
% 0.30/0.64      ! [D4: product_prod_num_num > $o,R2: product_prod_num_num > product_prod_num_num > $o,X3: product_prod_num_num,P: product_prod_num_num > $o] :
% 0.30/0.64        ( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R2 ) )
% 0.30/0.64       => ( ! [X5: product_prod_num_num,Z4: product_prod_num_num] :
% 0.30/0.64              ( ( D4 @ X5 )
% 0.30/0.64             => ( ( R2 @ Z4 @ X5 )
% 0.30/0.64               => ( D4 @ Z4 ) ) )
% 0.30/0.64         => ( ( D4 @ X3 )
% 0.30/0.64           => ( ! [X5: product_prod_num_num] :
% 0.30/0.64                  ( ( D4 @ X5 )
% 0.30/0.64                 => ( ! [Z3: product_prod_num_num] :
% 0.30/0.64                        ( ( R2 @ Z3 @ X5 )
% 0.30/0.64                       => ( P @ Z3 ) )
% 0.30/0.64                   => ( P @ X5 ) ) )
% 0.30/0.64             => ( P @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset_induct
% 0.30/0.64  thf(fact_7762_accp__subset__induct,axiom,
% 0.30/0.64      ! [D4: product_prod_nat_nat > $o,R2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.64        ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R2 ) )
% 0.30/0.64       => ( ! [X5: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 0.30/0.64              ( ( D4 @ X5 )
% 0.30/0.64             => ( ( R2 @ Z4 @ X5 )
% 0.30/0.64               => ( D4 @ Z4 ) ) )
% 0.30/0.64         => ( ( D4 @ X3 )
% 0.30/0.64           => ( ! [X5: product_prod_nat_nat] :
% 0.30/0.64                  ( ( D4 @ X5 )
% 0.30/0.64                 => ( ! [Z3: product_prod_nat_nat] :
% 0.30/0.64                        ( ( R2 @ Z3 @ X5 )
% 0.30/0.64                       => ( P @ Z3 ) )
% 0.30/0.64                   => ( P @ X5 ) ) )
% 0.30/0.64             => ( P @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset_induct
% 0.30/0.64  thf(fact_7763_accp__subset__induct,axiom,
% 0.30/0.64      ! [D4: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,P: product_prod_int_int > $o] :
% 0.30/0.64        ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R2 ) )
% 0.30/0.64       => ( ! [X5: product_prod_int_int,Z4: product_prod_int_int] :
% 0.30/0.64              ( ( D4 @ X5 )
% 0.30/0.64             => ( ( R2 @ Z4 @ X5 )
% 0.30/0.64               => ( D4 @ Z4 ) ) )
% 0.30/0.64         => ( ( D4 @ X3 )
% 0.30/0.64           => ( ! [X5: product_prod_int_int] :
% 0.30/0.64                  ( ( D4 @ X5 )
% 0.30/0.64                 => ( ! [Z3: product_prod_int_int] :
% 0.30/0.64                        ( ( R2 @ Z3 @ X5 )
% 0.30/0.64                       => ( P @ Z3 ) )
% 0.30/0.64                   => ( P @ X5 ) ) )
% 0.30/0.64             => ( P @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset_induct
% 0.30/0.64  thf(fact_7764_accp__subset__induct,axiom,
% 0.30/0.64      ! [D4: list_nat > $o,R2: list_nat > list_nat > $o,X3: list_nat,P: list_nat > $o] :
% 0.30/0.64        ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R2 ) )
% 0.30/0.64       => ( ! [X5: list_nat,Z4: list_nat] :
% 0.30/0.64              ( ( D4 @ X5 )
% 0.30/0.64             => ( ( R2 @ Z4 @ X5 )
% 0.30/0.64               => ( D4 @ Z4 ) ) )
% 0.30/0.64         => ( ( D4 @ X3 )
% 0.30/0.64           => ( ! [X5: list_nat] :
% 0.30/0.64                  ( ( D4 @ X5 )
% 0.30/0.64                 => ( ! [Z3: list_nat] :
% 0.30/0.64                        ( ( R2 @ Z3 @ X5 )
% 0.30/0.64                       => ( P @ Z3 ) )
% 0.30/0.64                   => ( P @ X5 ) ) )
% 0.30/0.64             => ( P @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset_induct
% 0.30/0.64  thf(fact_7765_accp__subset__induct,axiom,
% 0.30/0.64      ! [D4: nat > $o,R2: nat > nat > $o,X3: nat,P: nat > $o] :
% 0.30/0.64        ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R2 ) )
% 0.30/0.64       => ( ! [X5: nat,Z4: nat] :
% 0.30/0.64              ( ( D4 @ X5 )
% 0.30/0.64             => ( ( R2 @ Z4 @ X5 )
% 0.30/0.64               => ( D4 @ Z4 ) ) )
% 0.30/0.64         => ( ( D4 @ X3 )
% 0.30/0.64           => ( ! [X5: nat] :
% 0.30/0.64                  ( ( D4 @ X5 )
% 0.30/0.64                 => ( ! [Z3: nat] :
% 0.30/0.64                        ( ( R2 @ Z3 @ X5 )
% 0.30/0.64                       => ( P @ Z3 ) )
% 0.30/0.64                   => ( P @ X5 ) ) )
% 0.30/0.64             => ( P @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % accp_subset_induct
% 0.30/0.64  thf(fact_7766_sin__pi__divide__n__gt__0,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.64       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_pi_divide_n_gt_0
% 0.30/0.64  thf(fact_7767_sin__zero__lemma,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ( sin_real @ X3 )
% 0.30/0.64            = zero_zero_real )
% 0.30/0.64         => ? [N3: nat] :
% 0.30/0.64              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero_lemma
% 0.30/0.64  thf(fact_7768_sin__zero__iff,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        = ( ? [N2: nat] :
% 0.30/0.64              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.64          | ? [N2: nat] :
% 0.30/0.64              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero_iff
% 0.30/0.64  thf(fact_7769_cos__zero__lemma,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ( cos_real @ X3 )
% 0.30/0.64            = zero_zero_real )
% 0.30/0.64         => ? [N3: nat] :
% 0.30/0.64              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_zero_lemma
% 0.30/0.64  thf(fact_7770_cos__zero__iff,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        = ( ? [N2: nat] :
% 0.30/0.64              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.64          | ? [N2: nat] :
% 0.30/0.64              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64              & ( X3
% 0.30/0.64                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_zero_iff
% 0.30/0.64  thf(fact_7771_mono__SucI1,axiom,
% 0.30/0.64      ! [X9: nat > real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_7772_mono__SucI1,axiom,
% 0.30/0.64      ! [X9: nat > set_int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_7773_mono__SucI1,axiom,
% 0.30/0.64      ! [X9: nat > rat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_7774_mono__SucI1,axiom,
% 0.30/0.64      ! [X9: nat > num] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_num @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_7775_mono__SucI1,axiom,
% 0.30/0.64      ! [X9: nat > nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_7776_mono__SucI1,axiom,
% 0.30/0.64      ! [X9: nat > int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_7777_mono__SucI2,axiom,
% 0.30/0.64      ! [X9: nat > real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_7778_mono__SucI2,axiom,
% 0.30/0.64      ! [X9: nat > set_int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_7779_mono__SucI2,axiom,
% 0.30/0.64      ! [X9: nat > rat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_7780_mono__SucI2,axiom,
% 0.30/0.64      ! [X9: nat > num] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_num @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_7781_mono__SucI2,axiom,
% 0.30/0.64      ! [X9: nat > nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_7782_mono__SucI2,axiom,
% 0.30/0.64      ! [X9: nat > int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_7783_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo6980174941875973593q_real
% 0.30/0.64      = ( ^ [X7: nat > real] :
% 0.30/0.64            ( ! [N2: nat] : ( ord_less_eq_real @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 0.30/0.64            | ! [N2: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_7784_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo3100542954746470799et_int
% 0.30/0.64      = ( ^ [X7: nat > set_int] :
% 0.30/0.64            ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 0.30/0.64            | ! [N2: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_7785_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo4267028734544971653eq_rat
% 0.30/0.64      = ( ^ [X7: nat > rat] :
% 0.30/0.64            ( ! [N2: nat] : ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 0.30/0.64            | ! [N2: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_7786_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo1459490580787246023eq_num
% 0.30/0.64      = ( ^ [X7: nat > num] :
% 0.30/0.64            ( ! [N2: nat] : ( ord_less_eq_num @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 0.30/0.64            | ! [N2: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_7787_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo4902158794631467389eq_nat
% 0.30/0.64      = ( ^ [X7: nat > nat] :
% 0.30/0.64            ( ! [N2: nat] : ( ord_less_eq_nat @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 0.30/0.64            | ! [N2: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_7788_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo4899668324122417113eq_int
% 0.30/0.64      = ( ^ [X7: nat > int] :
% 0.30/0.64            ( ! [N2: nat] : ( ord_less_eq_int @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 0.30/0.64            | ! [N2: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_7789_monoI1,axiom,
% 0.30/0.64      ! [X9: nat > real] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_7790_monoI1,axiom,
% 0.30/0.64      ! [X9: nat > set_int] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_set_int @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_7791_monoI1,axiom,
% 0.30/0.64      ! [X9: nat > rat] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_rat @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_7792_monoI1,axiom,
% 0.30/0.64      ! [X9: nat > num] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_num @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_7793_monoI1,axiom,
% 0.30/0.64      ! [X9: nat > nat] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_nat @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_7794_monoI1,axiom,
% 0.30/0.64      ! [X9: nat > int] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_int @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_7795_monoI2,axiom,
% 0.30/0.64      ! [X9: nat > real] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_7796_monoI2,axiom,
% 0.30/0.64      ! [X9: nat > set_int] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_set_int @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_7797_monoI2,axiom,
% 0.30/0.64      ! [X9: nat > rat] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_7798_monoI2,axiom,
% 0.30/0.64      ! [X9: nat > num] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_num @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_7799_monoI2,axiom,
% 0.30/0.64      ! [X9: nat > nat] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_7800_monoI2,axiom,
% 0.30/0.64      ! [X9: nat > int] :
% 0.30/0.64        ( ! [M4: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.30/0.64           => ( ord_less_eq_int @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_7801_vebt__maxt_Opelims,axiom,
% 0.30/0.64      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.64        ( ( ( vEBT_vebt_maxt @ X3 )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
% 0.30/0.64         => ( ! [A3: $o,B3: $o] :
% 0.30/0.64                ( ( X3
% 0.30/0.64                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.64               => ( ( ( B3
% 0.30/0.64                     => ( Y3
% 0.30/0.64                        = ( some_nat @ one_one_nat ) ) )
% 0.30/0.64                    & ( ~ B3
% 0.30/0.64                     => ( ( A3
% 0.30/0.64                         => ( Y3
% 0.30/0.64                            = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.64                        & ( ~ A3
% 0.30/0.64                         => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.64                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 0.30/0.64           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.64                  ( ( X3
% 0.30/0.64                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.64                 => ( ( Y3 = none_nat )
% 0.30/0.64                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.64             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.64                    ( ( X3
% 0.30/0.64                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.64                   => ( ( Y3
% 0.30/0.64                        = ( some_nat @ Ma2 ) )
% 0.30/0.64                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % vebt_maxt.pelims
% 0.30/0.64  thf(fact_7802_vebt__mint_Opelims,axiom,
% 0.30/0.64      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.64        ( ( ( vEBT_vebt_mint @ X3 )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
% 0.30/0.64         => ( ! [A3: $o,B3: $o] :
% 0.30/0.64                ( ( X3
% 0.30/0.64                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.30/0.64               => ( ( ( A3
% 0.30/0.64                     => ( Y3
% 0.30/0.64                        = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.64                    & ( ~ A3
% 0.30/0.64                     => ( ( B3
% 0.30/0.64                         => ( Y3
% 0.30/0.64                            = ( some_nat @ one_one_nat ) ) )
% 0.30/0.64                        & ( ~ B3
% 0.30/0.64                         => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.64                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 0.30/0.64           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.64                  ( ( X3
% 0.30/0.64                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.64                 => ( ( Y3 = none_nat )
% 0.30/0.64                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.64             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.64                    ( ( X3
% 0.30/0.64                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.64                   => ( ( Y3
% 0.30/0.64                        = ( some_nat @ Mi2 ) )
% 0.30/0.64                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % vebt_mint.pelims
% 0.30/0.64  thf(fact_7803_Maclaurin__cos__expansion2,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64         => ? [T4: real] :
% 0.30/0.64              ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/0.64              & ( ord_less_real @ T4 @ 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                    @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_cos_expansion2
% 0.30/0.64  thf(fact_7804_Maclaurin__minus__cos__expansion,axiom,
% 0.30/0.64      ! [N: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.64         => ? [T4: real] :
% 0.30/0.64              ( ( ord_less_real @ X3 @ T4 )
% 0.30/0.64              & ( ord_less_real @ T4 @ 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                    @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_minus_cos_expansion
% 0.30/0.64  thf(fact_7805_fact__Suc,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.30/0.64        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_7806_fact__Suc,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.30/0.64        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_7807_fact__Suc,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.30/0.64        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_7808_fact__Suc,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.30/0.64        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_7809_fact__2,axiom,
% 0.30/0.64      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_2
% 0.30/0.64  thf(fact_7810_fact__2,axiom,
% 0.30/0.64      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_2
% 0.30/0.64  thf(fact_7811_fact__2,axiom,
% 0.30/0.64      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_2
% 0.30/0.64  thf(fact_7812_fact__2,axiom,
% 0.30/0.64      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_2
% 0.30/0.64  thf(fact_7813_fact__2,axiom,
% 0.30/0.64      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.64      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_2
% 0.30/0.64  thf(fact_7814_fact__ge__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_7815_fact__ge__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_7816_fact__ge__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_7817_fact__ge__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_7818_fact__not__neg,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_7819_fact__not__neg,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_7820_fact__not__neg,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_7821_fact__not__neg,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_7822_fact__gt__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_7823_fact__gt__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_7824_fact__gt__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_7825_fact__gt__zero,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_7826_fact__ge__1,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_7827_fact__ge__1,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_7828_fact__ge__1,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_7829_fact__ge__1,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_7830_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_7831_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_7832_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_7833_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_7834_fact__dvd,axiom,
% 0.30/0.64      ! [N: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_7835_fact__dvd,axiom,
% 0.30/0.64      ! [N: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_7836_fact__dvd,axiom,
% 0.30/0.64      ! [N: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_7837_fact__dvd,axiom,
% 0.30/0.64      ! [N: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.64       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_7838_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N )
% 0.30/0.64         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_7839_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N )
% 0.30/0.64         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_7840_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N )
% 0.30/0.64         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_7841_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N )
% 0.30/0.64         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_7842_fact__mod,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( 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_7843_fact__mod,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.30/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mod
% 0.30/0.64  thf(fact_7844_fact__mod,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( 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_7845_fact__le__power,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_7846_fact__le__power,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_7847_fact__le__power,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_7848_fact__le__power,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_7849_fact__numeral,axiom,
% 0.30/0.64      ! [K: num] :
% 0.30/0.64        ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64        = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_numeral
% 0.30/0.64  thf(fact_7850_fact__numeral,axiom,
% 0.30/0.64      ! [K: num] :
% 0.30/0.64        ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64        = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_numeral
% 0.30/0.64  thf(fact_7851_fact__numeral,axiom,
% 0.30/0.64      ! [K: num] :
% 0.30/0.64        ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64        = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_numeral
% 0.30/0.64  thf(fact_7852_fact__numeral,axiom,
% 0.30/0.64      ! [K: num] :
% 0.30/0.64        ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64        = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_numeral
% 0.30/0.64  thf(fact_7853_fact__numeral,axiom,
% 0.30/0.64      ! [K: num] :
% 0.30/0.64        ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.64        = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_numeral
% 0.30/0.64  thf(fact_7854_square__fact__le__2__fact,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % square_fact_le_2_fact
% 0.30/0.64  thf(fact_7855_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri5044797733671781792omplex
% 0.30/0.64      = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_7856_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri773545260158071498ct_rat
% 0.30/0.64      = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_7857_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri1406184849735516958ct_int
% 0.30/0.64      = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_7858_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri2265585572941072030t_real
% 0.30/0.64      = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_7859_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri1408675320244567234ct_nat
% 0.30/0.64      = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_7860_fact__reduce,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( semiri773545260158071498ct_rat @ N )
% 0.30/0.64          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_7861_fact__reduce,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( semiri1406184849735516958ct_int @ N )
% 0.30/0.64          = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_7862_fact__reduce,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( semiri2265585572941072030t_real @ N )
% 0.30/0.64          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_7863_fact__reduce,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( semiri1408675320244567234ct_nat @ N )
% 0.30/0.64          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_7864_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Diff: nat > complex > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 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_7865_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Diff: nat > real > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 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_7866_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Diff: nat > rat > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 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_7867_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Diff: nat > nat > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 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_7868_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N: nat,Diff: nat > int > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 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_7869_Maclaurin__lemma,axiom,
% 0.30/0.64      ! [H2: real,F: real > real,J: nat > real,N: 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                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.64                @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64              @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_lemma
% 0.30/0.64  thf(fact_7870_cos__paired,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.64        @ ( cos_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_paired
% 0.30/0.64  thf(fact_7871_cos__coeff__def,axiom,
% 0.30/0.64      ( cos_coeff
% 0.30/0.64      = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_coeff_def
% 0.30/0.64  thf(fact_7872_sin__paired,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ 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_7873_Maclaurin__cos__expansion,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64      ? [T4: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ 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              @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_cos_expansion
% 0.30/0.64  thf(fact_7874_Maclaurin__sin__expansion3,axiom,
% 0.30/0.64      ! [N: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ? [T4: real] :
% 0.30/0.64              ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/0.64              & ( ord_less_real @ T4 @ 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                    @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64                  @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_sin_expansion3
% 0.30/0.64  thf(fact_7875_Maclaurin__sin__expansion4,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ? [T4: real] :
% 0.30/0.64            ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/0.64            & ( ord_less_eq_real @ T4 @ 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                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64                  @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_sin_expansion4
% 0.30/0.64  thf(fact_7876_Maclaurin__sin__expansion2,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64      ? [T4: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ 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              @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_sin_expansion2
% 0.30/0.64  thf(fact_7877_Maclaurin__sin__expansion,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64      ? [T4: real] :
% 0.30/0.64        ( ( sin_real @ X3 )
% 0.30/0.64        = ( plus_plus_real
% 0.30/0.64          @ ( groups6591440286371151544t_real
% 0.30/0.64            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.64            @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_sin_expansion
% 0.30/0.64  thf(fact_7878_sin__coeff__def,axiom,
% 0.30/0.64      ( sin_coeff
% 0.30/0.64      = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_coeff_def
% 0.30/0.64  thf(fact_7879_fact__ge__self,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_self
% 0.30/0.64  thf(fact_7880_fact__mono__nat,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono_nat
% 0.30/0.64  thf(fact_7881_fact__less__mono__nat,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N )
% 0.30/0.64         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono_nat
% 0.30/0.64  thf(fact_7882_fact__ge__Suc__0__nat,axiom,
% 0.30/0.64      ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_Suc_0_nat
% 0.30/0.64  thf(fact_7883_dvd__fact,axiom,
% 0.30/0.64      ! [M: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.30/0.64       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.64         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % dvd_fact
% 0.30/0.64  thf(fact_7884_fact__diff__Suc,axiom,
% 0.30/0.64      ! [N: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.30/0.64       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.30/0.64          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_diff_Suc
% 0.30/0.64  thf(fact_7885_fact__div__fact__le__pow,axiom,
% 0.30/0.64      ! [R: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ R @ N )
% 0.30/0.64       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_div_fact_le_pow
% 0.30/0.64  thf(fact_7886_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_7887_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_7888_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_7889_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_7890_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.64       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_7891_fold__atLeastAtMost__nat_Opsimps,axiom,
% 0.30/0.64      ! [F: nat > nat > nat,A: nat,B2: nat,Acc: nat] :
% 0.30/0.64        ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) )
% 0.30/0.64       => ( ( ( ord_less_nat @ B2 @ A )
% 0.30/0.64           => ( ( set_fo2584398358068434914at_nat @ F @ A @ B2 @ Acc )
% 0.30/0.64              = Acc ) )
% 0.30/0.64          & ( ~ ( ord_less_nat @ B2 @ A )
% 0.30/0.64           => ( ( set_fo2584398358068434914at_nat @ F @ A @ B2 @ Acc )
% 0.30/0.64              = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B2 @ ( F @ A @ Acc ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fold_atLeastAtMost_nat.psimps
% 0.30/0.64  thf(fact_7892_fold__atLeastAtMost__nat_Opelims,axiom,
% 0.30/0.64      ! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y3: nat] :
% 0.30/0.64        ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) )
% 0.30/0.64         => ~ ( ( ( ( ord_less_nat @ Xb2 @ Xa2 )
% 0.30/0.64                 => ( Y3 = Xc ) )
% 0.30/0.64                & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
% 0.30/0.64                 => ( Y3
% 0.30/0.64                    = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) )
% 0.30/0.64             => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fold_atLeastAtMost_nat.pelims
% 0.30/0.64  thf(fact_7893_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.30/0.64      ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.30/0.64        ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb2 )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ Xa2 @ Xb2 ) ) )
% 0.30/0.64         => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.30/0.64             => ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.64               => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb2 ) ) ) ) )
% 0.30/0.64           => ( ! [V2: product_prod_nat_nat] :
% 0.30/0.64                  ( ( Xa2
% 0.30/0.64                    = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.30/0.64                 => ( ( Xb2 = none_P5556105721700978146at_nat )
% 0.30/0.64                   => ( ( Y3 = none_P5556105721700978146at_nat )
% 0.30/0.64                     => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
% 0.30/0.64             => ~ ! [A3: product_prod_nat_nat] :
% 0.30/0.64                    ( ( Xa2
% 0.30/0.64                      = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.64                   => ! [B3: product_prod_nat_nat] :
% 0.30/0.64                        ( ( Xb2
% 0.30/0.64                          = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.30/0.64                       => ( ( Y3
% 0.30/0.64                            = ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) )
% 0.30/0.64                         => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % VEBT_internal.option_shift.pelims
% 0.30/0.64  thf(fact_7894_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.30/0.64      ! [X3: num > num > num,Xa2: option_num,Xb2: option_num,Y3: option_num] :
% 0.30/0.64        ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb2 )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ Xa2 @ Xb2 ) ) )
% 0.30/0.64         => ( ( ( Xa2 = none_num )
% 0.30/0.64             => ( ( Y3 = none_num )
% 0.30/0.64               => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ none_num @ Xb2 ) ) ) ) )
% 0.30/0.64           => ( ! [V2: num] :
% 0.30/0.64                  ( ( Xa2
% 0.30/0.64                    = ( some_num @ V2 ) )
% 0.30/0.64                 => ( ( Xb2 = none_num )
% 0.30/0.64                   => ( ( Y3 = none_num )
% 0.30/0.64                     => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
% 0.30/0.64             => ~ ! [A3: num] :
% 0.30/0.64                    ( ( Xa2
% 0.30/0.64                      = ( some_num @ A3 ) )
% 0.30/0.64                   => ! [B3: num] :
% 0.30/0.64                        ( ( Xb2
% 0.30/0.64                          = ( some_num @ B3 ) )
% 0.30/0.64                       => ( ( Y3
% 0.30/0.64                            = ( some_num @ ( X3 @ A3 @ B3 ) ) )
% 0.30/0.64                         => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % VEBT_internal.option_shift.pelims
% 0.30/0.64  thf(fact_7895_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.30/0.64      ! [X3: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y3: option_nat] :
% 0.30/0.64        ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb2 )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb2 ) ) )
% 0.30/0.64         => ( ( ( Xa2 = none_nat )
% 0.30/0.64             => ( ( Y3 = none_nat )
% 0.30/0.64               => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ none_nat @ Xb2 ) ) ) ) )
% 0.30/0.64           => ( ! [V2: nat] :
% 0.30/0.64                  ( ( Xa2
% 0.30/0.64                    = ( some_nat @ V2 ) )
% 0.30/0.64                 => ( ( Xb2 = none_nat )
% 0.30/0.64                   => ( ( Y3 = none_nat )
% 0.30/0.64                     => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
% 0.30/0.64             => ~ ! [A3: nat] :
% 0.30/0.64                    ( ( Xa2
% 0.30/0.64                      = ( some_nat @ A3 ) )
% 0.30/0.64                   => ! [B3: nat] :
% 0.30/0.64                        ( ( Xb2
% 0.30/0.64                          = ( some_nat @ B3 ) )
% 0.30/0.64                       => ( ( Y3
% 0.30/0.64                            = ( some_nat @ ( X3 @ A3 @ B3 ) ) )
% 0.30/0.64                         => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % VEBT_internal.option_shift.pelims
% 0.30/0.64  thf(fact_7896_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_7897_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_7898_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_7899_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_7900_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_7901_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.30/0.64      ! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y3: nat] :
% 0.30/0.64        ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
% 0.30/0.64          = Y3 )
% 0.30/0.64       => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
% 0.30/0.64           => ( Y3 = Xc ) )
% 0.30/0.64          & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
% 0.30/0.64           => ( Y3
% 0.30/0.64              = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fold_atLeastAtMost_nat.elims
% 0.30/0.64  thf(fact_7902_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.30/0.64      ( set_fo2584398358068434914at_nat
% 0.30/0.64      = ( ^ [F3: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F3 @ A5 @ Acc2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fold_atLeastAtMost_nat.simps
% 0.30/0.64  thf(fact_7903_sum__atLeastAtMost__code,axiom,
% 0.30/0.64      ! [F: nat > complex,A: nat,B2: nat] :
% 0.30/0.64        ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.64        = ( set_fo1517530859248394432omplex
% 0.30/0.64          @ ^ [A5: nat] : ( plus_plus_complex @ ( F @ A5 ) )
% 0.30/0.64          @ A
% 0.30/0.64          @ B2
% 0.30/0.64          @ zero_zero_complex ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_atLeastAtMost_code
% 0.30/0.64  thf(fact_7904_sum__atLeastAtMost__code,axiom,
% 0.30/0.64      ! [F: nat > rat,A: nat,B2: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.64        = ( set_fo1949268297981939178at_rat
% 0.30/0.64          @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
% 0.30/0.64          @ A
% 0.30/0.64          @ B2
% 0.30/0.64          @ zero_zero_rat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_atLeastAtMost_code
% 0.30/0.64  thf(fact_7905_sum__atLeastAtMost__code,axiom,
% 0.30/0.64      ! [F: nat > int,A: nat,B2: nat] :
% 0.30/0.64        ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.64        = ( set_fo2581907887559384638at_int
% 0.30/0.64          @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
% 0.30/0.64          @ A
% 0.30/0.64          @ B2
% 0.30/0.64          @ zero_zero_int ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_atLeastAtMost_code
% 0.30/0.64  thf(fact_7906_sum__atLeastAtMost__code,axiom,
% 0.30/0.64      ! [F: nat > nat,A: nat,B2: nat] :
% 0.30/0.64        ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.64        = ( set_fo2584398358068434914at_nat
% 0.30/0.64          @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
% 0.30/0.64          @ A
% 0.30/0.64          @ B2
% 0.30/0.64          @ zero_zero_nat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_atLeastAtMost_code
% 0.30/0.64  thf(fact_7907_sum__atLeastAtMost__code,axiom,
% 0.30/0.64      ! [F: nat > real,A: nat,B2: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.64        = ( set_fo3111899725591712190t_real
% 0.30/0.64          @ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
% 0.30/0.64          @ A
% 0.30/0.64          @ B2
% 0.30/0.64          @ zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_atLeastAtMost_code
% 0.30/0.64  thf(fact_7908_fact__code,axiom,
% 0.30/0.64      ( semiri1406184849735516958ct_int
% 0.30/0.64      = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_code
% 0.30/0.64  thf(fact_7909_fact__code,axiom,
% 0.30/0.64      ( semiri2265585572941072030t_real
% 0.30/0.64      = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_code
% 0.30/0.64  thf(fact_7910_fact__code,axiom,
% 0.30/0.64      ( semiri1408675320244567234ct_nat
% 0.30/0.64      = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_code
% 0.30/0.64  thf(fact_7911_diffs__equiv,axiom,
% 0.30/0.64      ! [C: nat > complex,X3: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.64          @ ( suminf_complex
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_equiv
% 0.30/0.64  thf(fact_7912_diffs__equiv,axiom,
% 0.30/0.64      ! [C: nat > real,X3: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.64          @ ( suminf_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_equiv
% 0.30/0.64  thf(fact_7913_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_7914_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_7915_in__measure,axiom,
% 0.30/0.64      ! [X3: int,Y3: int,F: int > nat] :
% 0.30/0.64        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( measure_int @ F ) )
% 0.30/0.64        = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % in_measure
% 0.30/0.64  thf(fact_7916_complex__unimodular__polar,axiom,
% 0.30/0.64      ! [Z: complex] :
% 0.30/0.64        ( ( ( real_V1022390504157884413omplex @ Z )
% 0.30/0.64          = one_one_real )
% 0.30/0.64       => ~ ! [T4: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.64             => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64               => ( Z
% 0.30/0.64                 != ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % complex_unimodular_polar
% 0.30/0.64  thf(fact_7917_Maclaurin__exp__lt,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( X3 != zero_zero_real )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64         => ? [T4: real] :
% 0.30/0.64              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
% 0.30/0.64              & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( 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                    @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_exp_lt
% 0.30/0.64  thf(fact_7918_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_7919_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_7920_exp__le__cancel__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_le_cancel_iff
% 0.30/0.64  thf(fact_7921_tan__npi,axiom,
% 0.30/0.64      ! [N: nat] :
% 0.30/0.64        ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_npi
% 0.30/0.64  thf(fact_7922_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_7923_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_7924_one__le__exp__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_le_exp_iff
% 0.30/0.64  thf(fact_7925_exp__le__one__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( exp_real @ X3 ) @ one_one_real )
% 0.30/0.64        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_le_one_iff
% 0.30/0.64  thf(fact_7926_tan__periodic__n,axiom,
% 0.30/0.64      ! [X3: real,N: num] :
% 0.30/0.64        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.30/0.64        = ( tan_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_periodic_n
% 0.30/0.64  thf(fact_7927_tan__periodic__nat,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.30/0.64        = ( tan_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_periodic_nat
% 0.30/0.64  thf(fact_7928_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_7929_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_7930_tan__periodic,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.64        = ( tan_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_periodic
% 0.30/0.64  thf(fact_7931_norm__exp,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X3 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_exp
% 0.30/0.64  thf(fact_7932_norm__exp,axiom,
% 0.30/0.64      ! [X3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X3 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_exp
% 0.30/0.64  thf(fact_7933_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_7934_exp__times__arg__commute,axiom,
% 0.30/0.64      ! [A4: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( exp_complex @ A4 ) @ A4 )
% 0.30/0.64        = ( times_times_complex @ A4 @ ( exp_complex @ A4 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_times_arg_commute
% 0.30/0.64  thf(fact_7935_exp__times__arg__commute,axiom,
% 0.30/0.64      ! [A4: real] :
% 0.30/0.64        ( ( times_times_real @ ( exp_real @ A4 ) @ A4 )
% 0.30/0.64        = ( times_times_real @ A4 @ ( exp_real @ A4 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_times_arg_commute
% 0.30/0.64  thf(fact_7936_exp__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64       => ? [X5: real] :
% 0.30/0.64            ( ( exp_real @ X5 )
% 0.30/0.64            = Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_total
% 0.30/0.64  thf(fact_7937_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_7938_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_7939_exp__ge__zero,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ge_zero
% 0.30/0.64  thf(fact_7940_not__exp__le__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ~ ( ord_less_eq_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % not_exp_le_zero
% 0.30/0.64  thf(fact_7941_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_7942_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_7943_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_7944_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_7945_exp__diff,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( exp_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_diff
% 0.30/0.64  thf(fact_7946_exp__diff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( exp_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( divide_divide_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_diff
% 0.30/0.64  thf(fact_7947_Complex__eq__numeral,axiom,
% 0.30/0.64      ! [A: real,B2: real,W: num] :
% 0.30/0.64        ( ( ( complex2 @ A @ B2 )
% 0.30/0.64          = ( numera6690914467698888265omplex @ W ) )
% 0.30/0.64        = ( ( A
% 0.30/0.64            = ( numeral_numeral_real @ W ) )
% 0.30/0.64          & ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Complex_eq_numeral
% 0.30/0.64  thf(fact_7948_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_7949_exp__ge__add__one__self,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ge_add_one_self
% 0.30/0.64  thf(fact_7950_exp__minus__inverse,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_minus_inverse
% 0.30/0.64  thf(fact_7951_exp__minus__inverse,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) )
% 0.30/0.64        = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_minus_inverse
% 0.30/0.64  thf(fact_7952_exp__of__nat2__mult,axiom,
% 0.30/0.64      ! [X3: complex,N: nat] :
% 0.30/0.64        ( ( exp_complex @ ( times_times_complex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.30/0.64        = ( power_power_complex @ ( exp_complex @ X3 ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_of_nat2_mult
% 0.30/0.64  thf(fact_7953_exp__of__nat2__mult,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( exp_real @ ( times_times_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.64        = ( power_power_real @ ( exp_real @ X3 ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_of_nat2_mult
% 0.30/0.64  thf(fact_7954_exp__of__nat__mult,axiom,
% 0.30/0.64      ! [N: nat,X3: complex] :
% 0.30/0.64        ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X3 ) )
% 0.30/0.64        = ( power_power_complex @ ( exp_complex @ X3 ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_of_nat_mult
% 0.30/0.64  thf(fact_7955_exp__of__nat__mult,axiom,
% 0.30/0.64      ! [N: nat,X3: real] :
% 0.30/0.64        ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) )
% 0.30/0.64        = ( power_power_real @ ( exp_real @ X3 ) @ N ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_of_nat_mult
% 0.30/0.64  thf(fact_7956_Complex__eq__neg__numeral,axiom,
% 0.30/0.64      ! [A: real,B2: real,W: num] :
% 0.30/0.64        ( ( ( complex2 @ A @ B2 )
% 0.30/0.64          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.64        = ( ( A
% 0.30/0.64            = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.64          & ( B2 = zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Complex_eq_neg_numeral
% 0.30/0.64  thf(fact_7957_complex__mult,axiom,
% 0.30/0.64      ! [A: real,B2: real,C: real,D: real] :
% 0.30/0.64        ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 0.30/0.64        = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % complex_mult
% 0.30/0.64  thf(fact_7958_tan__def,axiom,
% 0.30/0.64      ( tan_complex
% 0.30/0.64      = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_def
% 0.30/0.64  thf(fact_7959_tan__def,axiom,
% 0.30/0.64      ( tan_real
% 0.30/0.64      = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_def
% 0.30/0.64  thf(fact_7960_exp__ge__add__one__self__aux,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_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ge_add_one_self_aux
% 0.30/0.64  thf(fact_7961_lemma__exp__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 0.30/0.64       => ? [X5: real] :
% 0.30/0.64            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64            & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 0.30/0.64            & ( ( exp_real @ X5 )
% 0.30/0.64              = Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_exp_total
% 0.30/0.64  thf(fact_7962_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_7963_ln__x__over__x__mono,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.64         => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ln_x_over_x_mono
% 0.30/0.64  thf(fact_7964_diffs__def,axiom,
% 0.30/0.64      ( diffs_rat
% 0.30/0.64      = ( ^ [C3: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_7965_diffs__def,axiom,
% 0.30/0.64      ( diffs_int
% 0.30/0.64      = ( ^ [C3: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_7966_diffs__def,axiom,
% 0.30/0.64      ( diffs_real
% 0.30/0.64      = ( ^ [C3: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_7967_exp__le,axiom,
% 0.30/0.64      ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_le
% 0.30/0.64  thf(fact_7968_termdiff__converges__all,axiom,
% 0.30/0.64      ! [C: nat > complex,X3: complex] :
% 0.30/0.64        ( ! [X5: complex] :
% 0.30/0.64            ( summable_complex
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) )
% 0.30/0.64       => ( summable_complex
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % termdiff_converges_all
% 0.30/0.64  thf(fact_7969_termdiff__converges__all,axiom,
% 0.30/0.64      ! [C: nat > real,X3: real] :
% 0.30/0.64        ( ! [X5: real] :
% 0.30/0.64            ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % termdiff_converges_all
% 0.30/0.64  thf(fact_7970_exp__divide__power__eq,axiom,
% 0.30/0.64      ! [N: nat,X3: complex] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 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_7971_exp__divide__power__eq,axiom,
% 0.30/0.64      ! [N: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 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_7972_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_7973_tanh__altdef,axiom,
% 0.30/0.64      ( tanh_complex
% 0.30/0.64      = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tanh_altdef
% 0.30/0.64  thf(fact_7974_exp__half__le2,axiom,
% 0.30/0.64      ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_half_le2
% 0.30/0.64  thf(fact_7975_tan__45,axiom,
% 0.30/0.64      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_45
% 0.30/0.64  thf(fact_7976_exp__double,axiom,
% 0.30/0.64      ! [Z: complex] :
% 0.30/0.64        ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 0.30/0.64        = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_double
% 0.30/0.64  thf(fact_7977_exp__double,axiom,
% 0.30/0.64      ! [Z: real] :
% 0.30/0.64        ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 0.30/0.64        = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_double
% 0.30/0.64  thf(fact_7978_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       => ? [X5: real] :
% 0.30/0.64            ( ( ord_less_real @ zero_zero_real @ X5 )
% 0.30/0.64            & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64            & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_tan_total
% 0.30/0.64  thf(fact_7979_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_7980_lemma__tan__total1,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64      ? [X5: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.30/0.64        & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        & ( ( tan_real @ X5 )
% 0.30/0.64          = Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_tan_total1
% 0.30/0.64  thf(fact_7981_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_7982_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_7983_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_7984_tan__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64      ? [X5: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.30/0.64        & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        & ( ( tan_real @ X5 )
% 0.30/0.64          = Y3 )
% 0.30/0.64        & ! [Y5: real] :
% 0.30/0.64            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.30/0.64              & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64              & ( ( tan_real @ Y5 )
% 0.30/0.64                = Y3 ) )
% 0.30/0.64           => ( Y5 = X5 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_total
% 0.30/0.64  thf(fact_7985_tan__minus__45,axiom,
% 0.30/0.64      ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.64      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_minus_45
% 0.30/0.64  thf(fact_7986_tan__inverse,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 0.30/0.64        = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_inverse
% 0.30/0.64  thf(fact_7987_exp__bound__half,axiom,
% 0.30/0.64      ! [Z: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_bound_half
% 0.30/0.64  thf(fact_7988_exp__bound__half,axiom,
% 0.30/0.64      ! [Z: complex] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_bound_half
% 0.30/0.64  thf(fact_7989_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_7990_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_7991_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       => ( ! [X5: real] :
% 0.30/0.64              ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
% 0.30/0.64             => ( summable_real
% 0.30/0.64                @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % termdiff_converges
% 0.30/0.64  thf(fact_7992_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       => ( ! [X5: complex] :
% 0.30/0.64              ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
% 0.30/0.64             => ( summable_complex
% 0.30/0.64                @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) ) )
% 0.30/0.64         => ( summable_complex
% 0.30/0.64            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % termdiff_converges
% 0.30/0.64  thf(fact_7993_exp__bound,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_eq_real @ X3 @ one_one_real )
% 0.30/0.64         => ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_bound
% 0.30/0.64  thf(fact_7994_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_7995_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       => ? [X5: real] :
% 0.30/0.64            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.30/0.64            & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64            & ( ( tan_real @ X5 )
% 0.30/0.64              = Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_total_pos
% 0.30/0.64  thf(fact_7996_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_7997_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_7998_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_7999_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_8000_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_8001_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_8002_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_8003_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_8004_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_8005_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_8006_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_8007_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_8008_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_8009_real__exp__bound__lemma,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_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_exp_bound_lemma
% 0.30/0.64  thf(fact_8010_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.30/0.64      ! [N: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( 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_8011_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64        ( ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.64         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( 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_8012_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       => ? [Z4: real] :
% 0.30/0.64            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 0.30/0.64            & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64            & ( ( tan_real @ Z4 )
% 0.30/0.64              = X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_total_pi4
% 0.30/0.64  thf(fact_8013_exp__bound__lemma,axiom,
% 0.30/0.64      ! [Z: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_bound_lemma
% 0.30/0.64  thf(fact_8014_exp__bound__lemma,axiom,
% 0.30/0.64      ! [Z: complex] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_bound_lemma
% 0.30/0.64  thf(fact_8015_Maclaurin__exp__le,axiom,
% 0.30/0.64      ! [X3: real,N: nat] :
% 0.30/0.64      ? [T4: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( 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              @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.64            @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_exp_le
% 0.30/0.64  thf(fact_8016_exp__lower__Taylor__quadratic,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_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( divide_divide_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_lower_Taylor_quadratic
% 0.30/0.64  thf(fact_8017_tanh__real__altdef,axiom,
% 0.30/0.64      ( tanh_real
% 0.30/0.64      = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tanh_real_altdef
% 0.30/0.64  thf(fact_8018_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_8019_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_8020_binomial__code,axiom,
% 0.30/0.64      ( binomial
% 0.30/0.64      = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_code
% 0.30/0.64  thf(fact_8021_in__finite__psubset,axiom,
% 0.30/0.64      ! [A4: set_nat,B5: set_nat] :
% 0.30/0.64        ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B5 ) @ finite_psubset_nat )
% 0.30/0.64        = ( ( ord_less_set_nat @ A4 @ B5 )
% 0.30/0.64          & ( finite_finite_nat @ B5 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % in_finite_psubset
% 0.30/0.64  thf(fact_8022_in__finite__psubset,axiom,
% 0.30/0.64      ! [A4: set_int,B5: set_int] :
% 0.30/0.64        ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A4 @ B5 ) @ finite_psubset_int )
% 0.30/0.64        = ( ( ord_less_set_int @ A4 @ B5 )
% 0.30/0.64          & ( finite_finite_int @ B5 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % in_finite_psubset
% 0.30/0.64  thf(fact_8023_in__finite__psubset,axiom,
% 0.30/0.64      ! [A4: set_complex,B5: set_complex] :
% 0.30/0.64        ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A4 @ B5 ) @ finite8643634255014194347omplex )
% 0.30/0.64        = ( ( ord_less_set_complex @ A4 @ B5 )
% 0.30/0.64          & ( finite3207457112153483333omplex @ B5 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % in_finite_psubset
% 0.30/0.64  thf(fact_8024_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_8025_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_8026_pochhammer__double,axiom,
% 0.30/0.64      ! [Z: complex,N: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 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 ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_double
% 0.30/0.64  thf(fact_8027_pochhammer__double,axiom,
% 0.30/0.64      ! [Z: rat,N: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 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 ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_double
% 0.30/0.64  thf(fact_8028_pochhammer__double,axiom,
% 0.30/0.64      ! [Z: real,N: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 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 ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_double
% 0.30/0.64  thf(fact_8029_real__sqrt__eq__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( sqrt @ X3 )
% 0.30/0.64          = ( sqrt @ Y3 ) )
% 0.30/0.64        = ( X3 = Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_eq_iff
% 0.30/0.64  thf(fact_8030_real__sqrt__zero,axiom,
% 0.30/0.64      ( ( sqrt @ zero_zero_real )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_zero
% 0.30/0.64  thf(fact_8031_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( sqrt @ X3 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_eq_zero_cancel_iff
% 0.30/0.64  thf(fact_8032_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_8033_real__sqrt__le__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_le_iff
% 0.30/0.64  thf(fact_8034_real__sqrt__one,axiom,
% 0.30/0.64      ( ( sqrt @ one_one_real )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_one
% 0.30/0.64  thf(fact_8035_real__sqrt__eq__1__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( sqrt @ X3 )
% 0.30/0.64          = one_one_real )
% 0.30/0.64        = ( X3 = one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_eq_1_iff
% 0.30/0.64  thf(fact_8036_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_8037_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_8038_real__sqrt__ge__0__iff,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_ge_0_iff
% 0.30/0.64  thf(fact_8039_real__sqrt__le__0__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ zero_zero_real )
% 0.30/0.64        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_le_0_iff
% 0.30/0.64  thf(fact_8040_binomial__eq__0__iff,axiom,
% 0.30/0.64      ! [N: nat,K: nat] :
% 0.30/0.64        ( ( ( binomial @ N @ K )
% 0.30/0.64          = zero_zero_nat )
% 0.30/0.64        = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_eq_0_iff
% 0.30/0.64  thf(fact_8041_binomial__Suc__Suc,axiom,
% 0.30/0.64      ! [N: nat,K: nat] :
% 0.30/0.64        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_Suc_Suc
% 0.30/0.64  thf(fact_8042_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_8043_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_8044_real__sqrt__ge__1__iff,axiom,
% 0.30/0.65      ! [Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.30/0.65        = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_ge_1_iff
% 0.30/0.65  thf(fact_8045_real__sqrt__le__1__iff,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( sqrt @ 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_sqrt_le_1_iff
% 0.30/0.65  thf(fact_8046_real__sqrt__mult__self,axiom,
% 0.30/0.65      ! [A: real] :
% 0.30/0.65        ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.30/0.65        = ( abs_abs_real @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_mult_self
% 0.30/0.65  thf(fact_8047_real__sqrt__abs2,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( sqrt @ ( times_times_real @ X3 @ X3 ) )
% 0.30/0.65        = ( abs_abs_real @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_abs2
% 0.30/0.65  thf(fact_8048_real__sqrt__four,axiom,
% 0.30/0.65      ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.65      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_four
% 0.30/0.65  thf(fact_8049_zero__less__binomial__iff,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.30/0.65        = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_less_binomial_iff
% 0.30/0.65  thf(fact_8050_real__sqrt__abs,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( sqrt @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.65        = ( abs_abs_real @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_abs
% 0.30/0.65  thf(fact_8051_real__sqrt__pow2,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65          = X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_pow2
% 0.30/0.65  thf(fact_8052_real__sqrt__pow2__iff,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65          = X3 )
% 0.30/0.65        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_pow2_iff
% 0.30/0.65  thf(fact_8053_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.30/0.65      ! [X3: real,Y3: real,Xa2: real,Ya: real] :
% 0.30/0.65        ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65        = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_mult_squared_eq
% 0.30/0.65  thf(fact_8054_real__sqrt__less__mono,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.65       => ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_less_mono
% 0.30/0.65  thf(fact_8055_real__sqrt__le__mono,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.65       => ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_le_mono
% 0.30/0.65  thf(fact_8056_real__sqrt__minus,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( sqrt @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.65        = ( uminus_uminus_real @ ( sqrt @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_minus
% 0.30/0.65  thf(fact_8057_real__sqrt__power,axiom,
% 0.30/0.65      ! [X3: real,K: nat] :
% 0.30/0.65        ( ( sqrt @ ( power_power_real @ X3 @ K ) )
% 0.30/0.65        = ( power_power_real @ ( sqrt @ X3 ) @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_power
% 0.30/0.65  thf(fact_8058_real__sqrt__mult,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( sqrt @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.65        = ( times_times_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_mult
% 0.30/0.65  thf(fact_8059_real__sqrt__divide,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( sqrt @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.65        = ( divide_divide_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_divide
% 0.30/0.65  thf(fact_8060_binomial__eq__0,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ N @ K )
% 0.30/0.65       => ( ( binomial @ N @ K )
% 0.30/0.65          = zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_eq_0
% 0.30/0.65  thf(fact_8061_Suc__times__binomial,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.30/0.65        = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_times_binomial
% 0.30/0.65  thf(fact_8062_Suc__times__binomial__eq,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.30/0.65        = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_times_binomial_eq
% 0.30/0.65  thf(fact_8063_binomial__symmetric,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( binomial @ N @ K )
% 0.30/0.65          = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_symmetric
% 0.30/0.65  thf(fact_8064_real__sqrt__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 @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_gt_zero
% 0.30/0.65  thf(fact_8065_real__sqrt__eq__zero__cancel,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ( sqrt @ 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_sqrt_eq_zero_cancel
% 0.30/0.65  thf(fact_8066_real__sqrt__ge__zero,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_ge_zero
% 0.30/0.65  thf(fact_8067_choose__mult__lemma,axiom,
% 0.30/0.65      ! [M: nat,R: nat,K: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.30/0.65        = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_mult_lemma
% 0.30/0.65  thf(fact_8068_real__sqrt__ge__one,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.65       => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_ge_one
% 0.30/0.65  thf(fact_8069_binomial__le__pow,axiom,
% 0.30/0.65      ! [R: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ R @ N )
% 0.30/0.65       => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_le_pow
% 0.30/0.65  thf(fact_8070_pochhammer__pos,axiom,
% 0.30/0.65      ! [X3: real,N: nat] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_pos
% 0.30/0.65  thf(fact_8071_pochhammer__pos,axiom,
% 0.30/0.65      ! [X3: rat,N: nat] :
% 0.30/0.65        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.65       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_pos
% 0.30/0.65  thf(fact_8072_pochhammer__pos,axiom,
% 0.30/0.65      ! [X3: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.65       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_pos
% 0.30/0.65  thf(fact_8073_pochhammer__pos,axiom,
% 0.30/0.65      ! [X3: int,N: nat] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ X3 )
% 0.30/0.65       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_pos
% 0.30/0.65  thf(fact_8074_pochhammer__neq__0__mono,axiom,
% 0.30/0.65      ! [A: complex,M: nat,N: nat] :
% 0.30/0.65        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.30/0.65         != zero_zero_complex )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65         => ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.30/0.65           != zero_zero_complex ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_neq_0_mono
% 0.30/0.65  thf(fact_8075_pochhammer__neq__0__mono,axiom,
% 0.30/0.65      ! [A: real,M: nat,N: nat] :
% 0.30/0.65        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.30/0.65         != zero_zero_real )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65         => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.30/0.65           != zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_neq_0_mono
% 0.30/0.65  thf(fact_8076_pochhammer__neq__0__mono,axiom,
% 0.30/0.65      ! [A: rat,M: nat,N: nat] :
% 0.30/0.65        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.30/0.65         != zero_zero_rat )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65         => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.30/0.65           != zero_zero_rat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_neq_0_mono
% 0.30/0.65  thf(fact_8077_pochhammer__eq__0__mono,axiom,
% 0.30/0.65      ! [A: complex,N: nat,M: nat] :
% 0.30/0.65        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.30/0.65          = zero_zero_complex )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.30/0.65            = zero_zero_complex ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_eq_0_mono
% 0.30/0.65  thf(fact_8078_pochhammer__eq__0__mono,axiom,
% 0.30/0.65      ! [A: real,N: nat,M: nat] :
% 0.30/0.65        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.30/0.65          = zero_zero_real )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.30/0.65            = zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_eq_0_mono
% 0.30/0.65  thf(fact_8079_pochhammer__eq__0__mono,axiom,
% 0.30/0.65      ! [A: rat,N: nat,M: nat] :
% 0.30/0.65        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.30/0.65          = zero_zero_rat )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.30/0.65            = zero_zero_rat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_eq_0_mono
% 0.30/0.65  thf(fact_8080_zero__less__binomial,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_less_binomial
% 0.30/0.65  thf(fact_8081_Suc__times__binomial__add,axiom,
% 0.30/0.65      ! [A: nat,B2: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
% 0.30/0.65        = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_times_binomial_add
% 0.30/0.65  thf(fact_8082_binomial__Suc__Suc__eq__times,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.30/0.65        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_Suc_Suc_eq_times
% 0.30/0.65  thf(fact_8083_choose__mult,axiom,
% 0.30/0.65      ! [K: nat,M: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.65       => ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65         => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.30/0.65            = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_mult
% 0.30/0.65  thf(fact_8084_real__div__sqrt,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( divide_divide_real @ X3 @ ( sqrt @ X3 ) )
% 0.30/0.65          = ( sqrt @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_div_sqrt
% 0.30/0.65  thf(fact_8085_binomial__absorb__comp,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.30/0.65        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_absorb_comp
% 0.30/0.65  thf(fact_8086_sqrt__add__le__add__sqrt,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_eq_real @ ( sqrt @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_add_le_add_sqrt
% 0.30/0.65  thf(fact_8087_le__real__sqrt__sumsq,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % le_real_sqrt_sumsq
% 0.30/0.65  thf(fact_8088_pochhammer__nonneg,axiom,
% 0.30/0.65      ! [X3: real,N: nat] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_nonneg
% 0.30/0.65  thf(fact_8089_pochhammer__nonneg,axiom,
% 0.30/0.65      ! [X3: rat,N: nat] :
% 0.30/0.65        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.65       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_nonneg
% 0.30/0.65  thf(fact_8090_pochhammer__nonneg,axiom,
% 0.30/0.65      ! [X3: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.65       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_nonneg
% 0.30/0.65  thf(fact_8091_pochhammer__nonneg,axiom,
% 0.30/0.65      ! [X3: int,N: nat] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ X3 )
% 0.30/0.65       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_nonneg
% 0.30/0.65  thf(fact_8092_sqrt2__less__2,axiom,
% 0.30/0.65      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt2_less_2
% 0.30/0.65  thf(fact_8093_binomial__absorption,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.30/0.65        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_absorption
% 0.30/0.65  thf(fact_8094_binomial__fact__lemma,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.30/0.65          = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_fact_lemma
% 0.30/0.65  thf(fact_8095_pochhammer__rec,axiom,
% 0.30/0.65      ! [A: complex,N: nat] :
% 0.30/0.65        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec
% 0.30/0.65  thf(fact_8096_pochhammer__rec,axiom,
% 0.30/0.65      ! [A: real,N: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec
% 0.30/0.65  thf(fact_8097_pochhammer__rec,axiom,
% 0.30/0.65      ! [A: rat,N: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec
% 0.30/0.65  thf(fact_8098_pochhammer__rec,axiom,
% 0.30/0.65      ! [A: nat,N: nat] :
% 0.30/0.65        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec
% 0.30/0.65  thf(fact_8099_pochhammer__rec,axiom,
% 0.30/0.65      ! [A: int,N: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec
% 0.30/0.65  thf(fact_8100_pochhammer__Suc,axiom,
% 0.30/0.65      ! [A: rat,N: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc
% 0.30/0.65  thf(fact_8101_pochhammer__Suc,axiom,
% 0.30/0.65      ! [A: int,N: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc
% 0.30/0.65  thf(fact_8102_pochhammer__Suc,axiom,
% 0.30/0.65      ! [A: real,N: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc
% 0.30/0.65  thf(fact_8103_pochhammer__Suc,axiom,
% 0.30/0.65      ! [A: nat,N: nat] :
% 0.30/0.65        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc
% 0.30/0.65  thf(fact_8104_pochhammer__rec_H,axiom,
% 0.30/0.65      ! [Z: rat,N: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec'
% 0.30/0.65  thf(fact_8105_pochhammer__rec_H,axiom,
% 0.30/0.65      ! [Z: int,N: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec'
% 0.30/0.65  thf(fact_8106_pochhammer__rec_H,axiom,
% 0.30/0.65      ! [Z: real,N: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec'
% 0.30/0.65  thf(fact_8107_pochhammer__rec_H,axiom,
% 0.30/0.65      ! [Z: nat,N: nat] :
% 0.30/0.65        ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 0.30/0.65        = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_rec'
% 0.30/0.65  thf(fact_8108_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ N @ K )
% 0.30/0.65       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.30/0.65          = zero_zero_complex ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65  thf(fact_8109_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ N @ K )
% 0.30/0.65       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.30/0.65          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65  thf(fact_8110_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ N @ K )
% 0.30/0.65       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.30/0.65          = zero_zero_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65  thf(fact_8111_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ N @ K )
% 0.30/0.65       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.30/0.65          = zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65  thf(fact_8112_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ N @ K )
% 0.30/0.65       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.30/0.65          = zero_zero_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.65  thf(fact_8113_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.30/0.65          = zero_zero_complex )
% 0.30/0.65        = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_iff
% 0.30/0.65  thf(fact_8114_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.30/0.65          = zero_z3403309356797280102nteger )
% 0.30/0.65        = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_iff
% 0.30/0.65  thf(fact_8115_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.30/0.65          = zero_zero_rat )
% 0.30/0.65        = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_iff
% 0.30/0.65  thf(fact_8116_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.30/0.65          = zero_zero_int )
% 0.30/0.65        = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_iff
% 0.30/0.65  thf(fact_8117_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.30/0.65          = zero_zero_real )
% 0.30/0.65        = ( ord_less_nat @ N @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_iff
% 0.30/0.65  thf(fact_8118_pochhammer__eq__0__iff,axiom,
% 0.30/0.65      ! [A: complex,N: nat] :
% 0.30/0.65        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.30/0.65          = zero_zero_complex )
% 0.30/0.65        = ( ? [K2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ K2 @ N )
% 0.30/0.65              & ( A
% 0.30/0.65                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_eq_0_iff
% 0.30/0.65  thf(fact_8119_pochhammer__eq__0__iff,axiom,
% 0.30/0.65      ! [A: rat,N: nat] :
% 0.30/0.65        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.30/0.65          = zero_zero_rat )
% 0.30/0.65        = ( ? [K2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ K2 @ N )
% 0.30/0.65              & ( A
% 0.30/0.65                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_eq_0_iff
% 0.30/0.65  thf(fact_8120_pochhammer__eq__0__iff,axiom,
% 0.30/0.65      ! [A: real,N: nat] :
% 0.30/0.65        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.30/0.65          = zero_zero_real )
% 0.30/0.65        = ( ? [K2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ K2 @ N )
% 0.30/0.65              & ( A
% 0.30/0.65                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_eq_0_iff
% 0.30/0.65  thf(fact_8121_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.30/0.65         != zero_zero_complex ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65  thf(fact_8122_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.30/0.65         != zero_z3403309356797280102nteger ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65  thf(fact_8123_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.30/0.65         != zero_zero_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65  thf(fact_8124_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.30/0.65         != zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65  thf(fact_8125_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.30/0.65         != zero_zero_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.65  thf(fact_8126_pochhammer__product_H,axiom,
% 0.30/0.65      ! [Z: rat,N: nat,M: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.65        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product'
% 0.30/0.65  thf(fact_8127_pochhammer__product_H,axiom,
% 0.30/0.65      ! [Z: int,N: nat,M: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.65        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product'
% 0.30/0.65  thf(fact_8128_pochhammer__product_H,axiom,
% 0.30/0.65      ! [Z: real,N: nat,M: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.65        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product'
% 0.30/0.65  thf(fact_8129_pochhammer__product_H,axiom,
% 0.30/0.65      ! [Z: nat,N: nat,M: nat] :
% 0.30/0.65        ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.30/0.65        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product'
% 0.30/0.65  thf(fact_8130_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_ge_n_over_k_pow_k
% 0.30/0.65  thf(fact_8131_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_ge_n_over_k_pow_k
% 0.30/0.65  thf(fact_8132_binomial__mono,axiom,
% 0.30/0.65      ! [K: nat,K6: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ K6 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.30/0.65         => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_mono
% 0.30/0.65  thf(fact_8133_binomial__maximum_H,axiom,
% 0.30/0.65      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_maximum'
% 0.30/0.65  thf(fact_8134_binomial__maximum,axiom,
% 0.30/0.65      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_maximum
% 0.30/0.65  thf(fact_8135_binomial__antimono,axiom,
% 0.30/0.65      ! [K: nat,K6: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ K6 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.30/0.65         => ( ( ord_less_eq_nat @ K6 @ N )
% 0.30/0.65           => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_antimono
% 0.30/0.65  thf(fact_8136_binomial__le__pow2,axiom,
% 0.30/0.65      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_le_pow2
% 0.30/0.65  thf(fact_8137_real__less__rsqrt,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.65       => ( ord_less_real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_less_rsqrt
% 0.30/0.65  thf(fact_8138_choose__reduce__nat,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.65         => ( ( binomial @ N @ K )
% 0.30/0.65            = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_reduce_nat
% 0.30/0.65  thf(fact_8139_sqrt__le__D,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y3 )
% 0.30/0.65       => ( ord_less_eq_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_le_D
% 0.30/0.65  thf(fact_8140_real__le__rsqrt,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.65       => ( ord_less_eq_real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_le_rsqrt
% 0.30/0.65  thf(fact_8141_times__binomial__minus1__eq,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.65       => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.30/0.65          = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % times_binomial_minus1_eq
% 0.30/0.65  thf(fact_8142_tan__60,axiom,
% 0.30/0.65      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.30/0.65      = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % tan_60
% 0.30/0.65  thf(fact_8143_binomial__altdef__nat,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( binomial @ N @ K )
% 0.30/0.65          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_altdef_nat
% 0.30/0.65  thf(fact_8144_pochhammer__product,axiom,
% 0.30/0.65      ! [M: nat,N: nat,Z: rat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 0.30/0.65          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product
% 0.30/0.65  thf(fact_8145_pochhammer__product,axiom,
% 0.30/0.65      ! [M: nat,N: nat,Z: int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 0.30/0.65          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product
% 0.30/0.65  thf(fact_8146_pochhammer__product,axiom,
% 0.30/0.65      ! [M: nat,N: nat,Z: real] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 0.30/0.65          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product
% 0.30/0.65  thf(fact_8147_pochhammer__product,axiom,
% 0.30/0.65      ! [M: nat,N: nat,Z: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 0.30/0.65          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_product
% 0.30/0.65  thf(fact_8148_binomial__less__binomial__Suc,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.65       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_less_binomial_Suc
% 0.30/0.65  thf(fact_8149_binomial__strict__mono,axiom,
% 0.30/0.65      ! [K: nat,K6: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ K @ K6 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.30/0.65         => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_strict_mono
% 0.30/0.65  thf(fact_8150_binomial__strict__antimono,axiom,
% 0.30/0.65      ! [K: nat,K6: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ K @ K6 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.30/0.65         => ( ( ord_less_eq_nat @ K6 @ N )
% 0.30/0.65           => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_strict_antimono
% 0.30/0.65  thf(fact_8151_central__binomial__odd,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65       => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65          = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % central_binomial_odd
% 0.30/0.65  thf(fact_8152_binomial__addition__formula,axiom,
% 0.30/0.65      ! [N: nat,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( binomial @ N @ ( suc @ K ) )
% 0.30/0.65          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_addition_formula
% 0.30/0.65  thf(fact_8153_binomial__fact,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 0.30/0.65          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_fact
% 0.30/0.65  thf(fact_8154_binomial__fact,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 0.30/0.65          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_fact
% 0.30/0.65  thf(fact_8155_binomial__fact,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 0.30/0.65          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_fact
% 0.30/0.65  thf(fact_8156_fact__binomial,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 0.30/0.65          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_binomial
% 0.30/0.65  thf(fact_8157_fact__binomial,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 0.30/0.65          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_binomial
% 0.30/0.65  thf(fact_8158_fact__binomial,axiom,
% 0.30/0.65      ! [K: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 0.30/0.65          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_binomial
% 0.30/0.65  thf(fact_8159_real__le__lsqrt,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_eq_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.65           => ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_le_lsqrt
% 0.30/0.65  thf(fact_8160_real__sqrt__unique,axiom,
% 0.30/0.65      ! [Y3: real,X3: real] :
% 0.30/0.65        ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65          = X3 )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65         => ( ( sqrt @ X3 )
% 0.30/0.65            = Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_unique
% 0.30/0.65  thf(fact_8161_lemma__real__divide__sqrt__less,axiom,
% 0.30/0.65      ! [U: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ U )
% 0.30/0.65       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.30/0.65  
% 0.30/0.65  % lemma_real_divide_sqrt_less
% 0.30/0.65  thf(fact_8162_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65          = X3 )
% 0.30/0.65       => ( Y3 = zero_zero_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_eq_cancel
% 0.30/0.65  thf(fact_8163_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( X3 = zero_zero_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_eq_cancel2
% 0.30/0.65  thf(fact_8164_real__sqrt__sum__squares__ge1,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_ge1
% 0.30/0.65  thf(fact_8165_real__sqrt__sum__squares__ge2,axiom,
% 0.30/0.65      ! [Y3: real,X3: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_ge2
% 0.30/0.65  thf(fact_8166_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.30/0.65      ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_triangle_ineq
% 0.30/0.65  thf(fact_8167_sqrt__ge__absD,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ Y3 ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_ge_absD
% 0.30/0.65  thf(fact_8168_cos__45,axiom,
% 0.30/0.65      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cos_45
% 0.30/0.65  thf(fact_8169_sin__45,axiom,
% 0.30/0.65      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sin_45
% 0.30/0.65  thf(fact_8170_pochhammer__absorb__comp,axiom,
% 0.30/0.65      ! [R: complex,K: nat] :
% 0.30/0.65        ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 0.30/0.65        = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_absorb_comp
% 0.30/0.65  thf(fact_8171_pochhammer__absorb__comp,axiom,
% 0.30/0.65      ! [R: code_integer,K: nat] :
% 0.30/0.65        ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 0.30/0.65        = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_absorb_comp
% 0.30/0.65  thf(fact_8172_pochhammer__absorb__comp,axiom,
% 0.30/0.65      ! [R: rat,K: nat] :
% 0.30/0.65        ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 0.30/0.65        = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_absorb_comp
% 0.30/0.65  thf(fact_8173_pochhammer__absorb__comp,axiom,
% 0.30/0.65      ! [R: int,K: nat] :
% 0.30/0.65        ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 0.30/0.65        = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_absorb_comp
% 0.30/0.65  thf(fact_8174_pochhammer__absorb__comp,axiom,
% 0.30/0.65      ! [R: real,K: nat] :
% 0.30/0.65        ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 0.30/0.65        = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_absorb_comp
% 0.30/0.65  thf(fact_8175_pochhammer__same,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 0.30/0.65        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_same
% 0.30/0.65  thf(fact_8176_pochhammer__same,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 0.30/0.65        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_same
% 0.30/0.65  thf(fact_8177_pochhammer__same,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 0.30/0.65        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_same
% 0.30/0.65  thf(fact_8178_pochhammer__same,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 0.30/0.65        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_same
% 0.30/0.65  thf(fact_8179_pochhammer__same,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 0.30/0.65        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_same
% 0.30/0.65  thf(fact_8180_choose__two,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65        = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_two
% 0.30/0.65  thf(fact_8181_real__less__lsqrt,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 @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.65           => ( ord_less_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_less_lsqrt
% 0.30/0.65  thf(fact_8182_sqrt__sum__squares__le__sum,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_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_sum_squares_le_sum
% 0.30/0.65  thf(fact_8183_tan__30,axiom,
% 0.30/0.65      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.30/0.65      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % tan_30
% 0.30/0.65  thf(fact_8184_sqrt__even__pow2,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65       => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_even_pow2
% 0.30/0.65  thf(fact_8185_real__sqrt__ge__abs1,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_ge_abs1
% 0.30/0.65  thf(fact_8186_real__sqrt__ge__abs2,axiom,
% 0.30/0.65      ! [Y3: real,X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_ge_abs2
% 0.30/0.65  thf(fact_8187_sqrt__sum__squares__le__sum__abs,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_sum_squares_le_sum_abs
% 0.30/0.65  thf(fact_8188_ln__sqrt,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 @ ( sqrt @ X3 ) )
% 0.30/0.65          = ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % ln_sqrt
% 0.30/0.65  thf(fact_8189_cos__30,axiom,
% 0.30/0.65      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.30/0.65      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cos_30
% 0.30/0.65  thf(fact_8190_sin__60,axiom,
% 0.30/0.65      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.30/0.65      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sin_60
% 0.30/0.65  thf(fact_8191_complex__norm,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( real_V1022390504157884413omplex @ ( complex2 @ X3 @ Y3 ) )
% 0.30/0.65        = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % complex_norm
% 0.30/0.65  thf(fact_8192_pochhammer__minus,axiom,
% 0.30/0.65      ! [B2: complex,K: nat] :
% 0.30/0.65        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
% 0.30/0.65        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus
% 0.30/0.65  thf(fact_8193_pochhammer__minus,axiom,
% 0.30/0.65      ! [B2: code_integer,K: nat] :
% 0.30/0.65        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
% 0.30/0.65        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus
% 0.30/0.65  thf(fact_8194_pochhammer__minus,axiom,
% 0.30/0.65      ! [B2: rat,K: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
% 0.30/0.65        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus
% 0.30/0.65  thf(fact_8195_pochhammer__minus,axiom,
% 0.30/0.65      ! [B2: int,K: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
% 0.30/0.65        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus
% 0.30/0.65  thf(fact_8196_pochhammer__minus,axiom,
% 0.30/0.65      ! [B2: real,K: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
% 0.30/0.65        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus
% 0.30/0.65  thf(fact_8197_pochhammer__minus_H,axiom,
% 0.30/0.65      ! [B2: complex,K: nat] :
% 0.30/0.65        ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.30/0.65        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus'
% 0.30/0.65  thf(fact_8198_pochhammer__minus_H,axiom,
% 0.30/0.65      ! [B2: code_integer,K: nat] :
% 0.30/0.65        ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.30/0.65        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus'
% 0.30/0.65  thf(fact_8199_pochhammer__minus_H,axiom,
% 0.30/0.65      ! [B2: rat,K: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.30/0.65        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus'
% 0.30/0.65  thf(fact_8200_pochhammer__minus_H,axiom,
% 0.30/0.65      ! [B2: int,K: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.30/0.65        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus'
% 0.30/0.65  thf(fact_8201_pochhammer__minus_H,axiom,
% 0.30/0.65      ! [B2: real,K: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.30/0.65        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_minus'
% 0.30/0.65  thf(fact_8202_finite__psubset__def,axiom,
% 0.30/0.65      ( finite_psubset_nat
% 0.30/0.65      = ( collec6662362479098859352et_nat
% 0.30/0.65        @ ( produc6247414631856714078_nat_o
% 0.30/0.65          @ ^ [A6: set_nat,B6: set_nat] :
% 0.30/0.65              ( ( ord_less_set_nat @ A6 @ B6 )
% 0.30/0.65              & ( finite_finite_nat @ B6 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_psubset_def
% 0.30/0.65  thf(fact_8203_finite__psubset__def,axiom,
% 0.30/0.65      ( finite_psubset_int
% 0.30/0.65      = ( collec957716948307931664et_int
% 0.30/0.65        @ ( produc4109468873575309990_int_o
% 0.30/0.65          @ ^ [A6: set_int,B6: set_int] :
% 0.30/0.65              ( ( ord_less_set_int @ A6 @ B6 )
% 0.30/0.65              & ( finite_finite_int @ B6 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_psubset_def
% 0.30/0.65  thf(fact_8204_finite__psubset__def,axiom,
% 0.30/0.65      ( finite8643634255014194347omplex
% 0.30/0.65      = ( collec5108298041176329748omplex
% 0.30/0.65        @ ( produc3914248068834153634plex_o
% 0.30/0.65          @ ^ [A6: set_complex,B6: set_complex] :
% 0.30/0.65              ( ( ord_less_set_complex @ A6 @ B6 )
% 0.30/0.65              & ( finite3207457112153483333omplex @ B6 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_psubset_def
% 0.30/0.65  thf(fact_8205_arsinh__real__aux,axiom,
% 0.30/0.65      ! [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 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arsinh_real_aux
% 0.30/0.65  thf(fact_8206_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.30/0.65      ! [X3: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_mult_ge_zero
% 0.30/0.65  thf(fact_8207_real__sqrt__power__even,axiom,
% 0.30/0.65      ! [N: nat,X3: real] :
% 0.30/0.65        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ( power_power_real @ ( sqrt @ X3 ) @ N )
% 0.30/0.65            = ( power_power_real @ X3 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_power_even
% 0.30/0.65  thf(fact_8208_arith__geo__mean__sqrt,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_eq_real @ ( sqrt @ ( times_times_real @ X3 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arith_geo_mean_sqrt
% 0.30/0.65  thf(fact_8209_cos__x__y__le__one,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.30/0.65  
% 0.30/0.65  % cos_x_y_le_one
% 0.30/0.65  thf(fact_8210_real__sqrt__sum__squares__less,axiom,
% 0.30/0.65      ! [X3: real,U: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65         => ( 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 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_sqrt_sum_squares_less
% 0.30/0.65  thf(fact_8211_cos__arctan,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( cos_real @ ( arctan @ X3 ) )
% 0.30/0.65        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cos_arctan
% 0.30/0.65  thf(fact_8212_sin__arctan,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( sin_real @ ( arctan @ X3 ) )
% 0.30/0.65        = ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sin_arctan
% 0.30/0.65  thf(fact_8213_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s2602460028002588243omplex
% 0.30/0.65      = ( ^ [A5: complex,N2: nat] :
% 0.30/0.65            ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 0.30/0.65            @ ( set_fo1517530859248394432omplex
% 0.30/0.65              @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N2 @ 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_8214_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s4028243227959126397er_rat
% 0.30/0.65      = ( ^ [A5: rat,N2: nat] :
% 0.30/0.65            ( if_rat @ ( N2 = 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 @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N2 @ 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_8215_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s4660882817536571857er_int
% 0.30/0.65      = ( ^ [A5: int,N2: nat] :
% 0.30/0.65            ( if_int @ ( N2 = 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 @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N2 @ 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_8216_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s7457072308508201937r_real
% 0.30/0.65      = ( ^ [A5: real,N2: nat] :
% 0.30/0.65            ( if_real @ ( N2 = 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 @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N2 @ 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_8217_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s4663373288045622133er_nat
% 0.30/0.65      = ( ^ [A5: nat,N2: nat] :
% 0.30/0.65            ( if_nat @ ( N2 = 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 @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N2 @ 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_8218_sqrt__sum__squares__half__less,axiom,
% 0.30/0.65      ! [X3: real,U: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 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 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sqrt_sum_squares_half_less
% 0.30/0.65  thf(fact_8219_sin__cos__sqrt,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) )
% 0.30/0.65       => ( ( sin_real @ X3 )
% 0.30/0.65          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sin_cos_sqrt
% 0.30/0.65  thf(fact_8220_arctan__half,axiom,
% 0.30/0.65      ( arctan
% 0.30/0.65      = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arctan_half
% 0.30/0.65  thf(fact_8221_fact__double,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65        = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_double
% 0.30/0.65  thf(fact_8222_fact__double,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65        = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_double
% 0.30/0.65  thf(fact_8223_fact__double,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.65        = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_double
% 0.30/0.65  thf(fact_8224_central__binomial__lower__bound,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % central_binomial_lower_bound
% 0.30/0.65  thf(fact_8225_arcosh__real__def,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.65       => ( ( arcosh_real @ X3 )
% 0.30/0.65          = ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arcosh_real_def
% 0.30/0.65  thf(fact_8226_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.65      ! [Z: complex,N: nat] :
% 0.30/0.65        ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.30/0.65        = ( groups6464643781859351333omplex
% 0.30/0.65          @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_times_pochhammer_half
% 0.30/0.65  thf(fact_8227_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.65      ! [Z: rat,N: nat] :
% 0.30/0.65        ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.30/0.65        = ( groups73079841787564623at_rat
% 0.30/0.65          @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( 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 ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_times_pochhammer_half
% 0.30/0.65  thf(fact_8228_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.65      ! [Z: real,N: nat] :
% 0.30/0.65        ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( 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 ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_times_pochhammer_half
% 0.30/0.65  thf(fact_8229_arsinh__real__def,axiom,
% 0.30/0.65      ( arsinh_real
% 0.30/0.65      = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arsinh_real_def
% 0.30/0.65  thf(fact_8230_choose__odd__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups2073611262835488442omplex
% 0.30/0.65              @ ^ [I4: nat] :
% 0.30/0.65                  ( if_complex
% 0.30/0.65                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.65                  @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
% 0.30/0.65                  @ zero_zero_complex )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_odd_sum
% 0.30/0.65  thf(fact_8231_choose__odd__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups2906978787729119204at_rat
% 0.30/0.65              @ ^ [I4: nat] :
% 0.30/0.65                  ( if_rat
% 0.30/0.65                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.65                  @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
% 0.30/0.65                  @ zero_zero_rat )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_odd_sum
% 0.30/0.65  thf(fact_8232_choose__odd__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups3539618377306564664at_int
% 0.30/0.65              @ ^ [I4: nat] :
% 0.30/0.65                  ( if_int
% 0.30/0.65                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.65                  @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
% 0.30/0.65                  @ zero_zero_int )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_odd_sum
% 0.30/0.65  thf(fact_8233_choose__odd__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] :
% 0.30/0.65                  ( if_real
% 0.30/0.65                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.65                  @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
% 0.30/0.65                  @ zero_zero_real )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_odd_sum
% 0.30/0.65  thf(fact_8234_choose__even__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups2073611262835488442omplex
% 0.30/0.65              @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_even_sum
% 0.30/0.65  thf(fact_8235_choose__even__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups2906978787729119204at_rat
% 0.30/0.65              @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_even_sum
% 0.30/0.65  thf(fact_8236_choose__even__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups3539618377306564664at_int
% 0.30/0.65              @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_even_sum
% 0.30/0.65  thf(fact_8237_choose__even__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.65       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.65            @ ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_even_sum
% 0.30/0.65  thf(fact_8238_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_8239_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_8240_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_8241_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_8242_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_8243_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_8244_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_8245_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_8246_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_8247_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_8248_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_8249_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_8250_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_8251_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_8252_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_8253_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_8254_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_8255_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8256_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8257_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8258_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8259_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8260_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8261_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8262_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8263_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8264_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8265_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8266_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8267_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N: nat,M: nat,G: nat > complex] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = one_one_complex ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8268_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N: nat,M: nat,G: nat > real] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = one_one_real ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8269_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N: nat,M: nat,G: nat > rat] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = one_one_rat ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8270_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N: nat,M: nat,G: nat > nat] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = one_one_nat ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8271_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N: nat,M: nat,G: nat > int] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = one_one_int ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.30/0.65         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8272_prod_Odistrib,axiom,
% 0.30/0.65      ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.65          @ A4 )
% 0.30/0.65        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A4 ) @ ( groups708209901874060359at_nat @ H2 @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.distrib
% 0.30/0.65  thf(fact_8273_prod_Odistrib,axiom,
% 0.30/0.65      ! [G: nat > int,H2: nat > int,A4: set_nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.65          @ A4 )
% 0.30/0.65        = ( times_times_int @ ( groups705719431365010083at_int @ G @ A4 ) @ ( groups705719431365010083at_int @ H2 @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.distrib
% 0.30/0.65  thf(fact_8274_prod_Odistrib,axiom,
% 0.30/0.65      ! [G: int > int,H2: int > int,A4: set_int] :
% 0.30/0.65        ( ( groups1705073143266064639nt_int
% 0.30/0.65          @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.65          @ A4 )
% 0.30/0.65        = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A4 ) @ ( groups1705073143266064639nt_int @ H2 @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.distrib
% 0.30/0.65  thf(fact_8275_prod__power__distrib,axiom,
% 0.30/0.65      ! [F: nat > nat,A4: set_nat,N: nat] :
% 0.30/0.65        ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ N )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_power_distrib
% 0.30/0.65  thf(fact_8276_prod__power__distrib,axiom,
% 0.30/0.65      ! [F: nat > int,A4: set_nat,N: nat] :
% 0.30/0.65        ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A4 ) @ N )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_power_distrib
% 0.30/0.65  thf(fact_8277_prod__power__distrib,axiom,
% 0.30/0.65      ! [F: int > int,A4: set_int,N: nat] :
% 0.30/0.65        ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ N )
% 0.30/0.65        = ( groups1705073143266064639nt_int
% 0.30/0.65          @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_power_distrib
% 0.30/0.65  thf(fact_8278_mod__prod__eq,axiom,
% 0.30/0.65      ! [F: nat > nat,A: nat,A4: set_nat] :
% 0.30/0.65        ( ( modulo_modulo_nat
% 0.30/0.65          @ ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 0.30/0.65            @ A4 )
% 0.30/0.65          @ A )
% 0.30/0.65        = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % mod_prod_eq
% 0.30/0.65  thf(fact_8279_mod__prod__eq,axiom,
% 0.30/0.65      ! [F: nat > int,A: int,A4: set_nat] :
% 0.30/0.65        ( ( modulo_modulo_int
% 0.30/0.65          @ ( groups705719431365010083at_int
% 0.30/0.65            @ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.30/0.65            @ A4 )
% 0.30/0.65          @ A )
% 0.30/0.65        = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A4 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % mod_prod_eq
% 0.30/0.65  thf(fact_8280_mod__prod__eq,axiom,
% 0.30/0.65      ! [F: int > int,A: int,A4: set_int] :
% 0.30/0.65        ( ( modulo_modulo_int
% 0.30/0.65          @ ( groups1705073143266064639nt_int
% 0.30/0.65            @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.30/0.65            @ A4 )
% 0.30/0.65          @ A )
% 0.30/0.65        = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % mod_prod_eq
% 0.30/0.65  thf(fact_8281_prod_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_real @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups129246275422532515t_real
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc_shift
% 0.30/0.65  thf(fact_8282_prod_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_rat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups73079841787564623at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc_shift
% 0.30/0.65  thf(fact_8283_prod_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_nat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc_shift
% 0.30/0.65  thf(fact_8284_prod_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_int @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups705719431365010083at_int
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc_shift
% 0.30/0.65  thf(fact_8285_atMost__def,axiom,
% 0.30/0.65      ( set_ord_atMost_real
% 0.30/0.65      = ( ^ [U2: real] :
% 0.30/0.65            ( collect_real
% 0.30/0.65            @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_def
% 0.30/0.65  thf(fact_8286_atMost__def,axiom,
% 0.30/0.65      ( set_or58775011639299419et_int
% 0.30/0.65      = ( ^ [U2: set_int] :
% 0.30/0.65            ( collect_set_int
% 0.30/0.65            @ ^ [X: set_int] : ( ord_less_eq_set_int @ X @ U2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_def
% 0.30/0.65  thf(fact_8287_atMost__def,axiom,
% 0.30/0.65      ( set_ord_atMost_rat
% 0.30/0.65      = ( ^ [U2: rat] :
% 0.30/0.65            ( collect_rat
% 0.30/0.65            @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_def
% 0.30/0.65  thf(fact_8288_atMost__def,axiom,
% 0.30/0.65      ( set_ord_atMost_num
% 0.30/0.65      = ( ^ [U2: num] :
% 0.30/0.65            ( collect_num
% 0.30/0.65            @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_def
% 0.30/0.65  thf(fact_8289_atMost__def,axiom,
% 0.30/0.65      ( set_ord_atMost_nat
% 0.30/0.65      = ( ^ [U2: nat] :
% 0.30/0.65            ( collect_nat
% 0.30/0.65            @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_def
% 0.30/0.65  thf(fact_8290_atMost__def,axiom,
% 0.30/0.65      ( set_ord_atMost_int
% 0.30/0.65      = ( ^ [U2: int] :
% 0.30/0.65            ( collect_int
% 0.30/0.65            @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_def
% 0.30/0.65  thf(fact_8291_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.65        ( ! [I3: nat] :
% 0.30/0.65            ( ( member_nat @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8292_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > real,G: real > real] :
% 0.30/0.65        ( ! [I3: real] :
% 0.30/0.65            ( ( member_real @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8293_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > real,G: int > real] :
% 0.30/0.65        ( ! [I3: int] :
% 0.30/0.65            ( ( member_int @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8294_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.65        ( ! [I3: complex] :
% 0.30/0.65            ( ( member_complex @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8295_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.65        ( ! [I3: nat] :
% 0.30/0.65            ( ( member_nat @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8296_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > rat,G: real > rat] :
% 0.30/0.65        ( ! [I3: real] :
% 0.30/0.65            ( ( member_real @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8297_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > rat,G: int > rat] :
% 0.30/0.65        ( ! [I3: int] :
% 0.30/0.65            ( ( member_int @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8298_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.65        ( ! [I3: complex] :
% 0.30/0.65            ( ( member_complex @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8299_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > nat,G: real > nat] :
% 0.30/0.65        ( ! [I3: real] :
% 0.30/0.65            ( ( member_real @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8300_prod__mono,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > nat,G: int > nat] :
% 0.30/0.65        ( ! [I3: int] :
% 0.30/0.65            ( ( member_int @ I3 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 0.30/0.65              & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono
% 0.30/0.65  thf(fact_8301_prod__nonneg,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > nat] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_nonneg
% 0.30/0.65  thf(fact_8302_prod__nonneg,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > int] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_nonneg
% 0.30/0.65  thf(fact_8303_prod__nonneg,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > int] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_nonneg
% 0.30/0.65  thf(fact_8304_prod__pos,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > nat] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_pos
% 0.30/0.65  thf(fact_8305_prod__pos,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > int] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_pos
% 0.30/0.65  thf(fact_8306_prod__pos,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > int] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_pos
% 0.30/0.65  thf(fact_8307_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > real] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8308_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > real] :
% 0.30/0.65        ( ! [X5: real] :
% 0.30/0.65            ( ( member_real @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8309_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > real] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8310_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > real] :
% 0.30/0.65        ( ! [X5: complex] :
% 0.30/0.65            ( ( member_complex @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8311_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > rat] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8312_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > rat] :
% 0.30/0.65        ( ! [X5: real] :
% 0.30/0.65            ( ( member_real @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8313_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > rat] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8314_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > rat] :
% 0.30/0.65        ( ! [X5: complex] :
% 0.30/0.65            ( ( member_complex @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8315_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > nat] :
% 0.30/0.65        ( ! [X5: real] :
% 0.30/0.65            ( ( member_real @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8316_prod__ge__1,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > nat] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_ge_1
% 0.30/0.65  thf(fact_8317_prod_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( times_times_real @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups129246275422532515t_real
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_shift
% 0.30/0.65  thf(fact_8318_prod_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( times_times_rat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups73079841787564623at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_shift
% 0.30/0.65  thf(fact_8319_prod_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( times_times_nat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_shift
% 0.30/0.65  thf(fact_8320_prod_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( times_times_int @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups705719431365010083at_int
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_shift
% 0.30/0.65  thf(fact_8321_not__Iic__le__Icc,axiom,
% 0.30/0.65      ! [H2: int,L3: int,H3: int] :
% 0.30/0.65        ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % not_Iic_le_Icc
% 0.30/0.65  thf(fact_8322_not__Iic__le__Icc,axiom,
% 0.30/0.65      ! [H2: real,L3: real,H3: real] :
% 0.30/0.65        ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % not_Iic_le_Icc
% 0.30/0.65  thf(fact_8323_power__sum,axiom,
% 0.30/0.65      ! [C: real,F: nat > nat,A4: set_nat] :
% 0.30/0.65        ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % power_sum
% 0.30/0.65  thf(fact_8324_power__sum,axiom,
% 0.30/0.65      ! [C: complex,F: nat > nat,A4: set_nat] :
% 0.30/0.65        ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 0.30/0.65        = ( groups6464643781859351333omplex
% 0.30/0.65          @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % power_sum
% 0.30/0.65  thf(fact_8325_power__sum,axiom,
% 0.30/0.65      ! [C: nat,F: nat > nat,A4: set_nat] :
% 0.30/0.65        ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % power_sum
% 0.30/0.65  thf(fact_8326_power__sum,axiom,
% 0.30/0.65      ! [C: int,F: nat > nat,A4: set_nat] :
% 0.30/0.65        ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % power_sum
% 0.30/0.65  thf(fact_8327_power__sum,axiom,
% 0.30/0.65      ! [C: int,F: int > nat,A4: set_int] :
% 0.30/0.65        ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
% 0.30/0.65        = ( groups1705073143266064639nt_int
% 0.30/0.65          @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
% 0.30/0.65          @ A4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % power_sum
% 0.30/0.65  thf(fact_8328_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.65      ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.shift_bounds_cl_nat_ivl
% 0.30/0.65  thf(fact_8329_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.65      ! [G: nat > int,M: nat,K: nat,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.shift_bounds_cl_nat_ivl
% 0.30/0.65  thf(fact_8330_finite__nat__iff__bounded__le,axiom,
% 0.30/0.65      ( finite_finite_nat
% 0.30/0.65      = ( ^ [S4: set_nat] :
% 0.30/0.65          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_nat_iff_bounded_le
% 0.30/0.65  thf(fact_8331_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > real] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8332_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > real] :
% 0.30/0.65        ( ! [X5: real] :
% 0.30/0.65            ( ( member_real @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8333_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > real] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8334_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > real] :
% 0.30/0.65        ( ! [X5: complex] :
% 0.30/0.65            ( ( member_complex @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 0.30/0.65       => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8335_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > rat] :
% 0.30/0.65        ( ! [X5: nat] :
% 0.30/0.65            ( ( member_nat @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8336_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > rat] :
% 0.30/0.65        ( ! [X5: real] :
% 0.30/0.65            ( ( member_real @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8337_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > rat] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8338_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > rat] :
% 0.30/0.65        ( ! [X5: complex] :
% 0.30/0.65            ( ( member_complex @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 0.30/0.65       => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8339_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > nat] :
% 0.30/0.65        ( ! [X5: real] :
% 0.30/0.65            ( ( member_real @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8340_prod__le__1,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > nat] :
% 0.30/0.65        ( ! [X5: int] :
% 0.30/0.65            ( ( member_int @ X5 @ A4 )
% 0.30/0.65           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
% 0.30/0.65              & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
% 0.30/0.65       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_le_1
% 0.30/0.65  thf(fact_8341_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.65        ( ( R2 @ one_one_complex @ one_one_complex )
% 0.30/0.65       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_nat @ S2 )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8342_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 0.30/0.65        ( ( R2 @ one_one_complex @ one_one_complex )
% 0.30/0.65       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_int @ S2 )
% 0.30/0.65           => ( ! [X5: int] :
% 0.30/0.65                  ( ( member_int @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8343_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 0.30/0.65        ( ( R2 @ one_one_complex @ one_one_complex )
% 0.30/0.65       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite3207457112153483333omplex @ S2 )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8344_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
% 0.30/0.65        ( ( R2 @ one_one_real @ one_one_real )
% 0.30/0.65       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_nat @ S2 )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8345_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 0.30/0.65        ( ( R2 @ one_one_real @ one_one_real )
% 0.30/0.65       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_int @ S2 )
% 0.30/0.65           => ( ! [X5: int] :
% 0.30/0.65                  ( ( member_int @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8346_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.65        ( ( R2 @ one_one_real @ one_one_real )
% 0.30/0.65       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite3207457112153483333omplex @ S2 )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8347_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.30/0.65        ( ( R2 @ one_one_rat @ one_one_rat )
% 0.30/0.65       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_nat @ S2 )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8348_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 0.30/0.65        ( ( R2 @ one_one_rat @ one_one_rat )
% 0.30/0.65       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_int @ S2 )
% 0.30/0.65           => ( ! [X5: int] :
% 0.30/0.65                  ( ( member_int @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8349_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.65        ( ( R2 @ one_one_rat @ one_one_rat )
% 0.30/0.65       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite3207457112153483333omplex @ S2 )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8350_prod_Orelated,axiom,
% 0.30/0.65      ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 0.30/0.65        ( ( R2 @ one_one_nat @ one_one_nat )
% 0.30/0.65       => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
% 0.30/0.65              ( ( ( R2 @ X15 @ X22 )
% 0.30/0.65                & ( R2 @ Y15 @ Y22 ) )
% 0.30/0.65             => ( R2 @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X22 @ Y22 ) ) )
% 0.30/0.65         => ( ( finite_finite_int @ S2 )
% 0.30/0.65           => ( ! [X5: int] :
% 0.30/0.65                  ( ( member_int @ X5 @ S2 )
% 0.30/0.65                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 0.30/0.65             => ( R2 @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.related
% 0.30/0.65  thf(fact_8351_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8352_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8353_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8354_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > code_integer] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8355_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8356_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > code_integer] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8357_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > nat] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8358_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > int] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8359_prod__dvd__prod__subset,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > int] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset
% 0.30/0.65  thf(fact_8360_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_real,A4: set_real,F: real > nat,G: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: real] :
% 0.30/0.65                ( ( member_real @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8361_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: complex] :
% 0.30/0.65                ( ( member_complex @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8362_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_real,A4: set_real,F: real > int,G: real > int] :
% 0.30/0.65        ( ( finite_finite_real @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: real] :
% 0.30/0.65                ( ( member_real @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8363_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: complex] :
% 0.30/0.65                ( ( member_complex @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8364_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_real,A4: set_real,F: real > code_integer,G: real > code_integer] :
% 0.30/0.65        ( ( finite_finite_real @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: real] :
% 0.30/0.65                ( ( member_real @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A4 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8365_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: nat] :
% 0.30/0.65                ( ( member_nat @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8366_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: complex] :
% 0.30/0.65                ( ( member_complex @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8367_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: int] :
% 0.30/0.65                ( ( member_int @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8368_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > code_integer,G: int > code_integer] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: int] :
% 0.30/0.65                ( ( member_int @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8369_prod__dvd__prod__subset2,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( ! [A3: nat] :
% 0.30/0.65                ( ( member_nat @ A3 @ A4 )
% 0.30/0.65               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.30/0.65           => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_dvd_prod_subset2
% 0.30/0.65  thf(fact_8370_prod_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs_0
% 0.30/0.65  thf(fact_8371_prod_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups73079841787564623at_rat
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs_0
% 0.30/0.65  thf(fact_8372_prod_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs_0
% 0.30/0.65  thf(fact_8373_prod_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs_0
% 0.30/0.65  thf(fact_8374_prod_Onat__diff__reindex,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.65          @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.65        = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.nat_diff_reindex
% 0.30/0.65  thf(fact_8375_prod_Onat__diff__reindex,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.30/0.65          @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.65        = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.nat_diff_reindex
% 0.30/0.65  thf(fact_8376_Iic__subset__Iio__iff,axiom,
% 0.30/0.65      ! [A: rat,B2: rat] :
% 0.30/0.65        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B2 ) )
% 0.30/0.65        = ( ord_less_rat @ A @ B2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Iic_subset_Iio_iff
% 0.30/0.65  thf(fact_8377_Iic__subset__Iio__iff,axiom,
% 0.30/0.65      ! [A: num,B2: num] :
% 0.30/0.65        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B2 ) )
% 0.30/0.65        = ( ord_less_num @ A @ B2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Iic_subset_Iio_iff
% 0.30/0.65  thf(fact_8378_Iic__subset__Iio__iff,axiom,
% 0.30/0.65      ! [A: nat,B2: nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B2 ) )
% 0.30/0.65        = ( ord_less_nat @ A @ B2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Iic_subset_Iio_iff
% 0.30/0.65  thf(fact_8379_Iic__subset__Iio__iff,axiom,
% 0.30/0.65      ! [A: int,B2: int] :
% 0.30/0.65        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B2 ) )
% 0.30/0.65        = ( ord_less_int @ A @ B2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Iic_subset_Iio_iff
% 0.30/0.65  thf(fact_8380_Iic__subset__Iio__iff,axiom,
% 0.30/0.65      ! [A: real,B2: real] :
% 0.30/0.65        ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B2 ) )
% 0.30/0.65        = ( ord_less_real @ A @ B2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Iic_subset_Iio_iff
% 0.30/0.65  thf(fact_8381_prod_OatLeastAtMost__rev,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat,M: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeastAtMost_rev
% 0.30/0.65  thf(fact_8382_prod_OatLeastAtMost__rev,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat,M: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeastAtMost_rev
% 0.30/0.65  thf(fact_8383_prod_Ozero__middle,axiom,
% 0.30/0.65      ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.65        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.65         => ( ( groups6464643781859351333omplex
% 0.30/0.65              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.65            = ( groups6464643781859351333omplex
% 0.30/0.65              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.zero_middle
% 0.30/0.65  thf(fact_8384_prod_Ozero__middle,axiom,
% 0.30/0.65      ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
% 0.30/0.65        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.65         => ( ( groups129246275422532515t_real
% 0.30/0.65              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.65            = ( groups129246275422532515t_real
% 0.30/0.65              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.zero_middle
% 0.30/0.65  thf(fact_8385_prod_Ozero__middle,axiom,
% 0.30/0.65      ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.65         => ( ( groups73079841787564623at_rat
% 0.30/0.65              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.65            = ( groups73079841787564623at_rat
% 0.30/0.65              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.zero_middle
% 0.30/0.65  thf(fact_8386_prod_Ozero__middle,axiom,
% 0.30/0.65      ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.65         => ( ( groups708209901874060359at_nat
% 0.30/0.65              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.65            = ( groups708209901874060359at_nat
% 0.30/0.65              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.zero_middle
% 0.30/0.65  thf(fact_8387_prod_Ozero__middle,axiom,
% 0.30/0.65      ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.65         => ( ( groups705719431365010083at_int
% 0.30/0.65              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.65            = ( groups705719431365010083at_int
% 0.30/0.65              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.zero_middle
% 0.30/0.65  thf(fact_8388_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_real,I: real,F: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ I5 )
% 0.30/0.65       => ( ( member_real @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: real] :
% 0.30/0.65                  ( ( member_real @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8389_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_nat,I: nat,F: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ I5 )
% 0.30/0.65       => ( ( member_nat @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: nat] :
% 0.30/0.65                  ( ( member_nat @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8390_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_int,I: int,F: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ I5 )
% 0.30/0.65       => ( ( member_int @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: int] :
% 0.30/0.65                  ( ( member_int @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8391_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_complex,I: complex,F: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.65       => ( ( member_complex @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: complex] :
% 0.30/0.65                  ( ( member_complex @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8392_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_real,I: real,F: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ I5 )
% 0.30/0.65       => ( ( member_real @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: real] :
% 0.30/0.65                  ( ( member_real @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8393_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_nat,I: nat,F: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ I5 )
% 0.30/0.65       => ( ( member_nat @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: nat] :
% 0.30/0.65                  ( ( member_nat @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8394_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_int,I: int,F: int > rat] :
% 0.30/0.65        ( ( finite_finite_int @ I5 )
% 0.30/0.65       => ( ( member_int @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: int] :
% 0.30/0.65                  ( ( member_int @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8395_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_complex,I: complex,F: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.65       => ( ( member_complex @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: complex] :
% 0.30/0.65                  ( ( member_complex @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8396_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_real,I: real,F: real > int] :
% 0.30/0.65        ( ( finite_finite_real @ I5 )
% 0.30/0.65       => ( ( member_real @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: real] :
% 0.30/0.65                  ( ( member_real @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8397_less__1__prod2,axiom,
% 0.30/0.65      ! [I5: set_complex,I: complex,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.65       => ( ( member_complex @ I @ I5 )
% 0.30/0.65         => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 0.30/0.65           => ( ! [I3: complex] :
% 0.30/0.65                  ( ( member_complex @ I3 @ I5 )
% 0.30/0.65                 => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
% 0.30/0.65             => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod2
% 0.30/0.65  thf(fact_8398_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_complex,F: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_complex )
% 0.30/0.65         => ( ! [I3: complex] :
% 0.30/0.65                ( ( member_complex @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8399_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_nat,F: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_nat )
% 0.30/0.65         => ( ! [I3: nat] :
% 0.30/0.65                ( ( member_nat @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8400_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_int,F: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_int )
% 0.30/0.65         => ( ! [I3: int] :
% 0.30/0.65                ( ( member_int @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8401_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_real,F: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_real )
% 0.30/0.65         => ( ! [I3: real] :
% 0.30/0.65                ( ( member_real @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8402_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_complex,F: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_complex )
% 0.30/0.65         => ( ! [I3: complex] :
% 0.30/0.65                ( ( member_complex @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8403_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_nat,F: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_nat )
% 0.30/0.65         => ( ! [I3: nat] :
% 0.30/0.65                ( ( member_nat @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8404_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_int,F: int > rat] :
% 0.30/0.65        ( ( finite_finite_int @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_int )
% 0.30/0.65         => ( ! [I3: int] :
% 0.30/0.65                ( ( member_int @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8405_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_real,F: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_real )
% 0.30/0.65         => ( ! [I3: real] :
% 0.30/0.65                ( ( member_real @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8406_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_complex,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_complex )
% 0.30/0.65         => ( ! [I3: complex] :
% 0.30/0.65                ( ( member_complex @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8407_less__1__prod,axiom,
% 0.30/0.65      ! [I5: set_real,F: real > int] :
% 0.30/0.65        ( ( finite_finite_real @ I5 )
% 0.30/0.65       => ( ( I5 != bot_bot_set_real )
% 0.30/0.65         => ( ! [I3: real] :
% 0.30/0.65                ( ( member_real @ I3 @ I5 )
% 0.30/0.65               => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 0.30/0.65           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_1_prod
% 0.30/0.65  thf(fact_8408_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,G: nat > real] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.65       => ( ( finite_finite_nat @ A4 )
% 0.30/0.65         => ( ( groups129246275422532515t_real @ G @ A4 )
% 0.30/0.65            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8409_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,G: complex > real] :
% 0.30/0.65        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.65       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65         => ( ( groups766887009212190081x_real @ G @ A4 )
% 0.30/0.65            = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8410_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,G: nat > rat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.65       => ( ( finite_finite_nat @ A4 )
% 0.30/0.65         => ( ( groups73079841787564623at_rat @ G @ A4 )
% 0.30/0.65            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8411_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,G: complex > rat] :
% 0.30/0.65        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.65       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65         => ( ( groups225925009352817453ex_rat @ G @ A4 )
% 0.30/0.65            = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8412_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,G: complex > nat] :
% 0.30/0.65        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.65       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65         => ( ( groups861055069439313189ex_nat @ G @ A4 )
% 0.30/0.65            = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8413_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,G: complex > int] :
% 0.30/0.65        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 0.30/0.65       => ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65         => ( ( groups858564598930262913ex_int @ G @ A4 )
% 0.30/0.65            = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8414_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,G: int > real] :
% 0.30/0.65        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.65       => ( ( finite_finite_int @ A4 )
% 0.30/0.65         => ( ( groups2316167850115554303t_real @ G @ A4 )
% 0.30/0.65            = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8415_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,G: int > rat] :
% 0.30/0.65        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.65       => ( ( finite_finite_int @ A4 )
% 0.30/0.65         => ( ( groups1072433553688619179nt_rat @ G @ A4 )
% 0.30/0.65            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8416_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,G: int > nat] :
% 0.30/0.65        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 0.30/0.65       => ( ( finite_finite_int @ A4 )
% 0.30/0.65         => ( ( groups1707563613775114915nt_nat @ G @ A4 )
% 0.30/0.65            = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8417_prod_Osubset__diff,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,G: nat > nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 0.30/0.65       => ( ( finite_finite_nat @ A4 )
% 0.30/0.65         => ( ( groups708209901874060359at_nat @ G @ A4 )
% 0.30/0.65            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.subset_diff
% 0.30/0.65  thf(fact_8418_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_complex ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_complex ) )
% 0.30/0.65               => ( ( ( groups713298508707869441omplex @ G @ A4 )
% 0.30/0.65                    = ( groups713298508707869441omplex @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups713298508707869441omplex @ G @ C4 )
% 0.30/0.65                    = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8419_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_complex ) )
% 0.30/0.65             => ( ! [B3: nat] :
% 0.30/0.65                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_complex ) )
% 0.30/0.65               => ( ( ( groups6464643781859351333omplex @ G @ A4 )
% 0.30/0.65                    = ( groups6464643781859351333omplex @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups6464643781859351333omplex @ G @ C4 )
% 0.30/0.65                    = ( groups6464643781859351333omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8420_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_complex ) )
% 0.30/0.65             => ( ! [B3: complex] :
% 0.30/0.65                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_complex ) )
% 0.30/0.65               => ( ( ( groups3708469109370488835omplex @ G @ A4 )
% 0.30/0.65                    = ( groups3708469109370488835omplex @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups3708469109370488835omplex @ G @ C4 )
% 0.30/0.65                    = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8421_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_real ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_real ) )
% 0.30/0.65               => ( ( ( groups1681761925125756287l_real @ G @ A4 )
% 0.30/0.65                    = ( groups1681761925125756287l_real @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups1681761925125756287l_real @ G @ C4 )
% 0.30/0.65                    = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8422_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_real ) )
% 0.30/0.65             => ( ! [B3: nat] :
% 0.30/0.65                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_real ) )
% 0.30/0.65               => ( ( ( groups129246275422532515t_real @ G @ A4 )
% 0.30/0.65                    = ( groups129246275422532515t_real @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups129246275422532515t_real @ G @ C4 )
% 0.30/0.65                    = ( groups129246275422532515t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8423_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_real ) )
% 0.30/0.65             => ( ! [B3: complex] :
% 0.30/0.65                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_real ) )
% 0.30/0.65               => ( ( ( groups766887009212190081x_real @ G @ A4 )
% 0.30/0.65                    = ( groups766887009212190081x_real @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups766887009212190081x_real @ G @ C4 )
% 0.30/0.65                    = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8424_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_rat ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_rat ) )
% 0.30/0.65               => ( ( ( groups4061424788464935467al_rat @ G @ A4 )
% 0.30/0.65                    = ( groups4061424788464935467al_rat @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups4061424788464935467al_rat @ G @ C4 )
% 0.30/0.65                    = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8425_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_rat ) )
% 0.30/0.65             => ( ! [B3: nat] :
% 0.30/0.65                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_rat ) )
% 0.30/0.65               => ( ( ( groups73079841787564623at_rat @ G @ A4 )
% 0.30/0.65                    = ( groups73079841787564623at_rat @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups73079841787564623at_rat @ G @ C4 )
% 0.30/0.65                    = ( groups73079841787564623at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8426_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_rat ) )
% 0.30/0.65             => ( ! [B3: complex] :
% 0.30/0.65                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_rat ) )
% 0.30/0.65               => ( ( ( groups225925009352817453ex_rat @ G @ A4 )
% 0.30/0.65                    = ( groups225925009352817453ex_rat @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups225925009352817453ex_rat @ G @ C4 )
% 0.30/0.65                    = ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8427_prod_Osame__carrier,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_nat ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_nat ) )
% 0.30/0.65               => ( ( ( groups4696554848551431203al_nat @ G @ A4 )
% 0.30/0.65                    = ( groups4696554848551431203al_nat @ H2 @ B5 ) )
% 0.30/0.65                  = ( ( groups4696554848551431203al_nat @ G @ C4 )
% 0.30/0.65                    = ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrier
% 0.30/0.65  thf(fact_8428_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_complex ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_complex ) )
% 0.30/0.65               => ( ( ( groups713298508707869441omplex @ G @ C4 )
% 0.30/0.65                    = ( groups713298508707869441omplex @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups713298508707869441omplex @ G @ A4 )
% 0.30/0.65                    = ( groups713298508707869441omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8429_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_complex ) )
% 0.30/0.65             => ( ! [B3: nat] :
% 0.30/0.65                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_complex ) )
% 0.30/0.65               => ( ( ( groups6464643781859351333omplex @ G @ C4 )
% 0.30/0.65                    = ( groups6464643781859351333omplex @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups6464643781859351333omplex @ G @ A4 )
% 0.30/0.65                    = ( groups6464643781859351333omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8430_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_complex ) )
% 0.30/0.65             => ( ! [B3: complex] :
% 0.30/0.65                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_complex ) )
% 0.30/0.65               => ( ( ( groups3708469109370488835omplex @ G @ C4 )
% 0.30/0.65                    = ( groups3708469109370488835omplex @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups3708469109370488835omplex @ G @ A4 )
% 0.30/0.65                    = ( groups3708469109370488835omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8431_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_real ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_real ) )
% 0.30/0.65               => ( ( ( groups1681761925125756287l_real @ G @ C4 )
% 0.30/0.65                    = ( groups1681761925125756287l_real @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups1681761925125756287l_real @ G @ A4 )
% 0.30/0.65                    = ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8432_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_real ) )
% 0.30/0.65             => ( ! [B3: nat] :
% 0.30/0.65                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_real ) )
% 0.30/0.65               => ( ( ( groups129246275422532515t_real @ G @ C4 )
% 0.30/0.65                    = ( groups129246275422532515t_real @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups129246275422532515t_real @ G @ A4 )
% 0.30/0.65                    = ( groups129246275422532515t_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8433_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_real ) )
% 0.30/0.65             => ( ! [B3: complex] :
% 0.30/0.65                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_real ) )
% 0.30/0.65               => ( ( ( groups766887009212190081x_real @ G @ C4 )
% 0.30/0.65                    = ( groups766887009212190081x_real @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups766887009212190081x_real @ G @ A4 )
% 0.30/0.65                    = ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8434_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_rat ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_rat ) )
% 0.30/0.65               => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
% 0.30/0.65                    = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups4061424788464935467al_rat @ G @ A4 )
% 0.30/0.65                    = ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8435_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_rat ) )
% 0.30/0.65             => ( ! [B3: nat] :
% 0.30/0.65                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_rat ) )
% 0.30/0.65               => ( ( ( groups73079841787564623at_rat @ G @ C4 )
% 0.30/0.65                    = ( groups73079841787564623at_rat @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups73079841787564623at_rat @ G @ A4 )
% 0.30/0.65                    = ( groups73079841787564623at_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8436_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_rat ) )
% 0.30/0.65             => ( ! [B3: complex] :
% 0.30/0.65                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_rat ) )
% 0.30/0.65               => ( ( ( groups225925009352817453ex_rat @ G @ C4 )
% 0.30/0.65                    = ( groups225925009352817453ex_rat @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups225925009352817453ex_rat @ G @ A4 )
% 0.30/0.65                    = ( groups225925009352817453ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8437_prod_Osame__carrierI,axiom,
% 0.30/0.65      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ C4 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 0.30/0.65         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 0.30/0.65                 => ( ( G @ A3 )
% 0.30/0.65                    = one_one_nat ) )
% 0.30/0.65             => ( ! [B3: real] :
% 0.30/0.65                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.30/0.65                   => ( ( H2 @ B3 )
% 0.30/0.65                      = one_one_nat ) )
% 0.30/0.65               => ( ( ( groups4696554848551431203al_nat @ G @ C4 )
% 0.30/0.65                    = ( groups4696554848551431203al_nat @ H2 @ C4 ) )
% 0.30/0.65                 => ( ( groups4696554848551431203al_nat @ G @ A4 )
% 0.30/0.65                    = ( groups4696554848551431203al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.same_carrierI
% 0.30/0.65  thf(fact_8438_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ( groups6464643781859351333omplex @ G @ S2 )
% 0.30/0.65              = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8439_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ( groups3708469109370488835omplex @ G @ S2 )
% 0.30/0.65              = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8440_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ( groups129246275422532515t_real @ G @ S2 )
% 0.30/0.65              = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8441_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ( groups766887009212190081x_real @ G @ S2 )
% 0.30/0.65              = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8442_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ( groups73079841787564623at_rat @ G @ S2 )
% 0.30/0.65              = ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8443_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ( groups225925009352817453ex_rat @ G @ S2 )
% 0.30/0.65              = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8444_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_nat ) )
% 0.30/0.65           => ( ( groups861055069439313189ex_nat @ G @ S2 )
% 0.30/0.65              = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8445_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_int ) )
% 0.30/0.65           => ( ( groups858564598930262913ex_int @ G @ S2 )
% 0.30/0.65              = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8446_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.30/0.65        ( ( finite_finite_int @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: int] :
% 0.30/0.65                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ( groups7440179247065528705omplex @ G @ S2 )
% 0.30/0.65              = ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8447_prod_Omono__neutral__left,axiom,
% 0.30/0.65      ! [T3: set_int,S2: set_int,G: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: int] :
% 0.30/0.65                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ( groups2316167850115554303t_real @ G @ S2 )
% 0.30/0.65              = ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_left
% 0.30/0.65  thf(fact_8448_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ( groups6464643781859351333omplex @ G @ T3 )
% 0.30/0.65              = ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8449_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ( groups3708469109370488835omplex @ G @ T3 )
% 0.30/0.65              = ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8450_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ( groups129246275422532515t_real @ G @ T3 )
% 0.30/0.65              = ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8451_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ( groups766887009212190081x_real @ G @ T3 )
% 0.30/0.65              = ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8452_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ( groups73079841787564623at_rat @ G @ T3 )
% 0.30/0.65              = ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8453_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 0.30/0.65              = ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8454_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_nat ) )
% 0.30/0.65           => ( ( groups861055069439313189ex_nat @ G @ T3 )
% 0.30/0.65              = ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8455_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_int ) )
% 0.30/0.65           => ( ( groups858564598930262913ex_int @ G @ T3 )
% 0.30/0.65              = ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8456_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.30/0.65        ( ( finite_finite_int @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: int] :
% 0.30/0.65                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ( groups7440179247065528705omplex @ G @ T3 )
% 0.30/0.65              = ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8457_prod_Omono__neutral__right,axiom,
% 0.30/0.65      ! [T3: set_int,S2: set_int,G: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: int] :
% 0.30/0.65                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ( groups2316167850115554303t_real @ G @ T3 )
% 0.30/0.65              = ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_right
% 0.30/0.65  thf(fact_8458_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups713298508707869441omplex @ G @ S2 )
% 0.30/0.65                = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8459_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups6464643781859351333omplex @ G @ S2 )
% 0.30/0.65                = ( groups6464643781859351333omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8460_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups3708469109370488835omplex @ G @ S2 )
% 0.30/0.65                = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8461_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups1681761925125756287l_real @ G @ S2 )
% 0.30/0.65                = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8462_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,H2: nat > real,G: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups129246275422532515t_real @ G @ S2 )
% 0.30/0.65                = ( groups129246275422532515t_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8463_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups766887009212190081x_real @ G @ S2 )
% 0.30/0.65                = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8464_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups4061424788464935467al_rat @ G @ S2 )
% 0.30/0.65                = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8465_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups73079841787564623at_rat @ G @ S2 )
% 0.30/0.65                = ( groups73079841787564623at_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8466_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups225925009352817453ex_rat @ G @ S2 )
% 0.30/0.65                = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8467_prod_Omono__neutral__cong__left,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( H2 @ X5 )
% 0.30/0.65                  = one_one_nat ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups4696554848551431203al_nat @ G @ S2 )
% 0.30/0.65                = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_left
% 0.30/0.65  thf(fact_8468_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups713298508707869441omplex @ G @ T3 )
% 0.30/0.65                = ( groups713298508707869441omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8469_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups6464643781859351333omplex @ G @ T3 )
% 0.30/0.65                = ( groups6464643781859351333omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8470_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > complex,H2: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_complex ) )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups3708469109370488835omplex @ G @ T3 )
% 0.30/0.65                = ( groups3708469109370488835omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8471_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups1681761925125756287l_real @ G @ T3 )
% 0.30/0.65                = ( groups1681761925125756287l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8472_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > real,H2: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups129246275422532515t_real @ G @ T3 )
% 0.30/0.65                = ( groups129246275422532515t_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8473_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_real ) )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups766887009212190081x_real @ G @ T3 )
% 0.30/0.65                = ( groups766887009212190081x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8474_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups4061424788464935467al_rat @ G @ T3 )
% 0.30/0.65                = ( groups4061424788464935467al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8475_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_nat,S2: set_nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: nat] :
% 0.30/0.65                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ! [X5: nat] :
% 0.30/0.65                  ( ( member_nat @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups73079841787564623at_rat @ G @ T3 )
% 0.30/0.65                = ( groups73079841787564623at_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8476_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: complex] :
% 0.30/0.65                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_rat ) )
% 0.30/0.65           => ( ! [X5: complex] :
% 0.30/0.65                  ( ( member_complex @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 0.30/0.65                = ( groups225925009352817453ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8477_prod_Omono__neutral__cong__right,axiom,
% 0.30/0.65      ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ T3 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.30/0.65         => ( ! [X5: real] :
% 0.30/0.65                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.30/0.65               => ( ( G @ X5 )
% 0.30/0.65                  = one_one_nat ) )
% 0.30/0.65           => ( ! [X5: real] :
% 0.30/0.65                  ( ( member_real @ X5 @ S2 )
% 0.30/0.65                 => ( ( G @ X5 )
% 0.30/0.65                    = ( H2 @ X5 ) ) )
% 0.30/0.65             => ( ( groups4696554848551431203al_nat @ G @ T3 )
% 0.30/0.65                = ( groups4696554848551431203al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.mono_neutral_cong_right
% 0.30/0.65  thf(fact_8478_prod_OatLeast0__atMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast0_atMost_Suc
% 0.30/0.65  thf(fact_8479_prod_OatLeast0__atMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast0_atMost_Suc
% 0.30/0.65  thf(fact_8480_prod_OatLeast0__atMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast0_atMost_Suc
% 0.30/0.65  thf(fact_8481_prod_OatLeast0__atMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast0_atMost_Suc
% 0.30/0.65  thf(fact_8482_prod_OatLeast__Suc__atMost,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast_Suc_atMost
% 0.30/0.65  thf(fact_8483_prod_OatLeast__Suc__atMost,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast_Suc_atMost
% 0.30/0.65  thf(fact_8484_prod_OatLeast__Suc__atMost,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast_Suc_atMost
% 0.30/0.65  thf(fact_8485_prod_OatLeast__Suc__atMost,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atLeast_Suc_atMost
% 0.30/0.65  thf(fact_8486_prod_Onat__ivl__Suc_H,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.65       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.nat_ivl_Suc'
% 0.30/0.65  thf(fact_8487_prod_Onat__ivl__Suc_H,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.65       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.nat_ivl_Suc'
% 0.30/0.65  thf(fact_8488_prod_Onat__ivl__Suc_H,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.65       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.nat_ivl_Suc'
% 0.30/0.65  thf(fact_8489_prod_Onat__ivl__Suc_H,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.65       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.nat_ivl_Suc'
% 0.30/0.65  thf(fact_8490_prod_OlessThan__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_real @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups129246275422532515t_real
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc_shift
% 0.30/0.65  thf(fact_8491_prod_OlessThan__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_rat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups73079841787564623at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc_shift
% 0.30/0.65  thf(fact_8492_prod_OlessThan__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_nat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc_shift
% 0.30/0.65  thf(fact_8493_prod_OlessThan__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( times_times_int @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups705719431365010083at_int
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc_shift
% 0.30/0.65  thf(fact_8494_prod_OSuc__reindex__ivl,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > real] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_real @ ( G @ M )
% 0.30/0.65            @ ( groups129246275422532515t_real
% 0.30/0.65              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.Suc_reindex_ivl
% 0.30/0.65  thf(fact_8495_prod_OSuc__reindex__ivl,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > rat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_rat @ ( G @ M )
% 0.30/0.65            @ ( groups73079841787564623at_rat
% 0.30/0.65              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.Suc_reindex_ivl
% 0.30/0.65  thf(fact_8496_prod_OSuc__reindex__ivl,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_nat @ ( G @ M )
% 0.30/0.65            @ ( groups708209901874060359at_nat
% 0.30/0.65              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.Suc_reindex_ivl
% 0.30/0.65  thf(fact_8497_prod_OSuc__reindex__ivl,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.30/0.65          = ( times_times_int @ ( G @ M )
% 0.30/0.65            @ ( groups705719431365010083at_int
% 0.30/0.65              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.Suc_reindex_ivl
% 0.30/0.65  thf(fact_8498_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups2906978787729119204at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc_shift
% 0.30/0.65  thf(fact_8499_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups3539618377306564664at_int
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc_shift
% 0.30/0.65  thf(fact_8500_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups3542108847815614940at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc_shift
% 0.30/0.65  thf(fact_8501_sum_OatMost__Suc__shift,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 0.30/0.65        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups6591440286371151544t_real
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc_shift
% 0.30/0.65  thf(fact_8502_sum__telescope,axiom,
% 0.30/0.65      ! [F: nat > complex,I: nat] :
% 0.30/0.65        ( ( groups2073611262835488442omplex
% 0.30/0.65          @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ I ) )
% 0.30/0.65        = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_telescope
% 0.30/0.65  thf(fact_8503_sum__telescope,axiom,
% 0.30/0.65      ! [F: nat > rat,I: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat
% 0.30/0.65          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ I ) )
% 0.30/0.65        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_telescope
% 0.30/0.65  thf(fact_8504_sum__telescope,axiom,
% 0.30/0.65      ! [F: nat > int,I: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int
% 0.30/0.65          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ I ) )
% 0.30/0.65        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_telescope
% 0.30/0.65  thf(fact_8505_sum__telescope,axiom,
% 0.30/0.65      ! [F: nat > real,I: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ I ) )
% 0.30/0.65        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_telescope
% 0.30/0.65  thf(fact_8506_polyfun__eq__coeffs,axiom,
% 0.30/0.65      ! [C: nat > complex,N: nat,D: nat > complex] :
% 0.30/0.65        ( ( ! [X: complex] :
% 0.30/0.65              ( ( groups2073611262835488442omplex
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65              = ( groups2073611262835488442omplex
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) ) ) )
% 0.30/0.65        = ( ! [I4: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I4 @ N )
% 0.30/0.65             => ( ( C @ I4 )
% 0.30/0.65                = ( D @ I4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_eq_coeffs
% 0.30/0.65  thf(fact_8507_polyfun__eq__coeffs,axiom,
% 0.30/0.65      ! [C: nat > real,N: nat,D: nat > real] :
% 0.30/0.65        ( ( ! [X: real] :
% 0.30/0.65              ( ( groups6591440286371151544t_real
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65              = ( groups6591440286371151544t_real
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) ) ) )
% 0.30/0.65        = ( ! [I4: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I4 @ N )
% 0.30/0.65             => ( ( C @ I4 )
% 0.30/0.65                = ( D @ I4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_eq_coeffs
% 0.30/0.65  thf(fact_8508_bounded__imp__summable,axiom,
% 0.30/0.65      ! [A: nat > int,B5: int] :
% 0.30/0.65        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
% 0.30/0.65       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
% 0.30/0.65         => ( summable_int @ A ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bounded_imp_summable
% 0.30/0.65  thf(fact_8509_bounded__imp__summable,axiom,
% 0.30/0.65      ! [A: nat > nat,B5: nat] :
% 0.30/0.65        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
% 0.30/0.65       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
% 0.30/0.65         => ( summable_nat @ A ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bounded_imp_summable
% 0.30/0.65  thf(fact_8510_bounded__imp__summable,axiom,
% 0.30/0.65      ! [A: nat > real,B5: real] :
% 0.30/0.65        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.65       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
% 0.30/0.65         => ( summable_real @ A ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bounded_imp_summable
% 0.30/0.65  thf(fact_8511_prod__atLeastAtMost__code,axiom,
% 0.30/0.65      ! [F: nat > complex,A: nat,B2: nat] :
% 0.30/0.65        ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.65        = ( set_fo1517530859248394432omplex
% 0.30/0.65          @ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
% 0.30/0.65          @ A
% 0.30/0.65          @ B2
% 0.30/0.65          @ one_one_complex ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_atLeastAtMost_code
% 0.30/0.65  thf(fact_8512_prod__atLeastAtMost__code,axiom,
% 0.30/0.65      ! [F: nat > real,A: nat,B2: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.65        = ( set_fo3111899725591712190t_real
% 0.30/0.65          @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
% 0.30/0.65          @ A
% 0.30/0.65          @ B2
% 0.30/0.65          @ one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_atLeastAtMost_code
% 0.30/0.65  thf(fact_8513_prod__atLeastAtMost__code,axiom,
% 0.30/0.65      ! [F: nat > rat,A: nat,B2: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.65        = ( set_fo1949268297981939178at_rat
% 0.30/0.65          @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
% 0.30/0.65          @ A
% 0.30/0.65          @ B2
% 0.30/0.65          @ one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_atLeastAtMost_code
% 0.30/0.65  thf(fact_8514_prod__atLeastAtMost__code,axiom,
% 0.30/0.65      ! [F: nat > nat,A: nat,B2: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.65        = ( set_fo2584398358068434914at_nat
% 0.30/0.65          @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
% 0.30/0.65          @ A
% 0.30/0.65          @ B2
% 0.30/0.65          @ one_one_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_atLeastAtMost_code
% 0.30/0.65  thf(fact_8515_prod__atLeastAtMost__code,axiom,
% 0.30/0.65      ! [F: nat > int,A: nat,B2: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 0.30/0.65        = ( set_fo2581907887559384638at_int
% 0.30/0.65          @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
% 0.30/0.65          @ A
% 0.30/0.65          @ B2
% 0.30/0.65          @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_atLeastAtMost_code
% 0.30/0.65  thf(fact_8516_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65       => ( ! [I3: complex] :
% 0.30/0.65              ( ( member_complex @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_complex )
% 0.30/0.65           => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8517_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ A4 )
% 0.30/0.65       => ( ! [I3: nat] :
% 0.30/0.65              ( ( member_nat @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_nat )
% 0.30/0.65           => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8518_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > real,G: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ A4 )
% 0.30/0.65       => ( ! [I3: int] :
% 0.30/0.65              ( ( member_int @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_int )
% 0.30/0.65           => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8519_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > real,G: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ A4 )
% 0.30/0.65       => ( ! [I3: real] :
% 0.30/0.65              ( ( member_real @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_real )
% 0.30/0.65           => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8520_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65       => ( ! [I3: complex] :
% 0.30/0.65              ( ( member_complex @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_complex )
% 0.30/0.65           => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8521_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ A4 )
% 0.30/0.65       => ( ! [I3: nat] :
% 0.30/0.65              ( ( member_nat @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_nat )
% 0.30/0.65           => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8522_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > rat,G: int > rat] :
% 0.30/0.65        ( ( finite_finite_int @ A4 )
% 0.30/0.65       => ( ! [I3: int] :
% 0.30/0.65              ( ( member_int @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_int )
% 0.30/0.65           => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8523_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_real,F: real > rat,G: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ A4 )
% 0.30/0.65       => ( ! [I3: real] :
% 0.30/0.65              ( ( member_real @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_real )
% 0.30/0.65           => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8524_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65       => ( ! [I3: complex] :
% 0.30/0.65              ( ( member_complex @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_complex )
% 0.30/0.65           => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8525_prod__mono__strict,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > nat,G: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ A4 )
% 0.30/0.65       => ( ! [I3: int] :
% 0.30/0.65              ( ( member_int @ I3 @ A4 )
% 0.30/0.65             => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 0.30/0.65                & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 0.30/0.65         => ( ( A4 != bot_bot_set_int )
% 0.30/0.65           => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono_strict
% 0.30/0.65  thf(fact_8526_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > code_integer] :
% 0.30/0.65        ( ( finite_finite_nat @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: nat] :
% 0.30/0.65                ( ( member_nat @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8527_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > code_integer] :
% 0.30/0.65        ( ( finite_finite_int @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3827104343326376752nteger @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8528_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > code_integer] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: complex] :
% 0.30/0.65                ( ( member_complex @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8529_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8530_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: complex] :
% 0.30/0.65                ( ( member_complex @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8531_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_complex,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: complex] :
% 0.30/0.65                ( ( member_complex @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8532_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > nat] :
% 0.30/0.65        ( ( finite_finite_nat @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: nat] :
% 0.30/0.65                ( ( member_nat @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8533_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_nat,F: nat > int] :
% 0.30/0.65        ( ( finite_finite_nat @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: nat] :
% 0.30/0.65                ( ( member_nat @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8534_even__prod__iff,axiom,
% 0.30/0.65      ! [A4: set_int,F: int > int] :
% 0.30/0.65        ( ( finite_finite_int @ A4 )
% 0.30/0.65       => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A4 )
% 0.30/0.65                & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % even_prod_iff
% 0.30/0.65  thf(fact_8535_sum__choose__lower,axiom,
% 0.30/0.65      ! [R: nat,N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_choose_lower
% 0.30/0.65  thf(fact_8536_choose__rising__sum_I2_J,axiom,
% 0.30/0.65      ! [N: nat,M: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_rising_sum(2)
% 0.30/0.65  thf(fact_8537_choose__rising__sum_I1_J,axiom,
% 0.30/0.65      ! [N: nat,M: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_rising_sum(1)
% 0.30/0.65  thf(fact_8538_prod_Oub__add__nat,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > real,P4: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.65       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.65          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.ub_add_nat
% 0.30/0.65  thf(fact_8539_prod_Oub__add__nat,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > rat,P4: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.65       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.65          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.ub_add_nat
% 0.30/0.65  thf(fact_8540_prod_Oub__add__nat,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > nat,P4: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.65       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.65          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.ub_add_nat
% 0.30/0.65  thf(fact_8541_prod_Oub__add__nat,axiom,
% 0.30/0.65      ! [M: nat,N: nat,G: nat > int,P4: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.30/0.65       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 0.30/0.65          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.ub_add_nat
% 0.30/0.65  thf(fact_8542_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_real,Z: real > real,W: real > real] :
% 0.30/0.65        ( ! [I3: real] :
% 0.30/0.65            ( ( member_real @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: real] :
% 0.30/0.65              ( ( member_real @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups8097168146408367636l_real
% 0.30/0.65              @ ^ [I4: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8543_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_int,Z: int > real,W: int > real] :
% 0.30/0.65        ( ! [I3: int] :
% 0.30/0.65            ( ( member_int @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: int] :
% 0.30/0.65              ( ( member_int @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups8778361861064173332t_real
% 0.30/0.65              @ ^ [I4: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8544_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_complex,Z: complex > real,W: complex > real] :
% 0.30/0.65        ( ! [I3: complex] :
% 0.30/0.65            ( ( member_complex @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: complex] :
% 0.30/0.65              ( ( member_complex @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I5 ) @ ( groups766887009212190081x_real @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups5808333547571424918x_real
% 0.30/0.65              @ ^ [I4: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8545_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_real,Z: real > complex,W: real > complex] :
% 0.30/0.65        ( ! [I3: real] :
% 0.30/0.65            ( ( member_real @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: real] :
% 0.30/0.65              ( ( member_real @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups8097168146408367636l_real
% 0.30/0.65              @ ^ [I4: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8546_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_int,Z: int > complex,W: int > complex] :
% 0.30/0.65        ( ! [I3: int] :
% 0.30/0.65            ( ( member_int @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: int] :
% 0.30/0.65              ( ( member_int @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups8778361861064173332t_real
% 0.30/0.65              @ ^ [I4: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8547_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_complex,Z: complex > complex,W: complex > complex] :
% 0.30/0.65        ( ! [I3: complex] :
% 0.30/0.65            ( ( member_complex @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: complex] :
% 0.30/0.65              ( ( member_complex @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I5 ) @ ( groups3708469109370488835omplex @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups5808333547571424918x_real
% 0.30/0.65              @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8548_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_nat,Z: nat > real,W: nat > real] :
% 0.30/0.65        ( ! [I3: nat] :
% 0.30/0.65            ( ( member_nat @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: nat] :
% 0.30/0.65              ( ( member_nat @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8549_norm__prod__diff,axiom,
% 0.30/0.65      ! [I5: set_nat,Z: nat > complex,W: nat > complex] :
% 0.30/0.65        ( ! [I3: nat] :
% 0.30/0.65            ( ( member_nat @ I3 @ I5 )
% 0.30/0.65           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 0.30/0.65       => ( ! [I3: nat] :
% 0.30/0.65              ( ( member_nat @ I3 @ I5 )
% 0.30/0.65             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 0.30/0.65         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W @ I5 ) ) )
% 0.30/0.65            @ ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 0.30/0.65              @ I5 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % norm_prod_diff
% 0.30/0.65  thf(fact_8550_polyfun__eq__0,axiom,
% 0.30/0.65      ! [C: nat > complex,N: nat] :
% 0.30/0.65        ( ( ! [X: complex] :
% 0.30/0.65              ( ( groups2073611262835488442omplex
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65              = zero_zero_complex ) )
% 0.30/0.65        = ( ! [I4: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I4 @ N )
% 0.30/0.65             => ( ( C @ I4 )
% 0.30/0.65                = zero_zero_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_eq_0
% 0.30/0.65  thf(fact_8551_polyfun__eq__0,axiom,
% 0.30/0.65      ! [C: nat > real,N: nat] :
% 0.30/0.65        ( ( ! [X: real] :
% 0.30/0.65              ( ( groups6591440286371151544t_real
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65              = zero_zero_real ) )
% 0.30/0.65        = ( ! [I4: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I4 @ N )
% 0.30/0.65             => ( ( C @ I4 )
% 0.30/0.65                = zero_zero_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_eq_0
% 0.30/0.65  thf(fact_8552_zero__polynom__imp__zero__coeffs,axiom,
% 0.30/0.65      ! [C: nat > complex,N: nat,K: nat] :
% 0.30/0.65        ( ! [W2: complex] :
% 0.30/0.65            ( ( groups2073611262835488442omplex
% 0.30/0.65              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ W2 @ I4 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65            = zero_zero_complex )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65         => ( ( C @ K )
% 0.30/0.65            = zero_zero_complex ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_polynom_imp_zero_coeffs
% 0.30/0.65  thf(fact_8553_zero__polynom__imp__zero__coeffs,axiom,
% 0.30/0.65      ! [C: nat > real,N: nat,K: nat] :
% 0.30/0.65        ( ! [W2: real] :
% 0.30/0.65            ( ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ W2 @ I4 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65            = zero_zero_real )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65         => ( ( C @ K )
% 0.30/0.65            = zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_polynom_imp_zero_coeffs
% 0.30/0.65  thf(fact_8554_sum_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups2906978787729119204at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_shift
% 0.30/0.65  thf(fact_8555_sum_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups3539618377306564664at_int
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_shift
% 0.30/0.65  thf(fact_8556_sum_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups3542108847815614940at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_shift
% 0.30/0.65  thf(fact_8557_sum_OatMost__shift,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.30/0.65          @ ( groups6591440286371151544t_real
% 0.30/0.65            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.30/0.65            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_shift
% 0.30/0.65  thf(fact_8558_sum__up__index__split,axiom,
% 0.30/0.65      ! [F: nat > rat,M: nat,N: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.65        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_up_index_split
% 0.30/0.65  thf(fact_8559_sum__up__index__split,axiom,
% 0.30/0.65      ! [F: nat > int,M: nat,N: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.65        = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_up_index_split
% 0.30/0.65  thf(fact_8560_sum__up__index__split,axiom,
% 0.30/0.65      ! [F: nat > nat,M: nat,N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.65        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_up_index_split
% 0.30/0.65  thf(fact_8561_sum__up__index__split,axiom,
% 0.30/0.65      ! [F: nat > real,M: nat,N: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 0.30/0.65        = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_up_index_split
% 0.30/0.65  thf(fact_8562_sum_Otriangle__reindex__eq,axiom,
% 0.30/0.65      ! [G: nat > nat > nat,N: nat] :
% 0.30/0.65        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 0.30/0.65          @ ( collec3392354462482085612at_nat
% 0.30/0.65            @ ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.65        = ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K2: nat] :
% 0.30/0.65              ( groups3542108847815614940at_nat
% 0.30/0.65              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.triangle_reindex_eq
% 0.30/0.65  thf(fact_8563_sum_Otriangle__reindex__eq,axiom,
% 0.30/0.65      ! [G: nat > nat > real,N: nat] :
% 0.30/0.65        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 0.30/0.65          @ ( collec3392354462482085612at_nat
% 0.30/0.65            @ ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.65        = ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [K2: nat] :
% 0.30/0.65              ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.triangle_reindex_eq
% 0.30/0.65  thf(fact_8564_fact__eq__fact__times,axiom,
% 0.30/0.65      ! [N: nat,M: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65       => ( ( semiri1408675320244567234ct_nat @ M )
% 0.30/0.65          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.30/0.65            @ ( groups708209901874060359at_nat
% 0.30/0.65              @ ^ [X: nat] : X
% 0.30/0.65              @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_eq_fact_times
% 0.30/0.65  thf(fact_8565_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_real,A4: set_real,F: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: real] :
% 0.30/0.65                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8566_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: nat] :
% 0.30/0.65                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8567_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: complex] :
% 0.30/0.65                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8568_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_real,A4: set_real,F: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: real] :
% 0.30/0.65                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8569_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_nat,A4: set_nat,F: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: nat] :
% 0.30/0.65                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: nat] :
% 0.30/0.65                  ( ( member_nat @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8570_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: complex] :
% 0.30/0.65                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8571_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_real,A4: set_real,F: real > int] :
% 0.30/0.65        ( ( finite_finite_real @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: real] :
% 0.30/0.65                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: real] :
% 0.30/0.65                  ( ( member_real @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8572_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_complex,A4: set_complex,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ B5 )
% 0.30/0.65       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: complex] :
% 0.30/0.65                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: complex] :
% 0.30/0.65                  ( ( member_complex @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8573_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: int] :
% 0.30/0.65                ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: int] :
% 0.30/0.65                  ( ( member_int @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8574_prod__mono2,axiom,
% 0.30/0.65      ! [B5: set_int,A4: set_int,F: int > rat] :
% 0.30/0.65        ( ( finite_finite_int @ B5 )
% 0.30/0.65       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 0.30/0.65         => ( ! [B3: int] :
% 0.30/0.65                ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
% 0.30/0.65               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 0.30/0.65           => ( ! [A3: int] :
% 0.30/0.65                  ( ( member_int @ A3 @ A4 )
% 0.30/0.65                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 0.30/0.65             => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_mono2
% 0.30/0.65  thf(fact_8575_sum__choose__diagonal,axiom,
% 0.30/0.65      ! [M: nat,N: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups3542108847815614940at_nat
% 0.30/0.65            @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65          = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_choose_diagonal
% 0.30/0.65  thf(fact_8576_vandermonde,axiom,
% 0.30/0.65      ! [M: nat,N: nat,R: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ R ) )
% 0.30/0.65        = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).
% 0.30/0.65  
% 0.30/0.65  % vandermonde
% 0.30/0.65  thf(fact_8577_sum__gp__basic,axiom,
% 0.30/0.65      ! [X3: complex,N: nat] :
% 0.30/0.65        ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_gp_basic
% 0.30/0.65  thf(fact_8578_sum__gp__basic,axiom,
% 0.30/0.65      ! [X3: rat,N: nat] :
% 0.30/0.65        ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65        = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_gp_basic
% 0.30/0.65  thf(fact_8579_sum__gp__basic,axiom,
% 0.30/0.65      ! [X3: int,N: nat] :
% 0.30/0.65        ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65        = ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_gp_basic
% 0.30/0.65  thf(fact_8580_sum__gp__basic,axiom,
% 0.30/0.65      ! [X3: real,N: nat] :
% 0.30/0.65        ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65        = ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_gp_basic
% 0.30/0.65  thf(fact_8581_polyfun__finite__roots,axiom,
% 0.30/0.65      ! [C: nat > complex,N: nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex
% 0.30/0.65          @ ( collect_complex
% 0.30/0.65            @ ^ [X: complex] :
% 0.30/0.65                ( ( groups2073611262835488442omplex
% 0.30/0.65                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.30/0.65                  @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                = zero_zero_complex ) ) )
% 0.30/0.65        = ( ? [I4: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I4 @ N )
% 0.30/0.65              & ( ( C @ I4 )
% 0.30/0.65               != zero_zero_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_finite_roots
% 0.30/0.65  thf(fact_8582_polyfun__finite__roots,axiom,
% 0.30/0.65      ! [C: nat > real,N: nat] :
% 0.30/0.65        ( ( finite_finite_real
% 0.30/0.65          @ ( collect_real
% 0.30/0.65            @ ^ [X: real] :
% 0.30/0.65                ( ( groups6591440286371151544t_real
% 0.30/0.65                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.30/0.65                  @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                = zero_zero_real ) ) )
% 0.30/0.65        = ( ? [I4: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I4 @ N )
% 0.30/0.65              & ( ( C @ I4 )
% 0.30/0.65               != zero_zero_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_finite_roots
% 0.30/0.65  thf(fact_8583_polyfun__roots__finite,axiom,
% 0.30/0.65      ! [C: nat > complex,K: nat,N: nat] :
% 0.30/0.65        ( ( ( C @ K )
% 0.30/0.65         != zero_zero_complex )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65         => ( finite3207457112153483333omplex
% 0.30/0.65            @ ( collect_complex
% 0.30/0.65              @ ^ [Z2: complex] :
% 0.30/0.65                  ( ( groups2073611262835488442omplex
% 0.30/0.65                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z2 @ I4 ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                  = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_roots_finite
% 0.30/0.65  thf(fact_8584_polyfun__roots__finite,axiom,
% 0.30/0.65      ! [C: nat > real,K: nat,N: nat] :
% 0.30/0.65        ( ( ( C @ K )
% 0.30/0.65         != zero_zero_real )
% 0.30/0.65       => ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.65         => ( finite_finite_real
% 0.30/0.65            @ ( collect_real
% 0.30/0.65              @ ^ [Z2: real] :
% 0.30/0.65                  ( ( groups6591440286371151544t_real
% 0.30/0.65                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z2 @ I4 ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                  = zero_zero_real ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_roots_finite
% 0.30/0.65  thf(fact_8585_pochhammer__Suc__prod,axiom,
% 0.30/0.65      ! [A: rat,N: nat] :
% 0.30/0.65        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.30/0.65        = ( groups73079841787564623at_rat
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc_prod
% 0.30/0.65  thf(fact_8586_pochhammer__Suc__prod,axiom,
% 0.30/0.65      ! [A: real,N: nat] :
% 0.30/0.65        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc_prod
% 0.30/0.65  thf(fact_8587_pochhammer__Suc__prod,axiom,
% 0.30/0.65      ! [A: nat,N: nat] :
% 0.30/0.65        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc_prod
% 0.30/0.65  thf(fact_8588_pochhammer__Suc__prod,axiom,
% 0.30/0.65      ! [A: int,N: nat] :
% 0.30/0.65        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_Suc_prod
% 0.30/0.65  thf(fact_8589_polyfun__linear__factor__root,axiom,
% 0.30/0.65      ! [C: nat > complex,A: complex,N: nat] :
% 0.30/0.65        ( ( ( groups2073611262835488442omplex
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65          = zero_zero_complex )
% 0.30/0.65       => ~ ! [B3: nat > complex] :
% 0.30/0.65              ~ ! [Z3: complex] :
% 0.30/0.65                  ( ( groups2073611262835488442omplex
% 0.30/0.65                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                  = ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
% 0.30/0.65                    @ ( groups2073611262835488442omplex
% 0.30/0.65                      @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 0.30/0.65                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor_root
% 0.30/0.65  thf(fact_8590_polyfun__linear__factor__root,axiom,
% 0.30/0.65      ! [C: nat > rat,A: rat,N: nat] :
% 0.30/0.65        ( ( ( groups2906978787729119204at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65          = zero_zero_rat )
% 0.30/0.65       => ~ ! [B3: nat > rat] :
% 0.30/0.65              ~ ! [Z3: rat] :
% 0.30/0.65                  ( ( groups2906978787729119204at_rat
% 0.30/0.65                    @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                  = ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
% 0.30/0.65                    @ ( groups2906978787729119204at_rat
% 0.30/0.65                      @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 0.30/0.65                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor_root
% 0.30/0.65  thf(fact_8591_polyfun__linear__factor__root,axiom,
% 0.30/0.65      ! [C: nat > int,A: int,N: nat] :
% 0.30/0.65        ( ( ( groups3539618377306564664at_int
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65          = zero_zero_int )
% 0.30/0.65       => ~ ! [B3: nat > int] :
% 0.30/0.65              ~ ! [Z3: int] :
% 0.30/0.65                  ( ( groups3539618377306564664at_int
% 0.30/0.65                    @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                  = ( times_times_int @ ( minus_minus_int @ Z3 @ A )
% 0.30/0.65                    @ ( groups3539618377306564664at_int
% 0.30/0.65                      @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 0.30/0.65                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor_root
% 0.30/0.65  thf(fact_8592_polyfun__linear__factor__root,axiom,
% 0.30/0.65      ! [C: nat > real,A: real,N: nat] :
% 0.30/0.65        ( ( ( groups6591440286371151544t_real
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65          = zero_zero_real )
% 0.30/0.65       => ~ ! [B3: nat > real] :
% 0.30/0.65              ~ ! [Z3: real] :
% 0.30/0.65                  ( ( groups6591440286371151544t_real
% 0.30/0.65                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65                  = ( times_times_real @ ( minus_minus_real @ Z3 @ A )
% 0.30/0.65                    @ ( groups6591440286371151544t_real
% 0.30/0.65                      @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 0.30/0.65                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor_root
% 0.30/0.65  thf(fact_8593_polyfun__linear__factor,axiom,
% 0.30/0.65      ! [C: nat > complex,N: nat,A: complex] :
% 0.30/0.65      ? [B3: nat > complex] :
% 0.30/0.65      ! [Z3: complex] :
% 0.30/0.65        ( ( groups2073611262835488442omplex
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_complex
% 0.30/0.65          @ ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
% 0.30/0.65            @ ( groups2073611262835488442omplex
% 0.30/0.65              @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 0.30/0.65              @ ( set_ord_lessThan_nat @ N ) ) )
% 0.30/0.65          @ ( groups2073611262835488442omplex
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor
% 0.30/0.65  thf(fact_8594_polyfun__linear__factor,axiom,
% 0.30/0.65      ! [C: nat > rat,N: nat,A: rat] :
% 0.30/0.65      ? [B3: nat > rat] :
% 0.30/0.65      ! [Z3: rat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_rat
% 0.30/0.65          @ ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
% 0.30/0.65            @ ( groups2906978787729119204at_rat
% 0.30/0.65              @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 0.30/0.65              @ ( set_ord_lessThan_nat @ N ) ) )
% 0.30/0.65          @ ( groups2906978787729119204at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor
% 0.30/0.65  thf(fact_8595_polyfun__linear__factor,axiom,
% 0.30/0.65      ! [C: nat > int,N: nat,A: int] :
% 0.30/0.65      ? [B3: nat > int] :
% 0.30/0.65      ! [Z3: int] :
% 0.30/0.65        ( ( groups3539618377306564664at_int
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_int
% 0.30/0.65          @ ( times_times_int @ ( minus_minus_int @ Z3 @ A )
% 0.30/0.65            @ ( groups3539618377306564664at_int
% 0.30/0.65              @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 0.30/0.65              @ ( set_ord_lessThan_nat @ N ) ) )
% 0.30/0.65          @ ( groups3539618377306564664at_int
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor
% 0.30/0.65  thf(fact_8596_polyfun__linear__factor,axiom,
% 0.30/0.65      ! [C: nat > real,N: nat,A: real] :
% 0.30/0.65      ? [B3: nat > real] :
% 0.30/0.65      ! [Z3: real] :
% 0.30/0.65        ( ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( plus_plus_real
% 0.30/0.65          @ ( times_times_real @ ( minus_minus_real @ Z3 @ A )
% 0.30/0.65            @ ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 0.30/0.65              @ ( set_ord_lessThan_nat @ N ) ) )
% 0.30/0.65          @ ( groups6591440286371151544t_real
% 0.30/0.65            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polyfun_linear_factor
% 0.30/0.65  thf(fact_8597_pochhammer__prod__rev,axiom,
% 0.30/0.65      ( comm_s4028243227959126397er_rat
% 0.30/0.65      = ( ^ [A5: rat,N2: nat] :
% 0.30/0.65            ( groups73079841787564623at_rat
% 0.30/0.65            @ ^ [I4: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 0.30/0.65            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_prod_rev
% 0.30/0.65  thf(fact_8598_pochhammer__prod__rev,axiom,
% 0.30/0.65      ( comm_s7457072308508201937r_real
% 0.30/0.65      = ( ^ [A5: real,N2: nat] :
% 0.30/0.65            ( groups129246275422532515t_real
% 0.30/0.65            @ ^ [I4: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 0.30/0.65            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_prod_rev
% 0.30/0.65  thf(fact_8599_pochhammer__prod__rev,axiom,
% 0.30/0.65      ( comm_s4663373288045622133er_nat
% 0.30/0.65      = ( ^ [A5: nat,N2: nat] :
% 0.30/0.65            ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [I4: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 0.30/0.65            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_prod_rev
% 0.30/0.65  thf(fact_8600_pochhammer__prod__rev,axiom,
% 0.30/0.65      ( comm_s4660882817536571857er_int
% 0.30/0.65      = ( ^ [A5: int,N2: nat] :
% 0.30/0.65            ( groups705719431365010083at_int
% 0.30/0.65            @ ^ [I4: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 0.30/0.65            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_prod_rev
% 0.30/0.65  thf(fact_8601_sum__power__shift,axiom,
% 0.30/0.65      ! [M: nat,N: nat,X3: complex] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_power_shift
% 0.30/0.65  thf(fact_8602_sum__power__shift,axiom,
% 0.30/0.65      ! [M: nat,N: nat,X3: rat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_power_shift
% 0.30/0.65  thf(fact_8603_sum__power__shift,axiom,
% 0.30/0.65      ! [M: nat,N: nat,X3: int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_power_shift
% 0.30/0.65  thf(fact_8604_sum__power__shift,axiom,
% 0.30/0.65      ! [M: nat,N: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.65       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.30/0.65          = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_power_shift
% 0.30/0.65  thf(fact_8605_sum_Otriangle__reindex,axiom,
% 0.30/0.65      ! [G: nat > nat > nat,N: nat] :
% 0.30/0.65        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 0.30/0.65          @ ( collec3392354462482085612at_nat
% 0.30/0.65            @ ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.65        = ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K2: nat] :
% 0.30/0.65              ( groups3542108847815614940at_nat
% 0.30/0.65              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.65          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.triangle_reindex
% 0.30/0.65  thf(fact_8606_sum_Otriangle__reindex,axiom,
% 0.30/0.65      ! [G: nat > nat > real,N: nat] :
% 0.30/0.65        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 0.30/0.65          @ ( collec3392354462482085612at_nat
% 0.30/0.65            @ ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.65        = ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [K2: nat] :
% 0.30/0.65              ( groups6591440286371151544t_real
% 0.30/0.65              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.65          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.triangle_reindex
% 0.30/0.65  thf(fact_8607_fact__div__fact,axiom,
% 0.30/0.65      ! [N: nat,M: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.65       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.30/0.65          = ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [X: nat] : X
% 0.30/0.65            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_div_fact
% 0.30/0.65  thf(fact_8608_summable__Cauchy__product,axiom,
% 0.30/0.65      ! [A: nat > complex,B2: nat > complex] :
% 0.30/0.65        ( ( summable_real
% 0.30/0.65          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
% 0.30/0.65       => ( ( summable_real
% 0.30/0.65            @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
% 0.30/0.65         => ( summable_complex
% 0.30/0.65            @ ^ [K2: nat] :
% 0.30/0.65                ( groups2073611262835488442omplex
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % summable_Cauchy_product
% 0.30/0.65  thf(fact_8609_summable__Cauchy__product,axiom,
% 0.30/0.65      ! [A: nat > real,B2: nat > real] :
% 0.30/0.65        ( ( summable_real
% 0.30/0.65          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
% 0.30/0.65       => ( ( summable_real
% 0.30/0.65            @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
% 0.30/0.65         => ( summable_real
% 0.30/0.65            @ ^ [K2: nat] :
% 0.30/0.65                ( groups6591440286371151544t_real
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % summable_Cauchy_product
% 0.30/0.65  thf(fact_8610_choose__row__sum,axiom,
% 0.30/0.65      ! [N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.65        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_row_sum
% 0.30/0.65  thf(fact_8611_Cauchy__product,axiom,
% 0.30/0.65      ! [A: nat > complex,B2: nat > complex] :
% 0.30/0.65        ( ( summable_real
% 0.30/0.65          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
% 0.30/0.65       => ( ( summable_real
% 0.30/0.65            @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
% 0.30/0.65         => ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B2 ) )
% 0.30/0.65            = ( suminf_complex
% 0.30/0.65              @ ^ [K2: nat] :
% 0.30/0.65                  ( groups2073611262835488442omplex
% 0.30/0.65                  @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 0.30/0.65                  @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Cauchy_product
% 0.30/0.65  thf(fact_8612_Cauchy__product,axiom,
% 0.30/0.65      ! [A: nat > real,B2: nat > real] :
% 0.30/0.65        ( ( summable_real
% 0.30/0.65          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
% 0.30/0.65       => ( ( summable_real
% 0.30/0.65            @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
% 0.30/0.65         => ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B2 ) )
% 0.30/0.65            = ( suminf_real
% 0.30/0.65              @ ^ [K2: nat] :
% 0.30/0.65                  ( groups6591440286371151544t_real
% 0.30/0.65                  @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 0.30/0.65                  @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Cauchy_product
% 0.30/0.65  thf(fact_8613_binomial,axiom,
% 0.30/0.65      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.65        ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 0.30/0.65        = ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial
% 0.30/0.65  thf(fact_8614_prod_Oin__pairs,axiom,
% 0.30/0.65      ! [G: nat > real,M: nat,N: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs
% 0.30/0.65  thf(fact_8615_prod_Oin__pairs,axiom,
% 0.30/0.65      ! [G: nat > rat,M: nat,N: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups73079841787564623at_rat
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs
% 0.30/0.65  thf(fact_8616_prod_Oin__pairs,axiom,
% 0.30/0.65      ! [G: nat > nat,M: nat,N: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs
% 0.30/0.65  thf(fact_8617_prod_Oin__pairs,axiom,
% 0.30/0.65      ! [G: nat > int,M: nat,N: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.in_pairs
% 0.30/0.65  thf(fact_8618_sum_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > rat,N: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups2906978787729119204at_rat
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.in_pairs_0
% 0.30/0.65  thf(fact_8619_sum_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > int,N: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups3539618377306564664at_int
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.in_pairs_0
% 0.30/0.65  thf(fact_8620_sum_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > nat,N: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.in_pairs_0
% 0.30/0.65  thf(fact_8621_sum_Oin__pairs__0,axiom,
% 0.30/0.65      ! [G: nat > real,N: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.65        = ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.in_pairs_0
% 0.30/0.65  thf(fact_8622_polynomial__product,axiom,
% 0.30/0.65      ! [M: nat,A: nat > complex,N: nat,B2: nat > complex,X3: complex] :
% 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_complex ) )
% 0.30/0.65       => ( ! [J2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ N @ J2 )
% 0.30/0.65             => ( ( B2 @ J2 )
% 0.30/0.65                = zero_zero_complex ) )
% 0.30/0.65         => ( ( times_times_complex
% 0.30/0.65              @ ( groups2073611262835488442omplex
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65              @ ( groups2073611262835488442omplex
% 0.30/0.65                @ ^ [J3: nat] : ( times_times_complex @ ( B2 @ J3 ) @ ( power_power_complex @ X3 @ J3 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65            = ( groups2073611262835488442omplex
% 0.30/0.65              @ ^ [R5: nat] :
% 0.30/0.65                  ( times_times_complex
% 0.30/0.65                  @ ( groups2073611262835488442omplex
% 0.30/0.65                    @ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.65                  @ ( power_power_complex @ X3 @ R5 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polynomial_product
% 0.30/0.65  thf(fact_8623_polynomial__product,axiom,
% 0.30/0.65      ! [M: nat,A: nat > rat,N: nat,B2: nat > rat,X3: rat] :
% 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_rat ) )
% 0.30/0.65       => ( ! [J2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ N @ J2 )
% 0.30/0.65             => ( ( B2 @ J2 )
% 0.30/0.65                = zero_zero_rat ) )
% 0.30/0.65         => ( ( times_times_rat
% 0.30/0.65              @ ( groups2906978787729119204at_rat
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65              @ ( groups2906978787729119204at_rat
% 0.30/0.65                @ ^ [J3: nat] : ( times_times_rat @ ( B2 @ J3 ) @ ( power_power_rat @ X3 @ J3 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.65            = ( groups2906978787729119204at_rat
% 0.30/0.65              @ ^ [R5: nat] :
% 0.30/0.65                  ( times_times_rat
% 0.30/0.65                  @ ( groups2906978787729119204at_rat
% 0.30/0.65                    @ ^ [K2: nat] : ( times_times_rat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.65                  @ ( power_power_rat @ X3 @ R5 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polynomial_product
% 0.30/0.65  thf(fact_8624_polynomial__product,axiom,
% 0.30/0.65      ! [M: nat,A: nat > int,N: nat,B2: nat > int,X3: int] :
% 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_int ) )
% 0.30/0.65       => ( ! [J2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ N @ J2 )
% 0.30/0.65             => ( ( B2 @ J2 )
% 0.30/0.65                = zero_zero_int ) )
% 0.30/0.65         => ( ( times_times_int
% 0.30/0.65              @ ( groups3539618377306564664at_int
% 0.30/0.65                @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66              @ ( groups3539618377306564664at_int
% 0.30/0.66                @ ^ [J3: nat] : ( times_times_int @ ( B2 @ J3 ) @ ( power_power_int @ X3 @ J3 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66            = ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [R5: nat] :
% 0.30/0.66                  ( times_times_int
% 0.30/0.66                  @ ( groups3539618377306564664at_int
% 0.30/0.66                    @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.30/0.66                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.66                  @ ( power_power_int @ X3 @ R5 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polynomial_product
% 0.30/0.66  thf(fact_8625_polynomial__product,axiom,
% 0.30/0.66      ! [M: nat,A: nat > real,N: nat,B2: nat > real,X3: real] :
% 0.30/0.66        ( ! [I3: nat] :
% 0.30/0.66            ( ( ord_less_nat @ M @ I3 )
% 0.30/0.66           => ( ( A @ I3 )
% 0.30/0.66              = zero_zero_real ) )
% 0.30/0.66       => ( ! [J2: nat] :
% 0.30/0.66              ( ( ord_less_nat @ N @ J2 )
% 0.30/0.66             => ( ( B2 @ J2 )
% 0.30/0.66                = zero_zero_real ) )
% 0.30/0.66         => ( ( times_times_real
% 0.30/0.66              @ ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66              @ ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [J3: nat] : ( times_times_real @ ( B2 @ J3 ) @ ( power_power_real @ X3 @ J3 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66            = ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [R5: nat] :
% 0.30/0.66                  ( times_times_real
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.30/0.66                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.66                  @ ( power_power_real @ X3 @ R5 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polynomial_product
% 0.30/0.66  thf(fact_8626_pochhammer__Suc__prod__rev,axiom,
% 0.30/0.66      ! [A: rat,N: nat] :
% 0.30/0.66        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.30/0.66        = ( groups73079841787564623at_rat
% 0.30/0.66          @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_Suc_prod_rev
% 0.30/0.66  thf(fact_8627_pochhammer__Suc__prod__rev,axiom,
% 0.30/0.66      ! [A: real,N: nat] :
% 0.30/0.66        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.30/0.66        = ( groups129246275422532515t_real
% 0.30/0.66          @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I4 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_Suc_prod_rev
% 0.30/0.66  thf(fact_8628_pochhammer__Suc__prod__rev,axiom,
% 0.30/0.66      ! [A: nat,N: nat] :
% 0.30/0.66        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.30/0.66        = ( groups708209901874060359at_nat
% 0.30/0.66          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_Suc_prod_rev
% 0.30/0.66  thf(fact_8629_pochhammer__Suc__prod__rev,axiom,
% 0.30/0.66      ! [A: int,N: nat] :
% 0.30/0.66        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.30/0.66        = ( groups705719431365010083at_int
% 0.30/0.66          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_Suc_prod_rev
% 0.30/0.66  thf(fact_8630_polyfun__eq__const,axiom,
% 0.30/0.66      ! [C: nat > complex,N: nat,K: complex] :
% 0.30/0.66        ( ( ! [X: complex] :
% 0.30/0.66              ( ( groups2073611262835488442omplex
% 0.30/0.66                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66              = K ) )
% 0.30/0.66        = ( ( ( C @ zero_zero_nat )
% 0.30/0.66            = K )
% 0.30/0.66          & ! [X: nat] :
% 0.30/0.66              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 0.30/0.66             => ( ( C @ X )
% 0.30/0.66                = zero_zero_complex ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_eq_const
% 0.30/0.66  thf(fact_8631_polyfun__eq__const,axiom,
% 0.30/0.66      ! [C: nat > real,N: nat,K: real] :
% 0.30/0.66        ( ( ! [X: real] :
% 0.30/0.66              ( ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66              = K ) )
% 0.30/0.66        = ( ( ( C @ zero_zero_nat )
% 0.30/0.66            = K )
% 0.30/0.66          & ! [X: nat] :
% 0.30/0.66              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 0.30/0.66             => ( ( C @ X )
% 0.30/0.66                = zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_eq_const
% 0.30/0.66  thf(fact_8632_binomial__ring,axiom,
% 0.30/0.66      ! [A: complex,B2: complex,N: nat] :
% 0.30/0.66        ( ( power_power_complex @ ( plus_plus_complex @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % binomial_ring
% 0.30/0.66  thf(fact_8633_binomial__ring,axiom,
% 0.30/0.66      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.66        ( ( power_power_rat @ ( plus_plus_rat @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( power_power_rat @ A @ K2 ) ) @ ( power_power_rat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % binomial_ring
% 0.30/0.66  thf(fact_8634_binomial__ring,axiom,
% 0.30/0.66      ! [A: int,B2: int,N: nat] :
% 0.30/0.66        ( ( power_power_int @ ( plus_plus_int @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups3539618377306564664at_int
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % binomial_ring
% 0.30/0.66  thf(fact_8635_binomial__ring,axiom,
% 0.30/0.66      ! [A: nat,B2: nat,N: nat] :
% 0.30/0.66        ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups3542108847815614940at_nat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % binomial_ring
% 0.30/0.66  thf(fact_8636_binomial__ring,axiom,
% 0.30/0.66      ! [A: real,B2: real,N: nat] :
% 0.30/0.66        ( ( power_power_real @ ( plus_plus_real @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % binomial_ring
% 0.30/0.66  thf(fact_8637_pochhammer__binomial__sum,axiom,
% 0.30/0.66      ! [A: rat,B2: rat,N: nat] :
% 0.30/0.66        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_binomial_sum
% 0.30/0.66  thf(fact_8638_pochhammer__binomial__sum,axiom,
% 0.30/0.66      ! [A: int,B2: int,N: nat] :
% 0.30/0.66        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups3539618377306564664at_int
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_binomial_sum
% 0.30/0.66  thf(fact_8639_pochhammer__binomial__sum,axiom,
% 0.30/0.66      ! [A: real,B2: real,N: nat] :
% 0.30/0.66        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B2 ) @ N )
% 0.30/0.66        = ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pochhammer_binomial_sum
% 0.30/0.66  thf(fact_8640_polynomial__product__nat,axiom,
% 0.30/0.66      ! [M: nat,A: nat > nat,N: nat,B2: nat > nat,X3: nat] :
% 0.30/0.66        ( ! [I3: nat] :
% 0.30/0.66            ( ( ord_less_nat @ M @ I3 )
% 0.30/0.66           => ( ( A @ I3 )
% 0.30/0.66              = zero_zero_nat ) )
% 0.30/0.66       => ( ! [J2: nat] :
% 0.30/0.66              ( ( ord_less_nat @ N @ J2 )
% 0.30/0.66             => ( ( B2 @ J2 )
% 0.30/0.66                = zero_zero_nat ) )
% 0.30/0.66         => ( ( times_times_nat
% 0.30/0.66              @ ( groups3542108847815614940at_nat
% 0.30/0.66                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X3 @ I4 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66              @ ( groups3542108847815614940at_nat
% 0.30/0.66                @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66            = ( groups3542108847815614940at_nat
% 0.30/0.66              @ ^ [R5: nat] :
% 0.30/0.66                  ( times_times_nat
% 0.30/0.66                  @ ( groups3542108847815614940at_nat
% 0.30/0.66                    @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.30/0.66                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.66                  @ ( power_power_nat @ X3 @ R5 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polynomial_product_nat
% 0.30/0.66  thf(fact_8641_choose__square__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( groups3542108847815614940at_nat
% 0.30/0.66          @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66        = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_square_sum
% 0.30/0.66  thf(fact_8642_Cauchy__product__sums,axiom,
% 0.30/0.66      ! [A: nat > complex,B2: nat > complex] :
% 0.30/0.66        ( ( summable_real
% 0.30/0.66          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
% 0.30/0.66       => ( ( summable_real
% 0.30/0.66            @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
% 0.30/0.66         => ( sums_complex
% 0.30/0.66            @ ^ [K2: nat] :
% 0.30/0.66                ( groups2073611262835488442omplex
% 0.30/0.66                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.66            @ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Cauchy_product_sums
% 0.30/0.66  thf(fact_8643_Cauchy__product__sums,axiom,
% 0.30/0.66      ! [A: nat > real,B2: nat > real] :
% 0.30/0.66        ( ( summable_real
% 0.30/0.66          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
% 0.30/0.66       => ( ( summable_real
% 0.30/0.66            @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
% 0.30/0.66         => ( sums_real
% 0.30/0.66            @ ^ [K2: nat] :
% 0.30/0.66                ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 0.30/0.66                @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.66            @ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Cauchy_product_sums
% 0.30/0.66  thf(fact_8644_sum_Ozero__middle,axiom,
% 0.30/0.66      ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.66       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.66         => ( ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.66            = ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum.zero_middle
% 0.30/0.66  thf(fact_8645_sum_Ozero__middle,axiom,
% 0.30/0.66      ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.66       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.66         => ( ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.66            = ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum.zero_middle
% 0.30/0.66  thf(fact_8646_sum_Ozero__middle,axiom,
% 0.30/0.66      ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.66       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.66         => ( ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.66            = ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum.zero_middle
% 0.30/0.66  thf(fact_8647_sum_Ozero__middle,axiom,
% 0.30/0.66      ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.66       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.66         => ( ( groups3542108847815614940at_nat
% 0.30/0.66              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.66            = ( groups3542108847815614940at_nat
% 0.30/0.66              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum.zero_middle
% 0.30/0.66  thf(fact_8648_sum_Ozero__middle,axiom,
% 0.30/0.66      ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 0.30/0.66       => ( ( ord_less_eq_nat @ K @ P4 )
% 0.30/0.66         => ( ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ P4 ) )
% 0.30/0.66            = ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum.zero_middle
% 0.30/0.66  thf(fact_8649_root__polyfun,axiom,
% 0.30/0.66      ! [N: nat,Z: int,A: int] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( ( power_power_int @ Z @ N )
% 0.30/0.66            = A )
% 0.30/0.66          = ( ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % root_polyfun
% 0.30/0.66  thf(fact_8650_root__polyfun,axiom,
% 0.30/0.66      ! [N: nat,Z: complex,A: complex] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( ( power_power_complex @ Z @ N )
% 0.30/0.66            = A )
% 0.30/0.66          = ( ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = zero_zero_complex ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % root_polyfun
% 0.30/0.66  thf(fact_8651_root__polyfun,axiom,
% 0.30/0.66      ! [N: nat,Z: code_integer,A: code_integer] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( ( power_8256067586552552935nteger @ Z @ N )
% 0.30/0.66            = A )
% 0.30/0.66          = ( ( groups7501900531339628137nteger
% 0.30/0.66              @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I4 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % root_polyfun
% 0.30/0.66  thf(fact_8652_root__polyfun,axiom,
% 0.30/0.66      ! [N: nat,Z: rat,A: rat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( ( power_power_rat @ Z @ N )
% 0.30/0.66            = A )
% 0.30/0.66          = ( ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = zero_zero_rat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % root_polyfun
% 0.30/0.66  thf(fact_8653_root__polyfun,axiom,
% 0.30/0.66      ! [N: nat,Z: real,A: real] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( ( power_power_real @ Z @ N )
% 0.30/0.66            = A )
% 0.30/0.66          = ( ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % root_polyfun
% 0.30/0.66  thf(fact_8654_sum__gp0,axiom,
% 0.30/0.66      ! [X3: complex,N: nat] :
% 0.30/0.66        ( ( ( X3 = one_one_complex )
% 0.30/0.66         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
% 0.30/0.66        & ( ( X3 != one_one_complex )
% 0.30/0.66         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum_gp0
% 0.30/0.66  thf(fact_8655_sum__gp0,axiom,
% 0.30/0.66      ! [X3: rat,N: nat] :
% 0.30/0.66        ( ( ( X3 = one_one_rat )
% 0.30/0.66         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
% 0.30/0.66        & ( ( X3 != one_one_rat )
% 0.30/0.66         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum_gp0
% 0.30/0.66  thf(fact_8656_sum__gp0,axiom,
% 0.30/0.66      ! [X3: real,N: nat] :
% 0.30/0.66        ( ( ( X3 = one_one_real )
% 0.30/0.66         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
% 0.30/0.66        & ( ( X3 != one_one_real )
% 0.30/0.66         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum_gp0
% 0.30/0.66  thf(fact_8657_choose__alternating__linear__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( N != one_one_nat )
% 0.30/0.66       => ( ( groups2073611262835488442omplex
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_linear_sum
% 0.30/0.66  thf(fact_8658_choose__alternating__linear__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( N != one_one_nat )
% 0.30/0.66       => ( ( groups7501900531339628137nteger
% 0.30/0.66            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_linear_sum
% 0.30/0.66  thf(fact_8659_choose__alternating__linear__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( N != one_one_nat )
% 0.30/0.66       => ( ( groups2906978787729119204at_rat
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_rat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_linear_sum
% 0.30/0.66  thf(fact_8660_choose__alternating__linear__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( N != one_one_nat )
% 0.30/0.66       => ( ( groups3539618377306564664at_int
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_linear_sum
% 0.30/0.66  thf(fact_8661_choose__alternating__linear__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( N != one_one_nat )
% 0.30/0.66       => ( ( groups6591440286371151544t_real
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_linear_sum
% 0.30/0.66  thf(fact_8662_polyfun__diff__alt,axiom,
% 0.30/0.66      ! [N: nat,A: nat > complex,X3: complex,Y3: complex] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_complex
% 0.30/0.66            @ ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 0.30/0.66            @ ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( groups2073611262835488442omplex
% 0.30/0.66                  @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y3 @ K2 ) ) @ ( power_power_complex @ X3 @ J3 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff_alt
% 0.30/0.66  thf(fact_8663_polyfun__diff__alt,axiom,
% 0.30/0.66      ! [N: nat,A: nat > rat,X3: rat,Y3: rat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_rat
% 0.30/0.66            @ ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 0.30/0.66            @ ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( groups2906978787729119204at_rat
% 0.30/0.66                  @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_rat @ Y3 @ K2 ) ) @ ( power_power_rat @ X3 @ J3 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff_alt
% 0.30/0.66  thf(fact_8664_polyfun__diff__alt,axiom,
% 0.30/0.66      ! [N: nat,A: nat > int,X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_int
% 0.30/0.66            @ ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 0.30/0.66            @ ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( groups3539618377306564664at_int
% 0.30/0.66                  @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y3 @ K2 ) ) @ ( power_power_int @ X3 @ J3 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff_alt
% 0.30/0.66  thf(fact_8665_polyfun__diff__alt,axiom,
% 0.30/0.66      ! [N: nat,A: nat > real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_real
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( groups6591440286371151544t_real
% 0.30/0.66                  @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y3 @ K2 ) ) @ ( power_power_real @ X3 @ J3 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff_alt
% 0.30/0.66  thf(fact_8666_binomial__r__part__sum,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % binomial_r_part_sum
% 0.30/0.66  thf(fact_8667_choose__linear__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( groups3542108847815614940at_nat
% 0.30/0.66          @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66        = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_linear_sum
% 0.30/0.66  thf(fact_8668_choose__alternating__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66       => ( ( groups2073611262835488442omplex
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_sum
% 0.30/0.66  thf(fact_8669_choose__alternating__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66       => ( ( groups7501900531339628137nteger
% 0.30/0.66            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_sum
% 0.30/0.66  thf(fact_8670_choose__alternating__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66       => ( ( groups2906978787729119204at_rat
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_rat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_sum
% 0.30/0.66  thf(fact_8671_choose__alternating__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66       => ( ( groups3539618377306564664at_int
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_sum
% 0.30/0.66  thf(fact_8672_choose__alternating__sum,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66       => ( ( groups6591440286371151544t_real
% 0.30/0.66            @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 0.30/0.66            @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66          = zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % choose_alternating_sum
% 0.30/0.66  thf(fact_8673_polyfun__extremal__lemma,axiom,
% 0.30/0.66      ! [E: real,C: nat > complex,N: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.66       => ? [M8: real] :
% 0.30/0.66          ! [Z3: complex] :
% 0.30/0.66            ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z3 ) )
% 0.30/0.66           => ( ord_less_eq_real
% 0.30/0.66              @ ( real_V1022390504157884413omplex
% 0.30/0.66                @ ( groups2073611262835488442omplex
% 0.30/0.66                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 0.30/0.66                  @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66              @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_extremal_lemma
% 0.30/0.66  thf(fact_8674_polyfun__extremal__lemma,axiom,
% 0.30/0.66      ! [E: real,C: nat > real,N: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.66       => ? [M8: real] :
% 0.30/0.66          ! [Z3: real] :
% 0.30/0.66            ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z3 ) )
% 0.30/0.66           => ( ord_less_eq_real
% 0.30/0.66              @ ( real_V7735802525324610683m_real
% 0.30/0.66                @ ( groups6591440286371151544t_real
% 0.30/0.66                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 0.30/0.66                  @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66              @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z3 ) @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_extremal_lemma
% 0.30/0.66  thf(fact_8675_polyfun__diff,axiom,
% 0.30/0.66      ! [N: nat,A: nat > complex,X3: complex,Y3: complex] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_complex
% 0.30/0.66            @ ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 0.30/0.66            @ ( groups2073611262835488442omplex
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( times_times_complex
% 0.30/0.66                  @ ( groups2073611262835488442omplex
% 0.30/0.66                    @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.30/0.66                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.66                  @ ( power_power_complex @ X3 @ J3 ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff
% 0.30/0.66  thf(fact_8676_polyfun__diff,axiom,
% 0.30/0.66      ! [N: nat,A: nat > rat,X3: rat,Y3: rat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_rat
% 0.30/0.66            @ ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 0.30/0.66            @ ( groups2906978787729119204at_rat
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( times_times_rat
% 0.30/0.66                  @ ( groups2906978787729119204at_rat
% 0.30/0.66                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.30/0.66                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.66                  @ ( power_power_rat @ X3 @ J3 ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff
% 0.30/0.66  thf(fact_8677_polyfun__diff,axiom,
% 0.30/0.66      ! [N: nat,A: nat > int,X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_int
% 0.30/0.66            @ ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 0.30/0.66            @ ( groups3539618377306564664at_int
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( times_times_int
% 0.30/0.66                  @ ( groups3539618377306564664at_int
% 0.30/0.66                    @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.30/0.66                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.66                  @ ( power_power_int @ X3 @ J3 ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff
% 0.30/0.66  thf(fact_8678_polyfun__diff,axiom,
% 0.30/0.66      ! [N: nat,A: nat > real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.30/0.66       => ( ( minus_minus_real
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ N ) ) )
% 0.30/0.66          = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [J3: nat] :
% 0.30/0.66                  ( times_times_real
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 0.30/0.66                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 0.30/0.66                  @ ( power_power_real @ X3 @ J3 ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % polyfun_diff
% 0.30/0.66  thf(fact_8679_gbinomial__partial__row__sum,axiom,
% 0.30/0.66      ! [A: complex,M: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_row_sum
% 0.30/0.66  thf(fact_8680_gbinomial__partial__row__sum,axiom,
% 0.30/0.66      ! [A: rat,M: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_row_sum
% 0.30/0.66  thf(fact_8681_gbinomial__partial__row__sum,axiom,
% 0.30/0.66      ! [A: real,M: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_row_sum
% 0.30/0.66  thf(fact_8682_gbinomial__r__part__sum,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_r_part_sum
% 0.30/0.66  thf(fact_8683_gbinomial__r__part__sum,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_r_part_sum
% 0.30/0.66  thf(fact_8684_gbinomial__r__part__sum,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_r_part_sum
% 0.30/0.66  thf(fact_8685_cos__arcsin,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( cos_real @ ( arcsin @ X3 ) )
% 0.30/0.66            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_arcsin
% 0.30/0.66  thf(fact_8686_sin__arccos__abs,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.66       => ( ( sin_real @ ( arccos @ Y3 ) )
% 0.30/0.66          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_arccos_abs
% 0.30/0.66  thf(fact_8687_sin__arccos,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( sin_real @ ( arccos @ X3 ) )
% 0.30/0.66            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_arccos
% 0.30/0.66  thf(fact_8688_of__nat__id,axiom,
% 0.30/0.66      ( semiri1316708129612266289at_nat
% 0.30/0.66      = ( ^ [N2: nat] : N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_id
% 0.30/0.66  thf(fact_8689_prod__pos__nat__iff,axiom,
% 0.30/0.66      ! [A4: set_int,F: int > nat] :
% 0.30/0.66        ( ( finite_finite_int @ A4 )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
% 0.30/0.66          = ( ! [X: int] :
% 0.30/0.66                ( ( member_int @ X @ A4 )
% 0.30/0.66               => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod_pos_nat_iff
% 0.30/0.66  thf(fact_8690_prod__pos__nat__iff,axiom,
% 0.30/0.66      ! [A4: set_complex,F: complex > nat] :
% 0.30/0.66        ( ( finite3207457112153483333omplex @ A4 )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
% 0.30/0.66          = ( ! [X: complex] :
% 0.30/0.66                ( ( member_complex @ X @ A4 )
% 0.30/0.66               => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod_pos_nat_iff
% 0.30/0.66  thf(fact_8691_prod__pos__nat__iff,axiom,
% 0.30/0.66      ! [A4: set_nat,F: nat > nat] :
% 0.30/0.66        ( ( finite_finite_nat @ A4 )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
% 0.30/0.66          = ( ! [X: nat] :
% 0.30/0.66                ( ( member_nat @ X @ A4 )
% 0.30/0.66               => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod_pos_nat_iff
% 0.30/0.66  thf(fact_8692_cos__arccos,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.30/0.66            = Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_arccos
% 0.30/0.66  thf(fact_8693_sin__arcsin,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.30/0.66            = Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_arcsin
% 0.30/0.66  thf(fact_8694_arccos__0,axiom,
% 0.30/0.66      ( ( arccos @ zero_zero_real )
% 0.30/0.66      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_0
% 0.30/0.66  thf(fact_8695_arcsin__1,axiom,
% 0.30/0.66      ( ( arcsin @ one_one_real )
% 0.30/0.66      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_1
% 0.30/0.66  thf(fact_8696_arcsin__minus__1,axiom,
% 0.30/0.66      ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.66      = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_minus_1
% 0.30/0.66  thf(fact_8697_gbinomial__Suc__Suc,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.30/0.66        = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc_Suc
% 0.30/0.66  thf(fact_8698_gbinomial__Suc__Suc,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.30/0.66        = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc_Suc
% 0.30/0.66  thf(fact_8699_gbinomial__Suc__Suc,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.30/0.66        = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc_Suc
% 0.30/0.66  thf(fact_8700_gbinomial__of__nat__symmetric,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ K @ N )
% 0.30/0.66       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.30/0.66          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_of_nat_symmetric
% 0.30/0.66  thf(fact_8701_arccos__le__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_eq_real @ X3 @ Y3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66           => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_le_arccos
% 0.30/0.66  thf(fact_8702_arccos__eq__iff,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       => ( ( ( arccos @ X3 )
% 0.30/0.66            = ( arccos @ Y3 ) )
% 0.30/0.66          = ( X3 = Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_eq_iff
% 0.30/0.66  thf(fact_8703_arccos__le__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_eq_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_le_mono
% 0.30/0.66  thf(fact_8704_arcsin__minus,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( arcsin @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.66            = ( uminus_uminus_real @ ( arcsin @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_minus
% 0.30/0.66  thf(fact_8705_arcsin__le__arcsin,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_eq_real @ X3 @ Y3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66           => ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_le_arcsin
% 0.30/0.66  thf(fact_8706_arcsin__eq__iff,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         => ( ( ( arcsin @ X3 )
% 0.30/0.66              = ( arcsin @ Y3 ) )
% 0.30/0.66            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_eq_iff
% 0.30/0.66  thf(fact_8707_arcsin__le__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_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_le_mono
% 0.30/0.66  thf(fact_8708_gbinomial__addition__formula,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.30/0.66        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_addition_formula
% 0.30/0.66  thf(fact_8709_gbinomial__addition__formula,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.30/0.66        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_addition_formula
% 0.30/0.66  thf(fact_8710_gbinomial__addition__formula,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.30/0.66        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_addition_formula
% 0.30/0.66  thf(fact_8711_gbinomial__absorb__comp,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 0.30/0.66        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorb_comp
% 0.30/0.66  thf(fact_8712_gbinomial__absorb__comp,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 0.30/0.66        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorb_comp
% 0.30/0.66  thf(fact_8713_gbinomial__absorb__comp,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 0.30/0.66        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorb_comp
% 0.30/0.66  thf(fact_8714_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.66      ! [K: nat,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.30/0.66       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_ge_n_over_k_pow_k
% 0.30/0.66  thf(fact_8715_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.66      ! [K: nat,A: rat] :
% 0.30/0.66        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.30/0.66       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_ge_n_over_k_pow_k
% 0.30/0.66  thf(fact_8716_gbinomial__mult__1,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 0.30/0.66        = ( 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 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_mult_1
% 0.30/0.66  thf(fact_8717_gbinomial__mult__1,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 0.30/0.66        = ( 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 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_mult_1
% 0.30/0.66  thf(fact_8718_gbinomial__mult__1_H,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 0.30/0.66        = ( 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 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_mult_1'
% 0.30/0.66  thf(fact_8719_gbinomial__mult__1_H,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 0.30/0.66        = ( 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 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_mult_1'
% 0.30/0.66  thf(fact_8720_prod__int__plus__eq,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.30/0.66        = ( groups1705073143266064639nt_int
% 0.30/0.66          @ ^ [X: int] : X
% 0.30/0.66          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod_int_plus_eq
% 0.30/0.66  thf(fact_8721_arccos__lbound,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_lbound
% 0.30/0.66  thf(fact_8722_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_8723_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_8724_arccos__ubound,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_ubound
% 0.30/0.66  thf(fact_8725_arccos__cos,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_eq_real @ X3 @ pi )
% 0.30/0.66         => ( ( arccos @ ( cos_real @ X3 ) )
% 0.30/0.66            = X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_cos
% 0.30/0.66  thf(fact_8726_arcsin__less__arcsin,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 @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_less_arcsin
% 0.30/0.66  thf(fact_8727_arcsin__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 @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
% 0.30/0.66            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_less_mono
% 0.30/0.66  thf(fact_8728_cos__arccos__abs,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.66       => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.30/0.66          = Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_arccos_abs
% 0.30/0.66  thf(fact_8729_arccos__cos__eq__abs,axiom,
% 0.30/0.66      ! [Theta: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.30/0.66       => ( ( arccos @ ( cos_real @ Theta ) )
% 0.30/0.66          = ( abs_abs_real @ Theta ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_cos_eq_abs
% 0.30/0.66  thf(fact_8730_Suc__times__gbinomial,axiom,
% 0.30/0.66      ! [K: nat,A: complex] :
% 0.30/0.66        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 0.30/0.66        = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_times_gbinomial
% 0.30/0.66  thf(fact_8731_Suc__times__gbinomial,axiom,
% 0.30/0.66      ! [K: nat,A: rat] :
% 0.30/0.66        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 0.30/0.66        = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_times_gbinomial
% 0.30/0.66  thf(fact_8732_Suc__times__gbinomial,axiom,
% 0.30/0.66      ! [K: nat,A: real] :
% 0.30/0.66        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 0.30/0.66        = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_times_gbinomial
% 0.30/0.66  thf(fact_8733_gbinomial__absorption,axiom,
% 0.30/0.66      ! [K: nat,A: complex] :
% 0.30/0.66        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 0.30/0.66        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorption
% 0.30/0.66  thf(fact_8734_gbinomial__absorption,axiom,
% 0.30/0.66      ! [K: nat,A: rat] :
% 0.30/0.66        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 0.30/0.66        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorption
% 0.30/0.66  thf(fact_8735_gbinomial__absorption,axiom,
% 0.30/0.66      ! [K: nat,A: real] :
% 0.30/0.66        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 0.30/0.66        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorption
% 0.30/0.66  thf(fact_8736_gbinomial__trinomial__revision,axiom,
% 0.30/0.66      ! [K: nat,M: nat,A: complex] :
% 0.30/0.66        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.66       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.30/0.66          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_trinomial_revision
% 0.30/0.66  thf(fact_8737_gbinomial__trinomial__revision,axiom,
% 0.30/0.66      ! [K: nat,M: nat,A: rat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.66       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.30/0.66          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_trinomial_revision
% 0.30/0.66  thf(fact_8738_gbinomial__trinomial__revision,axiom,
% 0.30/0.66      ! [K: nat,M: nat,A: real] :
% 0.30/0.66        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.66       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.30/0.66          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_trinomial_revision
% 0.30/0.66  thf(fact_8739_ln__prod,axiom,
% 0.30/0.66      ! [I5: set_real,F: real > real] :
% 0.30/0.66        ( ( finite_finite_real @ I5 )
% 0.30/0.66       => ( ! [I3: real] :
% 0.30/0.66              ( ( member_real @ I3 @ I5 )
% 0.30/0.66             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.66         => ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I5 ) )
% 0.30/0.66            = ( groups8097168146408367636l_real
% 0.30/0.66              @ ^ [X: real] : ( ln_ln_real @ ( F @ X ) )
% 0.30/0.66              @ I5 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_prod
% 0.30/0.66  thf(fact_8740_ln__prod,axiom,
% 0.30/0.66      ! [I5: set_int,F: int > real] :
% 0.30/0.66        ( ( finite_finite_int @ I5 )
% 0.30/0.66       => ( ! [I3: int] :
% 0.30/0.66              ( ( member_int @ I3 @ I5 )
% 0.30/0.66             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.66         => ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I5 ) )
% 0.30/0.66            = ( groups8778361861064173332t_real
% 0.30/0.66              @ ^ [X: int] : ( ln_ln_real @ ( F @ X ) )
% 0.30/0.66              @ I5 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_prod
% 0.30/0.66  thf(fact_8741_ln__prod,axiom,
% 0.30/0.66      ! [I5: set_complex,F: complex > real] :
% 0.30/0.66        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.66       => ( ! [I3: complex] :
% 0.30/0.66              ( ( member_complex @ I3 @ I5 )
% 0.30/0.66             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.66         => ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I5 ) )
% 0.30/0.66            = ( groups5808333547571424918x_real
% 0.30/0.66              @ ^ [X: complex] : ( ln_ln_real @ ( F @ X ) )
% 0.30/0.66              @ I5 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_prod
% 0.30/0.66  thf(fact_8742_ln__prod,axiom,
% 0.30/0.66      ! [I5: set_nat,F: nat > real] :
% 0.30/0.66        ( ( finite_finite_nat @ I5 )
% 0.30/0.66       => ( ! [I3: nat] :
% 0.30/0.66              ( ( member_nat @ I3 @ I5 )
% 0.30/0.66             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.66         => ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I5 ) )
% 0.30/0.66            = ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [X: nat] : ( ln_ln_real @ ( F @ X ) )
% 0.30/0.66              @ I5 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_prod
% 0.30/0.66  thf(fact_8743_gbinomial__parallel__sum,axiom,
% 0.30/0.66      ! [A: complex,N: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ K2 )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66        = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_parallel_sum
% 0.30/0.66  thf(fact_8744_gbinomial__parallel__sum,axiom,
% 0.30/0.66      ! [A: rat,N: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66        = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_parallel_sum
% 0.30/0.66  thf(fact_8745_gbinomial__parallel__sum,axiom,
% 0.30/0.66      ! [A: real,N: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) )
% 0.30/0.66        = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_parallel_sum
% 0.30/0.66  thf(fact_8746_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_8747_arccos__bounded,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.30/0.66            & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_bounded
% 0.30/0.66  thf(fact_8748_gbinomial__rec,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.30/0.66        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_rec
% 0.30/0.66  thf(fact_8749_gbinomial__rec,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.30/0.66        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_rec
% 0.30/0.66  thf(fact_8750_gbinomial__rec,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.30/0.66        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_rec
% 0.30/0.66  thf(fact_8751_gbinomial__factors,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.30/0.66        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_factors
% 0.30/0.66  thf(fact_8752_gbinomial__factors,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.30/0.66        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_factors
% 0.30/0.66  thf(fact_8753_gbinomial__factors,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.30/0.66        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_factors
% 0.30/0.66  thf(fact_8754_gbinomial__index__swap,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 0.30/0.66        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_index_swap
% 0.30/0.66  thf(fact_8755_gbinomial__index__swap,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
% 0.30/0.66        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_index_swap
% 0.30/0.66  thf(fact_8756_gbinomial__index__swap,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
% 0.30/0.66        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_index_swap
% 0.30/0.66  thf(fact_8757_gbinomial__negated__upper,axiom,
% 0.30/0.66      ( gbinomial_complex
% 0.30/0.66      = ( ^ [A5: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A5 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_negated_upper
% 0.30/0.66  thf(fact_8758_gbinomial__negated__upper,axiom,
% 0.30/0.66      ( gbinomial_rat
% 0.30/0.66      = ( ^ [A5: rat,K2: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A5 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_negated_upper
% 0.30/0.66  thf(fact_8759_gbinomial__negated__upper,axiom,
% 0.30/0.66      ( gbinomial_real
% 0.30/0.66      = ( ^ [A5: real,K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A5 ) @ one_one_real ) @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_negated_upper
% 0.30/0.66  thf(fact_8760_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_8761_arccos__cos2,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 0.30/0.66         => ( ( arccos @ ( cos_real @ X3 ) )
% 0.30/0.66            = ( uminus_uminus_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_cos2
% 0.30/0.66  thf(fact_8762_arccos__minus,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( arccos @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.66            = ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_minus
% 0.30/0.66  thf(fact_8763_cos__arcsin__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         => ( ( cos_real @ ( arcsin @ X3 ) )
% 0.30/0.66           != zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_arcsin_nonzero
% 0.30/0.66  thf(fact_8764_prod_Otriangle__reindex__eq,axiom,
% 0.30/0.66      ! [G: nat > nat > nat,N: nat] :
% 0.30/0.66        ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
% 0.30/0.66          @ ( collec3392354462482085612at_nat
% 0.30/0.66            @ ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.66        = ( groups708209901874060359at_nat
% 0.30/0.66          @ ^ [K2: nat] :
% 0.30/0.66              ( groups708209901874060359at_nat
% 0.30/0.66              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod.triangle_reindex_eq
% 0.30/0.66  thf(fact_8765_prod_Otriangle__reindex__eq,axiom,
% 0.30/0.66      ! [G: nat > nat > int,N: nat] :
% 0.30/0.66        ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
% 0.30/0.66          @ ( collec3392354462482085612at_nat
% 0.30/0.66            @ ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.66        = ( groups705719431365010083at_int
% 0.30/0.66          @ ^ [K2: nat] :
% 0.30/0.66              ( groups705719431365010083at_int
% 0.30/0.66              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod.triangle_reindex_eq
% 0.30/0.66  thf(fact_8766_gbinomial__minus,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 0.30/0.66        = ( 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 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_minus
% 0.30/0.66  thf(fact_8767_gbinomial__minus,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 0.30/0.66        = ( 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 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_minus
% 0.30/0.66  thf(fact_8768_gbinomial__minus,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 0.30/0.66        = ( 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 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_minus
% 0.30/0.66  thf(fact_8769_gbinomial__reduce__nat,axiom,
% 0.30/0.66      ! [K: nat,A: complex] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66       => ( ( gbinomial_complex @ A @ K )
% 0.30/0.66          = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_reduce_nat
% 0.30/0.66  thf(fact_8770_gbinomial__reduce__nat,axiom,
% 0.30/0.66      ! [K: nat,A: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66       => ( ( gbinomial_real @ A @ K )
% 0.30/0.66          = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_reduce_nat
% 0.30/0.66  thf(fact_8771_gbinomial__reduce__nat,axiom,
% 0.30/0.66      ! [K: nat,A: rat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66       => ( ( gbinomial_rat @ A @ K )
% 0.30/0.66          = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_reduce_nat
% 0.30/0.66  thf(fact_8772_arccos,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.30/0.66            & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 0.30/0.66            & ( ( cos_real @ ( arccos @ Y3 ) )
% 0.30/0.66              = Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos
% 0.30/0.66  thf(fact_8773_prod_Otriangle__reindex,axiom,
% 0.30/0.66      ! [G: nat > nat > nat,N: nat] :
% 0.30/0.66        ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
% 0.30/0.66          @ ( collec3392354462482085612at_nat
% 0.30/0.66            @ ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.66        = ( groups708209901874060359at_nat
% 0.30/0.66          @ ^ [K2: nat] :
% 0.30/0.66              ( groups708209901874060359at_nat
% 0.30/0.66              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.66          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod.triangle_reindex
% 0.30/0.66  thf(fact_8774_prod_Otriangle__reindex,axiom,
% 0.30/0.66      ! [G: nat > nat > int,N: nat] :
% 0.30/0.66        ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
% 0.30/0.66          @ ( collec3392354462482085612at_nat
% 0.30/0.66            @ ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 0.30/0.66        = ( groups705719431365010083at_int
% 0.30/0.66          @ ^ [K2: nat] :
% 0.30/0.66              ( groups705719431365010083at_int
% 0.30/0.66              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 0.30/0.66              @ ( set_ord_atMost_nat @ K2 ) )
% 0.30/0.66          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % prod.triangle_reindex
% 0.30/0.66  thf(fact_8775_gbinomial__pochhammer,axiom,
% 0.30/0.66      ( gbinomial_complex
% 0.30/0.66      = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_pochhammer
% 0.30/0.66  thf(fact_8776_gbinomial__pochhammer,axiom,
% 0.30/0.66      ( gbinomial_rat
% 0.30/0.66      = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_pochhammer
% 0.30/0.66  thf(fact_8777_gbinomial__pochhammer,axiom,
% 0.30/0.66      ( gbinomial_real
% 0.30/0.66      = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A5 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_pochhammer
% 0.30/0.66  thf(fact_8778_gbinomial__pochhammer_H,axiom,
% 0.30/0.66      ( gbinomial_complex
% 0.30/0.66      = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_pochhammer'
% 0.30/0.66  thf(fact_8779_gbinomial__pochhammer_H,axiom,
% 0.30/0.66      ( gbinomial_rat
% 0.30/0.66      = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_pochhammer'
% 0.30/0.66  thf(fact_8780_gbinomial__pochhammer_H,axiom,
% 0.30/0.66      ( gbinomial_real
% 0.30/0.66      = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_pochhammer'
% 0.30/0.66  thf(fact_8781_arccos__minus__abs,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       => ( ( arccos @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.66          = ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_minus_abs
% 0.30/0.66  thf(fact_8782_gbinomial__sum__lower__neg,axiom,
% 0.30/0.66      ! [A: complex,M: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_lower_neg
% 0.30/0.66  thf(fact_8783_gbinomial__sum__lower__neg,axiom,
% 0.30/0.66      ! [A: rat,M: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_lower_neg
% 0.30/0.66  thf(fact_8784_gbinomial__sum__lower__neg,axiom,
% 0.30/0.66      ! [A: real,M: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_lower_neg
% 0.30/0.66  thf(fact_8785_gbinomial__partial__sum__poly,axiom,
% 0.30/0.66      ! [M: nat,A: complex,X3: complex,Y3: complex] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_sum_poly
% 0.30/0.66  thf(fact_8786_gbinomial__partial__sum__poly,axiom,
% 0.30/0.66      ! [M: nat,A: rat,X3: rat,Y3: rat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_sum_poly
% 0.30/0.66  thf(fact_8787_gbinomial__partial__sum__poly,axiom,
% 0.30/0.66      ! [M: nat,A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ X3 ) @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_sum_poly
% 0.30/0.66  thf(fact_8788_gbinomial__sum__up__index,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.66        = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_up_index
% 0.30/0.66  thf(fact_8789_gbinomial__sum__up__index,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.66        = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_up_index
% 0.30/0.66  thf(fact_8790_gbinomial__sum__up__index,axiom,
% 0.30/0.66      ! [K: nat,N: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.30/0.66        = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_up_index
% 0.30/0.66  thf(fact_8791_gbinomial__Suc,axiom,
% 0.30/0.66      ! [A: complex,K: nat] :
% 0.30/0.66        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.30/0.66        = ( divide1717551699836669952omplex
% 0.30/0.66          @ ( groups6464643781859351333omplex
% 0.30/0.66            @ ^ [I4: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I4 ) )
% 0.30/0.66            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.30/0.66          @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc
% 0.30/0.66  thf(fact_8792_gbinomial__Suc,axiom,
% 0.30/0.66      ! [A: rat,K: nat] :
% 0.30/0.66        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.30/0.66        = ( divide_divide_rat
% 0.30/0.66          @ ( groups73079841787564623at_rat
% 0.30/0.66            @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
% 0.30/0.66            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.30/0.66          @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc
% 0.30/0.66  thf(fact_8793_gbinomial__Suc,axiom,
% 0.30/0.66      ! [A: real,K: nat] :
% 0.30/0.66        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.30/0.66        = ( divide_divide_real
% 0.30/0.66          @ ( groups129246275422532515t_real
% 0.30/0.66            @ ^ [I4: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
% 0.30/0.66            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.30/0.66          @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc
% 0.30/0.66  thf(fact_8794_gbinomial__Suc,axiom,
% 0.30/0.66      ! [A: nat,K: nat] :
% 0.30/0.66        ( ( gbinomial_nat @ A @ ( suc @ K ) )
% 0.30/0.66        = ( divide_divide_nat
% 0.30/0.66          @ ( groups708209901874060359at_nat
% 0.30/0.66            @ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
% 0.30/0.66            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.30/0.66          @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc
% 0.30/0.66  thf(fact_8795_gbinomial__Suc,axiom,
% 0.30/0.66      ! [A: int,K: nat] :
% 0.30/0.66        ( ( gbinomial_int @ A @ ( suc @ K ) )
% 0.30/0.66        = ( divide_divide_int
% 0.30/0.66          @ ( groups705719431365010083at_int
% 0.30/0.66            @ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
% 0.30/0.66            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 0.30/0.66          @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_Suc
% 0.30/0.66  thf(fact_8796_gbinomial__absorption_H,axiom,
% 0.30/0.66      ! [K: nat,A: complex] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66       => ( ( gbinomial_complex @ A @ K )
% 0.30/0.66          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorption'
% 0.30/0.66  thf(fact_8797_gbinomial__absorption_H,axiom,
% 0.30/0.66      ! [K: nat,A: rat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66       => ( ( gbinomial_rat @ A @ K )
% 0.30/0.66          = ( 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 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorption'
% 0.30/0.66  thf(fact_8798_gbinomial__absorption_H,axiom,
% 0.30/0.66      ! [K: nat,A: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66       => ( ( gbinomial_real @ A @ K )
% 0.30/0.66          = ( 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 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_absorption'
% 0.30/0.66  thf(fact_8799_arccos__le__pi2,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_le_pi2
% 0.30/0.66  thf(fact_8800_gbinomial__sum__nat__pow2,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_nat_pow2
% 0.30/0.66  thf(fact_8801_gbinomial__sum__nat__pow2,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_nat_pow2
% 0.30/0.66  thf(fact_8802_gbinomial__sum__nat__pow2,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_sum_nat_pow2
% 0.30/0.66  thf(fact_8803_gbinomial__partial__sum__poly__xpos,axiom,
% 0.30/0.66      ! [M: nat,A: complex,X3: complex,Y3: complex] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_sum_poly_xpos
% 0.30/0.66  thf(fact_8804_gbinomial__partial__sum__poly__xpos,axiom,
% 0.30/0.66      ! [M: nat,A: rat,X3: rat,Y3: rat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A ) @ one_one_rat ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_sum_poly_xpos
% 0.30/0.66  thf(fact_8805_gbinomial__partial__sum__poly__xpos,axiom,
% 0.30/0.66      ! [M: nat,A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.66        = ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 0.30/0.66          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_partial_sum_poly_xpos
% 0.30/0.66  thf(fact_8806_arcsin__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 @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.66            & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_lt_bounded
% 0.30/0.66  thf(fact_8807_arcsin__lbound,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_lbound
% 0.30/0.66  thf(fact_8808_arcsin__ubound,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_ubound
% 0.30/0.66  thf(fact_8809_arcsin__bounded,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.66            & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_bounded
% 0.30/0.66  thf(fact_8810_arcsin__sin,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66         => ( ( arcsin @ ( sin_real @ X3 ) )
% 0.30/0.66            = X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_sin
% 0.30/0.66  thf(fact_8811_arcsin,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.66            & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66            & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.30/0.66              = Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin
% 0.30/0.66  thf(fact_8812_arcsin__pi,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.66            & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 0.30/0.66            & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.30/0.66              = Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_pi
% 0.30/0.66  thf(fact_8813_arcsin__le__iff,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_eq_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.66           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66             => ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ Y3 )
% 0.30/0.66                = ( ord_less_eq_real @ X3 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_le_iff
% 0.30/0.66  thf(fact_8814_le__arcsin__iff,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_eq_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.66           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66             => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X3 ) )
% 0.30/0.66                = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_arcsin_iff
% 0.30/0.66  thf(fact_8815_gbinomial__code,axiom,
% 0.30/0.66      ( gbinomial_complex
% 0.30/0.66      = ( ^ [A5: complex,K2: nat] :
% 0.30/0.66            ( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
% 0.30/0.66            @ ( divide1717551699836669952omplex
% 0.30/0.66              @ ( set_fo1517530859248394432omplex
% 0.30/0.66                @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L2 ) ) )
% 0.30/0.66                @ zero_zero_nat
% 0.30/0.66                @ ( minus_minus_nat @ K2 @ one_one_nat )
% 0.30/0.66                @ one_one_complex )
% 0.30/0.66              @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_code
% 0.30/0.66  thf(fact_8816_gbinomial__code,axiom,
% 0.30/0.66      ( gbinomial_rat
% 0.30/0.66      = ( ^ [A5: rat,K2: nat] :
% 0.30/0.66            ( if_rat @ ( K2 = zero_zero_nat ) @ one_one_rat
% 0.30/0.66            @ ( divide_divide_rat
% 0.30/0.66              @ ( set_fo1949268297981939178at_rat
% 0.30/0.66                @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
% 0.30/0.66                @ zero_zero_nat
% 0.30/0.66                @ ( minus_minus_nat @ K2 @ one_one_nat )
% 0.30/0.66                @ one_one_rat )
% 0.30/0.66              @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_code
% 0.30/0.66  thf(fact_8817_gbinomial__code,axiom,
% 0.30/0.66      ( gbinomial_real
% 0.30/0.66      = ( ^ [A5: real,K2: nat] :
% 0.30/0.66            ( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
% 0.30/0.66            @ ( divide_divide_real
% 0.30/0.66              @ ( set_fo3111899725591712190t_real
% 0.30/0.66                @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L2 ) ) )
% 0.30/0.66                @ zero_zero_nat
% 0.30/0.66                @ ( minus_minus_nat @ K2 @ one_one_nat )
% 0.30/0.66                @ one_one_real )
% 0.30/0.66              @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gbinomial_code
% 0.30/0.66  thf(fact_8818_gchoose__row__sum__weighted,axiom,
% 0.30/0.66      ! [R: complex,M: nat] :
% 0.30/0.66        ( ( groups2073611262835488442omplex
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.30/0.66        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R @ ( suc @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gchoose_row_sum_weighted
% 0.30/0.66  thf(fact_8819_gchoose__row__sum__weighted,axiom,
% 0.30/0.66      ! [R: rat,M: nat] :
% 0.30/0.66        ( ( groups2906978787729119204at_rat
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ R @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.30/0.66        = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R @ ( suc @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gchoose_row_sum_weighted
% 0.30/0.66  thf(fact_8820_gchoose__row__sum__weighted,axiom,
% 0.30/0.66      ! [R: real,M: nat] :
% 0.30/0.66        ( ( groups6591440286371151544t_real
% 0.30/0.66          @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
% 0.30/0.66          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.30/0.66        = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R @ ( suc @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gchoose_row_sum_weighted
% 0.30/0.66  thf(fact_8821_of__nat__code,axiom,
% 0.30/0.66      ( semiri8010041392384452111omplex
% 0.30/0.66      = ( ^ [N2: nat] :
% 0.30/0.66            ( semiri2816024913162550771omplex
% 0.30/0.66            @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 0.30/0.66            @ N2
% 0.30/0.66            @ zero_zero_complex ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_code
% 0.30/0.66  thf(fact_8822_of__nat__code,axiom,
% 0.30/0.66      ( semiri681578069525770553at_rat
% 0.30/0.66      = ( ^ [N2: nat] :
% 0.30/0.66            ( semiri7787848453975740701ux_rat
% 0.30/0.66            @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 0.30/0.66            @ N2
% 0.30/0.66            @ zero_zero_rat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_code
% 0.30/0.66  thf(fact_8823_of__nat__code,axiom,
% 0.30/0.66      ( semiri1314217659103216013at_int
% 0.30/0.66      = ( ^ [N2: nat] :
% 0.30/0.66            ( semiri8420488043553186161ux_int
% 0.30/0.66            @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 0.30/0.66            @ N2
% 0.30/0.66            @ zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_code
% 0.30/0.66  thf(fact_8824_of__nat__code,axiom,
% 0.30/0.66      ( semiri5074537144036343181t_real
% 0.30/0.66      = ( ^ [N2: nat] :
% 0.30/0.66            ( semiri7260567687927622513x_real
% 0.30/0.66            @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 0.30/0.66            @ N2
% 0.30/0.66            @ zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_code
% 0.30/0.66  thf(fact_8825_of__nat__code,axiom,
% 0.30/0.66      ( semiri1316708129612266289at_nat
% 0.30/0.66      = ( ^ [N2: nat] :
% 0.30/0.66            ( semiri8422978514062236437ux_nat
% 0.30/0.66            @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 0.30/0.66            @ N2
% 0.30/0.66            @ zero_zero_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_code
% 0.30/0.66  thf(fact_8826_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.30/0.66      ! [X3: vEBT_VEBT,Y3: $o] :
% 0.30/0.66        ( ( ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 0.30/0.66         => ( ( ( X3
% 0.30/0.66                = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.66             => ( Y3
% 0.30/0.66               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.30/0.66           => ( ! [Uv2: $o] :
% 0.30/0.66                  ( ( X3
% 0.30/0.66                    = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.66                 => ( ~ Y3
% 0.30/0.66                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.30/0.66             => ( ! [Uu2: $o] :
% 0.30/0.66                    ( ( X3
% 0.30/0.66                      = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.66                   => ( ~ Y3
% 0.30/0.66                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.30/0.66               => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.66                      ( ( X3
% 0.30/0.66                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.66                     => ( Y3
% 0.30/0.66                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.30/0.66                 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.66                        ( ( X3
% 0.30/0.66                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.30/0.66                       => ( ~ Y3
% 0.30/0.66                         => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % VEBT_internal.minNull.pelims(1)
% 0.30/0.66  thf(fact_8827_sin__x__sin__y,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( sums_real
% 0.30/0.66        @ ^ [P5: nat] :
% 0.30/0.66            ( groups6591440286371151544t_real
% 0.30/0.66            @ ^ [N2: nat] :
% 0.30/0.66                ( if_real
% 0.30/0.66                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
% 0.30/0.66                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.66                @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) )
% 0.30/0.66                @ zero_zero_real )
% 0.30/0.66            @ ( set_ord_atMost_nat @ P5 ) )
% 0.30/0.66        @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_x_sin_y
% 0.30/0.66  thf(fact_8828_sin__x__sin__y,axiom,
% 0.30/0.66      ! [X3: complex,Y3: complex] :
% 0.30/0.66        ( sums_complex
% 0.30/0.66        @ ^ [P5: nat] :
% 0.30/0.66            ( groups2073611262835488442omplex
% 0.30/0.66            @ ^ [N2: nat] :
% 0.30/0.66                ( if_complex
% 0.30/0.66                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
% 0.30/0.66                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.66                @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X3 @ N2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) )
% 0.30/0.66                @ zero_zero_complex )
% 0.30/0.66            @ ( set_ord_atMost_nat @ P5 ) )
% 0.30/0.66        @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_x_sin_y
% 0.30/0.66  thf(fact_8829_Maclaurin__sin__bound,axiom,
% 0.30/0.66      ! [X3: real,N: nat] :
% 0.30/0.66        ( ord_less_eq_real
% 0.30/0.66        @ ( abs_abs_real
% 0.30/0.66          @ ( minus_minus_real @ ( sin_real @ X3 )
% 0.30/0.66            @ ( groups6591440286371151544t_real
% 0.30/0.66              @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.66              @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.30/0.66        @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X3 ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_sin_bound
% 0.30/0.66  thf(fact_8830_sums__cos__x__plus__y,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( sums_real
% 0.30/0.66        @ ^ [P5: nat] :
% 0.30/0.66            ( groups6591440286371151544t_real
% 0.30/0.66            @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X3 @ N2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_real )
% 0.30/0.66            @ ( set_ord_atMost_nat @ P5 ) )
% 0.30/0.66        @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sums_cos_x_plus_y
% 0.30/0.66  thf(fact_8831_sums__cos__x__plus__y,axiom,
% 0.30/0.66      ! [X3: complex,Y3: complex] :
% 0.30/0.66        ( sums_complex
% 0.30/0.66        @ ^ [P5: nat] :
% 0.30/0.66            ( groups2073611262835488442omplex
% 0.30/0.66            @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X3 @ N2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_complex )
% 0.30/0.66            @ ( set_ord_atMost_nat @ P5 ) )
% 0.30/0.66        @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sums_cos_x_plus_y
% 0.30/0.66  thf(fact_8832_mult__scaleR__left,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X3 ) @ Y3 )
% 0.30/0.66        = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_scaleR_left
% 0.30/0.66  thf(fact_8833_mult__scaleR__left,axiom,
% 0.30/0.66      ! [A: real,X3: complex,Y3: complex] :
% 0.30/0.66        ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X3 ) @ Y3 )
% 0.30/0.66        = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_scaleR_left
% 0.30/0.66  thf(fact_8834_mult__scaleR__right,axiom,
% 0.30/0.66      ! [X3: real,A: real,Y3: real] :
% 0.30/0.66        ( ( times_times_real @ X3 @ ( real_V1485227260804924795R_real @ A @ Y3 ) )
% 0.30/0.66        = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_scaleR_right
% 0.30/0.66  thf(fact_8835_mult__scaleR__right,axiom,
% 0.30/0.66      ! [X3: complex,A: real,Y3: complex] :
% 0.30/0.66        ( ( times_times_complex @ X3 @ ( real_V2046097035970521341omplex @ A @ Y3 ) )
% 0.30/0.66        = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_scaleR_right
% 0.30/0.66  thf(fact_8836_inverse__mult__distrib,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
% 0.30/0.66        = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_mult_distrib
% 0.30/0.66  thf(fact_8837_inverse__mult__distrib,axiom,
% 0.30/0.66      ! [A: complex,B2: complex] :
% 0.30/0.66        ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
% 0.30/0.66        = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_mult_distrib
% 0.30/0.66  thf(fact_8838_inverse__mult__distrib,axiom,
% 0.30/0.66      ! [A: rat,B2: rat] :
% 0.30/0.66        ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
% 0.30/0.66        = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_mult_distrib
% 0.30/0.66  thf(fact_8839_inverse__divide,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
% 0.30/0.66        = ( divide_divide_real @ B2 @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_divide
% 0.30/0.66  thf(fact_8840_inverse__divide,axiom,
% 0.30/0.66      ! [A: complex,B2: complex] :
% 0.30/0.66        ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.66        = ( divide1717551699836669952omplex @ B2 @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_divide
% 0.30/0.66  thf(fact_8841_inverse__divide,axiom,
% 0.30/0.66      ! [A: rat,B2: rat] :
% 0.30/0.66        ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B2 ) )
% 0.30/0.66        = ( divide_divide_rat @ B2 @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_divide
% 0.30/0.66  thf(fact_8842_scaleR__scaleR,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: real] :
% 0.30/0.66        ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B2 @ X3 ) )
% 0.30/0.66        = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B2 ) @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % scaleR_scaleR
% 0.30/0.66  thf(fact_8843_scaleR__scaleR,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: complex] :
% 0.30/0.66        ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B2 @ X3 ) )
% 0.30/0.66        = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B2 ) @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % scaleR_scaleR
% 0.30/0.66  thf(fact_8844_inverse__nonpositive__iff__nonpositive,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.30/0.66        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_nonpositive_iff_nonpositive
% 0.30/0.66  thf(fact_8845_inverse__nonpositive__iff__nonpositive,axiom,
% 0.30/0.66      ! [A: rat] :
% 0.30/0.66        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.30/0.66        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_nonpositive_iff_nonpositive
% 0.30/0.66  thf(fact_8846_inverse__nonnegative__iff__nonnegative,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.30/0.66        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_nonnegative_iff_nonnegative
% 0.30/0.66  thf(fact_8847_inverse__nonnegative__iff__nonnegative,axiom,
% 0.30/0.66      ! [A: rat] :
% 0.30/0.66        ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.30/0.66        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_nonnegative_iff_nonnegative
% 0.30/0.66  thf(fact_8848_inverse__positive__iff__positive,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 0.30/0.66        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_positive_iff_positive
% 0.30/0.66  thf(fact_8849_inverse__positive__iff__positive,axiom,
% 0.30/0.66      ! [A: rat] :
% 0.30/0.66        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 0.30/0.66        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_positive_iff_positive
% 0.30/0.66  thf(fact_8850_inverse__negative__iff__negative,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 0.30/0.66        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_negative_iff_negative
% 0.30/0.66  thf(fact_8851_inverse__negative__iff__negative,axiom,
% 0.30/0.66      ! [A: rat] :
% 0.30/0.66        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 0.30/0.66        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_negative_iff_negative
% 0.30/0.66  thf(fact_8852_inverse__less__iff__less__neg,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 0.30/0.66         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.30/0.66            = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_less_iff_less_neg
% 0.30/0.66  thf(fact_8853_inverse__less__iff__less__neg,axiom,
% 0.30/0.66      ! [A: rat,B2: rat] :
% 0.30/0.66        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.66       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 0.30/0.66         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.30/0.66            = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_less_iff_less_neg
% 0.30/0.66  thf(fact_8854_inverse__less__iff__less,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.66         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 0.30/0.66            = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_less_iff_less
% 0.30/0.66  thf(fact_8855_inverse__less__iff__less,axiom,
% 0.30/0.66      ! [A: rat,B2: rat] :
% 0.30/0.66        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.66       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 0.30/0.66         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 0.30/0.66            = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_less_iff_less
% 0.30/0.66  thf(fact_8856_of__int__le__iff,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.30/0.66        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_le_iff
% 0.30/0.66  thf(fact_8857_of__int__le__iff,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.30/0.66        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_le_iff
% 0.30/0.66  thf(fact_8858_of__int__le__iff,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.30/0.66        = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_le_iff
% 0.30/0.66  thf(fact_8859_of__int__eq__numeral__iff,axiom,
% 0.30/0.66      ! [Z: int,N: num] :
% 0.30/0.66        ( ( ( ring_17405671764205052669omplex @ Z )
% 0.30/0.66          = ( numera6690914467698888265omplex @ N ) )
% 0.30/0.66        = ( Z
% 0.30/0.66          = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_eq_numeral_iff
% 0.30/0.66  thf(fact_8860_of__int__eq__numeral__iff,axiom,
% 0.30/0.66      ! [Z: int,N: num] :
% 0.30/0.66        ( ( ( ring_1_of_int_real @ Z )
% 0.30/0.66          = ( numeral_numeral_real @ N ) )
% 0.30/0.66        = ( Z
% 0.30/0.66          = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_eq_numeral_iff
% 0.30/0.66  thf(fact_8861_of__int__eq__numeral__iff,axiom,
% 0.30/0.66      ! [Z: int,N: num] :
% 0.30/0.66        ( ( ( ring_1_of_int_rat @ Z )
% 0.30/0.66          = ( numeral_numeral_rat @ N ) )
% 0.30/0.66        = ( Z
% 0.30/0.66          = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_eq_numeral_iff
% 0.30/0.66  thf(fact_8862_of__int__eq__numeral__iff,axiom,
% 0.30/0.66      ! [Z: int,N: num] :
% 0.30/0.66        ( ( ( ring_1_of_int_int @ Z )
% 0.30/0.66          = ( numeral_numeral_int @ N ) )
% 0.30/0.66        = ( Z
% 0.30/0.66          = ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_eq_numeral_iff
% 0.30/0.66  thf(fact_8863_of__int__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.30/0.66        = ( numera6690914467698888265omplex @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_numeral
% 0.30/0.66  thf(fact_8864_of__int__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.30/0.66        = ( numeral_numeral_real @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_numeral
% 0.30/0.66  thf(fact_8865_of__int__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.30/0.66        = ( numeral_numeral_rat @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_numeral
% 0.30/0.66  thf(fact_8866_of__int__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.66        = ( numeral_numeral_int @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_int_numeral
% 0.30/0.66  thf(fact_8867_scaleR__eq__iff,axiom,
% 0.30/0.66      ! [B2: complex,U: real,A: complex] :
% 0.30/0.66        ( ( ( plus_plus_complex @ B2 @ ( real_V2046097035970521341omplex @ U @ A ) )
% 0.30/0.66          = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B2 ) ) )
% 0.30/0.66        = ( ( A = B2 )
% 0.30/0.66          | ( U = one_one_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % scaleR_eq_iff
% 0.30/0.66  thf(fact_8868_sin__npi__int,axiom,
% 0.30/0.66      ! [N: int] :
% 0.30/0.66        ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.66        = zero_zero_real ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_npi_int
% 0.30/0.66  thf(fact_8869_tan__periodic__int,axiom,
% 0.30/0.66      ! [X3: real,I: int] :
% 0.30/0.66        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 0.30/0.66        = ( tan_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % tan_periodic_int
% 0.30/0.66  thf(fact_8870_sin__int__2pin,axiom,
% 0.30/0.66      ! [N: int] :
% 0.30/0.66        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.66        = zero_zero_real ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_int_2pin
% 0.30/0.66  thf(fact_8871_cos__int__2pin,axiom,
% 0.30/0.66      ! [N: int] :
% 0.30/0.66        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.66        = one_one_real ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_int_2pin
% 0.30/0.66  thf(fact_8872_cos__npi__int,axiom,
% 0.30/0.66      ! [N: int] :
% 0.30/0.66        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.66         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.66            = one_one_real ) )
% 0.30/0.66        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.30/0.66         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.30/0.66            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_npi_int
% 0.30/0.66  thf(fact_8873_real__scaleR__def,axiom,
% 0.30/0.66      real_V1485227260804924795R_real = times_times_real ).
% 0.30/0.66  
% 0.30/0.66  % real_scaleR_def
% 0.30/0.66  thf(fact_8874_real__sqrt__inverse,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( sqrt @ ( inverse_inverse_real @ X3 ) )
% 0.30/0.66        = ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_sqrt_inverse
% 0.30/0.66  thf(fact_8875_divide__real__def,axiom,
% 0.30/0.66      ( divide_divide_real
% 0.30/0.66      = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_real_def
% 0.30/0.66  thf(fact_8876_complex__scaleR,axiom,
% 0.30/0.66      ! [R: real,A: real,B2: real] :
% 0.30/0.66        ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B2 ) )
% 0.30/0.66        = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_scaleR
% 0.30/0.66  thf(fact_8877_real__of__int__div4,axiom,
% 0.30/0.66      ! [N: int,X3: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_div4
% 0.30/0.66  thf(fact_8878_real__of__int__div,axiom,
% 0.30/0.66      ! [D: int,N: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ D @ N )
% 0.30/0.66       => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.30/0.66          = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_div
% 0.30/0.66  thf(fact_8879_forall__pos__mono__1,axiom,
% 0.30/0.66      ! [P: real > $o,E: real] :
% 0.30/0.66        ( ! [D3: real,E2: real] :
% 0.30/0.66            ( ( ord_less_real @ D3 @ E2 )
% 0.30/0.66           => ( ( P @ D3 )
% 0.30/0.66             => ( P @ E2 ) ) )
% 0.30/0.66       => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.66           => ( P @ E ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % forall_pos_mono_1
% 0.30/0.66  thf(fact_8880_forall__pos__mono,axiom,
% 0.30/0.66      ! [P: real > $o,E: real] :
% 0.30/0.66        ( ! [D3: real,E2: real] :
% 0.30/0.66            ( ( ord_less_real @ D3 @ E2 )
% 0.30/0.66           => ( ( P @ D3 )
% 0.30/0.66             => ( P @ E2 ) ) )
% 0.30/0.66       => ( ! [N3: nat] :
% 0.30/0.66              ( ( N3 != zero_zero_nat )
% 0.30/0.66             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.66           => ( P @ E ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % forall_pos_mono
% 0.30/0.66  thf(fact_8881_real__arch__inverse,axiom,
% 0.30/0.66      ! [E: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.66        = ( ? [N2: nat] :
% 0.30/0.66              ( ( N2 != zero_zero_nat )
% 0.30/0.66              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.30/0.66              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_arch_inverse
% 0.30/0.66  thf(fact_8882_int__le__real__less,axiom,
% 0.30/0.66      ( ord_less_eq_int
% 0.30/0.66      = ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_le_real_less
% 0.30/0.66  thf(fact_8883_sqrt__divide__self__eq,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( divide_divide_real @ ( sqrt @ X3 ) @ X3 )
% 0.30/0.66          = ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sqrt_divide_self_eq
% 0.30/0.66  thf(fact_8884_int__less__real__le,axiom,
% 0.30/0.66      ( ord_less_int
% 0.30/0.66      = ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_less_real_le
% 0.30/0.66  thf(fact_8885_ln__inverse,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
% 0.30/0.66          = ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_inverse
% 0.30/0.66  thf(fact_8886_sin__zero__iff__int2,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ( sin_real @ X3 )
% 0.30/0.66          = zero_zero_real )
% 0.30/0.66        = ( ? [I4: int] :
% 0.30/0.66              ( X3
% 0.30/0.66              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_zero_iff_int2
% 0.30/0.66  thf(fact_8887_real__of__int__div__aux,axiom,
% 0.30/0.66      ! [X3: int,D: int] :
% 0.30/0.66        ( ( divide_divide_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ D ) )
% 0.30/0.66        = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X3 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X3 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_div_aux
% 0.30/0.66  thf(fact_8888_real__of__int__div2,axiom,
% 0.30/0.66      ! [N: int,X3: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_div2
% 0.30/0.66  thf(fact_8889_real__of__int__div3,axiom,
% 0.30/0.66      ! [N: int,X3: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) @ one_one_real ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_div3
% 0.30/0.66  thf(fact_8890_exp__plus__inverse__exp,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % exp_plus_inverse_exp
% 0.30/0.66  thf(fact_8891_plus__inverse__ge__2,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % plus_inverse_ge_2
% 0.30/0.66  thf(fact_8892_real__inv__sqrt__pow2,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.66          = ( inverse_inverse_real @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_inv_sqrt_pow2
% 0.30/0.66  thf(fact_8893_cos__one__2pi__int,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ( cos_real @ X3 )
% 0.30/0.66          = one_one_real )
% 0.30/0.66        = ( ? [X: int] :
% 0.30/0.66              ( X3
% 0.30/0.66              = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_one_2pi_int
% 0.30/0.66  thf(fact_8894_tan__cot,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) )
% 0.30/0.66        = ( inverse_inverse_real @ ( tan_real @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % tan_cot
% 0.30/0.66  thf(fact_8895_real__le__x__sinh,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_eq_real @ X3 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_le_x_sinh
% 0.30/0.66  thf(fact_8896_real__le__abs__sinh,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_le_abs_sinh
% 0.30/0.66  thf(fact_8897_arccos__cos__eq__abs__2pi,axiom,
% 0.30/0.66      ! [Theta: real] :
% 0.30/0.66        ~ ! [K3: int] :
% 0.30/0.66            ( ( arccos @ ( cos_real @ Theta ) )
% 0.30/0.66           != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_cos_eq_abs_2pi
% 0.30/0.66  thf(fact_8898_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.30/0.66      ! [X3: vEBT_VEBT] :
% 0.30/0.66        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.66       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 0.30/0.66         => ( ! [Uv2: $o] :
% 0.30/0.66                ( ( X3
% 0.30/0.66                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.66               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.30/0.66           => ( ! [Uu2: $o] :
% 0.30/0.66                  ( ( X3
% 0.30/0.66                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.66                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.30/0.66             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.66                    ( ( X3
% 0.30/0.66                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.30/0.66                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % VEBT_internal.minNull.pelims(3)
% 0.30/0.66  thf(fact_8899_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.30/0.66      ! [X3: vEBT_VEBT] :
% 0.30/0.66        ( ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.66       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 0.30/0.66         => ( ( ( X3
% 0.30/0.66                = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.66             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.66           => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.66                  ( ( X3
% 0.30/0.66                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.66                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % VEBT_internal.minNull.pelims(2)
% 0.30/0.66  thf(fact_8900_cos__zero__iff__int,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ( cos_real @ X3 )
% 0.30/0.66          = zero_zero_real )
% 0.30/0.66        = ( ? [I4: int] :
% 0.30/0.66              ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.66              & ( X3
% 0.30/0.66                = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cos_zero_iff_int
% 0.30/0.66  thf(fact_8901_sin__zero__iff__int,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ( sin_real @ X3 )
% 0.30/0.66          = zero_zero_real )
% 0.30/0.66        = ( ? [I4: int] :
% 0.30/0.66              ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.30/0.66              & ( X3
% 0.30/0.66                = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_zero_iff_int
% 0.30/0.66  thf(fact_8902_sinh__real__less__iff,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
% 0.30/0.66        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_real_less_iff
% 0.30/0.66  thf(fact_8903_sinh__real__le__iff,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
% 0.30/0.66        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_real_le_iff
% 0.30/0.66  thf(fact_8904_sinh__real__neg__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
% 0.30/0.66        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_real_neg_iff
% 0.30/0.66  thf(fact_8905_sinh__real__pos__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
% 0.30/0.66        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_real_pos_iff
% 0.30/0.66  thf(fact_8906_sinh__real__nonneg__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X3 ) )
% 0.30/0.66        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_real_nonneg_iff
% 0.30/0.66  thf(fact_8907_sinh__real__nonpos__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ zero_zero_real )
% 0.30/0.66        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_real_nonpos_iff
% 0.30/0.66  thf(fact_8908_divide__complex__def,axiom,
% 0.30/0.66      ( divide1717551699836669952omplex
% 0.30/0.66      = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_complex_def
% 0.30/0.66  thf(fact_8909_complex__inverse,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B2 ) )
% 0.30/0.66        = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B2 ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_inverse
% 0.30/0.66  thf(fact_8910_sinh__ln__real,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( sinh_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.66          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_ln_real
% 0.30/0.66  thf(fact_8911_divmod__BitM__2__eq,axiom,
% 0.30/0.66      ! [M: num] :
% 0.30/0.66        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.30/0.66        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divmod_BitM_2_eq
% 0.30/0.66  thf(fact_8912_cot__less__zero,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.66         => ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cot_less_zero
% 0.30/0.66  thf(fact_8913_i__even__power,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % i_even_power
% 0.30/0.66  thf(fact_8914_complex__i__mult__minus,axiom,
% 0.30/0.66      ! [X3: complex] :
% 0.30/0.66        ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X3 ) )
% 0.30/0.66        = ( uminus1482373934393186551omplex @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_i_mult_minus
% 0.30/0.66  thf(fact_8915_pred__numeral__simps_I2_J,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( pred_numeral @ ( bit0 @ K ) )
% 0.30/0.66        = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pred_numeral_simps(2)
% 0.30/0.66  thf(fact_8916_divide__i,axiom,
% 0.30/0.66      ! [X3: complex] :
% 0.30/0.66        ( ( divide1717551699836669952omplex @ X3 @ imaginary_unit )
% 0.30/0.66        = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_i
% 0.30/0.66  thf(fact_8917_i__squared,axiom,
% 0.30/0.66      ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.30/0.66      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % i_squared
% 0.30/0.66  thf(fact_8918_cot__npi,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.30/0.66        = zero_zero_real ) ).
% 0.30/0.66  
% 0.30/0.66  % cot_npi
% 0.30/0.66  thf(fact_8919_divide__numeral__i,axiom,
% 0.30/0.66      ! [Z: complex,N: num] :
% 0.30/0.66        ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 0.30/0.66        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_numeral_i
% 0.30/0.66  thf(fact_8920_cot__periodic,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( cot_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.66        = ( cot_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cot_periodic
% 0.30/0.66  thf(fact_8921_power2__i,axiom,
% 0.30/0.66      ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.66      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % power2_i
% 0.30/0.66  thf(fact_8922_sinh__less__cosh__real,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_less_cosh_real
% 0.30/0.66  thf(fact_8923_sinh__le__cosh__real,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sinh_le_cosh_real
% 0.30/0.66  thf(fact_8924_semiring__norm_I26_J,axiom,
% 0.30/0.66      ( ( bitM @ one )
% 0.30/0.66      = one ) ).
% 0.30/0.66  
% 0.30/0.66  % semiring_norm(26)
% 0.30/0.66  thf(fact_8925_complex__i__not__numeral,axiom,
% 0.30/0.66      ! [W: num] :
% 0.30/0.66        ( imaginary_unit
% 0.30/0.66       != ( numera6690914467698888265omplex @ W ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_i_not_numeral
% 0.30/0.66  thf(fact_8926_cosh__real__pos,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_pos
% 0.30/0.66  thf(fact_8927_cosh__real__nonneg,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_nonneg
% 0.30/0.66  thf(fact_8928_cosh__real__nonneg__le__iff,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_nonneg_le_iff
% 0.30/0.66  thf(fact_8929_cosh__real__nonpos__le__iff,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_nonpos_le_iff
% 0.30/0.66  thf(fact_8930_cosh__real__ge__1,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_ge_1
% 0.30/0.66  thf(fact_8931_semiring__norm_I27_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bitM @ ( bit0 @ N ) )
% 0.30/0.66        = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % semiring_norm(27)
% 0.30/0.66  thf(fact_8932_semiring__norm_I28_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bitM @ ( bit1 @ N ) )
% 0.30/0.66        = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % semiring_norm(28)
% 0.30/0.66  thf(fact_8933_i__times__eq__iff,axiom,
% 0.30/0.66      ! [W: complex,Z: complex] :
% 0.30/0.66        ( ( ( times_times_complex @ imaginary_unit @ W )
% 0.30/0.66          = Z )
% 0.30/0.66        = ( W
% 0.30/0.66          = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % i_times_eq_iff
% 0.30/0.66  thf(fact_8934_complex__i__not__neg__numeral,axiom,
% 0.30/0.66      ! [W: num] :
% 0.30/0.66        ( imaginary_unit
% 0.30/0.66       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_i_not_neg_numeral
% 0.30/0.66  thf(fact_8935_cosh__real__strict__mono,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.66         => ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_strict_mono
% 0.30/0.66  thf(fact_8936_cosh__real__nonneg__less__iff,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.66         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.66            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_nonneg_less_iff
% 0.30/0.66  thf(fact_8937_cosh__real__nonpos__less__iff,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.66         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.66            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_real_nonpos_less_iff
% 0.30/0.66  thf(fact_8938_arcosh__cosh__real,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( arcosh_real @ ( cosh_real @ X3 ) )
% 0.30/0.66          = X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcosh_cosh_real
% 0.30/0.66  thf(fact_8939_eval__nat__numeral_I2_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.30/0.66        = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eval_nat_numeral(2)
% 0.30/0.66  thf(fact_8940_one__plus__BitM,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.30/0.66        = ( bit0 @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % one_plus_BitM
% 0.30/0.66  thf(fact_8941_BitM__plus__one,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.30/0.66        = ( bit0 @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % BitM_plus_one
% 0.30/0.66  thf(fact_8942_Complex__mult__i,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ imaginary_unit )
% 0.30/0.66        = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Complex_mult_i
% 0.30/0.66  thf(fact_8943_i__mult__Complex,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B2 ) )
% 0.30/0.66        = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % i_mult_Complex
% 0.30/0.66  thf(fact_8944_cosh__ln__real,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( cosh_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.66          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cosh_ln_real
% 0.30/0.66  thf(fact_8945_cot__gt__zero,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cot_gt_zero
% 0.30/0.66  thf(fact_8946_tan__cot_H,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) )
% 0.30/0.66        = ( cot_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % tan_cot'
% 0.30/0.66  thf(fact_8947_Arg__minus__ii,axiom,
% 0.30/0.66      ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.30/0.66      = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_minus_ii
% 0.30/0.66  thf(fact_8948_Arg__ii,axiom,
% 0.30/0.66      ( ( arg @ imaginary_unit )
% 0.30/0.66      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_ii
% 0.30/0.66  thf(fact_8949_cis__minus__pi__half,axiom,
% 0.30/0.66      ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66      = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_minus_pi_half
% 0.30/0.66  thf(fact_8950_log__base__10__eq1,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.66          = ( 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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_base_10_eq1
% 0.30/0.66  thf(fact_8951_log__eq__one,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( log @ A @ A )
% 0.30/0.66            = one_one_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_eq_one
% 0.30/0.66  thf(fact_8952_log__less__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.66           => ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 0.30/0.66              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_less_cancel_iff
% 0.30/0.66  thf(fact_8953_log__less__one__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
% 0.30/0.66            = ( ord_less_real @ X3 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_less_one_cancel_iff
% 0.30/0.66  thf(fact_8954_one__less__log__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
% 0.30/0.66            = ( ord_less_real @ A @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % one_less_log_cancel_iff
% 0.30/0.66  thf(fact_8955_log__less__zero__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 0.30/0.66            = ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_less_zero_cancel_iff
% 0.30/0.66  thf(fact_8956_zero__less__log__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 0.30/0.66            = ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zero_less_log_cancel_iff
% 0.30/0.66  thf(fact_8957_zero__le__log__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zero_le_log_cancel_iff
% 0.30/0.66  thf(fact_8958_log__le__zero__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 0.30/0.66            = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_le_zero_cancel_iff
% 0.30/0.66  thf(fact_8959_one__le__log__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ A @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % one_le_log_cancel_iff
% 0.30/0.66  thf(fact_8960_log__le__one__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
% 0.30/0.66            = ( ord_less_eq_real @ X3 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_le_one_cancel_iff
% 0.30/0.66  thf(fact_8961_log__le__cancel__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.66           => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 0.30/0.66              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_le_cancel_iff
% 0.30/0.66  thf(fact_8962_log__pow__cancel,axiom,
% 0.30/0.66      ! [A: real,B2: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( log @ A @ ( power_power_real @ A @ B2 ) )
% 0.30/0.66            = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_pow_cancel
% 0.30/0.66  thf(fact_8963_cis__pi__half,axiom,
% 0.30/0.66      ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66      = imaginary_unit ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_pi_half
% 0.30/0.66  thf(fact_8964_cis__2pi,axiom,
% 0.30/0.66      ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.66      = one_one_complex ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_2pi
% 0.30/0.66  thf(fact_8965_cis__mult,axiom,
% 0.30/0.66      ! [A: real,B2: real] :
% 0.30/0.66        ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B2 ) )
% 0.30/0.66        = ( cis @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_mult
% 0.30/0.66  thf(fact_8966_log__base__change,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( log @ B2 @ X3 )
% 0.30/0.66            = ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_base_change
% 0.30/0.66  thf(fact_8967_log__of__power__eq,axiom,
% 0.30/0.66      ! [M: nat,B2: real,N: nat] :
% 0.30/0.66        ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.66          = ( power_power_real @ B2 @ N ) )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ( semiri5074537144036343181t_real @ N )
% 0.30/0.66            = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_of_power_eq
% 0.30/0.66  thf(fact_8968_less__log__of__power,axiom,
% 0.30/0.66      ! [B2: real,N: nat,M: real] :
% 0.30/0.66        ( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_log_of_power
% 0.30/0.66  thf(fact_8969_DeMoivre,axiom,
% 0.30/0.66      ! [A: real,N: nat] :
% 0.30/0.66        ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.30/0.66        = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DeMoivre
% 0.30/0.66  thf(fact_8970_log__mult,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.66             => ( ( log @ A @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.66                = ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_mult
% 0.30/0.66  thf(fact_8971_log__divide,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.66             => ( ( log @ A @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.66                = ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_divide
% 0.30/0.66  thf(fact_8972_le__log__of__power,axiom,
% 0.30/0.66      ! [B2: real,N: nat,M: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_log_of_power
% 0.30/0.66  thf(fact_8973_log__base__pow,axiom,
% 0.30/0.66      ! [A: real,N: nat,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( log @ ( power_power_real @ A @ N ) @ X3 )
% 0.30/0.66          = ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_base_pow
% 0.30/0.66  thf(fact_8974_log__nat__power,axiom,
% 0.30/0.66      ! [X3: real,B2: real,N: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( log @ B2 @ ( power_power_real @ X3 @ N ) )
% 0.30/0.66          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_nat_power
% 0.30/0.66  thf(fact_8975_log__inverse,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
% 0.30/0.66              = ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_inverse
% 0.30/0.66  thf(fact_8976_log2__of__power__eq,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( M
% 0.30/0.66          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.66       => ( ( semiri5074537144036343181t_real @ N )
% 0.30/0.66          = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log2_of_power_eq
% 0.30/0.66  thf(fact_8977_log__of__power__less,axiom,
% 0.30/0.66      ! [M: nat,B2: real,N: nat] :
% 0.30/0.66        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.66           => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_of_power_less
% 0.30/0.66  thf(fact_8978_log__eq__div__ln__mult__log,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.66           => ( ( B2 != one_one_real )
% 0.30/0.66             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66               => ( ( log @ A @ X3 )
% 0.30/0.66                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_eq_div_ln_mult_log
% 0.30/0.66  thf(fact_8979_Arg__bounded,axiom,
% 0.30/0.66      ! [Z: complex] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.30/0.66        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_bounded
% 0.30/0.66  thf(fact_8980_log__of__power__le,axiom,
% 0.30/0.66      ! [M: nat,B2: real,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.66           => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_of_power_le
% 0.30/0.66  thf(fact_8981_less__log2__of__power,axiom,
% 0.30/0.66      ! [N: nat,M: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.30/0.66       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_log2_of_power
% 0.30/0.66  thf(fact_8982_le__log2__of__power,axiom,
% 0.30/0.66      ! [N: nat,M: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.30/0.66       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_log2_of_power
% 0.30/0.66  thf(fact_8983_log2__of__power__less,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.66         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log2_of_power_less
% 0.30/0.66  thf(fact_8984_log2__of__power__le,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.66         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log2_of_power_le
% 0.30/0.66  thf(fact_8985_log__base__10__eq2,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.66          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_base_10_eq2
% 0.30/0.66  thf(fact_8986_ceiling__log__nat__eq__powr__iff,axiom,
% 0.30/0.66      ! [B2: nat,K: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.66              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.30/0.66            = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.30/0.66              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ceiling_log_nat_eq_powr_iff
% 0.30/0.66  thf(fact_8987_bij__betw__roots__unity,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66       => ( bij_betw_nat_complex
% 0.30/0.66          @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.66          @ ( set_ord_lessThan_nat @ N )
% 0.30/0.66          @ ( collect_complex
% 0.30/0.66            @ ^ [Z2: complex] :
% 0.30/0.66                ( ( power_power_complex @ Z2 @ N )
% 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_8988_ceiling__log__nat__eq__if,axiom,
% 0.30/0.66      ! [B2: nat,N: nat,K: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.30/0.66       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.30/0.66         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.66           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.66              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ceiling_log_nat_eq_if
% 0.30/0.66  thf(fact_8989_ceiling__log2__div2,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.66          = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ceiling_log2_div2
% 0.30/0.66  thf(fact_8990_ceiling__divide__eq__div__numeral,axiom,
% 0.30/0.66      ! [A: num,B2: num] :
% 0.30/0.66        ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 0.30/0.66        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ceiling_divide_eq_div_numeral
% 0.30/0.66  thf(fact_8991_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.30/0.66      ! [A: num,B2: num] :
% 0.30/0.66        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 0.30/0.66        = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ceiling_minus_divide_eq_div_numeral
% 0.30/0.66  thf(fact_8992_ceiling__log__eq__powr__iff,axiom,
% 0.30/0.66      ! [X3: real,B2: real,K: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X3 ) )
% 0.30/0.66              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.30/0.66            = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X3 )
% 0.30/0.66              & ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ceiling_log_eq_powr_iff
% 0.30/0.66  thf(fact_8993_floor__log__nat__eq__powr__iff,axiom,
% 0.30/0.66      ! [B2: nat,K: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.66         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.66              = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.66            = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.30/0.66              & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_log_nat_eq_powr_iff
% 0.30/0.66  thf(fact_8994_int__ge__less__than2__def,axiom,
% 0.30/0.66      ( int_ge_less_than2
% 0.30/0.66      = ( ^ [D2: int] :
% 0.30/0.66            ( collec213857154873943460nt_int
% 0.30/0.66            @ ( produc4947309494688390418_int_o
% 0.30/0.66              @ ^ [Z6: int,Z2: int] :
% 0.30/0.66                  ( ( ord_less_eq_int @ D2 @ Z2 )
% 0.30/0.66                  & ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_ge_less_than2_def
% 0.30/0.66  thf(fact_8995_int__ge__less__than__def,axiom,
% 0.30/0.66      ( int_ge_less_than
% 0.30/0.66      = ( ^ [D2: int] :
% 0.30/0.66            ( collec213857154873943460nt_int
% 0.30/0.66            @ ( produc4947309494688390418_int_o
% 0.30/0.66              @ ^ [Z6: int,Z2: int] :
% 0.30/0.66                  ( ( ord_less_eq_int @ D2 @ Z6 )
% 0.30/0.66                  & ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_ge_less_than_def
% 0.30/0.66  thf(fact_8996_powr__gt__zero,axiom,
% 0.30/0.66      ! [X3: real,A: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
% 0.30/0.66        = ( X3 != zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_gt_zero
% 0.30/0.66  thf(fact_8997_powr__nonneg__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( powr_real @ A @ X3 ) @ zero_zero_real )
% 0.30/0.66        = ( A = zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_nonneg_iff
% 0.30/0.66  thf(fact_8998_powr__less__cancel__iff,axiom,
% 0.30/0.66      ! [X3: real,A: real,B2: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
% 0.30/0.66          = ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_less_cancel_iff
% 0.30/0.66  thf(fact_8999_powr__eq__one__iff,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.66       => ( ( ( powr_real @ A @ X3 )
% 0.30/0.66            = one_one_real )
% 0.30/0.66          = ( X3 = zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_eq_one_iff
% 0.30/0.66  thf(fact_9000_powr__one,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( powr_real @ X3 @ one_one_real )
% 0.30/0.66          = X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_one
% 0.30/0.66  thf(fact_9001_powr__one__gt__zero__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ( powr_real @ X3 @ one_one_real )
% 0.30/0.66          = X3 )
% 0.30/0.66        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_one_gt_zero_iff
% 0.30/0.66  thf(fact_9002_powr__le__cancel__iff,axiom,
% 0.30/0.66      ! [X3: real,A: real,B2: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
% 0.30/0.66          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_le_cancel_iff
% 0.30/0.66  thf(fact_9003_numeral__powr__numeral__real,axiom,
% 0.30/0.66      ! [M: num,N: num] :
% 0.30/0.66        ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.30/0.66        = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % numeral_powr_numeral_real
% 0.30/0.66  thf(fact_9004_powr__log__cancel,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( powr_real @ A @ ( log @ A @ X3 ) )
% 0.30/0.66              = X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_log_cancel
% 0.30/0.66  thf(fact_9005_log__powr__cancel,axiom,
% 0.30/0.66      ! [A: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 0.30/0.66            = Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_powr_cancel
% 0.30/0.66  thf(fact_9006_floor__divide__eq__div__numeral,axiom,
% 0.30/0.66      ! [A: num,B2: num] :
% 0.30/0.66        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 0.30/0.66        = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_divide_eq_div_numeral
% 0.30/0.66  thf(fact_9007_powr__numeral,axiom,
% 0.30/0.66      ! [X3: real,N: num] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( powr_real @ X3 @ ( numeral_numeral_real @ N ) )
% 0.30/0.66          = ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_numeral
% 0.30/0.66  thf(fact_9008_floor__one__divide__eq__div__numeral,axiom,
% 0.30/0.66      ! [B2: num] :
% 0.30/0.66        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) )
% 0.30/0.66        = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_one_divide_eq_div_numeral
% 0.30/0.66  thf(fact_9009_floor__minus__divide__eq__div__numeral,axiom,
% 0.30/0.66      ! [A: num,B2: num] :
% 0.30/0.66        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 0.30/0.66        = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_minus_divide_eq_div_numeral
% 0.30/0.66  thf(fact_9010_square__powr__half,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( powr_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66        = ( abs_abs_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % square_powr_half
% 0.30/0.66  thf(fact_9011_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.30/0.66      ! [B2: num] :
% 0.30/0.66        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) ) )
% 0.30/0.66        = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_minus_one_divide_eq_div_numeral
% 0.30/0.66  thf(fact_9012_powr__powr,axiom,
% 0.30/0.66      ! [X3: real,A: real,B2: real] :
% 0.30/0.66        ( ( powr_real @ ( powr_real @ X3 @ A ) @ B2 )
% 0.30/0.66        = ( powr_real @ X3 @ ( times_times_real @ A @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_powr
% 0.30/0.66  thf(fact_9013_powr__less__mono2__neg,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.66           => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_less_mono2_neg
% 0.30/0.66  thf(fact_9014_powr__non__neg,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_non_neg
% 0.30/0.66  thf(fact_9015_powr__ge__pzero,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X3 @ Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_ge_pzero
% 0.30/0.66  thf(fact_9016_powr__mono2,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.66           => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_mono2
% 0.30/0.66  thf(fact_9017_powr__less__cancel,axiom,
% 0.30/0.66      ! [X3: real,A: real,B2: real] :
% 0.30/0.66        ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66         => ( ord_less_real @ A @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_less_cancel
% 0.30/0.66  thf(fact_9018_powr__less__mono,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66         => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_less_mono
% 0.30/0.66  thf(fact_9019_powr__mono,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.66       => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.66         => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_mono
% 0.30/0.66  thf(fact_9020_powr__less__mono2,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.66           => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_less_mono2
% 0.30/0.66  thf(fact_9021_powr__mono2_H,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.66           => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_mono2'
% 0.30/0.66  thf(fact_9022_powr__inj,axiom,
% 0.30/0.66      ! [A: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( A != one_one_real )
% 0.30/0.66         => ( ( ( powr_real @ A @ X3 )
% 0.30/0.66              = ( powr_real @ A @ Y3 ) )
% 0.30/0.66            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_inj
% 0.30/0.66  thf(fact_9023_gr__one__powr,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.66         => ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gr_one_powr
% 0.30/0.66  thf(fact_9024_powr__le1,axiom,
% 0.30/0.66      ! [A: real,X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.66           => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ one_one_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_le1
% 0.30/0.66  thf(fact_9025_powr__mono__both,axiom,
% 0.30/0.66      ! [A: real,B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.66       => ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.66         => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.66           => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.66             => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ B2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_mono_both
% 0.30/0.66  thf(fact_9026_ge__one__powr__ge__zero,axiom,
% 0.30/0.66      ! [X3: real,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.66         => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X3 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ge_one_powr_ge_zero
% 0.30/0.66  thf(fact_9027_powr__divide,axiom,
% 0.30/0.66      ! [X3: real,Y3: real,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.66         => ( ( powr_real @ ( divide_divide_real @ X3 @ Y3 ) @ A )
% 0.30/0.66            = ( divide_divide_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_divide
% 0.30/0.66  thf(fact_9028_powr__mult,axiom,
% 0.30/0.66      ! [X3: real,Y3: real,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.66         => ( ( powr_real @ ( times_times_real @ X3 @ Y3 ) @ A )
% 0.30/0.66            = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_mult
% 0.30/0.66  thf(fact_9029_inverse__powr,axiom,
% 0.30/0.66      ! [Y3: real,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.66       => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 0.30/0.66          = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inverse_powr
% 0.30/0.66  thf(fact_9030_divide__powr__uminus,axiom,
% 0.30/0.66      ! [A: real,B2: real,C: real] :
% 0.30/0.66        ( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
% 0.30/0.66        = ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_powr_uminus
% 0.30/0.66  thf(fact_9031_ln__powr,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( X3 != zero_zero_real )
% 0.30/0.66       => ( ( ln_ln_real @ ( powr_real @ X3 @ Y3 ) )
% 0.30/0.66          = ( times_times_real @ Y3 @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_powr
% 0.30/0.66  thf(fact_9032_log__powr,axiom,
% 0.30/0.66      ! [X3: real,B2: real,Y3: real] :
% 0.30/0.66        ( ( X3 != zero_zero_real )
% 0.30/0.66       => ( ( log @ B2 @ ( powr_real @ X3 @ Y3 ) )
% 0.30/0.66          = ( times_times_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_powr
% 0.30/0.66  thf(fact_9033_floor__log__eq__powr__iff,axiom,
% 0.30/0.66      ! [X3: real,B2: real,K: int] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66         => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X3 ) )
% 0.30/0.66              = K )
% 0.30/0.66            = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X3 )
% 0.30/0.66              & ( ord_less_real @ X3 @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_log_eq_powr_iff
% 0.30/0.66  thf(fact_9034_powr__realpow,axiom,
% 0.30/0.66      ! [X3: real,N: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
% 0.30/0.66          = ( power_power_real @ X3 @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_realpow
% 0.30/0.66  thf(fact_9035_powr__less__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X3 )
% 0.30/0.66            = ( ord_less_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_less_iff
% 0.30/0.66  thf(fact_9036_less__powr__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ X3 @ ( powr_real @ B2 @ Y3 ) )
% 0.30/0.66            = ( ord_less_real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_powr_iff
% 0.30/0.66  thf(fact_9037_log__less__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ ( log @ B2 @ X3 ) @ Y3 )
% 0.30/0.66            = ( ord_less_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_less_iff
% 0.30/0.66  thf(fact_9038_less__log__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_real @ Y3 @ ( log @ B2 @ X3 ) )
% 0.30/0.66            = ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_log_iff
% 0.30/0.66  thf(fact_9039_floor__eq,axiom,
% 0.30/0.66      ! [N: int,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.30/0.66         => ( ( archim6058952711729229775r_real @ X3 )
% 0.30/0.66            = N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_eq
% 0.30/0.66  thf(fact_9040_real__of__int__floor__add__one__gt,axiom,
% 0.30/0.66      ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_floor_add_one_gt
% 0.30/0.66  thf(fact_9041_real__of__int__floor__add__one__ge,axiom,
% 0.30/0.66      ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_floor_add_one_ge
% 0.30/0.66  thf(fact_9042_real__of__int__floor__gt__diff__one,axiom,
% 0.30/0.66      ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_floor_gt_diff_one
% 0.30/0.66  thf(fact_9043_real__of__int__floor__ge__diff__one,axiom,
% 0.30/0.66      ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_of_int_floor_ge_diff_one
% 0.30/0.66  thf(fact_9044_powr__neg__one,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.66          = ( divide_divide_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_neg_one
% 0.30/0.66  thf(fact_9045_powr__mult__base,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( times_times_real @ X3 @ ( powr_real @ X3 @ Y3 ) )
% 0.30/0.66          = ( powr_real @ X3 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_mult_base
% 0.30/0.66  thf(fact_9046_le__log__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_log_iff
% 0.30/0.66  thf(fact_9047_log__le__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( log @ B2 @ X3 ) @ Y3 )
% 0.30/0.66            = ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_le_iff
% 0.30/0.66  thf(fact_9048_le__powr__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ Y3 ) )
% 0.30/0.66            = ( ord_less_eq_real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_powr_iff
% 0.30/0.66  thf(fact_9049_powr__le__iff,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X3 )
% 0.30/0.66            = ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_le_iff
% 0.30/0.66  thf(fact_9050_floor__eq2,axiom,
% 0.30/0.66      ! [N: int,X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.30/0.66         => ( ( archim6058952711729229775r_real @ X3 )
% 0.30/0.66            = N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_eq2
% 0.30/0.66  thf(fact_9051_floor__divide__real__eq__div,axiom,
% 0.30/0.66      ! [B2: int,A: real] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 0.30/0.66       => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B2 ) ) )
% 0.30/0.66          = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_divide_real_eq_div
% 0.30/0.66  thf(fact_9052_ln__powr__bound,axiom,
% 0.30/0.66      ! [X3: real,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66         => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_powr_bound
% 0.30/0.66  thf(fact_9053_ln__powr__bound2,axiom,
% 0.30/0.66      ! [X3: real,A: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.66         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_powr_bound2
% 0.30/0.66  thf(fact_9054_log__add__eq__powr,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.66       => ( ( B2 != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( plus_plus_real @ ( log @ B2 @ X3 ) @ Y3 )
% 0.30/0.66              = ( log @ B2 @ ( times_times_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_add_eq_powr
% 0.30/0.66  thf(fact_9055_add__log__eq__powr,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.66       => ( ( B2 != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( plus_plus_real @ Y3 @ ( log @ B2 @ X3 ) )
% 0.30/0.66              = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % add_log_eq_powr
% 0.30/0.66  thf(fact_9056_minus__log__eq__powr,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.66       => ( ( B2 != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( minus_minus_real @ Y3 @ ( log @ B2 @ X3 ) )
% 0.30/0.66              = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % minus_log_eq_powr
% 0.30/0.66  thf(fact_9057_log__minus__eq__powr,axiom,
% 0.30/0.66      ! [B2: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.66       => ( ( B2 != one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( ( minus_minus_real @ ( log @ B2 @ X3 ) @ Y3 )
% 0.30/0.66              = ( log @ B2 @ ( times_times_real @ X3 @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_minus_eq_powr
% 0.30/0.66  thf(fact_9058_powr__half__sqrt,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66          = ( sqrt @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_half_sqrt
% 0.30/0.66  thf(fact_9059_powr__neg__numeral,axiom,
% 0.30/0.66      ! [X3: real,N: num] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.30/0.66          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_neg_numeral
% 0.30/0.66  thf(fact_9060_floor__log2__div2,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.66       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.66          = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_log2_div2
% 0.30/0.66  thf(fact_9061_floor__log__nat__eq__if,axiom,
% 0.30/0.66      ! [B2: nat,N: nat,K: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 0.30/0.66       => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.30/0.66         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 0.30/0.66           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.66              = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_log_nat_eq_if
% 0.30/0.66  thf(fact_9062_upto_Opinduct,axiom,
% 0.30/0.66      ! [A0: int,A12: int,P: int > int > $o] :
% 0.30/0.66        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.30/0.66       => ( ! [I3: int,J2: int] :
% 0.30/0.66              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 0.30/0.66             => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 0.30/0.66                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 0.30/0.66               => ( P @ I3 @ J2 ) ) )
% 0.30/0.66         => ( P @ A0 @ A12 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto.pinduct
% 0.30/0.66  thf(fact_9063_exp__pi__i,axiom,
% 0.30/0.66      ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.30/0.66      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % exp_pi_i
% 0.30/0.66  thf(fact_9064_exp__pi__i_H,axiom,
% 0.30/0.66      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.30/0.66      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % exp_pi_i'
% 0.30/0.66  thf(fact_9065_exp__two__pi__i,axiom,
% 0.30/0.66      ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.30/0.66      = one_one_complex ) ).
% 0.30/0.66  
% 0.30/0.66  % exp_two_pi_i
% 0.30/0.66  thf(fact_9066_exp__two__pi__i_H,axiom,
% 0.30/0.66      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66      = one_one_complex ) ).
% 0.30/0.66  
% 0.30/0.66  % exp_two_pi_i'
% 0.30/0.66  thf(fact_9067_complex__exp__exists,axiom,
% 0.30/0.66      ! [Z: complex] :
% 0.30/0.66      ? [A3: complex,R3: real] :
% 0.30/0.66        ( Z
% 0.30/0.66        = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_exp_exists
% 0.30/0.66  thf(fact_9068_complex__of__real__mult__Complex,axiom,
% 0.30/0.66      ! [R: real,X3: real,Y3: real] :
% 0.30/0.66        ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X3 @ Y3 ) )
% 0.30/0.66        = ( complex2 @ ( times_times_real @ R @ X3 ) @ ( times_times_real @ R @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_of_real_mult_Complex
% 0.30/0.66  thf(fact_9069_Complex__mult__complex__of__real,axiom,
% 0.30/0.66      ! [X3: real,Y3: real,R: real] :
% 0.30/0.66        ( ( times_times_complex @ ( complex2 @ X3 @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.30/0.66        = ( complex2 @ ( times_times_real @ X3 @ R ) @ ( times_times_real @ Y3 @ R ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Complex_mult_complex_of_real
% 0.30/0.66  thf(fact_9070_cis__conv__exp,axiom,
% 0.30/0.66      ( cis
% 0.30/0.66      = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_conv_exp
% 0.30/0.66  thf(fact_9071_i__complex__of__real,axiom,
% 0.30/0.66      ! [R: real] :
% 0.30/0.66        ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
% 0.30/0.66        = ( complex2 @ zero_zero_real @ R ) ) ).
% 0.30/0.66  
% 0.30/0.66  % i_complex_of_real
% 0.30/0.66  thf(fact_9072_complex__of__real__i,axiom,
% 0.30/0.66      ! [R: real] :
% 0.30/0.66        ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
% 0.30/0.66        = ( complex2 @ zero_zero_real @ R ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_of_real_i
% 0.30/0.66  thf(fact_9073_Complex__eq,axiom,
% 0.30/0.66      ( complex2
% 0.30/0.66      = ( ^ [A5: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A5 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Complex_eq
% 0.30/0.66  thf(fact_9074_complex__split__polar,axiom,
% 0.30/0.66      ! [Z: complex] :
% 0.30/0.66      ? [R3: real,A3: real] :
% 0.30/0.66        ( Z
% 0.30/0.66        = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % complex_split_polar
% 0.30/0.66  thf(fact_9075_cmod__unit__one,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.30/0.66        = one_one_real ) ).
% 0.30/0.66  
% 0.30/0.66  % cmod_unit_one
% 0.30/0.66  thf(fact_9076_cmod__complex__polar,axiom,
% 0.30/0.66      ! [R: real,A: real] :
% 0.30/0.66        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 0.30/0.66        = ( abs_abs_real @ R ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cmod_complex_polar
% 0.30/0.66  thf(fact_9077_csqrt__ii,axiom,
% 0.30/0.66      ( ( csqrt @ imaginary_unit )
% 0.30/0.66      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % csqrt_ii
% 0.30/0.66  thf(fact_9078_arctan__def,axiom,
% 0.30/0.66      ( arctan
% 0.30/0.66      = ( ^ [Y: real] :
% 0.30/0.66            ( the_real
% 0.30/0.66            @ ^ [X: real] :
% 0.30/0.66                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.30/0.66                & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66                & ( ( tan_real @ X )
% 0.30/0.66                  = Y ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arctan_def
% 0.30/0.66  thf(fact_9079_arcsin__def,axiom,
% 0.30/0.66      ( arcsin
% 0.30/0.66      = ( ^ [Y: real] :
% 0.30/0.66            ( the_real
% 0.30/0.66            @ ^ [X: real] :
% 0.30/0.66                ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.30/0.66                & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.66                & ( ( sin_real @ X )
% 0.30/0.66                  = Y ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcsin_def
% 0.30/0.66  thf(fact_9080_even__set__encode__iff,axiom,
% 0.30/0.66      ! [A4: set_nat] :
% 0.30/0.66        ( ( finite_finite_nat @ A4 )
% 0.30/0.66       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
% 0.30/0.66          = ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % even_set_encode_iff
% 0.30/0.66  thf(fact_9081_power2__csqrt,axiom,
% 0.30/0.66      ! [Z: complex] :
% 0.30/0.66        ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.66        = Z ) ).
% 0.30/0.66  
% 0.30/0.66  % power2_csqrt
% 0.30/0.66  thf(fact_9082_ln__neg__is__const,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.66       => ( ( ln_ln_real @ X3 )
% 0.30/0.66          = ( the_real
% 0.30/0.66            @ ^ [X: real] : $false ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ln_neg_is_const
% 0.30/0.66  thf(fact_9083_of__real__sqrt,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( real_V4546457046886955230omplex @ ( sqrt @ X3 ) )
% 0.30/0.66          = ( csqrt @ ( real_V4546457046886955230omplex @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % of_real_sqrt
% 0.30/0.66  thf(fact_9084_arccos__def,axiom,
% 0.30/0.66      ( arccos
% 0.30/0.66      = ( ^ [Y: real] :
% 0.30/0.66            ( the_real
% 0.30/0.66            @ ^ [X: real] :
% 0.30/0.66                ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.66                & ( ord_less_eq_real @ X @ pi )
% 0.30/0.66                & ( ( cos_real @ X )
% 0.30/0.66                  = Y ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_def
% 0.30/0.66  thf(fact_9085_set__encode__def,axiom,
% 0.30/0.66      ( nat_set_encode
% 0.30/0.66      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % set_encode_def
% 0.30/0.66  thf(fact_9086_pi__half,axiom,
% 0.30/0.66      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.66      = ( the_real
% 0.30/0.66        @ ^ [X: real] :
% 0.30/0.66            ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.66            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.66            & ( ( cos_real @ X )
% 0.30/0.66              = zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pi_half
% 0.30/0.66  thf(fact_9087_pi__def,axiom,
% 0.30/0.66      ( pi
% 0.30/0.66      = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.66        @ ( the_real
% 0.30/0.66          @ ^ [X: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.66              & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.66              & ( ( cos_real @ X )
% 0.30/0.66                = zero_zero_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pi_def
% 0.30/0.66  thf(fact_9088_modulo__int__unfold,axiom,
% 0.30/0.66      ! [L: int,K: int,N: nat,M: nat] :
% 0.30/0.66        ( ( ( ( ( sgn_sgn_int @ L )
% 0.30/0.66              = zero_zero_int )
% 0.30/0.66            | ( ( sgn_sgn_int @ K )
% 0.30/0.66              = zero_zero_int )
% 0.30/0.66            | ( N = zero_zero_nat ) )
% 0.30/0.66         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.66            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.30/0.66        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.30/0.66                = zero_zero_int )
% 0.30/0.66              | ( ( sgn_sgn_int @ K )
% 0.30/0.66                = zero_zero_int )
% 0.30/0.66              | ( N = zero_zero_nat ) )
% 0.30/0.66         => ( ( ( ( sgn_sgn_int @ K )
% 0.30/0.66                = ( sgn_sgn_int @ L ) )
% 0.30/0.66             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.66                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.30/0.66            & ( ( ( sgn_sgn_int @ K )
% 0.30/0.66               != ( sgn_sgn_int @ L ) )
% 0.30/0.66             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.66                = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.30/0.66                  @ ( minus_minus_int
% 0.30/0.66                    @ ( semiri1314217659103216013at_int
% 0.30/0.66                      @ ( times_times_nat @ N
% 0.30/0.66                        @ ( zero_n2687167440665602831ol_nat
% 0.30/0.66                          @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.30/0.66                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % modulo_int_unfold
% 0.30/0.66  thf(fact_9089_num_Osize__gen_I3_J,axiom,
% 0.30/0.66      ! [X32: num] :
% 0.30/0.66        ( ( size_num @ ( bit1 @ X32 ) )
% 0.30/0.66        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % num.size_gen(3)
% 0.30/0.66  thf(fact_9090_powr__int,axiom,
% 0.30/0.66      ! [X3: real,I: int] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.66           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 0.30/0.66              = ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.66           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 0.30/0.66              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_int
% 0.30/0.66  thf(fact_9091_mask__nat__positive__iff,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.30/0.66        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mask_nat_positive_iff
% 0.30/0.66  thf(fact_9092_nat__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.30/0.66        = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_numeral
% 0.30/0.66  thf(fact_9093_nat__0__iff,axiom,
% 0.30/0.66      ! [I: int] :
% 0.30/0.66        ( ( ( nat2 @ I )
% 0.30/0.66          = zero_zero_nat )
% 0.30/0.66        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_0_iff
% 0.30/0.66  thf(fact_9094_nat__le__0,axiom,
% 0.30/0.66      ! [Z: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.30/0.66       => ( ( nat2 @ Z )
% 0.30/0.66          = zero_zero_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_le_0
% 0.30/0.66  thf(fact_9095_zless__nat__conj,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.66        = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.66          & ( ord_less_int @ W @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zless_nat_conj
% 0.30/0.66  thf(fact_9096_nat__neg__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.66        = zero_zero_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_neg_numeral
% 0.30/0.66  thf(fact_9097_int__nat__eq,axiom,
% 0.30/0.66      ! [Z: int] :
% 0.30/0.66        ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.30/0.66            = Z ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.30/0.66            = zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_nat_eq
% 0.30/0.66  thf(fact_9098_sgn__mult__dvd__iff,axiom,
% 0.30/0.66      ! [R: int,L: int,K: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L ) @ K )
% 0.30/0.66        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.66          & ( ( R = zero_zero_int )
% 0.30/0.66           => ( K = zero_zero_int ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sgn_mult_dvd_iff
% 0.30/0.66  thf(fact_9099_mult__sgn__dvd__iff,axiom,
% 0.30/0.66      ! [L: int,R: int,K: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R ) ) @ K )
% 0.30/0.66        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.66          & ( ( R = zero_zero_int )
% 0.30/0.66           => ( K = zero_zero_int ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_sgn_dvd_iff
% 0.30/0.66  thf(fact_9100_dvd__sgn__mult__iff,axiom,
% 0.30/0.66      ! [L: int,R: int,K: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
% 0.30/0.66        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.66          | ( R = zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % dvd_sgn_mult_iff
% 0.30/0.66  thf(fact_9101_dvd__mult__sgn__iff,axiom,
% 0.30/0.66      ! [L: int,K: int,R: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
% 0.30/0.66        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.66          | ( R = zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % dvd_mult_sgn_iff
% 0.30/0.66  thf(fact_9102_zero__less__nat__eq,axiom,
% 0.30/0.66      ! [Z: int] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.30/0.66        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zero_less_nat_eq
% 0.30/0.66  thf(fact_9103_diff__nat__numeral,axiom,
% 0.30/0.66      ! [V: num,V3: num] :
% 0.30/0.66        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 0.30/0.66        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % diff_nat_numeral
% 0.30/0.66  thf(fact_9104_nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.66      ! [Y3: int,X3: num,N: nat] :
% 0.30/0.66        ( ( ( nat2 @ Y3 )
% 0.30/0.66          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 0.30/0.66        = ( Y3
% 0.30/0.66          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_eq_numeral_power_cancel_iff
% 0.30/0.66  thf(fact_9105_numeral__power__eq__nat__cancel__iff,axiom,
% 0.30/0.66      ! [X3: num,N: nat,Y3: int] :
% 0.30/0.66        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 0.30/0.66          = ( nat2 @ Y3 ) )
% 0.30/0.66        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
% 0.30/0.66          = Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % numeral_power_eq_nat_cancel_iff
% 0.30/0.66  thf(fact_9106_nat__ceiling__le__eq,axiom,
% 0.30/0.66      ! [X3: real,A: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
% 0.30/0.66        = ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_ceiling_le_eq
% 0.30/0.66  thf(fact_9107_one__less__nat__eq,axiom,
% 0.30/0.66      ! [Z: int] :
% 0.30/0.66        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.30/0.66        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % one_less_nat_eq
% 0.30/0.66  thf(fact_9108_nat__numeral__diff__1,axiom,
% 0.30/0.66      ! [V: num] :
% 0.30/0.66        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.30/0.66        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_numeral_diff_1
% 0.30/0.66  thf(fact_9109_nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.66      ! [A: int,X3: num,N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 0.30/0.66        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_less_numeral_power_cancel_iff
% 0.30/0.66  thf(fact_9110_numeral__power__less__nat__cancel__iff,axiom,
% 0.30/0.66      ! [X3: num,N: nat,A: int] :
% 0.30/0.66        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
% 0.30/0.66        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % numeral_power_less_nat_cancel_iff
% 0.30/0.66  thf(fact_9111_nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.66      ! [A: int,X3: num,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 0.30/0.66        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_le_numeral_power_cancel_iff
% 0.30/0.66  thf(fact_9112_numeral__power__le__nat__cancel__iff,axiom,
% 0.30/0.66      ! [X3: num,N: nat,A: int] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
% 0.30/0.66        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% 0.30/0.66  
% 0.30/0.66  % numeral_power_le_nat_cancel_iff
% 0.30/0.66  thf(fact_9113_less__eq__mask,axiom,
% 0.30/0.66      ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_eq_mask
% 0.30/0.66  thf(fact_9114_nat__mask__eq,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.30/0.66        = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mask_eq
% 0.30/0.66  thf(fact_9115_nat__zero__as__int,axiom,
% 0.30/0.66      ( zero_zero_nat
% 0.30/0.66      = ( nat2 @ zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_zero_as_int
% 0.30/0.66  thf(fact_9116_nat__numeral__as__int,axiom,
% 0.30/0.66      ( numeral_numeral_nat
% 0.30/0.66      = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_numeral_as_int
% 0.30/0.66  thf(fact_9117_nat__mono,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.66       => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mono
% 0.30/0.66  thf(fact_9118_int__sgnE,axiom,
% 0.30/0.66      ! [K: int] :
% 0.30/0.66        ~ ! [N3: nat,L4: int] :
% 0.30/0.66            ( K
% 0.30/0.66           != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_sgnE
% 0.30/0.66  thf(fact_9119_ex__nat,axiom,
% 0.30/0.66      ( ( ^ [P2: nat > $o] :
% 0.30/0.66          ? [X4: nat] : ( P2 @ X4 ) )
% 0.30/0.66      = ( ^ [P3: nat > $o] :
% 0.30/0.66          ? [X: int] :
% 0.30/0.66            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.66            & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % ex_nat
% 0.30/0.66  thf(fact_9120_all__nat,axiom,
% 0.30/0.66      ( ( ^ [P2: nat > $o] :
% 0.30/0.66          ! [X4: nat] : ( P2 @ X4 ) )
% 0.30/0.66      = ( ^ [P3: nat > $o] :
% 0.30/0.66          ! [X: int] :
% 0.30/0.66            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.66           => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % all_nat
% 0.30/0.66  thf(fact_9121_eq__nat__nat__iff,axiom,
% 0.30/0.66      ! [Z: int,Z7: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.30/0.66         => ( ( ( nat2 @ Z )
% 0.30/0.66              = ( nat2 @ Z7 ) )
% 0.30/0.66            = ( Z = Z7 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eq_nat_nat_iff
% 0.30/0.66  thf(fact_9122_nat__one__as__int,axiom,
% 0.30/0.66      ( one_one_nat
% 0.30/0.66      = ( nat2 @ one_one_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_one_as_int
% 0.30/0.66  thf(fact_9123_unset__bit__nat__def,axiom,
% 0.30/0.66      ( bit_se4205575877204974255it_nat
% 0.30/0.66      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % unset_bit_nat_def
% 0.30/0.66  thf(fact_9124_mask__nonnegative__int,axiom,
% 0.30/0.66      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mask_nonnegative_int
% 0.30/0.66  thf(fact_9125_not__mask__negative__int,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.30/0.66  
% 0.30/0.66  % not_mask_negative_int
% 0.30/0.66  thf(fact_9126_nat__mono__iff,axiom,
% 0.30/0.66      ! [Z: int,W: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.66          = ( ord_less_int @ W @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mono_iff
% 0.30/0.66  thf(fact_9127_zless__nat__eq__int__zless,axiom,
% 0.30/0.66      ! [M: nat,Z: int] :
% 0.30/0.66        ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.30/0.66        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zless_nat_eq_int_zless
% 0.30/0.66  thf(fact_9128_nat__le__iff,axiom,
% 0.30/0.66      ! [X3: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
% 0.30/0.66        = ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_le_iff
% 0.30/0.66  thf(fact_9129_nat__0__le,axiom,
% 0.30/0.66      ! [Z: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.30/0.66          = Z ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_0_le
% 0.30/0.66  thf(fact_9130_int__eq__iff,axiom,
% 0.30/0.66      ! [M: nat,Z: int] :
% 0.30/0.66        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.66          = Z )
% 0.30/0.66        = ( ( M
% 0.30/0.66            = ( nat2 @ Z ) )
% 0.30/0.66          & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_eq_iff
% 0.30/0.66  thf(fact_9131_nat__int__add,axiom,
% 0.30/0.66      ! [A: nat,B2: nat] :
% 0.30/0.66        ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
% 0.30/0.66        = ( plus_plus_nat @ A @ B2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_int_add
% 0.30/0.66  thf(fact_9132_sgn__mod,axiom,
% 0.30/0.66      ! [L: int,K: int] :
% 0.30/0.66        ( ( L != zero_zero_int )
% 0.30/0.66       => ( ~ ( dvd_dvd_int @ L @ K )
% 0.30/0.66         => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 0.30/0.66            = ( sgn_sgn_int @ L ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sgn_mod
% 0.30/0.66  thf(fact_9133_int__minus,axiom,
% 0.30/0.66      ! [N: nat,M: nat] :
% 0.30/0.66        ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.30/0.66        = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_minus
% 0.30/0.66  thf(fact_9134_nat__abs__mult__distrib,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.30/0.66        = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_abs_mult_distrib
% 0.30/0.66  thf(fact_9135_nat__plus__as__int,axiom,
% 0.30/0.66      ( plus_plus_nat
% 0.30/0.66      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_plus_as_int
% 0.30/0.66  thf(fact_9136_nat__times__as__int,axiom,
% 0.30/0.66      ( times_times_nat
% 0.30/0.66      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_times_as_int
% 0.30/0.66  thf(fact_9137_real__nat__ceiling__ge,axiom,
% 0.30/0.66      ! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % real_nat_ceiling_ge
% 0.30/0.66  thf(fact_9138_less__mask,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.66       => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_mask
% 0.30/0.66  thf(fact_9139_nat__minus__as__int,axiom,
% 0.30/0.66      ( minus_minus_nat
% 0.30/0.66      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_minus_as_int
% 0.30/0.66  thf(fact_9140_nat__div__as__int,axiom,
% 0.30/0.66      ( divide_divide_nat
% 0.30/0.66      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_div_as_int
% 0.30/0.66  thf(fact_9141_nat__mod__as__int,axiom,
% 0.30/0.66      ( modulo_modulo_nat
% 0.30/0.66      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mod_as_int
% 0.30/0.66  thf(fact_9142_zsgn__def,axiom,
% 0.30/0.66      ( sgn_sgn_int
% 0.30/0.66      = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zsgn_def
% 0.30/0.66  thf(fact_9143_nat__less__eq__zless,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.66       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.66          = ( ord_less_int @ W @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_less_eq_zless
% 0.30/0.66  thf(fact_9144_nat__le__eq__zle,axiom,
% 0.30/0.66      ! [W: int,Z: int] :
% 0.30/0.66        ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.30/0.66          | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.30/0.66       => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.30/0.66          = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_le_eq_zle
% 0.30/0.66  thf(fact_9145_nat__eq__iff,axiom,
% 0.30/0.66      ! [W: int,M: nat] :
% 0.30/0.66        ( ( ( nat2 @ W )
% 0.30/0.66          = M )
% 0.30/0.66        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.66           => ( W
% 0.30/0.66              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.66           => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_eq_iff
% 0.30/0.66  thf(fact_9146_nat__eq__iff2,axiom,
% 0.30/0.66      ! [M: nat,W: int] :
% 0.30/0.66        ( ( M
% 0.30/0.66          = ( nat2 @ W ) )
% 0.30/0.66        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.66           => ( W
% 0.30/0.66              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.66           => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_eq_iff2
% 0.30/0.66  thf(fact_9147_split__nat,axiom,
% 0.30/0.66      ! [P: nat > $o,I: int] :
% 0.30/0.66        ( ( P @ ( nat2 @ I ) )
% 0.30/0.66        = ( ! [N2: nat] :
% 0.30/0.66              ( ( I
% 0.30/0.66                = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.66             => ( P @ N2 ) )
% 0.30/0.66          & ( ( ord_less_int @ I @ zero_zero_int )
% 0.30/0.66           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % split_nat
% 0.30/0.66  thf(fact_9148_le__nat__iff,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66       => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.30/0.66          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_nat_iff
% 0.30/0.66  thf(fact_9149_nat__add__distrib,axiom,
% 0.30/0.66      ! [Z: int,Z7: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.30/0.66         => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 0.30/0.66            = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_add_distrib
% 0.30/0.66  thf(fact_9150_div__sgn__abs__cancel,axiom,
% 0.30/0.66      ! [V: int,K: int,L: int] :
% 0.30/0.66        ( ( V != zero_zero_int )
% 0.30/0.66       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 0.30/0.66          = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % div_sgn_abs_cancel
% 0.30/0.66  thf(fact_9151_nat__mult__distrib,axiom,
% 0.30/0.66      ! [Z: int,Z7: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.30/0.66          = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mult_distrib
% 0.30/0.66  thf(fact_9152_Suc__as__int,axiom,
% 0.30/0.66      ( suc
% 0.30/0.66      = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_as_int
% 0.30/0.66  thf(fact_9153_nat__diff__distrib_H,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66         => ( ( nat2 @ ( minus_minus_int @ X3 @ Y3 ) )
% 0.30/0.66            = ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_diff_distrib'
% 0.30/0.66  thf(fact_9154_nat__diff__distrib,axiom,
% 0.30/0.66      ! [Z7: int,Z: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.30/0.66       => ( ( ord_less_eq_int @ Z7 @ Z )
% 0.30/0.66         => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 0.30/0.66            = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_diff_distrib
% 0.30/0.66  thf(fact_9155_nat__abs__triangle__ineq,axiom,
% 0.30/0.66      ! [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 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_abs_triangle_ineq
% 0.30/0.66  thf(fact_9156_nat__div__distrib_H,axiom,
% 0.30/0.66      ! [Y3: int,X3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66       => ( ( nat2 @ ( divide_divide_int @ X3 @ Y3 ) )
% 0.30/0.66          = ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_div_distrib'
% 0.30/0.66  thf(fact_9157_nat__div__distrib,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.66       => ( ( nat2 @ ( divide_divide_int @ X3 @ Y3 ) )
% 0.30/0.66          = ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_div_distrib
% 0.30/0.66  thf(fact_9158_div__dvd__sgn__abs,axiom,
% 0.30/0.66      ! [L: int,K: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ L @ K )
% 0.30/0.66       => ( ( divide_divide_int @ K @ L )
% 0.30/0.66          = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % div_dvd_sgn_abs
% 0.30/0.66  thf(fact_9159_nat__power__eq,axiom,
% 0.30/0.66      ! [Z: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.30/0.66          = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_power_eq
% 0.30/0.66  thf(fact_9160_nat__floor__neg,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.66       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.66          = zero_zero_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_floor_neg
% 0.30/0.66  thf(fact_9161_nat__mod__distrib,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66         => ( ( nat2 @ ( modulo_modulo_int @ X3 @ Y3 ) )
% 0.30/0.66            = ( modulo_modulo_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mod_distrib
% 0.30/0.66  thf(fact_9162_div__abs__eq__div__nat,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 0.30/0.66        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % div_abs_eq_div_nat
% 0.30/0.66  thf(fact_9163_floor__eq3,axiom,
% 0.30/0.66      ! [N: nat,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.30/0.66         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.66            = N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_eq3
% 0.30/0.66  thf(fact_9164_le__nat__floor,axiom,
% 0.30/0.66      ! [X3: nat,A: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
% 0.30/0.66       => ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_nat_floor
% 0.30/0.66  thf(fact_9165_divide__int__def,axiom,
% 0.30/0.66      ( divide_divide_int
% 0.30/0.66      = ( ^ [K2: int,L2: int] :
% 0.30/0.66            ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 0.30/0.66            @ ( if_int
% 0.30/0.66              @ ( ( sgn_sgn_int @ K2 )
% 0.30/0.66                = ( sgn_sgn_int @ L2 ) )
% 0.30/0.66              @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 0.30/0.66              @ ( uminus_uminus_int
% 0.30/0.66                @ ( semiri1314217659103216013at_int
% 0.30/0.66                  @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 0.30/0.66                    @ ( zero_n2687167440665602831ol_nat
% 0.30/0.66                      @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_int_def
% 0.30/0.66  thf(fact_9166_modulo__int__def,axiom,
% 0.30/0.66      ( modulo_modulo_int
% 0.30/0.66      = ( ^ [K2: int,L2: int] :
% 0.30/0.66            ( if_int @ ( L2 = zero_zero_int ) @ K2
% 0.30/0.66            @ ( if_int
% 0.30/0.66              @ ( ( sgn_sgn_int @ K2 )
% 0.30/0.66                = ( sgn_sgn_int @ L2 ) )
% 0.30/0.66              @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 0.30/0.66              @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.30/0.66                @ ( minus_minus_int
% 0.30/0.66                  @ ( times_times_int @ ( abs_abs_int @ L2 )
% 0.30/0.66                    @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                      @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
% 0.30/0.66                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % modulo_int_def
% 0.30/0.66  thf(fact_9167_nat__2,axiom,
% 0.30/0.66      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.66      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_2
% 0.30/0.66  thf(fact_9168_Suc__nat__eq__nat__zadd1,axiom,
% 0.30/0.66      ! [Z: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66       => ( ( suc @ ( nat2 @ Z ) )
% 0.30/0.66          = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_nat_eq_nat_zadd1
% 0.30/0.66  thf(fact_9169_nat__less__iff,axiom,
% 0.30/0.66      ! [W: int,M: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.30/0.66       => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.30/0.66          = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_less_iff
% 0.30/0.66  thf(fact_9170_nat__mult__distrib__neg,axiom,
% 0.30/0.66      ! [Z: int,Z7: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.30/0.66       => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.30/0.66          = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_mult_distrib_neg
% 0.30/0.66  thf(fact_9171_nat__abs__int__diff,axiom,
% 0.30/0.66      ! [A: nat,B2: nat] :
% 0.30/0.66        ( ( ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.66         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 0.30/0.66            = ( minus_minus_nat @ B2 @ A ) ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_nat @ A @ B2 )
% 0.30/0.66         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 0.30/0.66            = ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_abs_int_diff
% 0.30/0.66  thf(fact_9172_floor__eq4,axiom,
% 0.30/0.66      ! [N: nat,X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.30/0.66         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.66            = N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % floor_eq4
% 0.30/0.66  thf(fact_9173_num_Osize__gen_I1_J,axiom,
% 0.30/0.66      ( ( size_num @ one )
% 0.30/0.66      = zero_zero_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % num.size_gen(1)
% 0.30/0.66  thf(fact_9174_diff__nat__eq__if,axiom,
% 0.30/0.66      ! [Z7: int,Z: int] :
% 0.30/0.66        ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 0.30/0.66         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.30/0.66            = ( nat2 @ Z ) ) )
% 0.30/0.66        & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 0.30/0.66         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.30/0.66            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % diff_nat_eq_if
% 0.30/0.66  thf(fact_9175_Suc__mask__eq__exp,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.30/0.66        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_mask_eq_exp
% 0.30/0.66  thf(fact_9176_mask__nat__less__exp,axiom,
% 0.30/0.66      ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mask_nat_less_exp
% 0.30/0.66  thf(fact_9177_nat__dvd__iff,axiom,
% 0.30/0.66      ! [Z: int,M: nat] :
% 0.30/0.66        ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 0.30/0.66        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66           => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.30/0.66           => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_dvd_iff
% 0.30/0.66  thf(fact_9178_eucl__rel__int__remainderI,axiom,
% 0.30/0.66      ! [R: int,L: int,K: int,Q2: int] :
% 0.30/0.66        ( ( ( sgn_sgn_int @ R )
% 0.30/0.66          = ( sgn_sgn_int @ L ) )
% 0.30/0.66       => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
% 0.30/0.66         => ( ( K
% 0.30/0.66              = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R ) )
% 0.30/0.66           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eucl_rel_int_remainderI
% 0.30/0.66  thf(fact_9179_mask__nat__def,axiom,
% 0.30/0.66      ( bit_se2002935070580805687sk_nat
% 0.30/0.66      = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mask_nat_def
% 0.30/0.66  thf(fact_9180_mask__half__int,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.66        = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mask_half_int
% 0.30/0.66  thf(fact_9181_mask__int__def,axiom,
% 0.30/0.66      ( bit_se2000444600071755411sk_int
% 0.30/0.66      = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mask_int_def
% 0.30/0.66  thf(fact_9182_eucl__rel__int_Osimps,axiom,
% 0.30/0.66      ( eucl_rel_int
% 0.30/0.66      = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
% 0.30/0.66            ( ? [K2: int] :
% 0.30/0.66                ( ( A1 = K2 )
% 0.30/0.66                & ( A22 = zero_zero_int )
% 0.30/0.66                & ( A32
% 0.30/0.66                  = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 0.30/0.66            | ? [L2: int,K2: int,Q4: int] :
% 0.30/0.66                ( ( A1 = K2 )
% 0.30/0.66                & ( A22 = L2 )
% 0.30/0.66                & ( A32
% 0.30/0.66                  = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.30/0.66                & ( L2 != zero_zero_int )
% 0.30/0.66                & ( K2
% 0.30/0.66                  = ( times_times_int @ Q4 @ L2 ) ) )
% 0.30/0.66            | ? [R5: int,L2: int,K2: int,Q4: int] :
% 0.30/0.66                ( ( A1 = K2 )
% 0.30/0.66                & ( A22 = L2 )
% 0.30/0.66                & ( A32
% 0.30/0.66                  = ( product_Pair_int_int @ Q4 @ R5 ) )
% 0.30/0.66                & ( ( sgn_sgn_int @ R5 )
% 0.30/0.66                  = ( sgn_sgn_int @ L2 ) )
% 0.30/0.66                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 0.30/0.66                & ( K2
% 0.30/0.66                  = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eucl_rel_int.simps
% 0.30/0.66  thf(fact_9183_eucl__rel__int_Ocases,axiom,
% 0.30/0.66      ! [A12: int,A23: int,A33: product_prod_int_int] :
% 0.30/0.66        ( ( eucl_rel_int @ A12 @ A23 @ A33 )
% 0.30/0.66       => ( ( ( A23 = zero_zero_int )
% 0.30/0.66           => ( A33
% 0.30/0.66             != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 0.30/0.66         => ( ! [Q3: int] :
% 0.30/0.66                ( ( A33
% 0.30/0.66                  = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 0.30/0.66               => ( ( A23 != zero_zero_int )
% 0.30/0.66                 => ( A12
% 0.30/0.66                   != ( times_times_int @ Q3 @ A23 ) ) ) )
% 0.30/0.66           => ~ ! [R3: int,Q3: int] :
% 0.30/0.66                  ( ( A33
% 0.30/0.66                    = ( product_Pair_int_int @ Q3 @ R3 ) )
% 0.30/0.66                 => ( ( ( sgn_sgn_int @ R3 )
% 0.30/0.66                      = ( sgn_sgn_int @ A23 ) )
% 0.30/0.66                   => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 0.30/0.66                     => ( A12
% 0.30/0.66                       != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eucl_rel_int.cases
% 0.30/0.66  thf(fact_9184_even__nat__iff,axiom,
% 0.30/0.66      ! [K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.30/0.66          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % even_nat_iff
% 0.30/0.66  thf(fact_9185_powr__real__of__int,axiom,
% 0.30/0.66      ! [X3: real,N: int] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.30/0.66           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
% 0.30/0.66              = ( power_power_real @ X3 @ ( nat2 @ N ) ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.30/0.66           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
% 0.30/0.66              = ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_real_of_int
% 0.30/0.66  thf(fact_9186_num_Osize__gen_I2_J,axiom,
% 0.30/0.66      ! [X2: num] :
% 0.30/0.66        ( ( size_num @ ( bit0 @ X2 ) )
% 0.30/0.66        = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % num.size_gen(2)
% 0.30/0.66  thf(fact_9187_divide__int__unfold,axiom,
% 0.30/0.66      ! [L: int,K: int,N: nat,M: nat] :
% 0.30/0.66        ( ( ( ( ( sgn_sgn_int @ L )
% 0.30/0.66              = zero_zero_int )
% 0.30/0.66            | ( ( sgn_sgn_int @ K )
% 0.30/0.66              = zero_zero_int )
% 0.30/0.66            | ( N = zero_zero_nat ) )
% 0.30/0.66         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.66            = zero_zero_int ) )
% 0.30/0.66        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.30/0.66                = zero_zero_int )
% 0.30/0.66              | ( ( sgn_sgn_int @ K )
% 0.30/0.66                = zero_zero_int )
% 0.30/0.66              | ( N = zero_zero_nat ) )
% 0.30/0.66         => ( ( ( ( sgn_sgn_int @ K )
% 0.30/0.66                = ( sgn_sgn_int @ L ) )
% 0.30/0.66             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.66                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.30/0.66            & ( ( ( sgn_sgn_int @ K )
% 0.30/0.66               != ( sgn_sgn_int @ L ) )
% 0.30/0.66             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.30/0.66                = ( uminus_uminus_int
% 0.30/0.66                  @ ( semiri1314217659103216013at_int
% 0.30/0.66                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.30/0.66                      @ ( zero_n2687167440665602831ol_nat
% 0.30/0.66                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_int_unfold
% 0.30/0.66  thf(fact_9188_and__int__unfold,axiom,
% 0.30/0.66      ( bit_se725231765392027082nd_int
% 0.30/0.66      = ( ^ [K2: int,L2: int] :
% 0.30/0.66            ( if_int
% 0.30/0.66            @ ( ( K2 = zero_zero_int )
% 0.30/0.66              | ( L2 = zero_zero_int ) )
% 0.30/0.66            @ zero_zero_int
% 0.30/0.66            @ ( if_int
% 0.30/0.66              @ ( K2
% 0.30/0.66                = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.66              @ L2
% 0.30/0.66              @ ( if_int
% 0.30/0.66                @ ( L2
% 0.30/0.66                  = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.66                @ K2
% 0.30/0.66                @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int_unfold
% 0.30/0.66  thf(fact_9189_zero__le__sgn__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X3 ) )
% 0.30/0.66        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zero_le_sgn_iff
% 0.30/0.66  thf(fact_9190_sgn__le__0__iff,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( sgn_sgn_real @ X3 ) @ zero_zero_real )
% 0.30/0.66        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sgn_le_0_iff
% 0.30/0.66  thf(fact_9191_concat__bit__of__zero__2,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.30/0.66        = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % concat_bit_of_zero_2
% 0.30/0.66  thf(fact_9192_and__nonnegative__int__iff,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.30/0.66        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66          | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nonnegative_int_iff
% 0.30/0.66  thf(fact_9193_and__negative__int__iff,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.30/0.66        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.66          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_negative_int_iff
% 0.30/0.66  thf(fact_9194_take__bit__of__Suc__0,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.66        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_of_Suc_0
% 0.30/0.66  thf(fact_9195_and__minus__numerals_I2_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.66        = one_one_int ) ).
% 0.30/0.66  
% 0.30/0.66  % and_minus_numerals(2)
% 0.30/0.66  thf(fact_9196_and__minus__numerals_I6_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.30/0.66        = one_one_int ) ).
% 0.30/0.66  
% 0.30/0.66  % and_minus_numerals(6)
% 0.30/0.66  thf(fact_9197_and__minus__numerals_I5_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.30/0.66        = zero_zero_int ) ).
% 0.30/0.66  
% 0.30/0.66  % and_minus_numerals(5)
% 0.30/0.66  thf(fact_9198_and__minus__numerals_I1_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.66        = zero_zero_int ) ).
% 0.30/0.66  
% 0.30/0.66  % and_minus_numerals(1)
% 0.30/0.66  thf(fact_9199_take__bit__nat__eq,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66       => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.30/0.66          = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_eq
% 0.30/0.66  thf(fact_9200_nat__take__bit__eq,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66       => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.30/0.66          = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_take_bit_eq
% 0.30/0.66  thf(fact_9201_take__bit__nat__less__eq__self,axiom,
% 0.30/0.66      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_less_eq_self
% 0.30/0.66  thf(fact_9202_take__bit__tightened__less__eq__nat,axiom,
% 0.30/0.66      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.66       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_tightened_less_eq_nat
% 0.30/0.66  thf(fact_9203_take__bit__diff,axiom,
% 0.30/0.66      ! [N: nat,K: int,L: int] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.30/0.66        = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_diff
% 0.30/0.66  thf(fact_9204_take__bit__mult,axiom,
% 0.30/0.66      ! [N: nat,K: int,L: int] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 0.30/0.66        = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_mult
% 0.30/0.66  thf(fact_9205_take__bit__minus,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.30/0.66        = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_minus
% 0.30/0.66  thf(fact_9206_concat__bit__take__bit__eq,axiom,
% 0.30/0.66      ! [N: nat,B2: int] :
% 0.30/0.66        ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B2 ) )
% 0.30/0.66        = ( bit_concat_bit @ N @ B2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % concat_bit_take_bit_eq
% 0.30/0.66  thf(fact_9207_concat__bit__eq__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int,L: int,R: int,S: int] :
% 0.30/0.66        ( ( ( bit_concat_bit @ N @ K @ L )
% 0.30/0.66          = ( bit_concat_bit @ N @ R @ S ) )
% 0.30/0.66        = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66            = ( bit_se2923211474154528505it_int @ N @ R ) )
% 0.30/0.66          & ( L = S ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % concat_bit_eq_iff
% 0.30/0.66  thf(fact_9208_take__bit__tightened__less__eq__int,axiom,
% 0.30/0.66      ! [M: nat,N: nat,K: int] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.66       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_tightened_less_eq_int
% 0.30/0.66  thf(fact_9209_AND__lower,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.66       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_lower
% 0.30/0.66  thf(fact_9210_AND__upper1,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.66       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_upper1
% 0.30/0.66  thf(fact_9211_AND__upper2,axiom,
% 0.30/0.66      ! [Y3: int,X3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_upper2
% 0.30/0.66  thf(fact_9212_AND__upper1_H,axiom,
% 0.30/0.66      ! [Y3: int,Z: int,Ya: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.66         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_upper1'
% 0.30/0.66  thf(fact_9213_AND__upper2_H,axiom,
% 0.30/0.66      ! [Y3: int,Z: int,X3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66       => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.30/0.66         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_upper2'
% 0.30/0.66  thf(fact_9214_take__bit__nonnegative,axiom,
% 0.30/0.66      ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nonnegative
% 0.30/0.66  thf(fact_9215_take__bit__int__less__eq__self__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.30/0.66        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_less_eq_self_iff
% 0.30/0.66  thf(fact_9216_not__take__bit__negative,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.30/0.66  
% 0.30/0.66  % not_take_bit_negative
% 0.30/0.66  thf(fact_9217_take__bit__int__greater__self__iff,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.30/0.66        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_greater_self_iff
% 0.30/0.66  thf(fact_9218_pow_Osimps_I1_J,axiom,
% 0.30/0.66      ! [X3: num] :
% 0.30/0.66        ( ( pow @ X3 @ one )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  % pow.simps(1)
% 0.30/0.66  thf(fact_9219_AND__upper2_H_H,axiom,
% 0.30/0.66      ! [Y3: int,Z: int,X3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66       => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.66         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_upper2''
% 0.30/0.66  thf(fact_9220_AND__upper1_H_H,axiom,
% 0.30/0.66      ! [Y3: int,Z: int,Ya: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66       => ( ( ord_less_int @ Y3 @ Z )
% 0.30/0.66         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % AND_upper1''
% 0.30/0.66  thf(fact_9221_and__less__eq,axiom,
% 0.30/0.66      ! [L: int,K: int] :
% 0.30/0.66        ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.66       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_less_eq
% 0.30/0.66  thf(fact_9222_take__bit__eq__mask__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66          = ( bit_se2000444600071755411sk_int @ N ) )
% 0.30/0.66        = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.30/0.66          = zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_eq_mask_iff
% 0.30/0.66  thf(fact_9223_take__bit__decr__eq,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66         != zero_zero_int )
% 0.30/0.66       => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.30/0.66          = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_decr_eq
% 0.30/0.66  thf(fact_9224_sgn__real__def,axiom,
% 0.30/0.66      ( sgn_sgn_real
% 0.30/0.66      = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sgn_real_def
% 0.30/0.66  thf(fact_9225_even__and__iff__int,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.30/0.66        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.66          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % even_and_iff_int
% 0.30/0.66  thf(fact_9226_take__bit__nat__eq__self,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.66       => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.30/0.66          = M ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_eq_self
% 0.30/0.66  thf(fact_9227_take__bit__nat__less__exp,axiom,
% 0.30/0.66      ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_less_exp
% 0.30/0.66  thf(fact_9228_take__bit__nat__eq__self__iff,axiom,
% 0.30/0.66      ! [N: nat,M: nat] :
% 0.30/0.66        ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.30/0.66          = M )
% 0.30/0.66        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_eq_self_iff
% 0.30/0.66  thf(fact_9229_take__bit__nat__def,axiom,
% 0.30/0.66      ( bit_se2925701944663578781it_nat
% 0.30/0.66      = ( ^ [N2: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_def
% 0.30/0.66  thf(fact_9230_sgn__power__injE,axiom,
% 0.30/0.66      ! [A: real,N: nat,X3: real,B2: real] :
% 0.30/0.66        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.30/0.66          = X3 )
% 0.30/0.66       => ( ( X3
% 0.30/0.66            = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
% 0.30/0.66         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66           => ( A = B2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sgn_power_injE
% 0.30/0.66  thf(fact_9231_take__bit__int__less__exp,axiom,
% 0.30/0.66      ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_less_exp
% 0.30/0.66  thf(fact_9232_take__bit__int__def,axiom,
% 0.30/0.66      ( bit_se2923211474154528505it_int
% 0.30/0.66      = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_def
% 0.30/0.66  thf(fact_9233_take__bit__nat__less__self__iff,axiom,
% 0.30/0.66      ! [N: nat,M: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.30/0.66        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_nat_less_self_iff
% 0.30/0.66  thf(fact_9234_take__bit__Suc__minus__bit0,axiom,
% 0.30/0.66      ! [N: nat,K: num] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.66        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_Suc_minus_bit0
% 0.30/0.66  thf(fact_9235_take__bit__int__less__self__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.30/0.66        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_less_self_iff
% 0.30/0.66  thf(fact_9236_take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.30/0.66        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_greater_eq_self_iff
% 0.30/0.66  thf(fact_9237_cis__Arg__unique,axiom,
% 0.30/0.66      ! [Z: complex,X3: real] :
% 0.30/0.66        ( ( ( sgn_sgn_complex @ Z )
% 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 @ Z )
% 0.30/0.66              = X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_Arg_unique
% 0.30/0.66  thf(fact_9238_take__bit__int__eq__self,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.66         => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66            = K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_eq_self
% 0.30/0.66  thf(fact_9239_take__bit__int__eq__self__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66          = K )
% 0.30/0.66        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_eq_self_iff
% 0.30/0.66  thf(fact_9240_take__bit__numeral__minus__bit0,axiom,
% 0.30/0.66      ! [L: num,K: num] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.66        = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_numeral_minus_bit0
% 0.30/0.66  thf(fact_9241_take__bit__incr__eq,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66         != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.30/0.66       => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.30/0.66          = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_incr_eq
% 0.30/0.66  thf(fact_9242_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.30/0.66          = ( bit_se2000444600071755411sk_int @ N ) )
% 0.30/0.66        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_eq_mask_iff_exp_dvd
% 0.30/0.66  thf(fact_9243_Arg__correct,axiom,
% 0.30/0.66      ! [Z: complex] :
% 0.30/0.66        ( ( Z != zero_zero_complex )
% 0.30/0.66       => ( ( ( sgn_sgn_complex @ Z )
% 0.30/0.66            = ( cis @ ( arg @ Z ) ) )
% 0.30/0.66          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.30/0.66          & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_correct
% 0.30/0.66  thf(fact_9244_take__bit__int__less__eq,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.66         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_less_eq
% 0.30/0.66  thf(fact_9245_take__bit__int__greater__eq,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.66       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_int_greater_eq
% 0.30/0.66  thf(fact_9246_signed__take__bit__eq__take__bit__shift,axiom,
% 0.30/0.66      ( bit_ri631733984087533419it_int
% 0.30/0.66      = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % signed_take_bit_eq_take_bit_shift
% 0.30/0.66  thf(fact_9247_and__int__rec,axiom,
% 0.30/0.66      ( bit_se725231765392027082nd_int
% 0.30/0.66      = ( ^ [K2: int,L2: int] :
% 0.30/0.66            ( plus_plus_int
% 0.30/0.66            @ ( zero_n2684676970156552555ol_int
% 0.30/0.66              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.66                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.66            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int_rec
% 0.30/0.66  thf(fact_9248_take__bit__minus__small__eq,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.66       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.66         => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.30/0.66            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_minus_small_eq
% 0.30/0.66  thf(fact_9249_arctan__inverse,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( X3 != zero_zero_real )
% 0.30/0.66       => ( ( arctan @ ( divide_divide_real @ one_one_real @ X3 ) )
% 0.30/0.66          = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X3 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arctan_inverse
% 0.30/0.66  thf(fact_9250_take__bit__numeral__minus__bit1,axiom,
% 0.30/0.66      ! [L: num,K: num] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.66        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_numeral_minus_bit1
% 0.30/0.66  thf(fact_9251_and__int_Osimps,axiom,
% 0.30/0.66      ( bit_se725231765392027082nd_int
% 0.30/0.66      = ( ^ [K2: int,L2: int] :
% 0.30/0.66            ( if_int
% 0.30/0.66            @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66            @ ( uminus_uminus_int
% 0.30/0.66              @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.66                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.30/0.66            @ ( plus_plus_int
% 0.30/0.66              @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.66                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.66              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int.simps
% 0.30/0.66  thf(fact_9252_and__int_Oelims,axiom,
% 0.30/0.66      ! [X3: int,Xa2: int,Y3: int] :
% 0.30/0.66        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66           => ( Y3
% 0.30/0.66              = ( uminus_uminus_int
% 0.30/0.66                @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.66                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.30/0.66          & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66                & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66           => ( Y3
% 0.30/0.66              = ( plus_plus_int
% 0.30/0.66                @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.66                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.30/0.66                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int.elims
% 0.30/0.66  thf(fact_9253_take__bit__Suc__minus__bit1,axiom,
% 0.30/0.66      ! [N: nat,K: num] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.66        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_Suc_minus_bit1
% 0.30/0.66  thf(fact_9254_pred__numeral__inc,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( pred_numeral @ ( inc @ K ) )
% 0.30/0.66        = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pred_numeral_inc
% 0.30/0.66  thf(fact_9255_and__nat__numerals_I1_J,axiom,
% 0.30/0.66      ! [Y3: num] :
% 0.30/0.66        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.66        = zero_zero_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_numerals(1)
% 0.30/0.66  thf(fact_9256_and__nat__numerals_I3_J,axiom,
% 0.30/0.66      ! [X3: num] :
% 0.30/0.66        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.66        = zero_zero_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_numerals(3)
% 0.30/0.66  thf(fact_9257_and__nat__numerals_I4_J,axiom,
% 0.30/0.66      ! [X3: num] :
% 0.30/0.66        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.66        = one_one_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_numerals(4)
% 0.30/0.66  thf(fact_9258_and__nat__numerals_I2_J,axiom,
% 0.30/0.66      ! [Y3: num] :
% 0.30/0.66        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.66        = one_one_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_numerals(2)
% 0.30/0.66  thf(fact_9259_and__Suc__0__eq,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.66        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_Suc_0_eq
% 0.30/0.66  thf(fact_9260_Suc__0__and__eq,axiom,
% 0.30/0.66      ! [N: nat] :
% 0.30/0.66        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.66        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Suc_0_and_eq
% 0.30/0.66  thf(fact_9261_num__induct,axiom,
% 0.30/0.66      ! [P: num > $o,X3: num] :
% 0.30/0.66        ( ( P @ one )
% 0.30/0.66       => ( ! [X5: num] :
% 0.30/0.66              ( ( P @ X5 )
% 0.30/0.66             => ( P @ ( inc @ X5 ) ) )
% 0.30/0.66         => ( P @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % num_induct
% 0.30/0.66  thf(fact_9262_add__inc,axiom,
% 0.30/0.66      ! [X3: num,Y3: num] :
% 0.30/0.66        ( ( plus_plus_num @ X3 @ ( inc @ Y3 ) )
% 0.30/0.66        = ( inc @ ( plus_plus_num @ X3 @ Y3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % add_inc
% 0.30/0.66  thf(fact_9263_inc_Osimps_I1_J,axiom,
% 0.30/0.66      ( ( inc @ one )
% 0.30/0.66      = ( bit0 @ one ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inc.simps(1)
% 0.30/0.66  thf(fact_9264_inc_Osimps_I2_J,axiom,
% 0.30/0.66      ! [X3: num] :
% 0.30/0.66        ( ( inc @ ( bit0 @ X3 ) )
% 0.30/0.66        = ( bit1 @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inc.simps(2)
% 0.30/0.66  thf(fact_9265_inc_Osimps_I3_J,axiom,
% 0.30/0.66      ! [X3: num] :
% 0.30/0.66        ( ( inc @ ( bit1 @ X3 ) )
% 0.30/0.66        = ( bit0 @ ( inc @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inc.simps(3)
% 0.30/0.66  thf(fact_9266_add__One,axiom,
% 0.30/0.66      ! [X3: num] :
% 0.30/0.66        ( ( plus_plus_num @ X3 @ one )
% 0.30/0.66        = ( inc @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % add_One
% 0.30/0.66  thf(fact_9267_inc__BitM__eq,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( inc @ ( bitM @ N ) )
% 0.30/0.66        = ( bit0 @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inc_BitM_eq
% 0.30/0.66  thf(fact_9268_BitM__inc__eq,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bitM @ ( inc @ N ) )
% 0.30/0.66        = ( bit1 @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % BitM_inc_eq
% 0.30/0.66  thf(fact_9269_and__nat__def,axiom,
% 0.30/0.66      ( bit_se727722235901077358nd_nat
% 0.30/0.66      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_def
% 0.30/0.66  thf(fact_9270_mult__inc,axiom,
% 0.30/0.66      ! [X3: num,Y3: num] :
% 0.30/0.66        ( ( times_times_num @ X3 @ ( inc @ Y3 ) )
% 0.30/0.66        = ( plus_plus_num @ ( times_times_num @ X3 @ Y3 ) @ X3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_inc
% 0.30/0.66  thf(fact_9271_atLeastAtMostPlus1__int__conv,axiom,
% 0.30/0.66      ! [M: int,N: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.30/0.66       => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.30/0.66          = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastAtMostPlus1_int_conv
% 0.30/0.66  thf(fact_9272_simp__from__to,axiom,
% 0.30/0.66      ( set_or1266510415728281911st_int
% 0.30/0.66      = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % simp_from_to
% 0.30/0.66  thf(fact_9273_and__nat__unfold,axiom,
% 0.30/0.66      ( bit_se727722235901077358nd_nat
% 0.30/0.66      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.66            ( if_nat
% 0.30/0.66            @ ( ( M3 = zero_zero_nat )
% 0.30/0.66              | ( N2 = zero_zero_nat ) )
% 0.30/0.66            @ zero_zero_nat
% 0.30/0.66            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_unfold
% 0.30/0.66  thf(fact_9274_and__nat__rec,axiom,
% 0.30/0.66      ( bit_se727722235901077358nd_nat
% 0.30/0.66      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.66            ( plus_plus_nat
% 0.30/0.66            @ ( zero_n2687167440665602831ol_nat
% 0.30/0.66              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.30/0.66                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.66            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_nat_rec
% 0.30/0.66  thf(fact_9275_and__int_Opsimps,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.30/0.66       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.30/0.66              = ( uminus_uminus_int
% 0.30/0.66                @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.66                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.30/0.66          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.30/0.66              = ( plus_plus_int
% 0.30/0.66                @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.66                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.30/0.66                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int.psimps
% 0.30/0.66  thf(fact_9276_and__int_Opelims,axiom,
% 0.30/0.66      ! [X3: int,Xa2: int,Y3: int] :
% 0.30/0.66        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 0.30/0.66         => ~ ( ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66                 => ( Y3
% 0.30/0.66                    = ( uminus_uminus_int
% 0.30/0.66                      @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.66                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.30/0.66                & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66                      & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66                 => ( Y3
% 0.30/0.66                    = ( plus_plus_int
% 0.30/0.66                      @ ( zero_n2684676970156552555ol_int
% 0.30/0.66                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.66                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.30/0.66                      @ ( 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 ) ) ) ) ) ) ) ) )
% 0.30/0.66             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int.pelims
% 0.30/0.66  thf(fact_9277_and__int_Opinduct,axiom,
% 0.30/0.66      ! [A0: int,A12: int,P: int > int > $o] :
% 0.30/0.66        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.30/0.66       => ( ! [K3: int,L4: int] :
% 0.30/0.66              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
% 0.30/0.66             => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.66                      & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.66                 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66               => ( P @ K3 @ L4 ) ) )
% 0.30/0.66         => ( P @ A0 @ A12 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_int.pinduct
% 0.30/0.66  thf(fact_9278_Arg__def,axiom,
% 0.30/0.66      ( arg
% 0.30/0.66      = ( ^ [Z2: complex] :
% 0.30/0.66            ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
% 0.30/0.66            @ ( fChoice_real
% 0.30/0.66              @ ^ [A5: real] :
% 0.30/0.66                  ( ( ( sgn_sgn_complex @ Z2 )
% 0.30/0.66                    = ( cis @ A5 ) )
% 0.30/0.66                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
% 0.30/0.66                  & ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_def
% 0.30/0.66  thf(fact_9279_set__encode__insert,axiom,
% 0.30/0.66      ! [A4: set_nat,N: nat] :
% 0.30/0.66        ( ( finite_finite_nat @ A4 )
% 0.30/0.66       => ( ~ ( member_nat @ N @ A4 )
% 0.30/0.66         => ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
% 0.30/0.66            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % set_encode_insert
% 0.30/0.66  thf(fact_9280_atLeastAtMost__insertL,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.66       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.30/0.66          = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastAtMost_insertL
% 0.30/0.66  thf(fact_9281_atLeastAtMostSuc__conv,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.30/0.66       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.30/0.66          = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastAtMostSuc_conv
% 0.30/0.66  thf(fact_9282_Icc__eq__insert__lb__nat,axiom,
% 0.30/0.66      ! [M: nat,N: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.66       => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.30/0.66          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Icc_eq_insert_lb_nat
% 0.30/0.66  thf(fact_9283_lessThan__nat__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.66        = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % lessThan_nat_numeral
% 0.30/0.66  thf(fact_9284_atMost__nat__numeral,axiom,
% 0.30/0.66      ! [K: num] :
% 0.30/0.66        ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.66        = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atMost_nat_numeral
% 0.30/0.66  thf(fact_9285_set__decode__plus__power__2,axiom,
% 0.30/0.66      ! [N: nat,Z: nat] :
% 0.30/0.66        ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 0.30/0.66       => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 0.30/0.66          = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % set_decode_plus_power_2
% 0.30/0.66  thf(fact_9286_signed__take__bit__eq__take__bit__minus,axiom,
% 0.30/0.66      ( bit_ri631733984087533419it_int
% 0.30/0.66      = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % signed_take_bit_eq_take_bit_minus
% 0.30/0.66  thf(fact_9287_or__int__unfold,axiom,
% 0.30/0.66      ( bit_se1409905431419307370or_int
% 0.30/0.66      = ( ^ [K2: int,L2: int] :
% 0.30/0.66            ( if_int
% 0.30/0.66            @ ( ( K2
% 0.30/0.66                = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.66              | ( L2
% 0.30/0.66                = ( uminus_uminus_int @ one_one_int ) ) )
% 0.30/0.66            @ ( uminus_uminus_int @ one_one_int )
% 0.30/0.66            @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % or_int_unfold
% 0.30/0.66  thf(fact_9288_cis__multiple__2pi,axiom,
% 0.30/0.66      ! [N: real] :
% 0.30/0.66        ( ( member_real @ N @ ring_1_Ints_real )
% 0.30/0.66       => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.30/0.66          = one_one_complex ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_multiple_2pi
% 0.30/0.66  thf(fact_9289_take__bit__Suc__from__most,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.30/0.66        = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_bit_Suc_from_most
% 0.30/0.66  thf(fact_9290_or__nonnegative__int__iff,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.30/0.66        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.66          & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % or_nonnegative_int_iff
% 0.30/0.66  thf(fact_9291_or__negative__int__iff,axiom,
% 0.30/0.66      ! [K: int,L: int] :
% 0.30/0.66        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.30/0.66        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.66          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % or_negative_int_iff
% 0.30/0.66  thf(fact_9292_signed__take__bit__nonnegative__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.30/0.66        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % signed_take_bit_nonnegative_iff
% 0.30/0.66  thf(fact_9293_signed__take__bit__negative__iff,axiom,
% 0.30/0.66      ! [N: nat,K: int] :
% 0.30/0.66        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.66        = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % signed_take_bit_negative_iff
% 0.30/0.66  thf(fact_9294_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.66      ! [W: num,N: nat] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.30/0.66        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_minus_numeral_Bit0_Suc_iff
% 0.30/0.66  thf(fact_9295_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.66      ! [W: num,N: nat] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.30/0.66        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_minus_numeral_Bit1_Suc_iff
% 0.30/0.66  thf(fact_9296_or__minus__numerals_I6_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.30/0.66        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % or_minus_numerals(6)
% 0.30/0.66  thf(fact_9297_or__minus__numerals_I2_J,axiom,
% 0.30/0.66      ! [N: num] :
% 0.30/0.66        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.66        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % or_minus_numerals(2)
% 0.30/0.66  thf(fact_9298_bit__minus__numeral__int_I1_J,axiom,
% 0.30/0.66      ! [W: num,N: num] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.66        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_minus_numeral_int(1)
% 0.30/0.66  thf(fact_9299_bit__minus__numeral__int_I2_J,axiom,
% 0.30/0.66      ! [W: num,N: num] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.66        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_minus_numeral_int(2)
% 0.30/0.66  thf(fact_9300_bit__or__int__iff,axiom,
% 0.30/0.66      ! [K: int,L: int,N: nat] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 0.30/0.66        = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.66          | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_or_int_iff
% 0.30/0.66  thf(fact_9301_bit__and__int__iff,axiom,
% 0.30/0.66      ! [K: int,L: int,N: nat] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 0.30/0.66        = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.66          & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_and_int_iff
% 0.30/0.66  thf(fact_9302_or__greater__eq,axiom,
% 0.30/0.66      ! [L: int,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.30/0.66       => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % or_greater_eq
% 0.30/0.66  thf(fact_9303_OR__lower,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.66       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.66         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % OR_lower
% 0.30/0.66  thf(fact_9304_plus__and__or,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) )
% 0.30/0.66        = ( plus_plus_int @ X3 @ Y3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % plus_and_or
% 0.30/0.66  thf(fact_9305_bit__not__int__iff_H,axiom,
% 0.30/0.66      ! [K: int,N: nat] :
% 0.30/0.66        ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.30/0.66        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bit_not_int_iff'
% 0.30/0.67  thf(fact_9306_bit__imp__take__bit__positive,axiom,
% 0.30/0.67      ! [N: nat,M: nat,K: int] :
% 0.30/0.67        ( ( ord_less_nat @ N @ M )
% 0.30/0.67       => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.67         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_imp_take_bit_positive
% 0.30/0.67  thf(fact_9307_bit__concat__bit__iff,axiom,
% 0.30/0.67      ! [M: nat,K: int,L: int,N: nat] :
% 0.30/0.67        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.30/0.67        = ( ( ( ord_less_nat @ N @ M )
% 0.30/0.67            & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.30/0.67          | ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.67            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_concat_bit_iff
% 0.30/0.67  thf(fact_9308_sin__times__pi__eq__0,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67        ( ( ( sin_real @ ( times_times_real @ X3 @ pi ) )
% 0.30/0.67          = zero_zero_real )
% 0.30/0.67        = ( member_real @ X3 @ ring_1_Ints_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sin_times_pi_eq_0
% 0.30/0.67  thf(fact_9309_signed__take__bit__eq__concat__bit,axiom,
% 0.30/0.67      ( bit_ri631733984087533419it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % signed_take_bit_eq_concat_bit
% 0.30/0.67  thf(fact_9310_int__bit__bound,axiom,
% 0.30/0.67      ! [K: int] :
% 0.30/0.67        ~ ! [N3: nat] :
% 0.30/0.67            ( ! [M2: nat] :
% 0.30/0.67                ( ( ord_less_eq_nat @ N3 @ M2 )
% 0.30/0.67               => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 0.30/0.67                  = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 0.30/0.67           => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.67               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 0.30/0.67                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_bit_bound
% 0.30/0.67  thf(fact_9311_bit__int__def,axiom,
% 0.30/0.67      ( bit_se1146084159140164899it_int
% 0.30/0.67      = ( ^ [K2: int,N2: nat] :
% 0.30/0.67            ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_int_def
% 0.30/0.67  thf(fact_9312_OR__upper,axiom,
% 0.30/0.67      ! [X3: int,N: nat,Y3: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.67       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.67         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.67           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % OR_upper
% 0.30/0.67  thf(fact_9313_sin__integer__2pi,axiom,
% 0.30/0.67      ! [N: real] :
% 0.30/0.67        ( ( member_real @ N @ ring_1_Ints_real )
% 0.30/0.67       => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.30/0.67          = zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sin_integer_2pi
% 0.30/0.67  thf(fact_9314_cos__integer__2pi,axiom,
% 0.30/0.67      ! [N: real] :
% 0.30/0.67        ( ( member_real @ N @ ring_1_Ints_real )
% 0.30/0.67       => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.30/0.67          = one_one_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cos_integer_2pi
% 0.30/0.67  thf(fact_9315_or__int__rec,axiom,
% 0.30/0.67      ( bit_se1409905431419307370or_int
% 0.30/0.67      = ( ^ [K2: int,L2: int] :
% 0.30/0.67            ( plus_plus_int
% 0.30/0.67            @ ( zero_n2684676970156552555ol_int
% 0.30/0.67              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.30/0.67                | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.67            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_int_rec
% 0.30/0.67  thf(fact_9316_set__bit__eq,axiom,
% 0.30/0.67      ( bit_se7879613467334960850it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] :
% 0.30/0.67            ( plus_plus_int @ K2
% 0.30/0.67            @ ( times_times_int
% 0.30/0.67              @ ( zero_n2684676970156552555ol_int
% 0.30/0.67                @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
% 0.30/0.67              @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % set_bit_eq
% 0.30/0.67  thf(fact_9317_unset__bit__eq,axiom,
% 0.30/0.67      ( bit_se4203085406695923979it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % unset_bit_eq
% 0.30/0.67  thf(fact_9318_or__minus__numerals_I1_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_numerals(1)
% 0.30/0.67  thf(fact_9319_or__minus__numerals_I5_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_numerals(5)
% 0.30/0.67  thf(fact_9320_xor__Suc__0__eq,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.67          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.67            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_Suc_0_eq
% 0.30/0.67  thf(fact_9321_or__nat__numerals_I2_J,axiom,
% 0.30/0.67      ! [Y3: num] :
% 0.30/0.67        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_numerals(2)
% 0.30/0.67  thf(fact_9322_or__nat__numerals_I4_J,axiom,
% 0.30/0.67      ! [X3: num] :
% 0.30/0.67        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_numerals(4)
% 0.30/0.67  thf(fact_9323_or__nat__numerals_I3_J,axiom,
% 0.30/0.67      ! [X3: num] :
% 0.30/0.67        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_numerals(3)
% 0.30/0.67  thf(fact_9324_or__nat__numerals_I1_J,axiom,
% 0.30/0.67      ! [Y3: num] :
% 0.30/0.67        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_numerals(1)
% 0.30/0.67  thf(fact_9325_xor__nat__numerals_I4_J,axiom,
% 0.30/0.67      ! [X3: num] :
% 0.30/0.67        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_numerals(4)
% 0.30/0.67  thf(fact_9326_xor__nat__numerals_I3_J,axiom,
% 0.30/0.67      ! [X3: num] :
% 0.30/0.67        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_numerals(3)
% 0.30/0.67  thf(fact_9327_xor__nat__numerals_I2_J,axiom,
% 0.30/0.67      ! [Y3: num] :
% 0.30/0.67        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_numerals(2)
% 0.30/0.67  thf(fact_9328_xor__nat__numerals_I1_J,axiom,
% 0.30/0.67      ! [Y3: num] :
% 0.30/0.67        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.67        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_numerals(1)
% 0.30/0.67  thf(fact_9329_or__minus__numerals_I4_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_numerals(4)
% 0.30/0.67  thf(fact_9330_or__minus__numerals_I8_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_numerals(8)
% 0.30/0.67  thf(fact_9331_or__minus__numerals_I3_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_numerals(3)
% 0.30/0.67  thf(fact_9332_or__minus__numerals_I7_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_numerals(7)
% 0.30/0.67  thf(fact_9333_or__not__num__neg_Osimps_I1_J,axiom,
% 0.30/0.67      ( ( bit_or_not_num_neg @ one @ one )
% 0.30/0.67      = one ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(1)
% 0.30/0.67  thf(fact_9334_bit__Suc__0__iff,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.67        = ( N = zero_zero_nat ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_Suc_0_iff
% 0.30/0.67  thf(fact_9335_not__bit__Suc__0__Suc,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_bit_Suc_0_Suc
% 0.30/0.67  thf(fact_9336_or__not__num__neg_Osimps_I4_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.30/0.67        = ( bit0 @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(4)
% 0.30/0.67  thf(fact_9337_or__not__num__neg_Osimps_I6_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.30/0.67        = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(6)
% 0.30/0.67  thf(fact_9338_or__not__num__neg_Osimps_I7_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.30/0.67        = one ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(7)
% 0.30/0.67  thf(fact_9339_or__not__num__neg_Osimps_I3_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.30/0.67        = ( bit1 @ M ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(3)
% 0.30/0.67  thf(fact_9340_or__not__num__neg_Osimps_I5_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.30/0.67        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(5)
% 0.30/0.67  thf(fact_9341_not__bit__Suc__0__numeral,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_bit_Suc_0_numeral
% 0.30/0.67  thf(fact_9342_or__not__num__neg_Osimps_I9_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 0.30/0.67        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(9)
% 0.30/0.67  thf(fact_9343_or__nat__def,axiom,
% 0.30/0.67      ( bit_se1412395901928357646or_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_def
% 0.30/0.67  thf(fact_9344_or__not__num__neg_Osimps_I2_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.30/0.67        = ( bit1 @ M ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(2)
% 0.30/0.67  thf(fact_9345_or__not__num__neg_Osimps_I8_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.30/0.67        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.simps(8)
% 0.30/0.67  thf(fact_9346_bit__nat__iff,axiom,
% 0.30/0.67      ! [K: int,N: nat] :
% 0.30/0.67        ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.30/0.67        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.67          & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_nat_iff
% 0.30/0.67  thf(fact_9347_or__not__num__neg_Oelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: num] :
% 0.30/0.67        ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( X3 = one )
% 0.30/0.67           => ( ( Xa2 = one )
% 0.30/0.67             => ( Y3 != one ) ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ! [M4: num] :
% 0.30/0.67                  ( ( Xa2
% 0.30/0.67                    = ( bit0 @ M4 ) )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                   != ( bit1 @ M4 ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ! [M4: num] :
% 0.30/0.67                    ( ( Xa2
% 0.30/0.67                      = ( bit1 @ M4 ) )
% 0.30/0.67                   => ( Y3
% 0.30/0.67                     != ( bit1 @ M4 ) ) ) )
% 0.30/0.67             => ( ( ? [N3: num] :
% 0.30/0.67                      ( X3
% 0.30/0.67                      = ( bit0 @ N3 ) )
% 0.30/0.67                 => ( ( Xa2 = one )
% 0.30/0.67                   => ( Y3
% 0.30/0.67                     != ( bit0 @ one ) ) ) )
% 0.30/0.67               => ( ! [N3: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ N3 ) )
% 0.30/0.67                     => ! [M4: num] :
% 0.30/0.67                          ( ( Xa2
% 0.30/0.67                            = ( bit0 @ M4 ) )
% 0.30/0.67                         => ( Y3
% 0.30/0.67                           != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
% 0.30/0.67                 => ( ! [N3: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ N3 ) )
% 0.30/0.67                       => ! [M4: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ( Y3
% 0.30/0.67                             != ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
% 0.30/0.67                   => ( ( ? [N3: num] :
% 0.30/0.67                            ( X3
% 0.30/0.67                            = ( bit1 @ N3 ) )
% 0.30/0.67                       => ( ( Xa2 = one )
% 0.30/0.67                         => ( Y3 != one ) ) )
% 0.30/0.67                     => ( ! [N3: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ N3 ) )
% 0.30/0.67                           => ! [M4: num] :
% 0.30/0.67                                ( ( Xa2
% 0.30/0.67                                  = ( bit0 @ M4 ) )
% 0.30/0.67                               => ( Y3
% 0.30/0.67                                 != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
% 0.30/0.67                       => ~ ! [N3: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ N3 ) )
% 0.30/0.67                             => ! [M4: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit1 @ M4 ) )
% 0.30/0.67                                 => ( Y3
% 0.30/0.67                                   != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.elims
% 0.30/0.67  thf(fact_9348_bit__nat__def,axiom,
% 0.30/0.67      ( bit_se1148574629649215175it_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.67            ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_nat_def
% 0.30/0.67  thf(fact_9349_Suc__0__or__eq,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.67        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Suc_0_or_eq
% 0.30/0.67  thf(fact_9350_or__Suc__0__eq,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_Suc_0_eq
% 0.30/0.67  thf(fact_9351_or__nat__rec,axiom,
% 0.30/0.67      ( bit_se1412395901928357646or_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.67            ( plus_plus_nat
% 0.30/0.67            @ ( zero_n2687167440665602831ol_nat
% 0.30/0.67              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.30/0.67                | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.67            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_rec
% 0.30/0.67  thf(fact_9352_xor__nat__unfold,axiom,
% 0.30/0.67      ( bit_se6528837805403552850or_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_unfold
% 0.30/0.67  thf(fact_9353_xor__nat__rec,axiom,
% 0.30/0.67      ( bit_se6528837805403552850or_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.67            ( plus_plus_nat
% 0.30/0.67            @ ( zero_n2687167440665602831ol_nat
% 0.30/0.67              @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.30/0.67               != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.67            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_rec
% 0.30/0.67  thf(fact_9354_or__nat__unfold,axiom,
% 0.30/0.67      ( bit_se1412395901928357646or_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_nat_unfold
% 0.30/0.67  thf(fact_9355_Suc__0__xor__eq,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.30/0.67        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.67          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.67            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Suc_0_xor_eq
% 0.30/0.67  thf(fact_9356_or__not__num__neg_Opelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: num] :
% 0.30/0.67        ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ( ( Xa2 = one )
% 0.30/0.67               => ( ( Y3 = one )
% 0.30/0.67                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ! [M4: num] :
% 0.30/0.67                    ( ( Xa2
% 0.30/0.67                      = ( bit0 @ M4 ) )
% 0.30/0.67                   => ( ( Y3
% 0.30/0.67                        = ( bit1 @ M4 ) )
% 0.30/0.67                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 0.30/0.67             => ( ( ( X3 = one )
% 0.30/0.67                 => ! [M4: num] :
% 0.30/0.67                      ( ( Xa2
% 0.30/0.67                        = ( bit1 @ M4 ) )
% 0.30/0.67                     => ( ( Y3
% 0.30/0.67                          = ( bit1 @ M4 ) )
% 0.30/0.67                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 0.30/0.67               => ( ! [N3: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ N3 ) )
% 0.30/0.67                     => ( ( Xa2 = one )
% 0.30/0.67                       => ( ( Y3
% 0.30/0.67                            = ( bit0 @ one ) )
% 0.30/0.67                         => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
% 0.30/0.67                 => ( ! [N3: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ N3 ) )
% 0.30/0.67                       => ! [M4: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit0 @ M4 ) )
% 0.30/0.67                           => ( ( Y3
% 0.30/0.67                                = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 0.30/0.67                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.30/0.67                   => ( ! [N3: num] :
% 0.30/0.67                          ( ( X3
% 0.30/0.67                            = ( bit0 @ N3 ) )
% 0.30/0.67                         => ! [M4: num] :
% 0.30/0.67                              ( ( Xa2
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 0.30/0.67                     => ( ! [N3: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ N3 ) )
% 0.30/0.67                           => ( ( Xa2 = one )
% 0.30/0.67                             => ( ( Y3 = one )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
% 0.30/0.67                       => ( ! [N3: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ N3 ) )
% 0.30/0.67                             => ! [M4: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit0 @ M4 ) )
% 0.30/0.67                                 => ( ( Y3
% 0.30/0.67                                      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 0.30/0.67                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.30/0.67                         => ~ ! [N3: num] :
% 0.30/0.67                                ( ( X3
% 0.30/0.67                                  = ( bit1 @ N3 ) )
% 0.30/0.67                               => ! [M4: num] :
% 0.30/0.67                                    ( ( Xa2
% 0.30/0.67                                      = ( bit1 @ M4 ) )
% 0.30/0.67                                   => ( ( Y3
% 0.30/0.67                                        = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 0.30/0.67                                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_num_neg.pelims
% 0.30/0.67  thf(fact_9357_horner__sum__of__bool__2__less,axiom,
% 0.30/0.67      ! [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 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % horner_sum_of_bool_2_less
% 0.30/0.67  thf(fact_9358_push__bit__nonnegative__int__iff,axiom,
% 0.30/0.67      ! [N: nat,K: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.30/0.67        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_nonnegative_int_iff
% 0.30/0.67  thf(fact_9359_push__bit__negative__int__iff,axiom,
% 0.30/0.67      ! [N: nat,K: int] :
% 0.30/0.67        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.67        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_negative_int_iff
% 0.30/0.67  thf(fact_9360_concat__bit__of__zero__1,axiom,
% 0.30/0.67      ! [N: nat,L: int] :
% 0.30/0.67        ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 0.30/0.67        = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 0.30/0.67  
% 0.30/0.67  % concat_bit_of_zero_1
% 0.30/0.67  thf(fact_9361_xor__nonnegative__int__iff,axiom,
% 0.30/0.67      ! [K: int,L: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.30/0.67        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.67          = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nonnegative_int_iff
% 0.30/0.67  thf(fact_9362_xor__negative__int__iff,axiom,
% 0.30/0.67      ! [K: int,L: int] :
% 0.30/0.67        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.30/0.67        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.67         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_negative_int_iff
% 0.30/0.67  thf(fact_9363_push__bit__of__Suc__0,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_of_Suc_0
% 0.30/0.67  thf(fact_9364_bit__xor__int__iff,axiom,
% 0.30/0.67      ! [K: int,L: int,N: nat] :
% 0.30/0.67        ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 0.30/0.67        = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.30/0.67         != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_xor_int_iff
% 0.30/0.67  thf(fact_9365_flip__bit__int__def,axiom,
% 0.30/0.67      ( bit_se2159334234014336723it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % flip_bit_int_def
% 0.30/0.67  thf(fact_9366_push__bit__nat__eq,axiom,
% 0.30/0.67      ! [N: nat,K: int] :
% 0.30/0.67        ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 0.30/0.67        = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_nat_eq
% 0.30/0.67  thf(fact_9367_XOR__lower,axiom,
% 0.30/0.67      ! [X3: int,Y3: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.67       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.67         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % XOR_lower
% 0.30/0.67  thf(fact_9368_flip__bit__nat__def,axiom,
% 0.30/0.67      ( bit_se2161824704523386999it_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % flip_bit_nat_def
% 0.30/0.67  thf(fact_9369_set__bit__nat__def,axiom,
% 0.30/0.67      ( bit_se7882103937844011126it_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % set_bit_nat_def
% 0.30/0.67  thf(fact_9370_bit__push__bit__iff__int,axiom,
% 0.30/0.67      ! [M: nat,K: int,N: nat] :
% 0.30/0.67        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.30/0.67        = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.67          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_push_bit_iff_int
% 0.30/0.67  thf(fact_9371_xor__nat__def,axiom,
% 0.30/0.67      ( bit_se6528837805403552850or_nat
% 0.30/0.67      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_nat_def
% 0.30/0.67  thf(fact_9372_bit__push__bit__iff__nat,axiom,
% 0.30/0.67      ! [M: nat,Q2: nat,N: nat] :
% 0.30/0.67        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 0.30/0.67        = ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.67          & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_push_bit_iff_nat
% 0.30/0.67  thf(fact_9373_concat__bit__eq,axiom,
% 0.30/0.67      ( bit_concat_bit
% 0.30/0.67      = ( ^ [N2: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % concat_bit_eq
% 0.30/0.67  thf(fact_9374_concat__bit__def,axiom,
% 0.30/0.67      ( bit_concat_bit
% 0.30/0.67      = ( ^ [N2: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % concat_bit_def
% 0.30/0.67  thf(fact_9375_set__bit__int__def,axiom,
% 0.30/0.67      ( bit_se7879613467334960850it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % set_bit_int_def
% 0.30/0.67  thf(fact_9376_push__bit__int__def,axiom,
% 0.30/0.67      ( bit_se545348938243370406it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_int_def
% 0.30/0.67  thf(fact_9377_push__bit__nat__def,axiom,
% 0.30/0.67      ( bit_se547839408752420682it_nat
% 0.30/0.67      = ( ^ [N2: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_nat_def
% 0.30/0.67  thf(fact_9378_push__bit__minus__one,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % push_bit_minus_one
% 0.30/0.67  thf(fact_9379_XOR__upper,axiom,
% 0.30/0.67      ! [X3: int,N: nat,Y3: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.67       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.67         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.30/0.67           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % XOR_upper
% 0.30/0.67  thf(fact_9380_xor__int__rec,axiom,
% 0.30/0.67      ( bit_se6526347334894502574or_int
% 0.30/0.67      = ( ^ [K2: int,L2: int] :
% 0.30/0.67            ( plus_plus_int
% 0.30/0.67            @ ( zero_n2684676970156552555ol_int
% 0.30/0.67              @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 0.30/0.67               != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.30/0.67            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_int_rec
% 0.30/0.67  thf(fact_9381_xor__int__unfold,axiom,
% 0.30/0.67      ( bit_se6526347334894502574or_int
% 0.30/0.67      = ( ^ [K2: int,L2: int] :
% 0.30/0.67            ( if_int
% 0.30/0.67            @ ( K2
% 0.30/0.67              = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.67            @ ( bit_ri7919022796975470100ot_int @ L2 )
% 0.30/0.67            @ ( if_int
% 0.30/0.67              @ ( L2
% 0.30/0.67                = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.67              @ ( bit_ri7919022796975470100ot_int @ K2 )
% 0.30/0.67              @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_int_unfold
% 0.30/0.67  thf(fact_9382_Sum__Ico__nat,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( groups3542108847815614940at_nat
% 0.30/0.67          @ ^ [X: nat] : X
% 0.30/0.67          @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.30/0.67        = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Sum_Ico_nat
% 0.30/0.67  thf(fact_9383_Cauchy__iff2,axiom,
% 0.30/0.67      ( topolo4055970368930404560y_real
% 0.30/0.67      = ( ^ [X7: nat > real] :
% 0.30/0.67          ! [J3: nat] :
% 0.30/0.67          ? [M9: nat] :
% 0.30/0.67          ! [M3: nat] :
% 0.30/0.67            ( ( ord_less_eq_nat @ M9 @ M3 )
% 0.30/0.67           => ! [N2: nat] :
% 0.30/0.67                ( ( ord_less_eq_nat @ M9 @ N2 )
% 0.30/0.67               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Cauchy_iff2
% 0.30/0.67  thf(fact_9384_not__nonnegative__int__iff,axiom,
% 0.30/0.67      ! [K: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.30/0.67        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_nonnegative_int_iff
% 0.30/0.67  thf(fact_9385_not__negative__int__iff,axiom,
% 0.30/0.67      ! [K: int] :
% 0.30/0.67        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.30/0.67        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_negative_int_iff
% 0.30/0.67  thf(fact_9386_and__minus__minus__numerals,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_minus_minus_numerals
% 0.30/0.67  thf(fact_9387_or__minus__minus__numerals,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_minus_minus_numerals
% 0.30/0.67  thf(fact_9388_bit__not__int__iff,axiom,
% 0.30/0.67      ! [K: int,N: nat] :
% 0.30/0.67        ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 0.30/0.67        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_not_int_iff
% 0.30/0.67  thf(fact_9389_ex__nat__less__eq,axiom,
% 0.30/0.67      ! [N: nat,P: nat > $o] :
% 0.30/0.67        ( ( ? [M3: nat] :
% 0.30/0.67              ( ( ord_less_nat @ M3 @ N )
% 0.30/0.67              & ( P @ M3 ) ) )
% 0.30/0.67        = ( ? [X: nat] :
% 0.30/0.67              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.67              & ( P @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % ex_nat_less_eq
% 0.30/0.67  thf(fact_9390_all__nat__less__eq,axiom,
% 0.30/0.67      ! [N: nat,P: nat > $o] :
% 0.30/0.67        ( ( ! [M3: nat] :
% 0.30/0.67              ( ( ord_less_nat @ M3 @ N )
% 0.30/0.67             => ( P @ M3 ) ) )
% 0.30/0.67        = ( ! [X: nat] :
% 0.30/0.67              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.67             => ( P @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % all_nat_less_eq
% 0.30/0.67  thf(fact_9391_or__int__def,axiom,
% 0.30/0.67      ( bit_se1409905431419307370or_int
% 0.30/0.67      = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_int_def
% 0.30/0.67  thf(fact_9392_not__int__def,axiom,
% 0.30/0.67      ( bit_ri7919022796975470100ot_int
% 0.30/0.67      = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_int_def
% 0.30/0.67  thf(fact_9393_and__not__numerals_I1_J,axiom,
% 0.30/0.67      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.67      = zero_zero_int ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(1)
% 0.30/0.67  thf(fact_9394_or__not__numerals_I1_J,axiom,
% 0.30/0.67      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.67      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(1)
% 0.30/0.67  thf(fact_9395_unset__bit__int__def,axiom,
% 0.30/0.67      ( bit_se4203085406695923979it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % unset_bit_int_def
% 0.30/0.67  thf(fact_9396_xor__int__def,axiom,
% 0.30/0.67      ( bit_se6526347334894502574or_int
% 0.30/0.67      = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_int_def
% 0.30/0.67  thf(fact_9397_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.30/0.67      ! [N4: set_nat,N: nat] :
% 0.30/0.67        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.67       => ( finite_finite_nat @ N4 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % subset_eq_atLeast0_lessThan_finite
% 0.30/0.67  thf(fact_9398_not__int__div__2,axiom,
% 0.30/0.67      ! [K: int] :
% 0.30/0.67        ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_int_div_2
% 0.30/0.67  thf(fact_9399_even__not__iff__int,axiom,
% 0.30/0.67      ! [K: int] :
% 0.30/0.67        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.30/0.67        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % even_not_iff_int
% 0.30/0.67  thf(fact_9400_and__not__numerals_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = one_one_int ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(2)
% 0.30/0.67  thf(fact_9401_and__not__numerals_I4_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.67        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(4)
% 0.30/0.67  thf(fact_9402_or__not__numerals_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(2)
% 0.30/0.67  thf(fact_9403_or__not__numerals_I4_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(4)
% 0.30/0.67  thf(fact_9404_bit__minus__int__iff,axiom,
% 0.30/0.67      ! [K: int,N: nat] :
% 0.30/0.67        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.30/0.67        = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_minus_int_iff
% 0.30/0.67  thf(fact_9405_numeral__or__not__num__eq,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % numeral_or_not_num_eq
% 0.30/0.67  thf(fact_9406_int__numeral__not__or__num__neg,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_numeral_not_or_num_neg
% 0.30/0.67  thf(fact_9407_int__numeral__or__not__num__neg,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_numeral_or_not_num_neg
% 0.30/0.67  thf(fact_9408_atLeastLessThanSuc,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.67         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.30/0.67            = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.30/0.67         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.30/0.67            = bot_bot_set_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % atLeastLessThanSuc
% 0.30/0.67  thf(fact_9409_and__not__numerals_I5_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(5)
% 0.30/0.67  thf(fact_9410_and__not__numerals_I7_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.67        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(7)
% 0.30/0.67  thf(fact_9411_or__not__numerals_I3_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(3)
% 0.30/0.67  thf(fact_9412_and__not__numerals_I3_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = zero_zero_int ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(3)
% 0.30/0.67  thf(fact_9413_or__not__numerals_I7_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(7)
% 0.30/0.67  thf(fact_9414_atLeastLessThan__nat__numeral,axiom,
% 0.30/0.67      ! [M: nat,K: num] :
% 0.30/0.67        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.67         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.30/0.67            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.67         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.30/0.67            = bot_bot_set_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % atLeastLessThan_nat_numeral
% 0.30/0.67  thf(fact_9415_and__not__numerals_I9_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(9)
% 0.30/0.67  thf(fact_9416_and__not__numerals_I6_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(6)
% 0.30/0.67  thf(fact_9417_or__not__numerals_I6_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(6)
% 0.30/0.67  thf(fact_9418_or__not__numerals_I5_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(5)
% 0.30/0.67  thf(fact_9419_and__not__numerals_I8_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_numerals(8)
% 0.30/0.67  thf(fact_9420_or__not__numerals_I8_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(8)
% 0.30/0.67  thf(fact_9421_or__not__numerals_I9_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % or_not_numerals(9)
% 0.30/0.67  thf(fact_9422_not__int__rec,axiom,
% 0.30/0.67      ( bit_ri7919022796975470100ot_int
% 0.30/0.67      = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % not_int_rec
% 0.30/0.67  thf(fact_9423_sum__power2,axiom,
% 0.30/0.67      ! [K: nat] :
% 0.30/0.67        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.30/0.67        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sum_power2
% 0.30/0.67  thf(fact_9424_Chebyshev__sum__upper__nat,axiom,
% 0.30/0.67      ! [N: nat,A: nat > nat,B2: nat > nat] :
% 0.30/0.67        ( ! [I3: nat,J2: nat] :
% 0.30/0.67            ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.30/0.67           => ( ( ord_less_nat @ J2 @ N )
% 0.30/0.67             => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 0.30/0.67       => ( ! [I3: nat,J2: nat] :
% 0.30/0.67              ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.30/0.67             => ( ( ord_less_nat @ J2 @ N )
% 0.30/0.67               => ( ord_less_eq_nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
% 0.30/0.67         => ( ord_less_eq_nat
% 0.30/0.67            @ ( times_times_nat @ N
% 0.30/0.67              @ ( groups3542108847815614940at_nat
% 0.30/0.67                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B2 @ I4 ) )
% 0.30/0.67                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.67            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Chebyshev_sum_upper_nat
% 0.30/0.67  thf(fact_9425_valid__eq,axiom,
% 0.30/0.67      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.30/0.67  
% 0.30/0.67  % valid_eq
% 0.30/0.67  thf(fact_9426_valid__eq1,axiom,
% 0.30/0.67      ! [T: vEBT_VEBT,D: nat] :
% 0.30/0.67        ( ( vEBT_invar_vebt @ T @ D )
% 0.30/0.67       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.30/0.67  
% 0.30/0.67  % valid_eq1
% 0.30/0.67  thf(fact_9427_valid__eq2,axiom,
% 0.30/0.67      ! [T: vEBT_VEBT,D: nat] :
% 0.30/0.67        ( ( vEBT_VEBT_valid @ T @ D )
% 0.30/0.67       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.30/0.67  
% 0.30/0.67  % valid_eq2
% 0.30/0.67  thf(fact_9428_Code__Target__Int_Opositive__def,axiom,
% 0.30/0.67      code_Target_positive = numeral_numeral_int ).
% 0.30/0.67  
% 0.30/0.67  % Code_Target_Int.positive_def
% 0.30/0.67  thf(fact_9429_divmod__step__integer__def,axiom,
% 0.30/0.67      ( unique4921790084139445826nteger
% 0.30/0.67      = ( ^ [L2: num] :
% 0.30/0.67            ( produc6916734918728496179nteger
% 0.30/0.67            @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_step_integer_def
% 0.30/0.67  thf(fact_9430_times__integer__code_I2_J,axiom,
% 0.30/0.67      ! [L: code_integer] :
% 0.30/0.67        ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 0.30/0.67        = zero_z3403309356797280102nteger ) ).
% 0.30/0.67  
% 0.30/0.67  % times_integer_code(2)
% 0.30/0.67  thf(fact_9431_times__integer__code_I1_J,axiom,
% 0.30/0.67      ! [K: code_integer] :
% 0.30/0.67        ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.30/0.67        = zero_z3403309356797280102nteger ) ).
% 0.30/0.67  
% 0.30/0.67  % times_integer_code(1)
% 0.30/0.67  thf(fact_9432_exhaustive__integer_H_Ocases,axiom,
% 0.30/0.67      ! [X3: produc8763457246119570046nteger] :
% 0.30/0.67        ~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
% 0.30/0.67            ( X3
% 0.30/0.67           != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % exhaustive_integer'.cases
% 0.30/0.67  thf(fact_9433_full__exhaustive__integer_H_Ocases,axiom,
% 0.30/0.67      ! [X3: produc1908205239877642774nteger] :
% 0.30/0.67        ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
% 0.30/0.67            ( X3
% 0.30/0.67           != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % full_exhaustive_integer'.cases
% 0.30/0.67  thf(fact_9434_less__eq__integer__code_I1_J,axiom,
% 0.30/0.67      ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_integer_code(1)
% 0.30/0.67  thf(fact_9435_divmod__integer_H__def,axiom,
% 0.30/0.67      ( unique3479559517661332726nteger
% 0.30/0.67      = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_integer'_def
% 0.30/0.67  thf(fact_9436_sgn__integer__code,axiom,
% 0.30/0.67      ( sgn_sgn_Code_integer
% 0.30/0.67      = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sgn_integer_code
% 0.30/0.67  thf(fact_9437_integer__of__int__code,axiom,
% 0.30/0.67      ( code_integer_of_int
% 0.30/0.67      = ( ^ [K2: int] :
% 0.30/0.67            ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.30/0.67            @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.30/0.67              @ ( if_Code_integer
% 0.30/0.67                @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.67                  = zero_zero_int )
% 0.30/0.67                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_of_int_code
% 0.30/0.67  thf(fact_9438_Code__Numeral_Opositive__def,axiom,
% 0.30/0.67      code_positive = numera6620942414471956472nteger ).
% 0.30/0.67  
% 0.30/0.67  % Code_Numeral.positive_def
% 0.30/0.67  thf(fact_9439_abs__integer__code,axiom,
% 0.30/0.67      ( abs_abs_Code_integer
% 0.30/0.67      = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % abs_integer_code
% 0.30/0.67  thf(fact_9440_less__integer_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: int,X3: int] :
% 0.30/0.67        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 0.30/0.67        = ( ord_less_int @ Xa2 @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_integer.abs_eq
% 0.30/0.67  thf(fact_9441_less__integer__code_I1_J,axiom,
% 0.30/0.67      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.30/0.67  
% 0.30/0.67  % less_integer_code(1)
% 0.30/0.67  thf(fact_9442_times__integer_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: int,X3: int] :
% 0.30/0.67        ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 0.30/0.67        = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_integer.abs_eq
% 0.30/0.67  thf(fact_9443_less__eq__integer_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: int,X3: int] :
% 0.30/0.67        ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 0.30/0.67        = ( ord_less_eq_int @ Xa2 @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_integer.abs_eq
% 0.30/0.67  thf(fact_9444_int__of__integer__code,axiom,
% 0.30/0.67      ( code_int_of_integer
% 0.30/0.67      = ( ^ [K2: code_integer] :
% 0.30/0.67            ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 0.30/0.67            @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.30/0.67              @ ( produc1553301316500091796er_int
% 0.30/0.67                @ ^ [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 ) )
% 0.30/0.67                @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_of_integer_code
% 0.30/0.67  thf(fact_9445_bit__cut__integer__def,axiom,
% 0.30/0.67      ( code_bit_cut_integer
% 0.30/0.67      = ( ^ [K2: code_integer] :
% 0.30/0.67            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.67            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_cut_integer_def
% 0.30/0.67  thf(fact_9446_int__of__integer__numeral,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.30/0.67        = ( numeral_numeral_int @ K ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_of_integer_numeral
% 0.30/0.67  thf(fact_9447_times__integer_Orep__eq,axiom,
% 0.30/0.67      ! [X3: code_integer,Xa2: code_integer] :
% 0.30/0.67        ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X3 @ Xa2 ) )
% 0.30/0.67        = ( times_times_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_integer.rep_eq
% 0.30/0.67  thf(fact_9448_less__integer_Orep__eq,axiom,
% 0.30/0.67      ( ord_le6747313008572928689nteger
% 0.30/0.67      = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_integer.rep_eq
% 0.30/0.67  thf(fact_9449_integer__less__iff,axiom,
% 0.30/0.67      ( ord_le6747313008572928689nteger
% 0.30/0.67      = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_less_iff
% 0.30/0.67  thf(fact_9450_integer__less__eq__iff,axiom,
% 0.30/0.67      ( ord_le3102999989581377725nteger
% 0.30/0.67      = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_less_eq_iff
% 0.30/0.67  thf(fact_9451_less__eq__integer_Orep__eq,axiom,
% 0.30/0.67      ( ord_le3102999989581377725nteger
% 0.30/0.67      = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_integer.rep_eq
% 0.30/0.67  thf(fact_9452_divmod__integer__def,axiom,
% 0.30/0.67      ( code_divmod_integer
% 0.30/0.67      = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L2 ) @ ( modulo364778990260209775nteger @ K2 @ L2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_integer_def
% 0.30/0.67  thf(fact_9453_num__of__integer__code,axiom,
% 0.30/0.67      ( code_num_of_integer
% 0.30/0.67      = ( ^ [K2: code_integer] :
% 0.30/0.67            ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 0.30/0.67            @ ( produc7336495610019696514er_num
% 0.30/0.67              @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 0.30/0.67              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_integer_code
% 0.30/0.67  thf(fact_9454_bit__cut__integer__code,axiom,
% 0.30/0.67      ( code_bit_cut_integer
% 0.30/0.67      = ( ^ [K2: code_integer] :
% 0.30/0.67            ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.30/0.67            @ ( produc9125791028180074456eger_o
% 0.30/0.67              @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
% 0.30/0.67              @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bit_cut_integer_code
% 0.30/0.67  thf(fact_9455_nat__of__integer__code,axiom,
% 0.30/0.67      ( code_nat_of_integer
% 0.30/0.67      = ( ^ [K2: code_integer] :
% 0.30/0.67            ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.30/0.67            @ ( produc1555791787009142072er_nat
% 0.30/0.67              @ ^ [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 ) )
% 0.30/0.67              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat_of_integer_code
% 0.30/0.67  thf(fact_9456_nat__of__integer__non__positive,axiom,
% 0.30/0.67      ! [K: code_integer] :
% 0.30/0.67        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.30/0.67       => ( ( code_nat_of_integer @ K )
% 0.30/0.67          = zero_zero_nat ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat_of_integer_non_positive
% 0.30/0.67  thf(fact_9457_nat__of__integer__code__post_I3_J,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.30/0.67        = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat_of_integer_code_post(3)
% 0.30/0.67  thf(fact_9458_divmod__abs__code_I6_J,axiom,
% 0.30/0.67      ! [J: code_integer] :
% 0.30/0.67        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 0.30/0.67        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_abs_code(6)
% 0.30/0.67  thf(fact_9459_divmod__abs__code_I5_J,axiom,
% 0.30/0.67      ! [J: code_integer] :
% 0.30/0.67        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 0.30/0.67        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_abs_code(5)
% 0.30/0.67  thf(fact_9460_divmod__abs__def,axiom,
% 0.30/0.67      ( code_divmod_abs
% 0.30/0.67      = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_abs_def
% 0.30/0.67  thf(fact_9461_divmod__integer__code,axiom,
% 0.30/0.67      ( code_divmod_integer
% 0.30/0.67      = ( ^ [K2: code_integer,L2: code_integer] :
% 0.30/0.67            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.30/0.67            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.30/0.67              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
% 0.30/0.67                @ ( produc6916734918728496179nteger
% 0.30/0.67                  @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
% 0.30/0.67                  @ ( code_divmod_abs @ K2 @ L2 ) ) )
% 0.30/0.67              @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.30/0.67                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.30/0.67                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
% 0.30/0.67                    @ ( produc6916734918728496179nteger
% 0.30/0.67                      @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
% 0.30/0.67                      @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_integer_code
% 0.30/0.67  thf(fact_9462_csqrt_Osimps_I1_J,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( re @ ( csqrt @ Z ) )
% 0.30/0.67        = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt.simps(1)
% 0.30/0.67  thf(fact_9463_card__Collect__less__nat,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( finite_card_nat
% 0.30/0.67          @ ( collect_nat
% 0.30/0.67            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 0.30/0.67        = N ) ).
% 0.30/0.67  
% 0.30/0.67  % card_Collect_less_nat
% 0.30/0.67  thf(fact_9464_card__atLeastLessThan,axiom,
% 0.30/0.67      ! [L: nat,U: nat] :
% 0.30/0.67        ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 0.30/0.67        = ( minus_minus_nat @ U @ L ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_atLeastLessThan
% 0.30/0.67  thf(fact_9465_card__Collect__le__nat,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( finite_card_nat
% 0.30/0.67          @ ( collect_nat
% 0.30/0.67            @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 0.30/0.67        = ( suc @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_Collect_le_nat
% 0.30/0.67  thf(fact_9466_card__atLeastAtMost,axiom,
% 0.30/0.67      ! [L: nat,U: nat] :
% 0.30/0.67        ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.30/0.67        = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_atLeastAtMost
% 0.30/0.67  thf(fact_9467_complex__Re__numeral,axiom,
% 0.30/0.67      ! [V: num] :
% 0.30/0.67        ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.67        = ( numeral_numeral_real @ V ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_Re_numeral
% 0.30/0.67  thf(fact_9468_Re__divide__numeral,axiom,
% 0.30/0.67      ! [Z: complex,W: num] :
% 0.30/0.67        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.67        = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_divide_numeral
% 0.30/0.67  thf(fact_9469_complex__Re__le__cmod,axiom,
% 0.30/0.67      ! [X3: complex] : ( ord_less_eq_real @ ( re @ X3 ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_Re_le_cmod
% 0.30/0.67  thf(fact_9470_scaleR__complex_Osimps_I1_J,axiom,
% 0.30/0.67      ! [R: real,X3: complex] :
% 0.30/0.67        ( ( re @ ( real_V2046097035970521341omplex @ R @ X3 ) )
% 0.30/0.67        = ( times_times_real @ R @ ( re @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % scaleR_complex.simps(1)
% 0.30/0.67  thf(fact_9471_abs__Re__le__cmod,axiom,
% 0.30/0.67      ! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % abs_Re_le_cmod
% 0.30/0.67  thf(fact_9472_Re__csqrt,axiom,
% 0.30/0.67      ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_csqrt
% 0.30/0.67  thf(fact_9473_card__less__Suc2,axiom,
% 0.30/0.67      ! [M7: set_nat,I: nat] :
% 0.30/0.67        ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.67       => ( ( finite_card_nat
% 0.30/0.67            @ ( collect_nat
% 0.30/0.67              @ ^ [K2: nat] :
% 0.30/0.67                  ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 0.30/0.67                  & ( ord_less_nat @ K2 @ I ) ) ) )
% 0.30/0.67          = ( finite_card_nat
% 0.30/0.67            @ ( collect_nat
% 0.30/0.67              @ ^ [K2: nat] :
% 0.30/0.67                  ( ( member_nat @ K2 @ M7 )
% 0.30/0.67                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_less_Suc2
% 0.30/0.67  thf(fact_9474_card__less__Suc,axiom,
% 0.30/0.67      ! [M7: set_nat,I: nat] :
% 0.30/0.67        ( ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.67       => ( ( suc
% 0.30/0.67            @ ( finite_card_nat
% 0.30/0.67              @ ( collect_nat
% 0.30/0.67                @ ^ [K2: nat] :
% 0.30/0.67                    ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 0.30/0.67                    & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 0.30/0.67          = ( finite_card_nat
% 0.30/0.67            @ ( collect_nat
% 0.30/0.67              @ ^ [K2: nat] :
% 0.30/0.67                  ( ( member_nat @ K2 @ M7 )
% 0.30/0.67                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_less_Suc
% 0.30/0.67  thf(fact_9475_card__less,axiom,
% 0.30/0.67      ! [M7: set_nat,I: nat] :
% 0.30/0.67        ( ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.67       => ( ( finite_card_nat
% 0.30/0.67            @ ( collect_nat
% 0.30/0.67              @ ^ [K2: nat] :
% 0.30/0.67                  ( ( member_nat @ K2 @ M7 )
% 0.30/0.67                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 0.30/0.67         != zero_zero_nat ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_less
% 0.30/0.67  thf(fact_9476_subset__card__intvl__is__intvl,axiom,
% 0.30/0.67      ! [A4: set_nat,K: nat] :
% 0.30/0.67        ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) )
% 0.30/0.67       => ( A4
% 0.30/0.67          = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % subset_card_intvl_is_intvl
% 0.30/0.67  thf(fact_9477_subset__eq__atLeast0__lessThan__card,axiom,
% 0.30/0.67      ! [N4: set_nat,N: nat] :
% 0.30/0.67        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.30/0.67       => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % subset_eq_atLeast0_lessThan_card
% 0.30/0.67  thf(fact_9478_card__sum__le__nat__sum,axiom,
% 0.30/0.67      ! [S2: set_nat] :
% 0.30/0.67        ( ord_less_eq_nat
% 0.30/0.67        @ ( groups3542108847815614940at_nat
% 0.30/0.67          @ ^ [X: nat] : X
% 0.30/0.67          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 0.30/0.67        @ ( groups3542108847815614940at_nat
% 0.30/0.67          @ ^ [X: nat] : X
% 0.30/0.67          @ S2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_sum_le_nat_sum
% 0.30/0.67  thf(fact_9479_card__nth__roots,axiom,
% 0.30/0.67      ! [C: complex,N: nat] :
% 0.30/0.67        ( ( C != zero_zero_complex )
% 0.30/0.67       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67         => ( ( finite_card_complex
% 0.30/0.67              @ ( collect_complex
% 0.30/0.67                @ ^ [Z2: complex] :
% 0.30/0.67                    ( ( power_power_complex @ Z2 @ N )
% 0.30/0.67                    = C ) ) )
% 0.30/0.67            = N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_nth_roots
% 0.30/0.67  thf(fact_9480_card__roots__unity__eq,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( finite_card_complex
% 0.30/0.67            @ ( collect_complex
% 0.30/0.67              @ ^ [Z2: complex] :
% 0.30/0.67                  ( ( power_power_complex @ Z2 @ N )
% 0.30/0.67                  = one_one_complex ) ) )
% 0.30/0.67          = N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_roots_unity_eq
% 0.30/0.67  thf(fact_9481_cmod__plus__Re__le__0__iff,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.30/0.67        = ( ( re @ Z )
% 0.30/0.67          = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cmod_plus_Re_le_0_iff
% 0.30/0.67  thf(fact_9482_cos__n__Re__cis__pow__n,axiom,
% 0.30/0.67      ! [N: nat,A: real] :
% 0.30/0.67        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.30/0.67        = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cos_n_Re_cis_pow_n
% 0.30/0.67  thf(fact_9483_csqrt_Ocode,axiom,
% 0.30/0.67      ( csqrt
% 0.30/0.67      = ( ^ [Z2: complex] :
% 0.30/0.67            ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.67            @ ( times_times_real
% 0.30/0.67              @ ( if_real
% 0.30/0.67                @ ( ( im @ Z2 )
% 0.30/0.67                  = zero_zero_real )
% 0.30/0.67                @ one_one_real
% 0.30/0.67                @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
% 0.30/0.67              @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt.code
% 0.30/0.67  thf(fact_9484_csqrt_Osimps_I2_J,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( im @ ( csqrt @ Z ) )
% 0.30/0.67        = ( times_times_real
% 0.30/0.67          @ ( if_real
% 0.30/0.67            @ ( ( im @ Z )
% 0.30/0.67              = zero_zero_real )
% 0.30/0.67            @ one_one_real
% 0.30/0.67            @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.30/0.67          @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt.simps(2)
% 0.30/0.67  thf(fact_9485_csqrt__of__real__nonpos,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( ( im @ X3 )
% 0.30/0.67          = zero_zero_real )
% 0.30/0.67       => ( ( ord_less_eq_real @ ( re @ X3 ) @ zero_zero_real )
% 0.30/0.67         => ( ( csqrt @ X3 )
% 0.30/0.67            = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt_of_real_nonpos
% 0.30/0.67  thf(fact_9486_complex__Im__numeral,axiom,
% 0.30/0.67      ! [V: num] :
% 0.30/0.67        ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.67        = zero_zero_real ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_Im_numeral
% 0.30/0.67  thf(fact_9487_Im__i__times,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.30/0.67        = ( re @ Z ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_i_times
% 0.30/0.67  thf(fact_9488_Re__i__times,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.30/0.67        = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_i_times
% 0.30/0.67  thf(fact_9489_Im__divide__numeral,axiom,
% 0.30/0.67      ! [Z: complex,W: num] :
% 0.30/0.67        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.67        = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_divide_numeral
% 0.30/0.67  thf(fact_9490_csqrt__of__real__nonneg,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( ( im @ X3 )
% 0.30/0.67          = zero_zero_real )
% 0.30/0.67       => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) )
% 0.30/0.67         => ( ( csqrt @ X3 )
% 0.30/0.67            = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X3 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt_of_real_nonneg
% 0.30/0.67  thf(fact_9491_csqrt__minus,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( ( ord_less_real @ ( im @ X3 ) @ zero_zero_real )
% 0.30/0.67          | ( ( ( im @ X3 )
% 0.30/0.67              = zero_zero_real )
% 0.30/0.67            & ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) ) ) )
% 0.30/0.67       => ( ( csqrt @ ( uminus1482373934393186551omplex @ X3 ) )
% 0.30/0.67          = ( times_times_complex @ imaginary_unit @ ( csqrt @ X3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt_minus
% 0.30/0.67  thf(fact_9492_scaleR__complex_Osimps_I2_J,axiom,
% 0.30/0.67      ! [R: real,X3: complex] :
% 0.30/0.67        ( ( im @ ( real_V2046097035970521341omplex @ R @ X3 ) )
% 0.30/0.67        = ( times_times_real @ R @ ( im @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % scaleR_complex.simps(2)
% 0.30/0.67  thf(fact_9493_abs__Im__le__cmod,axiom,
% 0.30/0.67      ! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % abs_Im_le_cmod
% 0.30/0.67  thf(fact_9494_times__complex_Osimps_I2_J,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( im @ ( times_times_complex @ X3 @ Y3 ) )
% 0.30/0.67        = ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_complex.simps(2)
% 0.30/0.67  thf(fact_9495_cmod__Im__le__iff,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( ( re @ X3 )
% 0.30/0.67          = ( re @ Y3 ) )
% 0.30/0.67       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.30/0.67          = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cmod_Im_le_iff
% 0.30/0.67  thf(fact_9496_cmod__Re__le__iff,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( ( im @ X3 )
% 0.30/0.67          = ( im @ Y3 ) )
% 0.30/0.67       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.30/0.67          = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cmod_Re_le_iff
% 0.30/0.67  thf(fact_9497_times__complex_Osimps_I1_J,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( re @ ( times_times_complex @ X3 @ Y3 ) )
% 0.30/0.67        = ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_complex.simps(1)
% 0.30/0.67  thf(fact_9498_scaleR__complex_Ocode,axiom,
% 0.30/0.67      ( real_V2046097035970521341omplex
% 0.30/0.67      = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % scaleR_complex.code
% 0.30/0.67  thf(fact_9499_csqrt__principal,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.30/0.67        | ( ( ( re @ ( csqrt @ Z ) )
% 0.30/0.67            = zero_zero_real )
% 0.30/0.67          & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt_principal
% 0.30/0.67  thf(fact_9500_cmod__le,axiom,
% 0.30/0.67      ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cmod_le
% 0.30/0.67  thf(fact_9501_sin__n__Im__cis__pow__n,axiom,
% 0.30/0.67      ! [N: nat,A: real] :
% 0.30/0.67        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.30/0.67        = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sin_n_Im_cis_pow_n
% 0.30/0.67  thf(fact_9502_Re__exp,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( re @ ( exp_complex @ Z ) )
% 0.30/0.67        = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_exp
% 0.30/0.67  thf(fact_9503_Im__exp,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( im @ ( exp_complex @ Z ) )
% 0.30/0.67        = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_exp
% 0.30/0.67  thf(fact_9504_complex__eq,axiom,
% 0.30/0.67      ! [A: complex] :
% 0.30/0.67        ( A
% 0.30/0.67        = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_eq
% 0.30/0.67  thf(fact_9505_times__complex_Ocode,axiom,
% 0.30/0.67      ( times_times_complex
% 0.30/0.67      = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_complex.code
% 0.30/0.67  thf(fact_9506_exp__eq__polar,axiom,
% 0.30/0.67      ( exp_complex
% 0.30/0.67      = ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % exp_eq_polar
% 0.30/0.67  thf(fact_9507_cmod__power2,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.67        = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cmod_power2
% 0.30/0.67  thf(fact_9508_Im__power2,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( im @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.67        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X3 ) ) @ ( im @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_power2
% 0.30/0.67  thf(fact_9509_Re__power2,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( re @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.67        = ( minus_minus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_power2
% 0.30/0.67  thf(fact_9510_complex__eq__0,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( Z = zero_zero_complex )
% 0.30/0.67        = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.67          = zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_eq_0
% 0.30/0.67  thf(fact_9511_norm__complex__def,axiom,
% 0.30/0.67      ( real_V1022390504157884413omplex
% 0.30/0.67      = ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % norm_complex_def
% 0.30/0.67  thf(fact_9512_inverse__complex_Osimps_I1_J,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( re @ ( invers8013647133539491842omplex @ X3 ) )
% 0.30/0.67        = ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % inverse_complex.simps(1)
% 0.30/0.67  thf(fact_9513_complex__neq__0,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( Z != zero_zero_complex )
% 0.30/0.67        = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_neq_0
% 0.30/0.67  thf(fact_9514_Re__divide,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( re @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 0.30/0.67        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_divide
% 0.30/0.67  thf(fact_9515_csqrt__unique,axiom,
% 0.30/0.67      ! [W: complex,Z: complex] :
% 0.30/0.67        ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.67          = Z )
% 0.30/0.67       => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.30/0.67            | ( ( ( re @ W )
% 0.30/0.67                = zero_zero_real )
% 0.30/0.67              & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.30/0.67         => ( ( csqrt @ Z )
% 0.30/0.67            = W ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt_unique
% 0.30/0.67  thf(fact_9516_csqrt__square,axiom,
% 0.30/0.67      ! [B2: complex] :
% 0.30/0.67        ( ( ( ord_less_real @ zero_zero_real @ ( re @ B2 ) )
% 0.30/0.67          | ( ( ( re @ B2 )
% 0.30/0.67              = zero_zero_real )
% 0.30/0.67            & ( ord_less_eq_real @ zero_zero_real @ ( im @ B2 ) ) ) )
% 0.30/0.67       => ( ( csqrt @ ( power_power_complex @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.67          = B2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % csqrt_square
% 0.30/0.67  thf(fact_9517_inverse__complex_Osimps_I2_J,axiom,
% 0.30/0.67      ! [X3: complex] :
% 0.30/0.67        ( ( im @ ( invers8013647133539491842omplex @ X3 ) )
% 0.30/0.67        = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % inverse_complex.simps(2)
% 0.30/0.67  thf(fact_9518_Im__divide,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( im @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 0.30/0.67        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_divide
% 0.30/0.67  thf(fact_9519_complex__abs__le__norm,axiom,
% 0.30/0.67      ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_abs_le_norm
% 0.30/0.67  thf(fact_9520_complex__unit__circle,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( Z != zero_zero_complex )
% 0.30/0.67       => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.67          = one_one_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_unit_circle
% 0.30/0.67  thf(fact_9521_inverse__complex_Ocode,axiom,
% 0.30/0.67      ( invers8013647133539491842omplex
% 0.30/0.67      = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % inverse_complex.code
% 0.30/0.67  thf(fact_9522_Complex__divide,axiom,
% 0.30/0.67      ( divide1717551699836669952omplex
% 0.30/0.67      = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Complex_divide
% 0.30/0.67  thf(fact_9523_Im__Reals__divide,axiom,
% 0.30/0.67      ! [R: complex,Z: complex] :
% 0.30/0.67        ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.30/0.67       => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
% 0.30/0.67          = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_Reals_divide
% 0.30/0.67  thf(fact_9524_imaginary__eq__real__iff,axiom,
% 0.30/0.67      ! [Y3: complex,X3: complex] :
% 0.30/0.67        ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.30/0.67       => ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
% 0.30/0.67         => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 0.30/0.67              = X3 )
% 0.30/0.67            = ( ( X3 = zero_zero_complex )
% 0.30/0.67              & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % imaginary_eq_real_iff
% 0.30/0.67  thf(fact_9525_real__eq__imaginary__iff,axiom,
% 0.30/0.67      ! [Y3: complex,X3: complex] :
% 0.30/0.67        ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.30/0.67       => ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
% 0.30/0.67         => ( ( X3
% 0.30/0.67              = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 0.30/0.67            = ( ( X3 = zero_zero_complex )
% 0.30/0.67              & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_eq_imaginary_iff
% 0.30/0.67  thf(fact_9526_Re__Reals__divide,axiom,
% 0.30/0.67      ! [R: complex,Z: complex] :
% 0.30/0.67        ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.30/0.67       => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
% 0.30/0.67          = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_Reals_divide
% 0.30/0.67  thf(fact_9527_complex__mult__cnj,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.30/0.67        = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_mult_cnj
% 0.30/0.67  thf(fact_9528_cnj__add__mult__eq__Re,axiom,
% 0.30/0.67      ! [Z: complex,W: complex] :
% 0.30/0.67        ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.30/0.67        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % cnj_add_mult_eq_Re
% 0.30/0.67  thf(fact_9529_complex__div__cnj,axiom,
% 0.30/0.67      ( divide1717551699836669952omplex
% 0.30/0.67      = ( ^ [A5: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_div_cnj
% 0.30/0.67  thf(fact_9530_complex__cnj__mult,axiom,
% 0.30/0.67      ! [X3: complex,Y3: complex] :
% 0.30/0.67        ( ( cnj @ ( times_times_complex @ X3 @ Y3 ) )
% 0.30/0.67        = ( times_times_complex @ ( cnj @ X3 ) @ ( cnj @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_cnj_mult
% 0.30/0.67  thf(fact_9531_complex__cnj__numeral,axiom,
% 0.30/0.67      ! [W: num] :
% 0.30/0.67        ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.67        = ( numera6690914467698888265omplex @ W ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_cnj_numeral
% 0.30/0.67  thf(fact_9532_complex__cnj__neg__numeral,axiom,
% 0.30/0.67      ! [W: num] :
% 0.30/0.67        ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.67        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_cnj_neg_numeral
% 0.30/0.67  thf(fact_9533_complex__In__mult__cnj__zero,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.30/0.67        = zero_zero_real ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_In_mult_cnj_zero
% 0.30/0.67  thf(fact_9534_Re__complex__div__eq__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ( re @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.67          = zero_zero_real )
% 0.30/0.67        = ( ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 0.30/0.67          = zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_complex_div_eq_0
% 0.30/0.67  thf(fact_9535_Im__complex__div__eq__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ( im @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 0.30/0.67          = zero_zero_real )
% 0.30/0.67        = ( ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 0.30/0.67          = zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_complex_div_eq_0
% 0.30/0.67  thf(fact_9536_complex__mod__sqrt__Re__mult__cnj,axiom,
% 0.30/0.67      ( real_V1022390504157884413omplex
% 0.30/0.67      = ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_mod_sqrt_Re_mult_cnj
% 0.30/0.67  thf(fact_9537_Re__complex__div__lt__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.30/0.67        = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_complex_div_lt_0
% 0.30/0.67  thf(fact_9538_Re__complex__div__gt__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.67        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_complex_div_gt_0
% 0.30/0.67  thf(fact_9539_Re__complex__div__le__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.30/0.67        = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_complex_div_le_0
% 0.30/0.67  thf(fact_9540_Re__complex__div__ge__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.67        = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Re_complex_div_ge_0
% 0.30/0.67  thf(fact_9541_Im__complex__div__lt__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.30/0.67        = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_complex_div_lt_0
% 0.30/0.67  thf(fact_9542_Im__complex__div__gt__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.67        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_complex_div_gt_0
% 0.30/0.67  thf(fact_9543_Im__complex__div__le__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 0.30/0.67        = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_complex_div_le_0
% 0.30/0.67  thf(fact_9544_Im__complex__div__ge__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.67        = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Im_complex_div_ge_0
% 0.30/0.67  thf(fact_9545_complex__mod__mult__cnj,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.30/0.67        = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_mod_mult_cnj
% 0.30/0.67  thf(fact_9546_complex__div__gt__0,axiom,
% 0.30/0.67      ! [A: complex,B2: complex] :
% 0.30/0.67        ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.67          = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) )
% 0.30/0.67        & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 0.30/0.67          = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_div_gt_0
% 0.30/0.67  thf(fact_9547_complex__norm__square,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.67        = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_norm_square
% 0.30/0.67  thf(fact_9548_complex__add__cnj,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.30/0.67        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_add_cnj
% 0.30/0.67  thf(fact_9549_complex__diff__cnj,axiom,
% 0.30/0.67      ! [Z: complex] :
% 0.30/0.67        ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.30/0.67        = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.30/0.67  
% 0.30/0.67  % complex_diff_cnj
% 0.30/0.67  thf(fact_9550_nat_Odisc__eq__case_I2_J,axiom,
% 0.30/0.67      ! [Nat: nat] :
% 0.30/0.67        ( ( Nat != zero_zero_nat )
% 0.30/0.67        = ( case_nat_o @ $false
% 0.30/0.67          @ ^ [Uu3: nat] : $true
% 0.30/0.67          @ Nat ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat.disc_eq_case(2)
% 0.30/0.67  thf(fact_9551_nat_Odisc__eq__case_I1_J,axiom,
% 0.30/0.67      ! [Nat: nat] :
% 0.30/0.67        ( ( Nat = zero_zero_nat )
% 0.30/0.67        = ( case_nat_o @ $true
% 0.30/0.67          @ ^ [Uu3: nat] : $false
% 0.30/0.67          @ Nat ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat.disc_eq_case(1)
% 0.30/0.67  thf(fact_9552_less__eq__nat_Osimps_I2_J,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.30/0.67        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_nat.simps(2)
% 0.30/0.67  thf(fact_9553_max__Suc2,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.30/0.67        = ( case_nat_nat @ ( suc @ N )
% 0.30/0.67          @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
% 0.30/0.67          @ M ) ) ).
% 0.30/0.67  
% 0.30/0.67  % max_Suc2
% 0.30/0.67  thf(fact_9554_max__Suc1,axiom,
% 0.30/0.67      ! [N: nat,M: nat] :
% 0.30/0.67        ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.30/0.67        = ( case_nat_nat @ ( suc @ N )
% 0.30/0.67          @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
% 0.30/0.67          @ M ) ) ).
% 0.30/0.67  
% 0.30/0.67  % max_Suc1
% 0.30/0.67  thf(fact_9555_diff__Suc,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.30/0.67        = ( case_nat_nat @ zero_zero_nat
% 0.30/0.67          @ ^ [K2: nat] : K2
% 0.30/0.67          @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % diff_Suc
% 0.30/0.67  thf(fact_9556_pred__def,axiom,
% 0.30/0.67      ( pred
% 0.30/0.67      = ( case_nat_nat @ zero_zero_nat
% 0.30/0.67        @ ^ [X23: nat] : X23 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % pred_def
% 0.30/0.67  thf(fact_9557_floor__real__def,axiom,
% 0.30/0.67      ( archim6058952711729229775r_real
% 0.30/0.67      = ( ^ [X: real] :
% 0.30/0.67            ( the_int
% 0.30/0.67            @ ^ [Z2: int] :
% 0.30/0.67                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 0.30/0.67                & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % floor_real_def
% 0.30/0.67  thf(fact_9558_floor__rat__def,axiom,
% 0.30/0.67      ( archim3151403230148437115or_rat
% 0.30/0.67      = ( ^ [X: rat] :
% 0.30/0.67            ( the_int
% 0.30/0.67            @ ^ [Z2: int] :
% 0.30/0.67                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 0.30/0.67                & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % floor_rat_def
% 0.30/0.67  thf(fact_9559_bezw__0,axiom,
% 0.30/0.67      ! [X3: nat] :
% 0.30/0.67        ( ( bezw @ X3 @ zero_zero_nat )
% 0.30/0.67        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezw_0
% 0.30/0.67  thf(fact_9560_prod__decode__aux_Oelims,axiom,
% 0.30/0.67      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.30/0.67        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.67           => ( Y3
% 0.30/0.67              = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 0.30/0.67          & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.67           => ( Y3
% 0.30/0.67              = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % prod_decode_aux.elims
% 0.30/0.67  thf(fact_9561_less__eq__rat__def,axiom,
% 0.30/0.67      ( ord_less_eq_rat
% 0.30/0.67      = ( ^ [X: rat,Y: rat] :
% 0.30/0.67            ( ( ord_less_rat @ X @ Y )
% 0.30/0.67            | ( X = Y ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_rat_def
% 0.30/0.67  thf(fact_9562_obtain__pos__sum,axiom,
% 0.30/0.67      ! [R: rat] :
% 0.30/0.67        ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.30/0.67       => ~ ! [S3: rat] :
% 0.30/0.67              ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.30/0.67             => ! [T4: rat] :
% 0.30/0.67                  ( ( ord_less_rat @ zero_zero_rat @ T4 )
% 0.30/0.67                 => ( R
% 0.30/0.67                   != ( plus_plus_rat @ S3 @ T4 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % obtain_pos_sum
% 0.30/0.67  thf(fact_9563_sgn__rat__def,axiom,
% 0.30/0.67      ( sgn_sgn_rat
% 0.30/0.67      = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sgn_rat_def
% 0.30/0.67  thf(fact_9564_abs__rat__def,axiom,
% 0.30/0.67      ( abs_abs_rat
% 0.30/0.67      = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % abs_rat_def
% 0.30/0.67  thf(fact_9565_prod__decode__aux_Osimps,axiom,
% 0.30/0.67      ( nat_prod_decode_aux
% 0.30/0.67      = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % prod_decode_aux.simps
% 0.30/0.67  thf(fact_9566_prod__decode__aux_Opelims,axiom,
% 0.30/0.67      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.30/0.67        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 0.30/0.67         => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                    = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 0.30/0.67                & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                    = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
% 0.30/0.67             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % prod_decode_aux.pelims
% 0.30/0.67  thf(fact_9567_rat__inverse__code,axiom,
% 0.30/0.67      ! [P4: rat] :
% 0.30/0.67        ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_inverse_code
% 0.30/0.67  thf(fact_9568_normalize__negative,axiom,
% 0.30/0.67      ! [Q2: int,P4: int] :
% 0.30/0.67        ( ( ord_less_int @ Q2 @ zero_zero_int )
% 0.30/0.67       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.67          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % normalize_negative
% 0.30/0.67  thf(fact_9569_quotient__of__number_I3_J,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.30/0.67        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_number(3)
% 0.30/0.67  thf(fact_9570_normalize__denom__zero,axiom,
% 0.30/0.67      ! [P4: int] :
% 0.30/0.67        ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 0.30/0.67        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % normalize_denom_zero
% 0.30/0.67  thf(fact_9571_rat__one__code,axiom,
% 0.30/0.67      ( ( quotient_of @ one_one_rat )
% 0.30/0.67      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_one_code
% 0.30/0.67  thf(fact_9572_rat__zero__code,axiom,
% 0.30/0.67      ( ( quotient_of @ zero_zero_rat )
% 0.30/0.67      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_zero_code
% 0.30/0.67  thf(fact_9573_quotient__of__number_I5_J,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.67        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_number(5)
% 0.30/0.67  thf(fact_9574_quotient__of__number_I4_J,axiom,
% 0.30/0.67      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.67      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_number(4)
% 0.30/0.67  thf(fact_9575_divide__rat__def,axiom,
% 0.30/0.67      ( divide_divide_rat
% 0.30/0.67      = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divide_rat_def
% 0.30/0.67  thf(fact_9576_rat__divide__code,axiom,
% 0.30/0.67      ! [P4: rat,Q2: rat] :
% 0.30/0.67        ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int,C3: int] :
% 0.30/0.67              ( produc4245557441103728435nt_int
% 0.30/0.67              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
% 0.30/0.67              @ ( quotient_of @ Q2 ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_divide_code
% 0.30/0.67  thf(fact_9577_rat__times__code,axiom,
% 0.30/0.67      ! [P4: rat,Q2: rat] :
% 0.30/0.67        ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int,C3: int] :
% 0.30/0.67              ( produc4245557441103728435nt_int
% 0.30/0.67              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
% 0.30/0.67              @ ( quotient_of @ Q2 ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_times_code
% 0.30/0.67  thf(fact_9578_quotient__of__div,axiom,
% 0.30/0.67      ! [R: rat,N: int,D: int] :
% 0.30/0.67        ( ( ( quotient_of @ R )
% 0.30/0.67          = ( product_Pair_int_int @ N @ D ) )
% 0.30/0.67       => ( R
% 0.30/0.67          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_div
% 0.30/0.67  thf(fact_9579_rat__plus__code,axiom,
% 0.30/0.67      ! [P4: rat,Q2: rat] :
% 0.30/0.67        ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int,C3: int] :
% 0.30/0.67              ( produc4245557441103728435nt_int
% 0.30/0.67              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 0.30/0.67              @ ( quotient_of @ Q2 ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_plus_code
% 0.30/0.67  thf(fact_9580_rat__minus__code,axiom,
% 0.30/0.67      ! [P4: rat,Q2: rat] :
% 0.30/0.67        ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int,C3: int] :
% 0.30/0.67              ( produc4245557441103728435nt_int
% 0.30/0.67              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 0.30/0.67              @ ( quotient_of @ Q2 ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_minus_code
% 0.30/0.67  thf(fact_9581_quotient__of__denom__pos,axiom,
% 0.30/0.67      ! [R: rat,P4: int,Q2: int] :
% 0.30/0.67        ( ( ( quotient_of @ R )
% 0.30/0.67          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.67       => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_denom_pos
% 0.30/0.67  thf(fact_9582_rat__uminus__code,axiom,
% 0.30/0.67      ! [P4: rat] :
% 0.30/0.67        ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_uminus_code
% 0.30/0.67  thf(fact_9583_rat__abs__code,axiom,
% 0.30/0.67      ! [P4: rat] :
% 0.30/0.67        ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 0.30/0.67        = ( produc4245557441103728435nt_int
% 0.30/0.67          @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
% 0.30/0.67          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_abs_code
% 0.30/0.67  thf(fact_9584_normalize__denom__pos,axiom,
% 0.30/0.67      ! [R: product_prod_int_int,P4: int,Q2: int] :
% 0.30/0.67        ( ( ( normalize @ R )
% 0.30/0.67          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.67       => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % normalize_denom_pos
% 0.30/0.67  thf(fact_9585_normalize__crossproduct,axiom,
% 0.30/0.67      ! [Q2: int,S: int,P4: int,R: int] :
% 0.30/0.67        ( ( Q2 != zero_zero_int )
% 0.30/0.67       => ( ( S != zero_zero_int )
% 0.30/0.67         => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.67              = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
% 0.30/0.67           => ( ( times_times_int @ P4 @ S )
% 0.30/0.67              = ( times_times_int @ R @ Q2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % normalize_crossproduct
% 0.30/0.67  thf(fact_9586_rat__less__code,axiom,
% 0.30/0.67      ( ord_less_rat
% 0.30/0.67      = ( ^ [P5: rat,Q4: rat] :
% 0.30/0.67            ( produc4947309494688390418_int_o
% 0.30/0.67            @ ^ [A5: int,C3: int] :
% 0.30/0.67                ( produc4947309494688390418_int_o
% 0.30/0.67                @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 0.30/0.67                @ ( quotient_of @ Q4 ) )
% 0.30/0.67            @ ( quotient_of @ P5 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_less_code
% 0.30/0.67  thf(fact_9587_rat__less__eq__code,axiom,
% 0.30/0.67      ( ord_less_eq_rat
% 0.30/0.67      = ( ^ [P5: rat,Q4: rat] :
% 0.30/0.67            ( produc4947309494688390418_int_o
% 0.30/0.67            @ ^ [A5: int,C3: int] :
% 0.30/0.67                ( produc4947309494688390418_int_o
% 0.30/0.67                @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 0.30/0.67                @ ( quotient_of @ Q4 ) )
% 0.30/0.67            @ ( quotient_of @ P5 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_less_eq_code
% 0.30/0.67  thf(fact_9588_quotient__of__int,axiom,
% 0.30/0.67      ! [A: int] :
% 0.30/0.67        ( ( quotient_of @ ( of_int @ A ) )
% 0.30/0.67        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_int
% 0.30/0.67  thf(fact_9589_drop__bit__numeral__minus__bit1,axiom,
% 0.30/0.67      ! [L: num,K: num] :
% 0.30/0.67        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.67        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_numeral_minus_bit1
% 0.30/0.67  thf(fact_9590_Suc__0__div__numeral,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.67        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Suc_0_div_numeral
% 0.30/0.67  thf(fact_9591_drop__bit__nonnegative__int__iff,axiom,
% 0.30/0.67      ! [N: nat,K: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.30/0.67        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_nonnegative_int_iff
% 0.30/0.67  thf(fact_9592_drop__bit__negative__int__iff,axiom,
% 0.30/0.67      ! [N: nat,K: int] :
% 0.30/0.67        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.30/0.67        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_negative_int_iff
% 0.30/0.67  thf(fact_9593_drop__bit__minus__one,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.67        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_minus_one
% 0.30/0.67  thf(fact_9594_fst__divmod__nat,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.30/0.67        = ( divide_divide_nat @ M @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % fst_divmod_nat
% 0.30/0.67  thf(fact_9595_drop__bit__Suc__minus__bit0,axiom,
% 0.30/0.67      ! [N: nat,K: num] :
% 0.30/0.67        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.67        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_Suc_minus_bit0
% 0.30/0.67  thf(fact_9596_drop__bit__numeral__minus__bit0,axiom,
% 0.30/0.67      ! [L: num,K: num] :
% 0.30/0.67        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.67        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_numeral_minus_bit0
% 0.30/0.67  thf(fact_9597_drop__bit__Suc__minus__bit1,axiom,
% 0.30/0.67      ! [N: nat,K: num] :
% 0.30/0.67        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.67        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_Suc_minus_bit1
% 0.30/0.67  thf(fact_9598_drop__bit__push__bit__int,axiom,
% 0.30/0.67      ! [M: nat,N: nat,K: int] :
% 0.30/0.67        ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.30/0.67        = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_push_bit_int
% 0.30/0.67  thf(fact_9599_drop__bit__int__def,axiom,
% 0.30/0.67      ( bit_se8568078237143864401it_int
% 0.30/0.67      = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_int_def
% 0.30/0.67  thf(fact_9600_Frct__code__post_I5_J,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.67        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(5)
% 0.30/0.67  thf(fact_9601_Frct__code__post_I6_J,axiom,
% 0.30/0.67      ! [K: num,L: num] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.30/0.67        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(6)
% 0.30/0.67  thf(fact_9602_Suc__0__mod__numeral,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.67        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Suc_0_mod_numeral
% 0.30/0.67  thf(fact_9603_drop__bit__of__Suc__0,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.67        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_of_Suc_0
% 0.30/0.67  thf(fact_9604_drop__bit__nat__eq,axiom,
% 0.30/0.67      ! [N: nat,K: int] :
% 0.30/0.67        ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 0.30/0.67        = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_nat_eq
% 0.30/0.67  thf(fact_9605_drop__bit__nat__def,axiom,
% 0.30/0.67      ( bit_se8570568707652914677it_nat
% 0.30/0.67      = ( ^ [N2: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % drop_bit_nat_def
% 0.30/0.67  thf(fact_9606_Frct__code__post_I1_J,axiom,
% 0.30/0.67      ! [A: int] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 0.30/0.67        = zero_zero_rat ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(1)
% 0.30/0.67  thf(fact_9607_Frct__code__post_I2_J,axiom,
% 0.30/0.67      ! [A: int] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 0.30/0.67        = zero_zero_rat ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(2)
% 0.30/0.67  thf(fact_9608_Frct__code__post_I8_J,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B2 ) ) )
% 0.30/0.67        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(8)
% 0.30/0.67  thf(fact_9609_Frct__code__post_I7_J,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B2 ) )
% 0.30/0.67        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(7)
% 0.30/0.67  thf(fact_9610_Frct__code__post_I3_J,axiom,
% 0.30/0.67      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.30/0.67      = one_one_rat ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(3)
% 0.30/0.67  thf(fact_9611_rat__sgn__code,axiom,
% 0.30/0.67      ! [P4: rat] :
% 0.30/0.67        ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 0.30/0.67        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_sgn_code
% 0.30/0.67  thf(fact_9612_Frct__code__post_I4_J,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.30/0.67        = ( numeral_numeral_rat @ K ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Frct_code_post(4)
% 0.30/0.67  thf(fact_9613_quotient__of__denom__pos_H,axiom,
% 0.30/0.67      ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_denom_pos'
% 0.30/0.67  thf(fact_9614_bezw__non__0,axiom,
% 0.30/0.67      ! [Y3: nat,X3: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 0.30/0.67       => ( ( bezw @ X3 @ Y3 )
% 0.30/0.67          = ( 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 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezw_non_0
% 0.30/0.67  thf(fact_9615_bezw_Osimps,axiom,
% 0.30/0.67      ( bezw
% 0.30/0.67      = ( ^ [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 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezw.simps
% 0.30/0.67  thf(fact_9616_bezw_Oelims,axiom,
% 0.30/0.67      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.30/0.67        ( ( ( bezw @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.67           => ( Y3
% 0.30/0.67              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.30/0.67          & ( ( Xa2 != zero_zero_nat )
% 0.30/0.67           => ( Y3
% 0.30/0.67              = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezw.elims
% 0.30/0.67  thf(fact_9617_bezw_Opelims,axiom,
% 0.30/0.67      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.30/0.67        ( ( ( bezw @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 0.30/0.67         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.30/0.67                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                    = ( 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 ) ) ) ) ) ) ) )
% 0.30/0.67             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezw.pelims
% 0.30/0.67  thf(fact_9618_minus__one__mod__numeral,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % minus_one_mod_numeral
% 0.30/0.67  thf(fact_9619_one__mod__minus__numeral,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % one_mod_minus_numeral
% 0.30/0.67  thf(fact_9620_minus__numeral__mod__numeral,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % minus_numeral_mod_numeral
% 0.30/0.67  thf(fact_9621_numeral__mod__minus__numeral,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % numeral_mod_minus_numeral
% 0.30/0.67  thf(fact_9622_normalize__def,axiom,
% 0.30/0.67      ( normalize
% 0.30/0.67      = ( ^ [P5: product_prod_int_int] :
% 0.30/0.67            ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
% 0.30/0.67            @ ( if_Pro3027730157355071871nt_int
% 0.30/0.67              @ ( ( product_snd_int_int @ P5 )
% 0.30/0.67                = zero_zero_int )
% 0.30/0.67              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.67              @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % normalize_def
% 0.30/0.67  thf(fact_9623_gcd__pos__int,axiom,
% 0.30/0.67      ! [M: int,N: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.30/0.67        = ( ( M != zero_zero_int )
% 0.30/0.67          | ( N != zero_zero_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_pos_int
% 0.30/0.67  thf(fact_9624_gcd__neg__numeral__2__int,axiom,
% 0.30/0.67      ! [X3: int,N: num] :
% 0.30/0.67        ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( gcd_gcd_int @ X3 @ ( numeral_numeral_int @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_neg_numeral_2_int
% 0.30/0.67  thf(fact_9625_gcd__neg__numeral__1__int,axiom,
% 0.30/0.67      ! [N: num,X3: int] :
% 0.30/0.67        ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X3 )
% 0.30/0.67        = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_neg_numeral_1_int
% 0.30/0.67  thf(fact_9626_gcd__ge__0__int,axiom,
% 0.30/0.67      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_ge_0_int
% 0.30/0.67  thf(fact_9627_bezout__int,axiom,
% 0.30/0.67      ! [X3: int,Y3: int] :
% 0.30/0.67      ? [U3: int,V2: int] :
% 0.30/0.67        ( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y3 ) )
% 0.30/0.67        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezout_int
% 0.30/0.67  thf(fact_9628_gcd__mult__distrib__int,axiom,
% 0.30/0.67      ! [K: int,M: int,N: int] :
% 0.30/0.67        ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 0.30/0.67        = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_mult_distrib_int
% 0.30/0.67  thf(fact_9629_gcd__le2__int,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_le2_int
% 0.30/0.67  thf(fact_9630_gcd__le1__int,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.67       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ A ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_le1_int
% 0.30/0.67  thf(fact_9631_gcd__cases__int,axiom,
% 0.30/0.67      ! [X3: int,Y3: int,P: int > $o] :
% 0.30/0.67        ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.67         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.67           => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) )
% 0.30/0.67       => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.67           => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.67             => ( P @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.30/0.67         => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 0.30/0.67             => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.67               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 ) ) ) )
% 0.30/0.67           => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 0.30/0.67               => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.67                 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.30/0.67             => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_cases_int
% 0.30/0.67  thf(fact_9632_gcd__unique__int,axiom,
% 0.30/0.67      ! [D: int,A: int,B2: int] :
% 0.30/0.67        ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.30/0.67          & ( dvd_dvd_int @ D @ A )
% 0.30/0.67          & ( dvd_dvd_int @ D @ B2 )
% 0.30/0.67          & ! [E3: int] :
% 0.30/0.67              ( ( ( dvd_dvd_int @ E3 @ A )
% 0.30/0.67                & ( dvd_dvd_int @ E3 @ B2 ) )
% 0.30/0.67             => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.30/0.67        = ( D
% 0.30/0.67          = ( gcd_gcd_int @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_unique_int
% 0.30/0.67  thf(fact_9633_gcd__non__0__int,axiom,
% 0.30/0.67      ! [Y3: int,X3: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ Y3 )
% 0.30/0.67       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 0.30/0.67          = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_non_0_int
% 0.30/0.67  thf(fact_9634_nat__descend__induct,axiom,
% 0.30/0.67      ! [N: nat,P: nat > $o,M: nat] :
% 0.30/0.67        ( ! [K3: nat] :
% 0.30/0.67            ( ( ord_less_nat @ N @ K3 )
% 0.30/0.67           => ( P @ K3 ) )
% 0.30/0.67       => ( ! [K3: nat] :
% 0.30/0.67              ( ( ord_less_eq_nat @ K3 @ N )
% 0.30/0.67             => ( ! [I2: nat] :
% 0.30/0.67                    ( ( ord_less_nat @ K3 @ I2 )
% 0.30/0.67                   => ( P @ I2 ) )
% 0.30/0.67               => ( P @ K3 ) ) )
% 0.30/0.67         => ( P @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat_descend_induct
% 0.30/0.67  thf(fact_9635_finite__enumerate,axiom,
% 0.30/0.67      ! [S2: set_nat] :
% 0.30/0.67        ( ( finite_finite_nat @ S2 )
% 0.30/0.67       => ? [R3: nat > nat] :
% 0.30/0.67            ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 0.30/0.67            & ! [N9: nat] :
% 0.30/0.67                ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S2 ) )
% 0.30/0.67               => ( member_nat @ ( R3 @ N9 ) @ S2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % finite_enumerate
% 0.30/0.67  thf(fact_9636_divmod__integer__eq__cases,axiom,
% 0.30/0.67      ( code_divmod_integer
% 0.30/0.67      = ( ^ [K2: code_integer,L2: code_integer] :
% 0.30/0.67            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.30/0.67            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.30/0.67              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 0.30/0.67                @ ( if_Pro6119634080678213985nteger
% 0.30/0.67                  @ ( ( sgn_sgn_Code_integer @ K2 )
% 0.30/0.67                    = ( sgn_sgn_Code_integer @ L2 ) )
% 0.30/0.67                  @ ( code_divmod_abs @ K2 @ L2 )
% 0.30/0.67                  @ ( produc6916734918728496179nteger
% 0.30/0.67                    @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
% 0.30/0.67                    @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divmod_integer_eq_cases
% 0.30/0.67  thf(fact_9637_gcd__pos__nat,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.30/0.67        = ( ( M != zero_zero_nat )
% 0.30/0.67          | ( N != zero_zero_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_pos_nat
% 0.30/0.67  thf(fact_9638_gcd__mult__distrib__nat,axiom,
% 0.30/0.67      ! [K: nat,M: nat,N: nat] :
% 0.30/0.67        ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 0.30/0.67        = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_mult_distrib_nat
% 0.30/0.67  thf(fact_9639_gcd__le2__nat,axiom,
% 0.30/0.67      ! [B2: nat,A: nat] :
% 0.30/0.67        ( ( B2 != zero_zero_nat )
% 0.30/0.67       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ B2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_le2_nat
% 0.30/0.67  thf(fact_9640_gcd__le1__nat,axiom,
% 0.30/0.67      ! [A: nat,B2: nat] :
% 0.30/0.67        ( ( A != zero_zero_nat )
% 0.30/0.67       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ A ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_le1_nat
% 0.30/0.67  thf(fact_9641_gcd__diff1__nat,axiom,
% 0.30/0.67      ! [N: nat,M: nat] :
% 0.30/0.67        ( ( ord_less_eq_nat @ N @ M )
% 0.30/0.67       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.30/0.67          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_diff1_nat
% 0.30/0.67  thf(fact_9642_gcd__diff2__nat,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.67       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.30/0.67          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_diff2_nat
% 0.30/0.67  thf(fact_9643_bezout__nat,axiom,
% 0.30/0.67      ! [A: nat,B2: nat] :
% 0.30/0.67        ( ( A != zero_zero_nat )
% 0.30/0.67       => ? [X5: nat,Y4: nat] :
% 0.30/0.67            ( ( times_times_nat @ A @ X5 )
% 0.30/0.67            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezout_nat
% 0.30/0.67  thf(fact_9644_bezout__gcd__nat_H,axiom,
% 0.30/0.67      ! [B2: nat,A: nat] :
% 0.30/0.67      ? [X5: nat,Y4: nat] :
% 0.30/0.67        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
% 0.30/0.67          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 0.30/0.67            = ( gcd_gcd_nat @ A @ B2 ) ) )
% 0.30/0.67        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B2 @ X5 ) )
% 0.30/0.67          & ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.30/0.67            = ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezout_gcd_nat'
% 0.30/0.67  thf(fact_9645_bezw__aux,axiom,
% 0.30/0.67      ! [X3: nat,Y3: nat] :
% 0.30/0.67        ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y3 ) )
% 0.30/0.67        = ( 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 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bezw_aux
% 0.30/0.67  thf(fact_9646_gcd__nat_Opelims,axiom,
% 0.30/0.67      ! [X3: nat,Xa2: nat,Y3: nat] :
% 0.30/0.67        ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 0.30/0.67         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.67                 => ( Y3 = X3 ) )
% 0.30/0.67                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                    = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
% 0.30/0.67             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % gcd_nat.pelims
% 0.30/0.67  thf(fact_9647_Code__Numeral_Onegative__def,axiom,
% 0.30/0.67      ( code_negative
% 0.30/0.67      = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Code_Numeral.negative_def
% 0.30/0.67  thf(fact_9648_Code__Target__Int_Onegative__def,axiom,
% 0.30/0.67      ( code_Target_negative
% 0.30/0.67      = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Code_Target_Int.negative_def
% 0.30/0.67  thf(fact_9649_root__powr__inverse,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67         => ( ( root @ N @ X3 )
% 0.30/0.67            = ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % root_powr_inverse
% 0.30/0.67  thf(fact_9650_real__root__zero,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( root @ N @ zero_zero_real )
% 0.30/0.67        = zero_zero_real ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_zero
% 0.30/0.67  thf(fact_9651_real__root__Suc__0,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67        ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
% 0.30/0.67        = X3 ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_Suc_0
% 0.30/0.67  thf(fact_9652_root__0,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67        ( ( root @ zero_zero_nat @ X3 )
% 0.30/0.67        = zero_zero_real ) ).
% 0.30/0.67  
% 0.30/0.67  % root_0
% 0.30/0.67  thf(fact_9653_real__root__eq__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ( root @ N @ X3 )
% 0.30/0.67            = ( root @ N @ Y3 ) )
% 0.30/0.67          = ( X3 = Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_eq_iff
% 0.30/0.67  thf(fact_9654_real__root__eq__0__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ( root @ N @ X3 )
% 0.30/0.67            = zero_zero_real )
% 0.30/0.67          = ( X3 = zero_zero_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_eq_0_iff
% 0.30/0.67  thf(fact_9655_real__root__less__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
% 0.30/0.67          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_less_iff
% 0.30/0.67  thf(fact_9656_real__root__le__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
% 0.30/0.67          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_le_iff
% 0.30/0.67  thf(fact_9657_real__root__one,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( root @ N @ one_one_real )
% 0.30/0.67          = one_one_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_one
% 0.30/0.67  thf(fact_9658_real__root__eq__1__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ( root @ N @ X3 )
% 0.30/0.67            = one_one_real )
% 0.30/0.67          = ( X3 = one_one_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_eq_1_iff
% 0.30/0.67  thf(fact_9659_real__root__lt__0__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ ( root @ N @ X3 ) @ zero_zero_real )
% 0.30/0.67          = ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_lt_0_iff
% 0.30/0.67  thf(fact_9660_real__root__gt__0__iff,axiom,
% 0.30/0.67      ! [N: nat,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.30/0.67          = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_gt_0_iff
% 0.30/0.67  thf(fact_9661_real__root__le__0__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ zero_zero_real )
% 0.30/0.67          = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_le_0_iff
% 0.30/0.67  thf(fact_9662_real__root__ge__0__iff,axiom,
% 0.30/0.67      ! [N: nat,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.30/0.67          = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_ge_0_iff
% 0.30/0.67  thf(fact_9663_real__root__gt__1__iff,axiom,
% 0.30/0.67      ! [N: nat,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.30/0.67          = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_gt_1_iff
% 0.30/0.67  thf(fact_9664_real__root__lt__1__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ ( root @ N @ X3 ) @ one_one_real )
% 0.30/0.67          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_lt_1_iff
% 0.30/0.67  thf(fact_9665_real__root__le__1__iff,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ one_one_real )
% 0.30/0.67          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_le_1_iff
% 0.30/0.67  thf(fact_9666_real__root__ge__1__iff,axiom,
% 0.30/0.67      ! [N: nat,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.30/0.67          = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_ge_1_iff
% 0.30/0.67  thf(fact_9667_real__root__pow__pos2,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.67         => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
% 0.30/0.67            = X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_pow_pos2
% 0.30/0.67  thf(fact_9668_real__root__inverse,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( root @ N @ ( inverse_inverse_real @ X3 ) )
% 0.30/0.67        = ( inverse_inverse_real @ ( root @ N @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_inverse
% 0.30/0.67  thf(fact_9669_real__root__minus,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( root @ N @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.67        = ( uminus_uminus_real @ ( root @ N @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_minus
% 0.30/0.67  thf(fact_9670_real__root__commute,axiom,
% 0.30/0.67      ! [M: nat,N: nat,X3: real] :
% 0.30/0.67        ( ( root @ M @ ( root @ N @ X3 ) )
% 0.30/0.67        = ( root @ N @ ( root @ M @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_commute
% 0.30/0.67  thf(fact_9671_real__root__mult,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( root @ N @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.67        = ( times_times_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_mult
% 0.30/0.67  thf(fact_9672_real__root__mult__exp,axiom,
% 0.30/0.67      ! [M: nat,N: nat,X3: real] :
% 0.30/0.67        ( ( root @ ( times_times_nat @ M @ N ) @ X3 )
% 0.30/0.67        = ( root @ M @ ( root @ N @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_mult_exp
% 0.30/0.67  thf(fact_9673_real__root__divide,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( root @ N @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.67        = ( divide_divide_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_divide
% 0.30/0.67  thf(fact_9674_real__root__pos__pos__le,axiom,
% 0.30/0.67      ! [X3: real,N: nat] :
% 0.30/0.67        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.67       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_pos_pos_le
% 0.30/0.67  thf(fact_9675_real__root__less__mono,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.67         => ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_less_mono
% 0.30/0.67  thf(fact_9676_real__root__le__mono,axiom,
% 0.30/0.67      ! [N: nat,X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.67         => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_le_mono
% 0.30/0.67  thf(fact_9677_real__root__power,axiom,
% 0.30/0.67      ! [N: nat,X3: real,K: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( root @ N @ ( power_power_real @ X3 @ K ) )
% 0.30/0.67          = ( power_power_real @ ( root @ N @ X3 ) @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_power
% 0.30/0.67  thf(fact_9678_real__root__abs,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( root @ N @ ( abs_abs_real @ X3 ) )
% 0.30/0.67          = ( abs_abs_real @ ( root @ N @ X3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_abs
% 0.30/0.67  thf(fact_9679_sgn__root,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( sgn_sgn_real @ ( root @ N @ X3 ) )
% 0.30/0.67          = ( sgn_sgn_real @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sgn_root
% 0.30/0.67  thf(fact_9680_real__root__gt__zero,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67         => ( ord_less_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_gt_zero
% 0.30/0.67  thf(fact_9681_real__root__strict__decreasing,axiom,
% 0.30/0.67      ! [N: nat,N4: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.67         => ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.67           => ( ord_less_real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_strict_decreasing
% 0.30/0.67  thf(fact_9682_sqrt__def,axiom,
% 0.30/0.67      ( sqrt
% 0.30/0.67      = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sqrt_def
% 0.30/0.67  thf(fact_9683_root__abs__power,axiom,
% 0.30/0.67      ! [N: nat,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 0.30/0.67          = ( abs_abs_real @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % root_abs_power
% 0.30/0.67  thf(fact_9684_real__root__pos__pos,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_pos_pos
% 0.30/0.67  thf(fact_9685_real__root__strict__increasing,axiom,
% 0.30/0.67      ! [N: nat,N4: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_nat @ N @ N4 )
% 0.30/0.67         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67           => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.67             => ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_strict_increasing
% 0.30/0.67  thf(fact_9686_real__root__decreasing,axiom,
% 0.30/0.67      ! [N: nat,N4: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.67         => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.67           => ( ord_less_eq_real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_decreasing
% 0.30/0.67  thf(fact_9687_real__root__pow__pos,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67         => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
% 0.30/0.67            = X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_pow_pos
% 0.30/0.67  thf(fact_9688_real__root__pos__unique,axiom,
% 0.30/0.67      ! [N: nat,Y3: real,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.67         => ( ( ( power_power_real @ Y3 @ N )
% 0.30/0.67              = X3 )
% 0.30/0.67           => ( ( root @ N @ X3 )
% 0.30/0.67              = Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_pos_unique
% 0.30/0.67  thf(fact_9689_real__root__power__cancel,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.67         => ( ( root @ N @ ( power_power_real @ X3 @ N ) )
% 0.30/0.67            = X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_power_cancel
% 0.30/0.67  thf(fact_9690_odd__real__root__power__cancel,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67       => ( ( root @ N @ ( power_power_real @ X3 @ N ) )
% 0.30/0.67          = X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % odd_real_root_power_cancel
% 0.30/0.67  thf(fact_9691_odd__real__root__unique,axiom,
% 0.30/0.67      ! [N: nat,Y3: real,X3: real] :
% 0.30/0.67        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67       => ( ( ( power_power_real @ Y3 @ N )
% 0.30/0.67            = X3 )
% 0.30/0.67         => ( ( root @ N @ X3 )
% 0.30/0.67            = Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % odd_real_root_unique
% 0.30/0.67  thf(fact_9692_odd__real__root__pow,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67       => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
% 0.30/0.67          = X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % odd_real_root_pow
% 0.30/0.67  thf(fact_9693_real__root__increasing,axiom,
% 0.30/0.67      ! [N: nat,N4: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_eq_nat @ N @ N4 )
% 0.30/0.67         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.67           => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.67             => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % real_root_increasing
% 0.30/0.67  thf(fact_9694_sgn__power__root,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X3 ) ) @ N ) )
% 0.30/0.67          = X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sgn_power_root
% 0.30/0.67  thf(fact_9695_root__sgn__power,axiom,
% 0.30/0.67      ! [N: nat,Y3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 0.30/0.67          = Y3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % root_sgn_power
% 0.30/0.67  thf(fact_9696_ln__root,axiom,
% 0.30/0.67      ! [N: nat,B2: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.67         => ( ( ln_ln_real @ ( root @ N @ B2 ) )
% 0.30/0.67            = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % ln_root
% 0.30/0.67  thf(fact_9697_log__root,axiom,
% 0.30/0.67      ! [N: nat,A: real,B2: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.67         => ( ( log @ B2 @ ( root @ N @ A ) )
% 0.30/0.67            = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % log_root
% 0.30/0.67  thf(fact_9698_log__base__root,axiom,
% 0.30/0.67      ! [N: nat,B2: real,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 0.30/0.67         => ( ( log @ ( root @ N @ B2 ) @ X3 )
% 0.30/0.67            = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % log_base_root
% 0.30/0.67  thf(fact_9699_split__root,axiom,
% 0.30/0.67      ! [P: real > $o,N: nat,X3: real] :
% 0.30/0.67        ( ( P @ ( root @ N @ X3 ) )
% 0.30/0.67        = ( ( ( N = zero_zero_nat )
% 0.30/0.67           => ( P @ zero_zero_real ) )
% 0.30/0.67          & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67           => ! [Y: real] :
% 0.30/0.67                ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.30/0.67                  = X3 )
% 0.30/0.67               => ( P @ Y ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % split_root
% 0.30/0.67  thf(fact_9700_bij__betw__nth__root__unity,axiom,
% 0.30/0.67      ! [C: complex,N: nat] :
% 0.30/0.67        ( ( C != zero_zero_complex )
% 0.30/0.67       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.30/0.67            @ ( collect_complex
% 0.30/0.67              @ ^ [Z2: complex] :
% 0.30/0.67                  ( ( power_power_complex @ Z2 @ N )
% 0.30/0.67                  = one_one_complex ) )
% 0.30/0.67            @ ( collect_complex
% 0.30/0.67              @ ^ [Z2: complex] :
% 0.30/0.67                  ( ( power_power_complex @ Z2 @ N )
% 0.30/0.67                  = C ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bij_betw_nth_root_unity
% 0.30/0.67  thf(fact_9701_xor__minus__numerals_I2_J,axiom,
% 0.30/0.67      ! [K: int,N: num] :
% 0.30/0.67        ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_minus_numerals(2)
% 0.30/0.67  thf(fact_9702_xor__minus__numerals_I1_J,axiom,
% 0.30/0.67      ! [N: num,K: int] :
% 0.30/0.67        ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.30/0.67        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_minus_numerals(1)
% 0.30/0.67  thf(fact_9703_sub__BitM__One__eq,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.30/0.67        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sub_BitM_One_eq
% 0.30/0.67  thf(fact_9704_Suc__funpow,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( compow_nat_nat @ N @ suc )
% 0.30/0.67        = ( plus_plus_nat @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Suc_funpow
% 0.30/0.67  thf(fact_9705_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.67      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.30/0.67      @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.30/0.67      @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.67  
% 0.30/0.67  % max_nat.semilattice_neutr_order_axioms
% 0.30/0.67  thf(fact_9706_times__int_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( abs_Integ
% 0.30/0.67          @ ( produc27273713700761075at_nat
% 0.30/0.67            @ ^ [X: nat,Y: nat] :
% 0.30/0.67                ( produc2626176000494625587at_nat
% 0.30/0.67                @ ^ [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 ) ) ) )
% 0.30/0.67            @ Xa2
% 0.30/0.67            @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_int.abs_eq
% 0.30/0.67  thf(fact_9707_eq__Abs__Integ,axiom,
% 0.30/0.67      ! [Z: int] :
% 0.30/0.67        ~ ! [X5: nat,Y4: nat] :
% 0.30/0.67            ( Z
% 0.30/0.67           != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % eq_Abs_Integ
% 0.30/0.67  thf(fact_9708_nat_Oabs__eq,axiom,
% 0.30/0.67      ! [X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( nat2 @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat.abs_eq
% 0.30/0.67  thf(fact_9709_zero__int__def,axiom,
% 0.30/0.67      ( zero_zero_int
% 0.30/0.67      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % zero_int_def
% 0.30/0.67  thf(fact_9710_int__def,axiom,
% 0.30/0.67      ( semiri1314217659103216013at_int
% 0.30/0.67      = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_def
% 0.30/0.67  thf(fact_9711_uminus__int_Oabs__eq,axiom,
% 0.30/0.67      ! [X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( abs_Integ
% 0.30/0.67          @ ( produc2626176000494625587at_nat
% 0.30/0.67            @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
% 0.30/0.67            @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % uminus_int.abs_eq
% 0.30/0.67  thf(fact_9712_one__int__def,axiom,
% 0.30/0.67      ( one_one_int
% 0.30/0.67      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % one_int_def
% 0.30/0.67  thf(fact_9713_less__int_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( produc8739625826339149834_nat_o
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc6081775807080527818_nat_o
% 0.30/0.67              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.30/0.67          @ Xa2
% 0.30/0.67          @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_int.abs_eq
% 0.30/0.67  thf(fact_9714_less__eq__int_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( produc8739625826339149834_nat_o
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc6081775807080527818_nat_o
% 0.30/0.67              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.30/0.67          @ Xa2
% 0.30/0.67          @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_int.abs_eq
% 0.30/0.67  thf(fact_9715_plus__int_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( abs_Integ
% 0.30/0.67          @ ( produc27273713700761075at_nat
% 0.30/0.67            @ ^ [X: nat,Y: nat] :
% 0.30/0.67                ( produc2626176000494625587at_nat
% 0.30/0.67                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 0.30/0.67            @ Xa2
% 0.30/0.67            @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % plus_int.abs_eq
% 0.30/0.67  thf(fact_9716_minus__int_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.67        ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.67        = ( abs_Integ
% 0.30/0.67          @ ( produc27273713700761075at_nat
% 0.30/0.67            @ ^ [X: nat,Y: nat] :
% 0.30/0.67                ( produc2626176000494625587at_nat
% 0.30/0.67                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
% 0.30/0.67            @ Xa2
% 0.30/0.67            @ X3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % minus_int.abs_eq
% 0.30/0.67  thf(fact_9717_less__eq__int_Orep__eq,axiom,
% 0.30/0.67      ( ord_less_eq_int
% 0.30/0.67      = ( ^ [X: int,Xa4: int] :
% 0.30/0.67            ( produc8739625826339149834_nat_o
% 0.30/0.67            @ ^ [Y: nat,Z2: nat] :
% 0.30/0.67                ( produc6081775807080527818_nat_o
% 0.30/0.67                @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 0.30/0.67            @ ( rep_Integ @ X )
% 0.30/0.67            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_int.rep_eq
% 0.30/0.67  thf(fact_9718_less__int_Orep__eq,axiom,
% 0.30/0.67      ( ord_less_int
% 0.30/0.67      = ( ^ [X: int,Xa4: int] :
% 0.30/0.67            ( produc8739625826339149834_nat_o
% 0.30/0.67            @ ^ [Y: nat,Z2: nat] :
% 0.30/0.67                ( produc6081775807080527818_nat_o
% 0.30/0.67                @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 0.30/0.67            @ ( rep_Integ @ X )
% 0.30/0.67            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_int.rep_eq
% 0.30/0.67  thf(fact_9719_nat_Orep__eq,axiom,
% 0.30/0.67      ( nat2
% 0.30/0.67      = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat.rep_eq
% 0.30/0.67  thf(fact_9720_prod__encode__def,axiom,
% 0.30/0.67      ( nat_prod_encode
% 0.30/0.67      = ( produc6842872674320459806at_nat
% 0.30/0.67        @ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % prod_encode_def
% 0.30/0.67  thf(fact_9721_uminus__int__def,axiom,
% 0.30/0.67      ( uminus_uminus_int
% 0.30/0.67      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.30/0.67        @ ( produc2626176000494625587at_nat
% 0.30/0.67          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % uminus_int_def
% 0.30/0.67  thf(fact_9722_le__prod__encode__1,axiom,
% 0.30/0.67      ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % le_prod_encode_1
% 0.30/0.67  thf(fact_9723_le__prod__encode__2,axiom,
% 0.30/0.67      ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % le_prod_encode_2
% 0.30/0.67  thf(fact_9724_prod__encode__prod__decode__aux,axiom,
% 0.30/0.67      ! [K: nat,M: nat] :
% 0.30/0.67        ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.30/0.67        = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.30/0.67  
% 0.30/0.67  % prod_encode_prod_decode_aux
% 0.30/0.67  thf(fact_9725_times__int__def,axiom,
% 0.30/0.67      ( times_times_int
% 0.30/0.67      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.67        @ ( produc27273713700761075at_nat
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc2626176000494625587at_nat
% 0.30/0.67              @ ^ [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 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_int_def
% 0.30/0.67  thf(fact_9726_minus__int__def,axiom,
% 0.30/0.67      ( minus_minus_int
% 0.30/0.67      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.67        @ ( produc27273713700761075at_nat
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc2626176000494625587at_nat
% 0.30/0.67              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % minus_int_def
% 0.30/0.67  thf(fact_9727_plus__int__def,axiom,
% 0.30/0.67      ( plus_plus_int
% 0.30/0.67      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.67        @ ( produc27273713700761075at_nat
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc2626176000494625587at_nat
% 0.30/0.67              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % plus_int_def
% 0.30/0.67  thf(fact_9728_num__of__nat_Osimps_I2_J,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67         => ( ( num_of_nat @ ( suc @ N ) )
% 0.30/0.67            = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67         => ( ( num_of_nat @ ( suc @ N ) )
% 0.30/0.67            = one ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_nat.simps(2)
% 0.30/0.67  thf(fact_9729_pow_Osimps_I3_J,axiom,
% 0.30/0.67      ! [X3: num,Y3: num] :
% 0.30/0.67        ( ( pow @ X3 @ ( bit1 @ Y3 ) )
% 0.30/0.67        = ( times_times_num @ ( sqr @ ( pow @ X3 @ Y3 ) ) @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % pow.simps(3)
% 0.30/0.67  thf(fact_9730_num__of__nat__numeral__eq,axiom,
% 0.30/0.67      ! [Q2: num] :
% 0.30/0.67        ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.67        = Q2 ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_nat_numeral_eq
% 0.30/0.67  thf(fact_9731_sqr_Osimps_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( sqr @ ( bit0 @ N ) )
% 0.30/0.67        = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sqr.simps(2)
% 0.30/0.67  thf(fact_9732_sqr_Osimps_I1_J,axiom,
% 0.30/0.67      ( ( sqr @ one )
% 0.30/0.67      = one ) ).
% 0.30/0.67  
% 0.30/0.67  % sqr.simps(1)
% 0.30/0.67  thf(fact_9733_sqr__conv__mult,axiom,
% 0.30/0.67      ( sqr
% 0.30/0.67      = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sqr_conv_mult
% 0.30/0.67  thf(fact_9734_num__of__nat_Osimps_I1_J,axiom,
% 0.30/0.67      ( ( num_of_nat @ zero_zero_nat )
% 0.30/0.67      = one ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_nat.simps(1)
% 0.30/0.67  thf(fact_9735_numeral__num__of__nat,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.30/0.67          = N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % numeral_num_of_nat
% 0.30/0.67  thf(fact_9736_num__of__nat__One,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.30/0.67       => ( ( num_of_nat @ N )
% 0.30/0.67          = one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_nat_One
% 0.30/0.67  thf(fact_9737_pow_Osimps_I2_J,axiom,
% 0.30/0.67      ! [X3: num,Y3: num] :
% 0.30/0.67        ( ( pow @ X3 @ ( bit0 @ Y3 ) )
% 0.30/0.67        = ( sqr @ ( pow @ X3 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % pow.simps(2)
% 0.30/0.67  thf(fact_9738_num__of__nat__double,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.30/0.67          = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_nat_double
% 0.30/0.67  thf(fact_9739_num__of__nat__plus__distrib,axiom,
% 0.30/0.67      ! [M: nat,N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.67       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.30/0.67            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % num_of_nat_plus_distrib
% 0.30/0.67  thf(fact_9740_sqr_Osimps_I3_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( sqr @ ( bit1 @ N ) )
% 0.30/0.67        = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sqr.simps(3)
% 0.30/0.67  thf(fact_9741_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.30/0.67      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.67        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.67       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.67                ( X3
% 0.30/0.67                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.67           => ( Xa2 = one_one_nat ) )
% 0.30/0.67         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.67                ( ( X3
% 0.30/0.67                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.67               => ( ( Deg2 = Xa2 )
% 0.30/0.67                  & ! [X5: vEBT_VEBT] :
% 0.30/0.67                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                     => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.67                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                  & ( case_o184042715313410164at_nat
% 0.30/0.67                    @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 0.30/0.67                      & ! [X: vEBT_VEBT] :
% 0.30/0.67                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                         => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                    @ ( produc6081775807080527818_nat_o
% 0.30/0.67                      @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                          & ! [I4: nat] :
% 0.30/0.67                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                             => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 0.30/0.67                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.67                          & ( ( Mi3 = Ma3 )
% 0.30/0.67                           => ! [X: vEBT_VEBT] :
% 0.30/0.67                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                          & ( ( Mi3 != Ma3 )
% 0.30/0.67                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.67                              & ! [X: nat] :
% 0.30/0.67                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.67                                   => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67                    @ Mima ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.elims(3)
% 0.30/0.67  thf(fact_9742_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.30/0.67      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.67        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.67       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.67                ( X3
% 0.30/0.67                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.67           => ( Xa2 != one_one_nat ) )
% 0.30/0.67         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.67                ( ( X3
% 0.30/0.67                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.67               => ~ ( ( Deg2 = Xa2 )
% 0.30/0.67                    & ! [X6: vEBT_VEBT] :
% 0.30/0.67                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                       => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.67                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                    & ( case_o184042715313410164at_nat
% 0.30/0.67                      @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 0.30/0.67                        & ! [X: vEBT_VEBT] :
% 0.30/0.67                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                           => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                      @ ( produc6081775807080527818_nat_o
% 0.30/0.67                        @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                            & ! [I4: nat] :
% 0.30/0.67                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                               => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 0.30/0.67                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.67                            & ( ( Mi3 = Ma3 )
% 0.30/0.67                             => ! [X: vEBT_VEBT] :
% 0.30/0.67                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                            & ( ( Mi3 != Ma3 )
% 0.30/0.67                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.67                                & ! [X: nat] :
% 0.30/0.67                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.67                                     => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67                      @ Mima ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.elims(2)
% 0.30/0.67  thf(fact_9743_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.30/0.67      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.67        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.67                ( X3
% 0.30/0.67                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.67           => ( Y3
% 0.30/0.67              = ( Xa2 != one_one_nat ) ) )
% 0.30/0.67         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.67                ( ( X3
% 0.30/0.67                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.67               => ( Y3
% 0.30/0.67                  = ( ~ ( ( Deg2 = Xa2 )
% 0.30/0.67                        & ! [X: vEBT_VEBT] :
% 0.30/0.67                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.67                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                        & ( case_o184042715313410164at_nat
% 0.30/0.67                          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 0.30/0.67                            & ! [X: vEBT_VEBT] :
% 0.30/0.67                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                          @ ( produc6081775807080527818_nat_o
% 0.30/0.67                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                & ! [I4: nat] :
% 0.30/0.67                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 0.30/0.67                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.67                                & ( ( Mi3 = Ma3 )
% 0.30/0.67                                 => ! [X: vEBT_VEBT] :
% 0.30/0.67                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                                & ( ( Mi3 != Ma3 )
% 0.30/0.67                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.67                                    & ! [X: nat] :
% 0.30/0.67                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.67                                         => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67                          @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.elims(1)
% 0.30/0.67  thf(fact_9744_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.30/0.67      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.30/0.67        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
% 0.30/0.67        = ( ( Deg = Deg4 )
% 0.30/0.67          & ! [X: vEBT_VEBT] :
% 0.30/0.67              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.67             => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67          & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.67            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67          & ( case_o184042715313410164at_nat
% 0.30/0.67            @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
% 0.30/0.67              & ! [X: vEBT_VEBT] :
% 0.30/0.67                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.67                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67            @ ( produc6081775807080527818_nat_o
% 0.30/0.67              @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                  ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                  & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.67                  & ! [I4: nat] :
% 0.30/0.67                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
% 0.30/0.67                        = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.30/0.67                  & ( ( Mi3 = Ma3 )
% 0.30/0.67                   => ! [X: vEBT_VEBT] :
% 0.30/0.67                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.67                       => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                  & ( ( Mi3 != Ma3 )
% 0.30/0.67                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.67                      & ! [X: nat] :
% 0.30/0.67                          ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.67                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.67                           => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67            @ Mima2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.simps(2)
% 0.30/0.67  thf(fact_9745_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.30/0.67      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.67        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.67       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.67                ( ( X3
% 0.30/0.67                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.67               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.30/0.67                 => ( Xa2 = one_one_nat ) ) )
% 0.30/0.67           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.67                  ( ( X3
% 0.30/0.67                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.67                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.30/0.67                   => ( ( Deg2 = Xa2 )
% 0.30/0.67                      & ! [X5: vEBT_VEBT] :
% 0.30/0.67                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                         => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.67                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                      & ( case_o184042715313410164at_nat
% 0.30/0.67                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 0.30/0.67                          & ! [X: vEBT_VEBT] :
% 0.30/0.67                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                        @ ( produc6081775807080527818_nat_o
% 0.30/0.67                          @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                              & ! [I4: nat] :
% 0.30/0.67                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 0.30/0.67                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.67                              & ( ( Mi3 = Ma3 )
% 0.30/0.67                               => ! [X: vEBT_VEBT] :
% 0.30/0.67                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                              & ( ( Mi3 != Ma3 )
% 0.30/0.67                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.67                                  & ! [X: nat] :
% 0.30/0.67                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.67                                       => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67                        @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.pelims(3)
% 0.30/0.67  thf(fact_9746_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.30/0.67      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.67        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.67       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.67                ( ( X3
% 0.30/0.67                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.67               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.30/0.67                 => ( Xa2 != one_one_nat ) ) )
% 0.30/0.67           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.67                  ( ( X3
% 0.30/0.67                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.67                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.30/0.67                   => ~ ( ( Deg2 = Xa2 )
% 0.30/0.67                        & ! [X6: vEBT_VEBT] :
% 0.30/0.67                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                           => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.67                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                        & ( case_o184042715313410164at_nat
% 0.30/0.67                          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 0.30/0.67                            & ! [X: vEBT_VEBT] :
% 0.30/0.67                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                          @ ( produc6081775807080527818_nat_o
% 0.30/0.67                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                & ! [I4: nat] :
% 0.30/0.67                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 0.30/0.67                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.67                                & ( ( Mi3 = Ma3 )
% 0.30/0.67                                 => ! [X: vEBT_VEBT] :
% 0.30/0.67                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                                & ( ( Mi3 != Ma3 )
% 0.30/0.67                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.67                                    & ! [X: nat] :
% 0.30/0.67                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.67                                         => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67                          @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.pelims(2)
% 0.30/0.67  thf(fact_9747_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.30/0.67      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.67        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.67                ( ( X3
% 0.30/0.67                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.67               => ( ( Y3
% 0.30/0.67                    = ( Xa2 = one_one_nat ) )
% 0.30/0.67                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.30/0.67           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.67                  ( ( X3
% 0.30/0.67                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.67                 => ( ( Y3
% 0.30/0.67                      = ( ( Deg2 = Xa2 )
% 0.30/0.67                        & ! [X: vEBT_VEBT] :
% 0.30/0.67                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.67                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.67                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                        & ( case_o184042715313410164at_nat
% 0.30/0.67                          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 0.30/0.67                            & ! [X: vEBT_VEBT] :
% 0.30/0.67                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                          @ ( produc6081775807080527818_nat_o
% 0.30/0.67                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.67                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.67                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                & ! [I4: nat] :
% 0.30/0.67                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.67                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 0.30/0.67                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.30/0.67                                & ( ( Mi3 = Ma3 )
% 0.30/0.67                                 => ! [X: vEBT_VEBT] :
% 0.30/0.67                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.67                                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 0.30/0.67                                & ( ( Mi3 != Ma3 )
% 0.30/0.67                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.67                                    & ! [X: nat] :
% 0.30/0.67                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.67                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.67                                         => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.67                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.67                          @ Mima ) ) )
% 0.30/0.67                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % VEBT_internal.valid'.pelims(1)
% 0.30/0.67  thf(fact_9748_Sup__int__def,axiom,
% 0.30/0.67      ( complete_Sup_Sup_int
% 0.30/0.67      = ( ^ [X7: set_int] :
% 0.30/0.67            ( the_int
% 0.30/0.67            @ ^ [X: int] :
% 0.30/0.67                ( ( member_int @ X @ X7 )
% 0.30/0.67                & ! [Y: int] :
% 0.30/0.67                    ( ( member_int @ Y @ X7 )
% 0.30/0.67                   => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Sup_int_def
% 0.30/0.67  thf(fact_9749_take__bit__numeral__minus__numeral__int,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int
% 0.30/0.67          @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
% 0.30/0.67          @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_numeral_minus_numeral_int
% 0.30/0.67  thf(fact_9750_and__minus__numerals_I3_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_minus_numerals(3)
% 0.30/0.67  thf(fact_9751_take__bit__num__simps_I1_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.30/0.67        = none_num ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(1)
% 0.30/0.67  thf(fact_9752_take__bit__num__simps_I2_J,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.30/0.67        = ( some_num @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(2)
% 0.30/0.67  thf(fact_9753_take__bit__num__simps_I5_J,axiom,
% 0.30/0.67      ! [R: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 0.30/0.67        = ( some_num @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(5)
% 0.30/0.67  thf(fact_9754_take__bit__num__simps_I3_J,axiom,
% 0.30/0.67      ! [N: nat,M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.30/0.67        = ( case_o6005452278849405969um_num @ none_num
% 0.30/0.67          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.67          @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(3)
% 0.30/0.67  thf(fact_9755_take__bit__num__simps_I4_J,axiom,
% 0.30/0.67      ! [N: nat,M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.30/0.67        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(4)
% 0.30/0.67  thf(fact_9756_take__bit__num__simps_I6_J,axiom,
% 0.30/0.67      ! [R: num,M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 0.30/0.67        = ( case_o6005452278849405969um_num @ none_num
% 0.30/0.67          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.67          @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(6)
% 0.30/0.67  thf(fact_9757_take__bit__num__simps_I7_J,axiom,
% 0.30/0.67      ! [R: num,M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 0.30/0.67        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_simps(7)
% 0.30/0.67  thf(fact_9758_and__minus__numerals_I4_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_minus_numerals(4)
% 0.30/0.67  thf(fact_9759_and__minus__numerals_I8_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_minus_numerals(8)
% 0.30/0.67  thf(fact_9760_and__minus__numerals_I7_J,axiom,
% 0.30/0.67      ! [N: num,M: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_minus_numerals(7)
% 0.30/0.67  thf(fact_9761_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.30/0.67      ! [N: nat,M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.30/0.67        = ( case_nat_option_num @ none_num
% 0.30/0.67          @ ^ [N2: nat] :
% 0.30/0.67              ( case_o6005452278849405969um_num @ none_num
% 0.30/0.67              @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.67              @ ( bit_take_bit_num @ N2 @ M ) )
% 0.30/0.67          @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Code_Abstract_Nat.take_bit_num_code(2)
% 0.30/0.67  thf(fact_9762_and__not__num_Osimps_I1_J,axiom,
% 0.30/0.67      ( ( bit_and_not_num @ one @ one )
% 0.30/0.67      = none_num ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(1)
% 0.30/0.67  thf(fact_9763_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( bit_take_bit_num @ N @ one )
% 0.30/0.67        = ( case_nat_option_num @ none_num
% 0.30/0.67          @ ^ [N2: nat] : ( some_num @ one )
% 0.30/0.67          @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Code_Abstract_Nat.take_bit_num_code(1)
% 0.30/0.67  thf(fact_9764_and__not__num_Osimps_I4_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.30/0.67        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(4)
% 0.30/0.67  thf(fact_9765_and__not__num_Osimps_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.30/0.67        = ( some_num @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(2)
% 0.30/0.67  thf(fact_9766_GreatestI__ex__nat,axiom,
% 0.30/0.67      ! [P: nat > $o,B2: nat] :
% 0.30/0.67        ( ? [X_12: nat] : ( P @ X_12 )
% 0.30/0.67       => ( ! [Y4: nat] :
% 0.30/0.67              ( ( P @ Y4 )
% 0.30/0.67             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.30/0.67         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % GreatestI_ex_nat
% 0.30/0.67  thf(fact_9767_Greatest__le__nat,axiom,
% 0.30/0.67      ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.67        ( ( P @ K )
% 0.30/0.67       => ( ! [Y4: nat] :
% 0.30/0.67              ( ( P @ Y4 )
% 0.30/0.67             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.30/0.67         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Greatest_le_nat
% 0.30/0.67  thf(fact_9768_GreatestI__nat,axiom,
% 0.30/0.67      ! [P: nat > $o,K: nat,B2: nat] :
% 0.30/0.67        ( ( P @ K )
% 0.30/0.67       => ( ! [Y4: nat] :
% 0.30/0.67              ( ( P @ Y4 )
% 0.30/0.67             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 0.30/0.67         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % GreatestI_nat
% 0.30/0.67  thf(fact_9769_and__not__num_Osimps_I3_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.30/0.67        = none_num ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(3)
% 0.30/0.67  thf(fact_9770_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.30/0.67      ! [N: nat,M: num] :
% 0.30/0.67        ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.30/0.67        = ( case_nat_option_num @ none_num
% 0.30/0.67          @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 0.30/0.67          @ N ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Code_Abstract_Nat.take_bit_num_code(3)
% 0.30/0.67  thf(fact_9771_and__not__num_Osimps_I7_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.30/0.67        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(7)
% 0.30/0.67  thf(fact_9772_and__not__num__eq__Some__iff,axiom,
% 0.30/0.67      ! [M: num,N: num,Q2: num] :
% 0.30/0.67        ( ( ( bit_and_not_num @ M @ N )
% 0.30/0.67          = ( some_num @ Q2 ) )
% 0.30/0.67        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67          = ( numeral_numeral_int @ Q2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num_eq_Some_iff
% 0.30/0.67  thf(fact_9773_and__not__num_Osimps_I8_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67          @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(8)
% 0.30/0.67  thf(fact_9774_and__not__num__eq__None__iff,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( ( bit_and_not_num @ M @ N )
% 0.30/0.67          = none_num )
% 0.30/0.67        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67          = zero_zero_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num_eq_None_iff
% 0.30/0.67  thf(fact_9775_int__numeral__not__and__num,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_numeral_not_and_num
% 0.30/0.67  thf(fact_9776_int__numeral__and__not__num,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % int_numeral_and_not_num
% 0.30/0.67  thf(fact_9777_take__bit__num__def,axiom,
% 0.30/0.67      ( bit_take_bit_num
% 0.30/0.67      = ( ^ [N2: nat,M3: num] :
% 0.30/0.67            ( if_option_num
% 0.30/0.67            @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
% 0.30/0.67              = zero_zero_nat )
% 0.30/0.67            @ none_num
% 0.30/0.67            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % take_bit_num_def
% 0.30/0.67  thf(fact_9778_Bit__Operations_Otake__bit__num__code,axiom,
% 0.30/0.67      ( bit_take_bit_num
% 0.30/0.67      = ( ^ [N2: nat,M3: num] :
% 0.30/0.67            ( produc478579273971653890on_num
% 0.30/0.67            @ ^ [A5: nat,X: num] :
% 0.30/0.67                ( case_nat_option_num @ none_num
% 0.30/0.67                @ ^ [O: nat] :
% 0.30/0.67                    ( case_num_option_num @ ( some_num @ one )
% 0.30/0.67                    @ ^ [P5: num] :
% 0.30/0.67                        ( case_o6005452278849405969um_num @ none_num
% 0.30/0.67                        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.30/0.67                        @ ( bit_take_bit_num @ O @ P5 ) )
% 0.30/0.67                    @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
% 0.30/0.67                    @ X )
% 0.30/0.67                @ A5 )
% 0.30/0.67            @ ( product_Pair_nat_num @ N2 @ M3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Bit_Operations.take_bit_num_code
% 0.30/0.67  thf(fact_9779_integer__of__num__triv_I2_J,axiom,
% 0.30/0.67      ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.30/0.67      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_of_num_triv(2)
% 0.30/0.67  thf(fact_9780_integer__of__num__def,axiom,
% 0.30/0.67      code_integer_of_num = numera6620942414471956472nteger ).
% 0.30/0.67  
% 0.30/0.67  % integer_of_num_def
% 0.30/0.67  thf(fact_9781_integer__of__num__triv_I1_J,axiom,
% 0.30/0.67      ( ( code_integer_of_num @ one )
% 0.30/0.67      = one_one_Code_integer ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_of_num_triv(1)
% 0.30/0.67  thf(fact_9782_integer__of__num_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.30/0.67        = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_of_num(2)
% 0.30/0.67  thf(fact_9783_and__not__num_Oelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.67        ( ( ( bit_and_not_num @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( X3 = one )
% 0.30/0.67           => ( ( Xa2 = one )
% 0.30/0.67             => ( Y3 != none_num ) ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ( ? [N3: num] :
% 0.30/0.67                    ( Xa2
% 0.30/0.67                    = ( bit0 @ N3 ) )
% 0.30/0.67               => ( Y3
% 0.30/0.67                 != ( some_num @ one ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ( ? [N3: num] :
% 0.30/0.67                      ( Xa2
% 0.30/0.67                      = ( bit1 @ N3 ) )
% 0.30/0.67                 => ( Y3 != none_num ) ) )
% 0.30/0.67             => ( ! [M4: num] :
% 0.30/0.67                    ( ( X3
% 0.30/0.67                      = ( bit0 @ M4 ) )
% 0.30/0.67                   => ( ( Xa2 = one )
% 0.30/0.67                     => ( Y3
% 0.30/0.67                       != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.30/0.67               => ( ! [M4: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ M4 ) )
% 0.30/0.67                     => ! [N3: num] :
% 0.30/0.67                          ( ( Xa2
% 0.30/0.67                            = ( bit0 @ N3 ) )
% 0.30/0.67                         => ( Y3
% 0.30/0.67                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                 => ( ! [M4: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ M4 ) )
% 0.30/0.67                       => ! [N3: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit1 @ N3 ) )
% 0.30/0.67                           => ( Y3
% 0.30/0.67                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                   => ( ! [M4: num] :
% 0.30/0.67                          ( ( X3
% 0.30/0.67                            = ( bit1 @ M4 ) )
% 0.30/0.67                         => ( ( Xa2 = one )
% 0.30/0.67                           => ( Y3
% 0.30/0.67                             != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.30/0.67                     => ( ! [M4: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ! [N3: num] :
% 0.30/0.67                                ( ( Xa2
% 0.30/0.67                                  = ( bit0 @ N3 ) )
% 0.30/0.67                               => ( Y3
% 0.30/0.67                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67                                    @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                       => ~ ! [M4: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ! [N3: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit1 @ N3 ) )
% 0.30/0.67                                 => ( Y3
% 0.30/0.67                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.elims
% 0.30/0.67  thf(fact_9784_xor__num_Osimps_I6_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(6)
% 0.30/0.67  thf(fact_9785_xor__num_Osimps_I8_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(8)
% 0.30/0.67  thf(fact_9786_xor__num_Osimps_I5_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(5)
% 0.30/0.67  thf(fact_9787_xor__num_Osimps_I9_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(9)
% 0.30/0.67  thf(fact_9788_and__not__num_Osimps_I5_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(5)
% 0.30/0.67  thf(fact_9789_and__not__num_Osimps_I6_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(6)
% 0.30/0.67  thf(fact_9790_and__not__num_Osimps_I9_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.simps(9)
% 0.30/0.67  thf(fact_9791_xor__num_Osimps_I1_J,axiom,
% 0.30/0.67      ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.30/0.67      = none_num ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(1)
% 0.30/0.67  thf(fact_9792_xor__num_Oelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.67        ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( X3 = one )
% 0.30/0.67           => ( ( Xa2 = one )
% 0.30/0.67             => ( Y3 != none_num ) ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ! [N3: num] :
% 0.30/0.67                  ( ( Xa2
% 0.30/0.67                    = ( bit0 @ N3 ) )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                   != ( some_num @ ( bit1 @ N3 ) ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ! [N3: num] :
% 0.30/0.67                    ( ( Xa2
% 0.30/0.67                      = ( bit1 @ N3 ) )
% 0.30/0.67                   => ( Y3
% 0.30/0.67                     != ( some_num @ ( bit0 @ N3 ) ) ) ) )
% 0.30/0.67             => ( ! [M4: num] :
% 0.30/0.67                    ( ( X3
% 0.30/0.67                      = ( bit0 @ M4 ) )
% 0.30/0.67                   => ( ( Xa2 = one )
% 0.30/0.67                     => ( Y3
% 0.30/0.67                       != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 0.30/0.67               => ( ! [M4: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ M4 ) )
% 0.30/0.67                     => ! [N3: num] :
% 0.30/0.67                          ( ( Xa2
% 0.30/0.67                            = ( bit0 @ N3 ) )
% 0.30/0.67                         => ( Y3
% 0.30/0.67                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                 => ( ! [M4: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ M4 ) )
% 0.30/0.67                       => ! [N3: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit1 @ N3 ) )
% 0.30/0.67                           => ( Y3
% 0.30/0.67                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
% 0.30/0.67                   => ( ! [M4: num] :
% 0.30/0.67                          ( ( X3
% 0.30/0.67                            = ( bit1 @ M4 ) )
% 0.30/0.67                         => ( ( Xa2 = one )
% 0.30/0.67                           => ( Y3
% 0.30/0.67                             != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.30/0.67                     => ( ! [M4: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ! [N3: num] :
% 0.30/0.67                                ( ( Xa2
% 0.30/0.67                                  = ( bit0 @ N3 ) )
% 0.30/0.67                               => ( Y3
% 0.30/0.67                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
% 0.30/0.67                       => ~ ! [M4: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ! [N3: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit1 @ N3 ) )
% 0.30/0.67                                 => ( Y3
% 0.30/0.67                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.elims
% 0.30/0.67  thf(fact_9793_xor__num_Osimps_I7_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.30/0.67        = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(7)
% 0.30/0.67  thf(fact_9794_xor__num_Osimps_I4_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.30/0.67        = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(4)
% 0.30/0.67  thf(fact_9795_xor__num_Osimps_I3_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.30/0.67        = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(3)
% 0.30/0.67  thf(fact_9796_xor__num_Osimps_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.30/0.67        = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.simps(2)
% 0.30/0.67  thf(fact_9797_and__not__num_Opelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.67        ( ( ( bit_and_not_num @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ( ( Xa2 = one )
% 0.30/0.67               => ( ( Y3 = none_num )
% 0.30/0.67                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ! [N3: num] :
% 0.30/0.67                    ( ( Xa2
% 0.30/0.67                      = ( bit0 @ N3 ) )
% 0.30/0.67                   => ( ( Y3
% 0.30/0.67                        = ( some_num @ one ) )
% 0.30/0.67                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67             => ( ( ( X3 = one )
% 0.30/0.67                 => ! [N3: num] :
% 0.30/0.67                      ( ( Xa2
% 0.30/0.67                        = ( bit1 @ N3 ) )
% 0.30/0.67                     => ( ( Y3 = none_num )
% 0.30/0.67                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 0.30/0.67               => ( ! [M4: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ M4 ) )
% 0.30/0.67                     => ( ( Xa2 = one )
% 0.30/0.67                       => ( ( Y3
% 0.30/0.67                            = ( some_num @ ( bit0 @ M4 ) ) )
% 0.30/0.67                         => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67                 => ( ! [M4: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ M4 ) )
% 0.30/0.67                       => ! [N3: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit0 @ N3 ) )
% 0.30/0.67                           => ( ( Y3
% 0.30/0.67                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 0.30/0.67                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67                   => ( ! [M4: num] :
% 0.30/0.67                          ( ( X3
% 0.30/0.67                            = ( bit0 @ M4 ) )
% 0.30/0.67                         => ! [N3: num] :
% 0.30/0.67                              ( ( Xa2
% 0.30/0.67                                = ( bit1 @ N3 ) )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.30/0.67                     => ( ! [M4: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ( ( Xa2 = one )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( some_num @ ( bit0 @ M4 ) ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67                       => ( ! [M4: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ! [N3: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit0 @ N3 ) )
% 0.30/0.67                                 => ( ( Y3
% 0.30/0.67                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67                                        @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 0.30/0.67                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67                         => ~ ! [M4: num] :
% 0.30/0.67                                ( ( X3
% 0.30/0.67                                  = ( bit1 @ M4 ) )
% 0.30/0.67                               => ! [N3: num] :
% 0.30/0.67                                    ( ( Xa2
% 0.30/0.67                                      = ( bit1 @ N3 ) )
% 0.30/0.67                                   => ( ( Y3
% 0.30/0.67                                        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 0.30/0.67                                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_not_num.pelims
% 0.30/0.67  thf(fact_9798_xor__num_Opelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.67        ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ( ( Xa2 = one )
% 0.30/0.67               => ( ( Y3 = none_num )
% 0.30/0.67                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ! [N3: num] :
% 0.30/0.67                    ( ( Xa2
% 0.30/0.67                      = ( bit0 @ N3 ) )
% 0.30/0.67                   => ( ( Y3
% 0.30/0.67                        = ( some_num @ ( bit1 @ N3 ) ) )
% 0.30/0.67                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67             => ( ( ( X3 = one )
% 0.30/0.67                 => ! [N3: num] :
% 0.30/0.67                      ( ( Xa2
% 0.30/0.67                        = ( bit1 @ N3 ) )
% 0.30/0.67                     => ( ( Y3
% 0.30/0.67                          = ( some_num @ ( bit0 @ N3 ) ) )
% 0.30/0.67                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 0.30/0.67               => ( ! [M4: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ M4 ) )
% 0.30/0.67                     => ( ( Xa2 = one )
% 0.30/0.67                       => ( ( Y3
% 0.30/0.67                            = ( some_num @ ( bit1 @ M4 ) ) )
% 0.30/0.67                         => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67                 => ( ! [M4: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ M4 ) )
% 0.30/0.67                       => ! [N3: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit0 @ N3 ) )
% 0.30/0.67                           => ( ( Y3
% 0.30/0.67                                = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
% 0.30/0.67                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67                   => ( ! [M4: num] :
% 0.30/0.67                          ( ( X3
% 0.30/0.67                            = ( bit0 @ M4 ) )
% 0.30/0.67                         => ! [N3: num] :
% 0.30/0.67                              ( ( Xa2
% 0.30/0.67                                = ( bit1 @ N3 ) )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.30/0.67                     => ( ! [M4: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ( ( Xa2 = one )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( some_num @ ( bit0 @ M4 ) ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67                       => ( ! [M4: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ! [N3: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit0 @ N3 ) )
% 0.30/0.67                                 => ( ( Y3
% 0.30/0.67                                      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
% 0.30/0.67                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67                         => ~ ! [M4: num] :
% 0.30/0.67                                ( ( X3
% 0.30/0.67                                  = ( bit1 @ M4 ) )
% 0.30/0.67                               => ! [N3: num] :
% 0.30/0.67                                    ( ( Xa2
% 0.30/0.67                                      = ( bit1 @ N3 ) )
% 0.30/0.67                                   => ( ( Y3
% 0.30/0.67                                        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
% 0.30/0.67                                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % xor_num.pelims
% 0.30/0.67  thf(fact_9799_and__num_Oelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.67        ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( X3 = one )
% 0.30/0.67           => ( ( Xa2 = one )
% 0.30/0.67             => ( Y3
% 0.30/0.67               != ( some_num @ one ) ) ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ( ? [N3: num] :
% 0.30/0.67                    ( Xa2
% 0.30/0.67                    = ( bit0 @ N3 ) )
% 0.30/0.67               => ( Y3 != none_num ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ( ? [N3: num] :
% 0.30/0.67                      ( Xa2
% 0.30/0.67                      = ( bit1 @ N3 ) )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                   != ( some_num @ one ) ) ) )
% 0.30/0.67             => ( ( ? [M4: num] :
% 0.30/0.67                      ( X3
% 0.30/0.67                      = ( bit0 @ M4 ) )
% 0.30/0.67                 => ( ( Xa2 = one )
% 0.30/0.67                   => ( Y3 != none_num ) ) )
% 0.30/0.67               => ( ! [M4: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ M4 ) )
% 0.30/0.67                     => ! [N3: num] :
% 0.30/0.67                          ( ( Xa2
% 0.30/0.67                            = ( bit0 @ N3 ) )
% 0.30/0.67                         => ( Y3
% 0.30/0.67                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                 => ( ! [M4: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ M4 ) )
% 0.30/0.67                       => ! [N3: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit1 @ N3 ) )
% 0.30/0.67                           => ( Y3
% 0.30/0.67                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                   => ( ( ? [M4: num] :
% 0.30/0.67                            ( X3
% 0.30/0.67                            = ( bit1 @ M4 ) )
% 0.30/0.67                       => ( ( Xa2 = one )
% 0.30/0.67                         => ( Y3
% 0.30/0.67                           != ( some_num @ one ) ) ) )
% 0.30/0.67                     => ( ! [M4: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ! [N3: num] :
% 0.30/0.67                                ( ( Xa2
% 0.30/0.67                                  = ( bit0 @ N3 ) )
% 0.30/0.67                               => ( Y3
% 0.30/0.67                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
% 0.30/0.67                       => ~ ! [M4: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ! [N3: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit1 @ N3 ) )
% 0.30/0.67                                 => ( Y3
% 0.30/0.67                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67                                      @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.elims
% 0.30/0.67  thf(fact_9800_xor__num__rel__dict,axiom,
% 0.30/0.67      bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 0.30/0.67  
% 0.30/0.67  % xor_num_rel_dict
% 0.30/0.67  thf(fact_9801_xor__num__dict,axiom,
% 0.30/0.67      bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.30/0.67  
% 0.30/0.67  % xor_num_dict
% 0.30/0.67  thf(fact_9802_and__num_Osimps_I1_J,axiom,
% 0.30/0.67      ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.30/0.67      = ( some_num @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(1)
% 0.30/0.67  thf(fact_9803_and__num_Osimps_I5_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(5)
% 0.30/0.67  thf(fact_9804_and__num_Osimps_I3_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.30/0.67        = ( some_num @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(3)
% 0.30/0.67  thf(fact_9805_and__num_Osimps_I7_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.30/0.67        = ( some_num @ one ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(7)
% 0.30/0.67  thf(fact_9806_and__num_Osimps_I4_J,axiom,
% 0.30/0.67      ! [M: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.30/0.67        = none_num ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(4)
% 0.30/0.67  thf(fact_9807_and__num_Osimps_I2_J,axiom,
% 0.30/0.67      ! [N: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.30/0.67        = none_num ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(2)
% 0.30/0.67  thf(fact_9808_and__num_Osimps_I6_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(6)
% 0.30/0.67  thf(fact_9809_and__num_Osimps_I8_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.30/0.67        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(8)
% 0.30/0.67  thf(fact_9810_and__num_Osimps_I9_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.30/0.67        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67          @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.simps(9)
% 0.30/0.67  thf(fact_9811_and__num_Opelims,axiom,
% 0.30/0.67      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.67        ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 0.30/0.67         => ( ( ( X3 = one )
% 0.30/0.67             => ( ( Xa2 = one )
% 0.30/0.67               => ( ( Y3
% 0.30/0.67                    = ( some_num @ one ) )
% 0.30/0.67                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.30/0.67           => ( ( ( X3 = one )
% 0.30/0.67               => ! [N3: num] :
% 0.30/0.67                    ( ( Xa2
% 0.30/0.67                      = ( bit0 @ N3 ) )
% 0.30/0.67                   => ( ( Y3 = none_num )
% 0.30/0.67                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67             => ( ( ( X3 = one )
% 0.30/0.67                 => ! [N3: num] :
% 0.30/0.67                      ( ( Xa2
% 0.30/0.67                        = ( bit1 @ N3 ) )
% 0.30/0.67                     => ( ( Y3
% 0.30/0.67                          = ( some_num @ one ) )
% 0.30/0.67                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 0.30/0.67               => ( ! [M4: num] :
% 0.30/0.67                      ( ( X3
% 0.30/0.67                        = ( bit0 @ M4 ) )
% 0.30/0.67                     => ( ( Xa2 = one )
% 0.30/0.67                       => ( ( Y3 = none_num )
% 0.30/0.67                         => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67                 => ( ! [M4: num] :
% 0.30/0.67                        ( ( X3
% 0.30/0.67                          = ( bit0 @ M4 ) )
% 0.30/0.67                       => ! [N3: num] :
% 0.30/0.67                            ( ( Xa2
% 0.30/0.67                              = ( bit0 @ N3 ) )
% 0.30/0.67                           => ( ( Y3
% 0.30/0.67                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 0.30/0.67                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67                   => ( ! [M4: num] :
% 0.30/0.67                          ( ( X3
% 0.30/0.67                            = ( bit0 @ M4 ) )
% 0.30/0.67                         => ! [N3: num] :
% 0.30/0.67                              ( ( Xa2
% 0.30/0.67                                = ( bit1 @ N3 ) )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.30/0.67                     => ( ! [M4: num] :
% 0.30/0.67                            ( ( X3
% 0.30/0.67                              = ( bit1 @ M4 ) )
% 0.30/0.67                           => ( ( Xa2 = one )
% 0.30/0.67                             => ( ( Y3
% 0.30/0.67                                  = ( some_num @ one ) )
% 0.30/0.67                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.30/0.67                       => ( ! [M4: num] :
% 0.30/0.67                              ( ( X3
% 0.30/0.67                                = ( bit1 @ M4 ) )
% 0.30/0.67                             => ! [N3: num] :
% 0.30/0.67                                  ( ( Xa2
% 0.30/0.67                                    = ( bit0 @ N3 ) )
% 0.30/0.67                                 => ( ( Y3
% 0.30/0.67                                      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 0.30/0.67                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.30/0.67                         => ~ ! [M4: num] :
% 0.30/0.67                                ( ( X3
% 0.30/0.67                                  = ( bit1 @ M4 ) )
% 0.30/0.67                               => ! [N3: num] :
% 0.30/0.67                                    ( ( Xa2
% 0.30/0.67                                      = ( bit1 @ N3 ) )
% 0.30/0.67                                   => ( ( Y3
% 0.30/0.67                                        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.67                                          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.67                                          @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 0.30/0.67                                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % and_num.pelims
% 0.30/0.67  thf(fact_9812_and__num__dict,axiom,
% 0.30/0.67      bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.30/0.67  
% 0.30/0.67  % and_num_dict
% 0.30/0.67  thf(fact_9813_and__num__rel__dict,axiom,
% 0.30/0.67      bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 0.30/0.67  
% 0.30/0.67  % and_num_rel_dict
% 0.30/0.67  thf(fact_9814_Rats__eq__int__div__nat,axiom,
% 0.30/0.67      ( field_5140801741446780682s_real
% 0.30/0.67      = ( collect_real
% 0.30/0.67        @ ^ [Uu3: real] :
% 0.30/0.67          ? [I4: int,N2: nat] :
% 0.30/0.67            ( ( Uu3
% 0.30/0.67              = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.30/0.67            & ( N2 != zero_zero_nat ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rats_eq_int_div_nat
% 0.30/0.67  thf(fact_9815_rat__floor__lemma,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B2 ) ) @ ( fract @ A @ B2 ) )
% 0.30/0.67        & ( ord_less_rat @ ( fract @ A @ B2 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_floor_lemma
% 0.30/0.67  thf(fact_9816_Rats__abs__iff,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67        ( ( member_real @ ( abs_abs_real @ X3 ) @ field_5140801741446780682s_real )
% 0.30/0.67        = ( member_real @ X3 @ field_5140801741446780682s_real ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rats_abs_iff
% 0.30/0.67  thf(fact_9817_mult__rat,axiom,
% 0.30/0.67      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.67        ( ( times_times_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.30/0.67        = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % mult_rat
% 0.30/0.67  thf(fact_9818_divide__rat,axiom,
% 0.30/0.67      ! [A: int,B2: int,C: int,D: int] :
% 0.30/0.67        ( ( divide_divide_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.30/0.67        = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % divide_rat
% 0.30/0.67  thf(fact_9819_less__rat,axiom,
% 0.30/0.67      ! [B2: int,D: int,A: int,C: int] :
% 0.30/0.67        ( ( B2 != zero_zero_int )
% 0.30/0.67       => ( ( D != zero_zero_int )
% 0.30/0.67         => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.30/0.67            = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_rat
% 0.30/0.67  thf(fact_9820_add__rat,axiom,
% 0.30/0.67      ! [B2: int,D: int,A: int,C: int] :
% 0.30/0.67        ( ( B2 != zero_zero_int )
% 0.30/0.67       => ( ( D != zero_zero_int )
% 0.30/0.67         => ( ( plus_plus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.30/0.67            = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % add_rat
% 0.30/0.67  thf(fact_9821_le__rat,axiom,
% 0.30/0.67      ! [B2: int,D: int,A: int,C: int] :
% 0.30/0.67        ( ( B2 != zero_zero_int )
% 0.30/0.67       => ( ( D != zero_zero_int )
% 0.30/0.67         => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.30/0.67            = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % le_rat
% 0.30/0.67  thf(fact_9822_diff__rat,axiom,
% 0.30/0.67      ! [B2: int,D: int,A: int,C: int] :
% 0.30/0.67        ( ( B2 != zero_zero_int )
% 0.30/0.67       => ( ( D != zero_zero_int )
% 0.30/0.67         => ( ( minus_minus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 0.30/0.67            = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % diff_rat
% 0.30/0.67  thf(fact_9823_sgn__rat,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( sgn_sgn_rat @ ( fract @ A @ B2 ) )
% 0.30/0.67        = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % sgn_rat
% 0.30/0.67  thf(fact_9824_Rats__no__top__le,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67      ? [X5: real] :
% 0.30/0.67        ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.30/0.67        & ( ord_less_eq_real @ X3 @ X5 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rats_no_top_le
% 0.30/0.67  thf(fact_9825_Rats__no__bot__less,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67      ? [X5: real] :
% 0.30/0.67        ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.30/0.67        & ( ord_less_real @ X5 @ X3 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rats_no_bot_less
% 0.30/0.67  thf(fact_9826_Rats__dense__in__real,axiom,
% 0.30/0.67      ! [X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.67       => ? [X5: real] :
% 0.30/0.67            ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.30/0.67            & ( ord_less_real @ X3 @ X5 )
% 0.30/0.67            & ( ord_less_real @ X5 @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rats_dense_in_real
% 0.30/0.67  thf(fact_9827_Rat__induct__pos,axiom,
% 0.30/0.67      ! [P: rat > $o,Q2: rat] :
% 0.30/0.67        ( ! [A3: int,B3: int] :
% 0.30/0.67            ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.30/0.67           => ( P @ ( fract @ A3 @ B3 ) ) )
% 0.30/0.67       => ( P @ Q2 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rat_induct_pos
% 0.30/0.67  thf(fact_9828_eq__rat_I1_J,axiom,
% 0.30/0.67      ! [B2: int,D: int,A: int,C: int] :
% 0.30/0.67        ( ( B2 != zero_zero_int )
% 0.30/0.67       => ( ( D != zero_zero_int )
% 0.30/0.67         => ( ( ( fract @ A @ B2 )
% 0.30/0.67              = ( fract @ C @ D ) )
% 0.30/0.67            = ( ( times_times_int @ A @ D )
% 0.30/0.67              = ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % eq_rat(1)
% 0.30/0.67  thf(fact_9829_mult__rat__cancel,axiom,
% 0.30/0.67      ! [C: int,A: int,B2: int] :
% 0.30/0.67        ( ( C != zero_zero_int )
% 0.30/0.67       => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 0.30/0.67          = ( fract @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % mult_rat_cancel
% 0.30/0.67  thf(fact_9830_quotient__of__eq,axiom,
% 0.30/0.67      ! [A: int,B2: int,P4: int,Q2: int] :
% 0.30/0.67        ( ( ( quotient_of @ ( fract @ A @ B2 ) )
% 0.30/0.67          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.67       => ( ( fract @ P4 @ Q2 )
% 0.30/0.67          = ( fract @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_eq
% 0.30/0.67  thf(fact_9831_normalize__eq,axiom,
% 0.30/0.67      ! [A: int,B2: int,P4: int,Q2: int] :
% 0.30/0.67        ( ( ( normalize @ ( product_Pair_int_int @ A @ B2 ) )
% 0.30/0.67          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.67       => ( ( fract @ P4 @ Q2 )
% 0.30/0.67          = ( fract @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % normalize_eq
% 0.30/0.67  thf(fact_9832_rat__number__expand_I3_J,axiom,
% 0.30/0.67      ( numeral_numeral_rat
% 0.30/0.67      = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_number_expand(3)
% 0.30/0.67  thf(fact_9833_rat__number__collapse_I3_J,axiom,
% 0.30/0.67      ! [W: num] :
% 0.30/0.67        ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 0.30/0.67        = ( numeral_numeral_rat @ W ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_number_collapse(3)
% 0.30/0.67  thf(fact_9834_quotient__of__Fract,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( quotient_of @ ( fract @ A @ B2 ) )
% 0.30/0.67        = ( normalize @ ( product_Pair_int_int @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % quotient_of_Fract
% 0.30/0.67  thf(fact_9835_zero__less__Fract__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 0.30/0.67          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % zero_less_Fract_iff
% 0.30/0.67  thf(fact_9836_Fract__less__zero__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 0.30/0.67          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Fract_less_zero_iff
% 0.30/0.67  thf(fact_9837_one__less__Fract__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 0.30/0.67          = ( ord_less_int @ B2 @ A ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % one_less_Fract_iff
% 0.30/0.67  thf(fact_9838_Fract__less__one__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 0.30/0.67          = ( ord_less_int @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Fract_less_one_iff
% 0.30/0.67  thf(fact_9839_Rats__eq__int__div__int,axiom,
% 0.30/0.67      ( field_5140801741446780682s_real
% 0.30/0.67      = ( collect_real
% 0.30/0.67        @ ^ [Uu3: real] :
% 0.30/0.67          ? [I4: int,J3: int] :
% 0.30/0.67            ( ( Uu3
% 0.30/0.67              = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 0.30/0.67            & ( J3 != zero_zero_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rats_eq_int_div_int
% 0.30/0.67  thf(fact_9840_Fract__le__zero__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 0.30/0.67          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Fract_le_zero_iff
% 0.30/0.67  thf(fact_9841_zero__le__Fract__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 0.30/0.67          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % zero_le_Fract_iff
% 0.30/0.67  thf(fact_9842_Fract__le__one__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 0.30/0.67          = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Fract_le_one_iff
% 0.30/0.67  thf(fact_9843_one__le__Fract__iff,axiom,
% 0.30/0.67      ! [B2: int,A: int] :
% 0.30/0.67        ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.30/0.67       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 0.30/0.67          = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % one_le_Fract_iff
% 0.30/0.67  thf(fact_9844_rat__number__expand_I5_J,axiom,
% 0.30/0.67      ! [K: num] :
% 0.30/0.67        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 0.30/0.67        = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_number_expand(5)
% 0.30/0.67  thf(fact_9845_rat__number__collapse_I4_J,axiom,
% 0.30/0.67      ! [W: num] :
% 0.30/0.67        ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 0.30/0.67        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % rat_number_collapse(4)
% 0.30/0.67  thf(fact_9846_image__minus__const__atLeastLessThan__nat,axiom,
% 0.30/0.67      ! [C: nat,Y3: nat,X3: nat] :
% 0.30/0.67        ( ( ( ord_less_nat @ C @ Y3 )
% 0.30/0.67         => ( ( image_nat_nat
% 0.30/0.67              @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.30/0.67              @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 0.30/0.67            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_nat @ C @ Y3 )
% 0.30/0.67         => ( ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.67             => ( ( image_nat_nat
% 0.30/0.67                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.30/0.67                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 0.30/0.67                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.30/0.67            & ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.67             => ( ( image_nat_nat
% 0.30/0.67                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.30/0.67                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 0.30/0.67                = bot_bot_set_nat ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % image_minus_const_atLeastLessThan_nat
% 0.30/0.67  thf(fact_9847_of__nat__eq__id,axiom,
% 0.30/0.67      semiri1316708129612266289at_nat = id_nat ).
% 0.30/0.67  
% 0.30/0.67  % of_nat_eq_id
% 0.30/0.67  thf(fact_9848_bij__betw__Suc,axiom,
% 0.30/0.67      ! [M7: set_nat,N4: set_nat] :
% 0.30/0.67        ( ( bij_betw_nat_nat @ suc @ M7 @ N4 )
% 0.30/0.67        = ( ( image_nat_nat @ suc @ M7 )
% 0.30/0.67          = N4 ) ) ).
% 0.30/0.67  
% 0.30/0.67  % bij_betw_Suc
% 0.30/0.67  thf(fact_9849_less__int__def,axiom,
% 0.30/0.67      ( ord_less_int
% 0.30/0.67      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.30/0.67        @ ( produc8739625826339149834_nat_o
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc6081775807080527818_nat_o
% 0.30/0.67              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_int_def
% 0.30/0.67  thf(fact_9850_less__eq__int__def,axiom,
% 0.30/0.67      ( ord_less_eq_int
% 0.30/0.67      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.30/0.67        @ ( produc8739625826339149834_nat_o
% 0.30/0.67          @ ^ [X: nat,Y: nat] :
% 0.30/0.67              ( produc6081775807080527818_nat_o
% 0.30/0.67              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_eq_int_def
% 0.30/0.67  thf(fact_9851_nat__def,axiom,
% 0.30/0.67      ( nat2
% 0.30/0.67      = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat_def
% 0.30/0.67  thf(fact_9852_positive__rat,axiom,
% 0.30/0.67      ! [A: int,B2: int] :
% 0.30/0.67        ( ( positive @ ( fract @ A @ B2 ) )
% 0.30/0.67        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % positive_rat
% 0.30/0.67  thf(fact_9853_Inf__real__def,axiom,
% 0.30/0.67      ( comple4887499456419720421f_real
% 0.30/0.67      = ( ^ [X7: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X7 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Inf_real_def
% 0.30/0.67  thf(fact_9854_Rat_Opositive__mult,axiom,
% 0.30/0.67      ! [X3: rat,Y3: rat] :
% 0.30/0.67        ( ( positive @ X3 )
% 0.30/0.67       => ( ( positive @ Y3 )
% 0.30/0.67         => ( positive @ ( times_times_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rat.positive_mult
% 0.30/0.67  thf(fact_9855_finite__int__iff__bounded__le,axiom,
% 0.30/0.67      ( finite_finite_int
% 0.30/0.67      = ( ^ [S4: set_int] :
% 0.30/0.67          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % finite_int_iff_bounded_le
% 0.30/0.67  thf(fact_9856_finite__int__iff__bounded,axiom,
% 0.30/0.67      ( finite_finite_int
% 0.30/0.67      = ( ^ [S4: set_int] :
% 0.30/0.67          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % finite_int_iff_bounded
% 0.30/0.67  thf(fact_9857_suminf__eq__SUP__real,axiom,
% 0.30/0.67      ! [X9: nat > real] :
% 0.30/0.67        ( ( summable_real @ X9 )
% 0.30/0.67       => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X9 @ I3 ) )
% 0.30/0.67         => ( ( suminf_real @ X9 )
% 0.30/0.67            = ( comple1385675409528146559p_real
% 0.30/0.67              @ ( image_nat_real
% 0.30/0.67                @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X9 @ ( set_ord_lessThan_nat @ I4 ) )
% 0.30/0.67                @ top_top_set_nat ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % suminf_eq_SUP_real
% 0.30/0.67  thf(fact_9858_less__rat__def,axiom,
% 0.30/0.67      ( ord_less_rat
% 0.30/0.67      = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % less_rat_def
% 0.30/0.67  thf(fact_9859_image__atLeastZeroLessThan__int,axiom,
% 0.30/0.67      ! [U: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.30/0.67       => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.30/0.67          = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % image_atLeastZeroLessThan_int
% 0.30/0.67  thf(fact_9860_range__mod,axiom,
% 0.30/0.67      ! [N: nat] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( image_nat_nat
% 0.30/0.67            @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 0.30/0.67            @ top_top_set_nat )
% 0.30/0.67          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % range_mod
% 0.30/0.67  thf(fact_9861_Rat_Opositive_Orep__eq,axiom,
% 0.30/0.67      ( positive
% 0.30/0.67      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rat.positive.rep_eq
% 0.30/0.67  thf(fact_9862_card__UNIV__bool,axiom,
% 0.30/0.67      ( ( finite_card_o @ top_top_set_o )
% 0.30/0.67      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_UNIV_bool
% 0.30/0.67  thf(fact_9863_range__mult,axiom,
% 0.30/0.67      ! [A: real] :
% 0.30/0.67        ( ( ( A = zero_zero_real )
% 0.30/0.67         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.30/0.67            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.30/0.67        & ( ( A != zero_zero_real )
% 0.30/0.67         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.30/0.67            = top_top_set_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % range_mult
% 0.30/0.67  thf(fact_9864_Rat_Opositive__def,axiom,
% 0.30/0.67      ( positive
% 0.30/0.67      = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.30/0.67        @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rat.positive_def
% 0.30/0.67  thf(fact_9865_root__def,axiom,
% 0.30/0.67      ( root
% 0.30/0.67      = ( ^ [N2: nat,X: real] :
% 0.30/0.67            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.30/0.67            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.30/0.67              @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 0.30/0.67              @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % root_def
% 0.30/0.67  thf(fact_9866_card__UNIV__char,axiom,
% 0.30/0.67      ( ( finite_card_char @ top_top_set_char )
% 0.30/0.67      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % card_UNIV_char
% 0.30/0.67  thf(fact_9867_UNIV__char__of__nat,axiom,
% 0.30/0.67      ( top_top_set_char
% 0.30/0.67      = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % UNIV_char_of_nat
% 0.30/0.67  thf(fact_9868_nat__of__char__less__256,axiom,
% 0.30/0.67      ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nat_of_char_less_256
% 0.30/0.67  thf(fact_9869_range__nat__of__char,axiom,
% 0.30/0.67      ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.30/0.67      = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % range_nat_of_char
% 0.30/0.67  thf(fact_9870_integer__of__char__code,axiom,
% 0.30/0.67      ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.30/0.67        ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.30/0.67        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % integer_of_char_code
% 0.30/0.67  thf(fact_9871_of__rat__dense,axiom,
% 0.30/0.67      ! [X3: real,Y3: real] :
% 0.30/0.67        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.67       => ? [Q3: rat] :
% 0.30/0.67            ( ( ord_less_real @ X3 @ ( field_7254667332652039916t_real @ Q3 ) )
% 0.30/0.67            & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % of_rat_dense
% 0.30/0.67  thf(fact_9872_plus__rat__def,axiom,
% 0.30/0.67      ( plus_plus_rat
% 0.30/0.67      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.30/0.67        @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % plus_rat_def
% 0.30/0.67  thf(fact_9873_inverse__rat__def,axiom,
% 0.30/0.67      ( inverse_inverse_rat
% 0.30/0.67      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.30/0.67        @ ^ [X: product_prod_int_int] :
% 0.30/0.67            ( if_Pro3027730157355071871nt_int
% 0.30/0.67            @ ( ( product_fst_int_int @ X )
% 0.30/0.67              = zero_zero_int )
% 0.30/0.67            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.67            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % inverse_rat_def
% 0.30/0.67  thf(fact_9874_String_Ochar__of__ascii__of,axiom,
% 0.30/0.67      ! [C: char] :
% 0.30/0.67        ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.30/0.67        = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % String.char_of_ascii_of
% 0.30/0.67  thf(fact_9875_one__rat__def,axiom,
% 0.30/0.67      ( one_one_rat
% 0.30/0.67      = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % one_rat_def
% 0.30/0.67  thf(fact_9876_Fract_Oabs__eq,axiom,
% 0.30/0.67      ( fract
% 0.30/0.67      = ( ^ [Xa4: int,X: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Fract.abs_eq
% 0.30/0.67  thf(fact_9877_zero__rat__def,axiom,
% 0.30/0.67      ( zero_zero_rat
% 0.30/0.67      = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % zero_rat_def
% 0.30/0.67  thf(fact_9878_uminus__rat__def,axiom,
% 0.30/0.67      ( uminus_uminus_rat
% 0.30/0.67      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 0.30/0.67        @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % uminus_rat_def
% 0.30/0.67  thf(fact_9879_times__rat__def,axiom,
% 0.30/0.67      ( times_times_rat
% 0.30/0.67      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 0.30/0.67        @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_rat_def
% 0.30/0.67  thf(fact_9880_upto__aux__rec,axiom,
% 0.30/0.67      ( upto_aux
% 0.30/0.67      = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_aux_rec
% 0.30/0.67  thf(fact_9881_upto_Opelims,axiom,
% 0.30/0.67      ! [X3: int,Xa2: int,Y3: list_int] :
% 0.30/0.67        ( ( ( upto @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 0.30/0.67         => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.67                 => ( Y3
% 0.30/0.67                    = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 0.30/0.67                & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.67                 => ( Y3 = nil_int ) ) )
% 0.30/0.67             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto.pelims
% 0.30/0.67  thf(fact_9882_upto__empty,axiom,
% 0.30/0.67      ! [J: int,I: int] :
% 0.30/0.67        ( ( ord_less_int @ J @ I )
% 0.30/0.67       => ( ( upto @ I @ J )
% 0.30/0.67          = nil_int ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_empty
% 0.30/0.67  thf(fact_9883_upto__Nil2,axiom,
% 0.30/0.67      ! [I: int,J: int] :
% 0.30/0.67        ( ( nil_int
% 0.30/0.67          = ( upto @ I @ J ) )
% 0.30/0.67        = ( ord_less_int @ J @ I ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_Nil2
% 0.30/0.67  thf(fact_9884_upto__Nil,axiom,
% 0.30/0.67      ! [I: int,J: int] :
% 0.30/0.67        ( ( ( upto @ I @ J )
% 0.30/0.67          = nil_int )
% 0.30/0.67        = ( ord_less_int @ J @ I ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_Nil
% 0.30/0.67  thf(fact_9885_nth__upto,axiom,
% 0.30/0.67      ! [I: int,K: nat,J: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.30/0.67       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.30/0.67          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % nth_upto
% 0.30/0.67  thf(fact_9886_upto__rec__numeral_I1_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67            = nil_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_rec_numeral(1)
% 0.30/0.67  thf(fact_9887_upto__rec__numeral_I4_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67            = nil_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_rec_numeral(4)
% 0.30/0.67  thf(fact_9888_upto__rec__numeral_I3_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.30/0.67            = nil_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_rec_numeral(3)
% 0.30/0.67  thf(fact_9889_upto__rec__numeral_I2_J,axiom,
% 0.30/0.67      ! [M: num,N: num] :
% 0.30/0.67        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.30/0.67        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.30/0.67            = nil_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_rec_numeral(2)
% 0.30/0.67  thf(fact_9890_atLeastAtMost__upto,axiom,
% 0.30/0.67      ( set_or1266510415728281911st_int
% 0.30/0.67      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % atLeastAtMost_upto
% 0.30/0.67  thf(fact_9891_upto__split2,axiom,
% 0.30/0.67      ! [I: int,J: int,K: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.67       => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.67         => ( ( upto @ I @ K )
% 0.30/0.67            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_split2
% 0.30/0.67  thf(fact_9892_upto__split1,axiom,
% 0.30/0.67      ! [I: int,J: int,K: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.67       => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.67         => ( ( upto @ I @ K )
% 0.30/0.67            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_split1
% 0.30/0.67  thf(fact_9893_atLeastLessThan__upto,axiom,
% 0.30/0.67      ( set_or4662586982721622107an_int
% 0.30/0.67      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % atLeastLessThan_upto
% 0.30/0.67  thf(fact_9894_upto__rec1,axiom,
% 0.30/0.67      ! [I: int,J: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.67       => ( ( upto @ I @ J )
% 0.30/0.67          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_rec1
% 0.30/0.67  thf(fact_9895_upto_Osimps,axiom,
% 0.30/0.67      ( upto
% 0.30/0.67      = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto.simps
% 0.30/0.67  thf(fact_9896_upto_Oelims,axiom,
% 0.30/0.67      ! [X3: int,Xa2: int,Y3: list_int] :
% 0.30/0.67        ( ( ( upto @ X3 @ Xa2 )
% 0.30/0.67          = Y3 )
% 0.30/0.67       => ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.67           => ( Y3
% 0.30/0.67              = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 0.30/0.67          & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.67           => ( Y3 = nil_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto.elims
% 0.30/0.67  thf(fact_9897_upto__rec2,axiom,
% 0.30/0.67      ! [I: int,J: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.67       => ( ( upto @ I @ J )
% 0.30/0.67          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_rec2
% 0.30/0.67  thf(fact_9898_upto__split3,axiom,
% 0.30/0.67      ! [I: int,J: int,K: int] :
% 0.30/0.67        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.67       => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.67         => ( ( upto @ I @ K )
% 0.30/0.67            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto_split3
% 0.30/0.67  thf(fact_9899_upto_Opsimps,axiom,
% 0.30/0.67      ! [I: int,J: int] :
% 0.30/0.67        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.30/0.67       => ( ( ( ord_less_eq_int @ I @ J )
% 0.30/0.67           => ( ( upto @ I @ J )
% 0.30/0.67              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.30/0.67          & ( ~ ( ord_less_eq_int @ I @ J )
% 0.30/0.67           => ( ( upto @ I @ J )
% 0.30/0.67              = nil_int ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % upto.psimps
% 0.30/0.67  thf(fact_9900_plus__rat_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
% 0.30/0.67        ( ( ratrel @ Xa2 @ Xa2 )
% 0.30/0.67       => ( ( ratrel @ X3 @ X3 )
% 0.30/0.67         => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
% 0.30/0.67            = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % plus_rat.abs_eq
% 0.30/0.67  thf(fact_9901_ratrel__iff,axiom,
% 0.30/0.67      ( ratrel
% 0.30/0.67      = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.30/0.67            ( ( ( product_snd_int_int @ X )
% 0.30/0.67             != zero_zero_int )
% 0.30/0.67            & ( ( product_snd_int_int @ Y )
% 0.30/0.67             != zero_zero_int )
% 0.30/0.67            & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 0.30/0.67              = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % ratrel_iff
% 0.30/0.67  thf(fact_9902_one__rat_Orsp,axiom,
% 0.30/0.67      ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 0.30/0.67  
% 0.30/0.67  % one_rat.rsp
% 0.30/0.67  thf(fact_9903_zero__rat_Orsp,axiom,
% 0.30/0.67      ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 0.30/0.67  
% 0.30/0.67  % zero_rat.rsp
% 0.30/0.67  thf(fact_9904_ratrel__def,axiom,
% 0.30/0.67      ( ratrel
% 0.30/0.67      = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.30/0.67            ( ( ( product_snd_int_int @ X )
% 0.30/0.67             != zero_zero_int )
% 0.30/0.67            & ( ( product_snd_int_int @ Y )
% 0.30/0.67             != zero_zero_int )
% 0.30/0.67            & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 0.30/0.67              = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % ratrel_def
% 0.30/0.67  thf(fact_9905_uminus__rat_Oabs__eq,axiom,
% 0.30/0.67      ! [X3: product_prod_int_int] :
% 0.30/0.67        ( ( ratrel @ X3 @ X3 )
% 0.30/0.67       => ( ( uminus_uminus_rat @ ( abs_Rat @ X3 ) )
% 0.30/0.67          = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % uminus_rat.abs_eq
% 0.30/0.67  thf(fact_9906_times__rat_Oabs__eq,axiom,
% 0.30/0.67      ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
% 0.30/0.67        ( ( ratrel @ Xa2 @ Xa2 )
% 0.30/0.67       => ( ( ratrel @ X3 @ X3 )
% 0.30/0.67         => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
% 0.30/0.67            = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % times_rat.abs_eq
% 0.30/0.67  thf(fact_9907_Rat_Opositive_Oabs__eq,axiom,
% 0.30/0.67      ! [X3: product_prod_int_int] :
% 0.30/0.67        ( ( ratrel @ X3 @ X3 )
% 0.30/0.67       => ( ( positive @ ( abs_Rat @ X3 ) )
% 0.30/0.67          = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % Rat.positive.abs_eq
% 0.30/0.67  thf(fact_9908_inverse__rat_Oabs__eq,axiom,
% 0.30/0.67      ! [X3: product_prod_int_int] :
% 0.30/0.67        ( ( ratrel @ X3 @ X3 )
% 0.30/0.67       => ( ( inverse_inverse_rat @ ( abs_Rat @ X3 ) )
% 0.30/0.67          = ( abs_Rat
% 0.30/0.67            @ ( if_Pro3027730157355071871nt_int
% 0.30/0.67              @ ( ( product_fst_int_int @ X3 )
% 0.30/0.67                = zero_zero_int )
% 0.30/0.67              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.67              @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % inverse_rat.abs_eq
% 0.30/0.67  thf(fact_9909_DERIV__even__real__root,axiom,
% 0.30/0.67      ! [N: nat,X3: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67         => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.67           => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_even_real_root
% 0.30/0.67  thf(fact_9910_DERIV__real__root__generic,axiom,
% 0.30/0.67      ! [N: nat,X3: real,D4: real] :
% 0.30/0.67        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.67       => ( ( X3 != zero_zero_real )
% 0.30/0.67         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67               => ( D4
% 0.30/0.67                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.30/0.67           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67               => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.67                 => ( D4
% 0.30/0.67                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.30/0.67             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.67                 => ( D4
% 0.30/0.67                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.30/0.67               => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_real_root_generic
% 0.30/0.67  thf(fact_9911_DERIV__neg__imp__decreasing,axiom,
% 0.30/0.67      ! [A: real,B2: real,F: real > real] :
% 0.30/0.67        ( ( ord_less_real @ A @ B2 )
% 0.30/0.67       => ( ! [X5: real] :
% 0.30/0.67              ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.67             => ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.67               => ? [Y5: real] :
% 0.30/0.67                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.67                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.30/0.67         => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_neg_imp_decreasing
% 0.30/0.67  thf(fact_9912_DERIV__pos__imp__increasing,axiom,
% 0.30/0.67      ! [A: real,B2: real,F: real > real] :
% 0.30/0.67        ( ( ord_less_real @ A @ B2 )
% 0.30/0.67       => ( ! [X5: real] :
% 0.30/0.67              ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.67             => ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.67               => ? [Y5: real] :
% 0.30/0.67                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.67                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.30/0.67         => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_pos_imp_increasing
% 0.30/0.67  thf(fact_9913_deriv__nonneg__imp__mono,axiom,
% 0.30/0.67      ! [A: real,B2: real,G: real > real,G2: real > real] :
% 0.30/0.67        ( ! [X5: real] :
% 0.30/0.67            ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.30/0.67           => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.67       => ( ! [X5: real] :
% 0.30/0.67              ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.30/0.67             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X5 ) ) )
% 0.30/0.67         => ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.67           => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B2 ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % deriv_nonneg_imp_mono
% 0.30/0.67  thf(fact_9914_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.30/0.67      ! [A: real,B2: real,F: real > real] :
% 0.30/0.67        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.67       => ( ! [X5: real] :
% 0.30/0.67              ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.67             => ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.67               => ? [Y5: real] :
% 0.30/0.67                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.67                    & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.30/0.67         => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_nonneg_imp_nondecreasing
% 0.30/0.67  thf(fact_9915_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.30/0.67      ! [A: real,B2: real,F: real > real] :
% 0.30/0.67        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.67       => ( ! [X5: real] :
% 0.30/0.67              ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.67             => ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.67               => ? [Y5: real] :
% 0.30/0.67                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.67                    & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.30/0.67         => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_nonpos_imp_nonincreasing
% 0.30/0.67  thf(fact_9916_DERIV__const__ratio__const,axiom,
% 0.30/0.67      ! [A: real,B2: real,F: real > real,K: real] :
% 0.30/0.67        ( ( A != B2 )
% 0.30/0.67       => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.67         => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.30/0.67            = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ K ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_const_ratio_const
% 0.30/0.67  thf(fact_9917_has__real__derivative__pos__inc__left,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
% 0.30/0.67                   => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % has_real_derivative_pos_inc_left
% 0.30/0.67  thf(fact_9918_has__real__derivative__neg__dec__left,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 0.30/0.67       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
% 0.30/0.67                   => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % has_real_derivative_neg_dec_left
% 0.30/0.67  thf(fact_9919_has__real__derivative__pos__inc__right,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
% 0.30/0.67                   => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % has_real_derivative_pos_inc_right
% 0.30/0.67  thf(fact_9920_has__real__derivative__neg__dec__right,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 0.30/0.67       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
% 0.30/0.67                   => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % has_real_derivative_neg_dec_right
% 0.30/0.67  thf(fact_9921_DERIV__pos__inc__right,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_pos_inc_right
% 0.30/0.67  thf(fact_9922_DERIV__neg__dec__right,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_neg_dec_right
% 0.30/0.67  thf(fact_9923_DERIV__neg__dec__left,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_neg_dec_left
% 0.30/0.67  thf(fact_9924_DERIV__pos__inc__left,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.67         => ? [D3: real] :
% 0.30/0.67              ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.67              & ! [H4: real] :
% 0.30/0.67                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.67                 => ( ( ord_less_real @ H4 @ D3 )
% 0.30/0.67                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_pos_inc_left
% 0.30/0.67  thf(fact_9925_MVT2,axiom,
% 0.30/0.67      ! [A: real,B2: real,F: real > real,F4: real > real] :
% 0.30/0.67        ( ( ord_less_real @ A @ B2 )
% 0.30/0.67       => ( ! [X5: real] :
% 0.30/0.67              ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.67             => ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.67               => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.67         => ? [Z4: real] :
% 0.30/0.67              ( ( ord_less_real @ A @ Z4 )
% 0.30/0.67              & ( ord_less_real @ Z4 @ B2 )
% 0.30/0.67              & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.30/0.67                = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ ( F4 @ Z4 ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % MVT2
% 0.30/0.67  thf(fact_9926_DERIV__local__const,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,D: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.67         => ( ! [Y4: real] :
% 0.30/0.67                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
% 0.30/0.67               => ( ( F @ X3 )
% 0.30/0.67                  = ( F @ Y4 ) ) )
% 0.30/0.67           => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_local_const
% 0.30/0.67  thf(fact_9927_DERIV__ln,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_ln
% 0.30/0.67  thf(fact_9928_DERIV__const__average,axiom,
% 0.30/0.67      ! [A: real,B2: real,V: real > real,K: real] :
% 0.30/0.67        ( ( A != B2 )
% 0.30/0.67       => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.67         => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.67            = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_const_average
% 0.30/0.67  thf(fact_9929_DERIV__local__max,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,D: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.67         => ( ! [Y4: real] :
% 0.30/0.67                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
% 0.30/0.67               => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 0.30/0.67           => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_local_max
% 0.30/0.67  thf(fact_9930_DERIV__local__min,axiom,
% 0.30/0.67      ! [F: real > real,L: real,X3: real,D: real] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.67         => ( ! [Y4: real] :
% 0.30/0.67                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
% 0.30/0.67               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.30/0.67           => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_local_min
% 0.30/0.67  thf(fact_9931_DERIV__ln__divide,axiom,
% 0.30/0.67      ! [X3: real] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_ln_divide
% 0.30/0.67  thf(fact_9932_DERIV__pow,axiom,
% 0.30/0.67      ! [N: nat,X3: real,S: set_real] :
% 0.30/0.67        ( has_fi5821293074295781190e_real
% 0.30/0.67        @ ^ [X: real] : ( power_power_real @ X @ N )
% 0.30/0.67        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.67        @ ( topolo2177554685111907308n_real @ X3 @ S ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_pow
% 0.30/0.67  thf(fact_9933_DERIV__fun__pow,axiom,
% 0.30/0.67      ! [G: real > real,M: real,X3: real,N: nat] :
% 0.30/0.67        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.67       => ( has_fi5821293074295781190e_real
% 0.30/0.67          @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
% 0.30/0.67          @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.30/0.67          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % DERIV_fun_pow
% 0.30/0.67  thf(fact_9934_has__real__derivative__powr,axiom,
% 0.30/0.67      ! [Z: real,R: real] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ Z )
% 0.30/0.67       => ( has_fi5821293074295781190e_real
% 0.30/0.67          @ ^ [Z2: real] : ( powr_real @ Z2 @ R )
% 0.30/0.67          @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
% 0.30/0.67          @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.30/0.67  
% 0.30/0.67  % has_real_derivative_powr
% 0.30/0.67  thf(fact_9935_DERIV__log,axiom,
% 0.30/0.67      ! [X3: real,B2: real] :
% 0.30/0.67        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.67       => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_log
% 0.30/0.68  thf(fact_9936_DERIV__fun__powr,axiom,
% 0.30/0.68      ! [G: real > real,M: real,X3: real,R: real] :
% 0.30/0.68        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.68       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 0.30/0.68         => ( has_fi5821293074295781190e_real
% 0.30/0.68            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R )
% 0.30/0.68            @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.30/0.68            @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_fun_powr
% 0.30/0.68  thf(fact_9937_DERIV__powr,axiom,
% 0.30/0.68      ! [G: real > real,M: real,X3: real,F: real > real,R: real] :
% 0.30/0.68        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.68       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 0.30/0.68         => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.68           => ( has_fi5821293074295781190e_real
% 0.30/0.68              @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 0.30/0.68              @ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
% 0.30/0.68              @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_powr
% 0.30/0.68  thf(fact_9938_DERIV__real__sqrt,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.68       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_real_sqrt
% 0.30/0.68  thf(fact_9939_DERIV__arctan,axiom,
% 0.30/0.68      ! [X3: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_arctan
% 0.30/0.68  thf(fact_9940_arsinh__real__has__field__derivative,axiom,
% 0.30/0.68      ! [X3: real,A4: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A4 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % arsinh_real_has_field_derivative
% 0.30/0.68  thf(fact_9941_DERIV__real__sqrt__generic,axiom,
% 0.30/0.68      ! [X3: real,D4: real] :
% 0.30/0.68        ( ( X3 != zero_zero_real )
% 0.30/0.68       => ( ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.68           => ( D4
% 0.30/0.68              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.68         => ( ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.68             => ( D4
% 0.30/0.68                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.68           => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_real_sqrt_generic
% 0.30/0.68  thf(fact_9942_arcosh__real__has__field__derivative,axiom,
% 0.30/0.68      ! [X3: real,A4: set_real] :
% 0.30/0.68        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.68       => ( 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 @ A4 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % arcosh_real_has_field_derivative
% 0.30/0.68  thf(fact_9943_artanh__real__has__field__derivative,axiom,
% 0.30/0.68      ! [X3: real,A4: set_real] :
% 0.30/0.68        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.68       => ( 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 @ A4 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % artanh_real_has_field_derivative
% 0.30/0.68  thf(fact_9944_DERIV__real__root,axiom,
% 0.30/0.68      ! [N: nat,X3: real] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.68         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_real_root
% 0.30/0.68  thf(fact_9945_DERIV__arccos,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.68         => ( 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 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_arccos
% 0.30/0.68  thf(fact_9946_DERIV__arcsin,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.68         => ( 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 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_arcsin
% 0.30/0.68  thf(fact_9947_Maclaurin__all__le,axiom,
% 0.30/0.68      ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
% 0.30/0.68        ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68          = F )
% 0.30/0.68       => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68         => ? [T4: real] :
% 0.30/0.68              ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.68              & ( ( F @ X3 )
% 0.30/0.68                = ( plus_plus_real
% 0.30/0.68                  @ ( groups6591440286371151544t_real
% 0.30/0.68                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.68                    @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin_all_le
% 0.30/0.68  thf(fact_9948_Maclaurin__all__le__objl,axiom,
% 0.30/0.68      ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
% 0.30/0.68        ( ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68            = F )
% 0.30/0.68          & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.68       => ? [T4: real] :
% 0.30/0.68            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.68            & ( ( F @ X3 )
% 0.30/0.68              = ( plus_plus_real
% 0.30/0.68                @ ( groups6591440286371151544t_real
% 0.30/0.68                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.68                  @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin_all_le_objl
% 0.30/0.68  thf(fact_9949_DERIV__odd__real__root,axiom,
% 0.30/0.68      ! [N: nat,X3: real] :
% 0.30/0.68        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.68       => ( ( X3 != zero_zero_real )
% 0.30/0.68         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_odd_real_root
% 0.30/0.68  thf(fact_9950_Maclaurin,axiom,
% 0.30/0.68      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.68       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68         => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68              = F )
% 0.30/0.68           => ( ! [M4: nat,T4: real] :
% 0.30/0.68                  ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                    & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.68                    & ( ord_less_eq_real @ T4 @ H2 ) )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68             => ? [T4: real] :
% 0.30/0.68                  ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/0.68                  & ( ord_less_real @ T4 @ H2 )
% 0.30/0.68                  & ( ( F @ H2 )
% 0.30/0.68                    = ( plus_plus_real
% 0.30/0.68                      @ ( groups6591440286371151544t_real
% 0.30/0.68                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.68                        @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin
% 0.30/0.68  thf(fact_9951_Maclaurin2,axiom,
% 0.30/0.68      ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.30/0.68        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.68       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68            = F )
% 0.30/0.68         => ( ! [M4: nat,T4: real] :
% 0.30/0.68                ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                  & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.68                  & ( ord_less_eq_real @ T4 @ H2 ) )
% 0.30/0.68               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68           => ? [T4: real] :
% 0.30/0.68                ( ( ord_less_real @ zero_zero_real @ T4 )
% 0.30/0.68                & ( ord_less_eq_real @ T4 @ H2 )
% 0.30/0.68                & ( ( F @ H2 )
% 0.30/0.68                  = ( plus_plus_real
% 0.30/0.68                    @ ( groups6591440286371151544t_real
% 0.30/0.68                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.68                      @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin2
% 0.30/0.68  thf(fact_9952_Maclaurin__minus,axiom,
% 0.30/0.68      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.30/0.68       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68         => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68              = F )
% 0.30/0.68           => ( ! [M4: nat,T4: real] :
% 0.30/0.68                  ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                    & ( ord_less_eq_real @ H2 @ T4 )
% 0.30/0.68                    & ( ord_less_eq_real @ T4 @ zero_zero_real ) )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68             => ? [T4: real] :
% 0.30/0.68                  ( ( ord_less_real @ H2 @ T4 )
% 0.30/0.68                  & ( ord_less_real @ T4 @ zero_zero_real )
% 0.30/0.68                  & ( ( F @ H2 )
% 0.30/0.68                    = ( plus_plus_real
% 0.30/0.68                      @ ( groups6591440286371151544t_real
% 0.30/0.68                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.30/0.68                        @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin_minus
% 0.30/0.68  thf(fact_9953_Maclaurin__all__lt,axiom,
% 0.30/0.68      ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
% 0.30/0.68        ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68          = F )
% 0.30/0.68       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68         => ( ( X3 != zero_zero_real )
% 0.30/0.68           => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68             => ? [T4: real] :
% 0.30/0.68                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
% 0.30/0.68                  & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.68                  & ( ( F @ X3 )
% 0.30/0.68                    = ( plus_plus_real
% 0.30/0.68                      @ ( groups6591440286371151544t_real
% 0.30/0.68                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.68                        @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin_all_lt
% 0.30/0.68  thf(fact_9954_Maclaurin__bi__le,axiom,
% 0.30/0.68      ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
% 0.30/0.68        ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68          = F )
% 0.30/0.68       => ( ! [M4: nat,T4: real] :
% 0.30/0.68              ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                & ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) ) )
% 0.30/0.68             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68         => ? [T4: real] :
% 0.30/0.68              ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.68              & ( ( F @ X3 )
% 0.30/0.68                = ( plus_plus_real
% 0.30/0.68                  @ ( groups6591440286371151544t_real
% 0.30/0.68                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 0.30/0.68                    @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin_bi_le
% 0.30/0.68  thf(fact_9955_Taylor,axiom,
% 0.30/0.68      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X3: real] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68            = F )
% 0.30/0.68         => ( ! [M4: nat,T4: real] :
% 0.30/0.68                ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                  & ( ord_less_eq_real @ A @ T4 )
% 0.30/0.68                  & ( ord_less_eq_real @ T4 @ B2 ) )
% 0.30/0.68               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68           => ( ( ord_less_eq_real @ A @ C )
% 0.30/0.68             => ( ( ord_less_eq_real @ C @ B2 )
% 0.30/0.68               => ( ( ord_less_eq_real @ A @ X3 )
% 0.30/0.68                 => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.30/0.68                   => ( ( X3 != C )
% 0.30/0.68                     => ? [T4: real] :
% 0.30/0.68                          ( ( ( ord_less_real @ X3 @ C )
% 0.30/0.68                           => ( ( ord_less_real @ X3 @ T4 )
% 0.30/0.68                              & ( ord_less_real @ T4 @ C ) ) )
% 0.30/0.68                          & ( ~ ( ord_less_real @ X3 @ C )
% 0.30/0.68                           => ( ( ord_less_real @ C @ T4 )
% 0.30/0.68                              & ( ord_less_real @ T4 @ X3 ) ) )
% 0.30/0.68                          & ( ( F @ X3 )
% 0.30/0.68                            = ( plus_plus_real
% 0.30/0.68                              @ ( groups6591440286371151544t_real
% 0.30/0.68                                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ M3 ) )
% 0.30/0.68                                @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Taylor
% 0.30/0.68  thf(fact_9956_Taylor__up,axiom,
% 0.30/0.68      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68            = F )
% 0.30/0.68         => ( ! [M4: nat,T4: real] :
% 0.30/0.68                ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                  & ( ord_less_eq_real @ A @ T4 )
% 0.30/0.68                  & ( ord_less_eq_real @ T4 @ B2 ) )
% 0.30/0.68               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68           => ( ( ord_less_eq_real @ A @ C )
% 0.30/0.68             => ( ( ord_less_real @ C @ B2 )
% 0.30/0.68               => ? [T4: real] :
% 0.30/0.68                    ( ( ord_less_real @ C @ T4 )
% 0.30/0.68                    & ( ord_less_real @ T4 @ B2 )
% 0.30/0.68                    & ( ( F @ B2 )
% 0.30/0.68                      = ( plus_plus_real
% 0.30/0.68                        @ ( groups6591440286371151544t_real
% 0.30/0.68                          @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M3 ) )
% 0.30/0.68                          @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Taylor_up
% 0.30/0.68  thf(fact_9957_Taylor__down,axiom,
% 0.30/0.68      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.68            = F )
% 0.30/0.68         => ( ! [M4: nat,T4: real] :
% 0.30/0.68                ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68                  & ( ord_less_eq_real @ A @ T4 )
% 0.30/0.68                  & ( ord_less_eq_real @ T4 @ B2 ) )
% 0.30/0.68               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68           => ( ( ord_less_real @ A @ C )
% 0.30/0.68             => ( ( ord_less_eq_real @ C @ B2 )
% 0.30/0.68               => ? [T4: real] :
% 0.30/0.68                    ( ( ord_less_real @ A @ T4 )
% 0.30/0.68                    & ( ord_less_real @ T4 @ C )
% 0.30/0.68                    & ( ( F @ A )
% 0.30/0.68                      = ( plus_plus_real
% 0.30/0.68                        @ ( groups6591440286371151544t_real
% 0.30/0.68                          @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 0.30/0.68                          @ ( set_ord_lessThan_nat @ N ) )
% 0.30/0.68                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Taylor_down
% 0.30/0.68  thf(fact_9958_Maclaurin__lemma2,axiom,
% 0.30/0.68      ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
% 0.30/0.68        ( ! [M4: nat,T4: real] :
% 0.30/0.68            ( ( ( ord_less_nat @ M4 @ N )
% 0.30/0.68              & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 0.30/0.68              & ( ord_less_eq_real @ T4 @ H2 ) )
% 0.30/0.68           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 0.30/0.68       => ( ( N
% 0.30/0.68            = ( suc @ K ) )
% 0.30/0.68         => ! [M2: nat,T5: real] :
% 0.30/0.68              ( ( ( ord_less_nat @ M2 @ N )
% 0.30/0.68                & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.30/0.68                & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.30/0.68             => ( has_fi5821293074295781190e_real
% 0.30/0.68                @ ^ [U2: real] :
% 0.30/0.68                    ( minus_minus_real @ ( Diff @ M2 @ U2 )
% 0.30/0.68                    @ ( plus_plus_real
% 0.30/0.68                      @ ( groups6591440286371151544t_real
% 0.30/0.68                        @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
% 0.30/0.68                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 0.30/0.68                      @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 0.30/0.68                @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T5 )
% 0.30/0.68                  @ ( plus_plus_real
% 0.30/0.68                    @ ( groups6591440286371151544t_real
% 0.30/0.68                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T5 @ P5 ) )
% 0.30/0.68                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 0.30/0.68                    @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T5 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 0.30/0.68                @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Maclaurin_lemma2
% 0.30/0.68  thf(fact_9959_DERIV__arctan__series,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.68       => ( has_fi5821293074295781190e_real
% 0.30/0.68          @ ^ [X10: real] :
% 0.30/0.68              ( suminf_real
% 0.30/0.68              @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.30/0.68          @ ( suminf_real
% 0.30/0.68            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.68          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_arctan_series
% 0.30/0.68  thf(fact_9960_DERIV__power__series_H,axiom,
% 0.30/0.68      ! [R2: real,F: nat > real,X0: real] :
% 0.30/0.68        ( ! [X5: real] :
% 0.30/0.68            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 0.30/0.68           => ( summable_real
% 0.30/0.68              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 0.30/0.68       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 0.30/0.68         => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.30/0.68           => ( has_fi5821293074295781190e_real
% 0.30/0.68              @ ^ [X: real] :
% 0.30/0.68                  ( suminf_real
% 0.30/0.68                  @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
% 0.30/0.68              @ ( suminf_real
% 0.30/0.68                @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 0.30/0.68              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_power_series'
% 0.30/0.68  thf(fact_9961_DERIV__isconst3,axiom,
% 0.30/0.68      ! [A: real,B2: real,X3: real,Y3: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68         => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68           => ( ! [X5: real] :
% 0.30/0.68                  ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.68             => ( ( F @ X3 )
% 0.30/0.68                = ( F @ Y3 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_isconst3
% 0.30/0.68  thf(fact_9962_DERIV__series_H,axiom,
% 0.30/0.68      ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B2: real,L5: nat > real] :
% 0.30/0.68        ( ! [N3: nat] :
% 0.30/0.68            ( has_fi5821293074295781190e_real
% 0.30/0.68            @ ^ [X: real] : ( F @ X @ N3 )
% 0.30/0.68            @ ( F4 @ X0 @ N3 )
% 0.30/0.68            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68             => ( summable_real @ ( F @ X5 ) ) )
% 0.30/0.68         => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68           => ( ( summable_real @ ( F4 @ X0 ) )
% 0.30/0.68             => ( ( summable_real @ L5 )
% 0.30/0.68               => ( ! [N3: nat,X5: real,Y4: real] :
% 0.30/0.68                      ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68                     => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68                       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y4 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real
% 0.30/0.68                    @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 0.30/0.68                    @ ( suminf_real @ ( F4 @ X0 ) )
% 0.30/0.68                    @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_series'
% 0.30/0.68  thf(fact_9963_card__greaterThanLessThan,axiom,
% 0.30/0.68      ! [L: nat,U: nat] :
% 0.30/0.68        ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.30/0.68        = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % card_greaterThanLessThan
% 0.30/0.68  thf(fact_9964_isCont__Lb__Ub,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.68                & ( ord_less_eq_real @ X5 @ B2 ) )
% 0.30/0.68             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 0.30/0.68         => ? [L6: real,M8: real] :
% 0.30/0.68              ( ! [X6: real] :
% 0.30/0.68                  ( ( ( ord_less_eq_real @ A @ X6 )
% 0.30/0.68                    & ( ord_less_eq_real @ X6 @ B2 ) )
% 0.30/0.68                 => ( ( ord_less_eq_real @ L6 @ ( F @ X6 ) )
% 0.30/0.68                    & ( ord_less_eq_real @ ( F @ X6 ) @ M8 ) ) )
% 0.30/0.68              & ! [Y5: real] :
% 0.30/0.68                  ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 0.30/0.68                    & ( ord_less_eq_real @ Y5 @ M8 ) )
% 0.30/0.68                 => ? [X5: real] :
% 0.30/0.68                      ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.68                      & ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.68                      & ( ( F @ X5 )
% 0.30/0.68                        = Y5 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_Lb_Ub
% 0.30/0.68  thf(fact_9965_LIM__fun__less__zero,axiom,
% 0.30/0.68      ! [F: real > real,L: real,C: real] :
% 0.30/0.68        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.68       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.68         => ? [R3: real] :
% 0.30/0.68              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.68              & ! [X6: real] :
% 0.30/0.68                  ( ( ( X6 != C )
% 0.30/0.68                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
% 0.30/0.68                 => ( ord_less_real @ ( F @ X6 ) @ zero_zero_real ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIM_fun_less_zero
% 0.30/0.68  thf(fact_9966_LIM__fun__not__zero,axiom,
% 0.30/0.68      ! [F: real > real,L: real,C: real] :
% 0.30/0.68        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.68       => ( ( L != zero_zero_real )
% 0.30/0.68         => ? [R3: real] :
% 0.30/0.68              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.68              & ! [X6: real] :
% 0.30/0.68                  ( ( ( X6 != C )
% 0.30/0.68                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
% 0.30/0.68                 => ( ( F @ X6 )
% 0.30/0.68                   != zero_zero_real ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIM_fun_not_zero
% 0.30/0.68  thf(fact_9967_LIM__fun__gt__zero,axiom,
% 0.30/0.68      ! [F: real > real,L: real,C: real] :
% 0.30/0.68        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.68       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.68         => ? [R3: real] :
% 0.30/0.68              ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.30/0.68              & ! [X6: real] :
% 0.30/0.68                  ( ( ( X6 != C )
% 0.30/0.68                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
% 0.30/0.68                 => ( ord_less_real @ zero_zero_real @ ( F @ X6 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIM_fun_gt_zero
% 0.30/0.68  thf(fact_9968_isCont__real__sqrt,axiom,
% 0.30/0.68      ! [X3: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ sqrt ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_real_sqrt
% 0.30/0.68  thf(fact_9969_isCont__real__root,axiom,
% 0.30/0.68      ! [X3: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ ( root @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_real_root
% 0.30/0.68  thf(fact_9970_isCont__inverse__function2,axiom,
% 0.30/0.68      ! [A: real,X3: real,B2: real,G: real > real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ B2 )
% 0.30/0.68         => ( ! [Z4: real] :
% 0.30/0.68                ( ( ord_less_eq_real @ A @ Z4 )
% 0.30/0.68               => ( ( ord_less_eq_real @ Z4 @ B2 )
% 0.30/0.68                 => ( ( G @ ( F @ Z4 ) )
% 0.30/0.68                    = Z4 ) ) )
% 0.30/0.68           => ( ! [Z4: real] :
% 0.30/0.68                  ( ( ord_less_eq_real @ A @ Z4 )
% 0.30/0.68                 => ( ( ord_less_eq_real @ Z4 @ B2 )
% 0.30/0.68                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.30/0.68             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_inverse_function2
% 0.30/0.68  thf(fact_9971_isCont__arcosh,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.68       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_arcosh
% 0.30/0.68  thf(fact_9972_LIM__cos__div__sin,axiom,
% 0.30/0.68      ( filterlim_real_real
% 0.30/0.68      @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
% 0.30/0.68      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68      @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIM_cos_div_sin
% 0.30/0.68  thf(fact_9973_DERIV__inverse__function,axiom,
% 0.30/0.68      ! [F: real > real,D4: real,G: real > real,X3: real,A: real,B2: real] :
% 0.30/0.68        ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
% 0.30/0.68       => ( ( D4 != zero_zero_real )
% 0.30/0.68         => ( ( ord_less_real @ A @ X3 )
% 0.30/0.68           => ( ( ord_less_real @ X3 @ B2 )
% 0.30/0.68             => ( ! [Y4: real] :
% 0.30/0.68                    ( ( ord_less_real @ A @ Y4 )
% 0.30/0.68                   => ( ( ord_less_real @ Y4 @ B2 )
% 0.30/0.68                     => ( ( F @ ( G @ Y4 ) )
% 0.30/0.68                        = Y4 ) ) )
% 0.30/0.68               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_inverse_function
% 0.30/0.68  thf(fact_9974_isCont__arccos,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.68         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arccos ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_arccos
% 0.30/0.68  thf(fact_9975_isCont__arcsin,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.68         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_arcsin
% 0.30/0.68  thf(fact_9976_LIM__less__bound,axiom,
% 0.30/0.68      ! [B2: real,X3: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ B2 @ X3 )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B2 @ X3 ) )
% 0.30/0.68             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.30/0.68         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
% 0.30/0.68           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIM_less_bound
% 0.30/0.68  thf(fact_9977_isCont__artanh,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.68         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_artanh
% 0.30/0.68  thf(fact_9978_greaterThanLessThan__upto,axiom,
% 0.30/0.68      ( set_or5832277885323065728an_int
% 0.30/0.68      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % greaterThanLessThan_upto
% 0.30/0.68  thf(fact_9979_isCont__inverse__function,axiom,
% 0.30/0.68      ! [D: real,X3: real,G: real > real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.68       => ( ! [Z4: real] :
% 0.30/0.68              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X3 ) ) @ D )
% 0.30/0.68             => ( ( G @ ( F @ Z4 ) )
% 0.30/0.68                = Z4 ) )
% 0.30/0.68         => ( ! [Z4: real] :
% 0.30/0.68                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X3 ) ) @ D )
% 0.30/0.68               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 0.30/0.68           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % isCont_inverse_function
% 0.30/0.68  thf(fact_9980_GMVT_H,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ! [Z4: real] :
% 0.30/0.68              ( ( ord_less_eq_real @ A @ Z4 )
% 0.30/0.68             => ( ( ord_less_eq_real @ Z4 @ B2 )
% 0.30/0.68               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.30/0.68         => ( ! [Z4: real] :
% 0.30/0.68                ( ( ord_less_eq_real @ A @ Z4 )
% 0.30/0.68               => ( ( ord_less_eq_real @ Z4 @ B2 )
% 0.30/0.68                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
% 0.30/0.68           => ( ! [Z4: real] :
% 0.30/0.68                  ( ( ord_less_real @ A @ Z4 )
% 0.30/0.68                 => ( ( ord_less_real @ Z4 @ B2 )
% 0.30/0.68                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.30/0.68             => ( ! [Z4: real] :
% 0.30/0.68                    ( ( ord_less_real @ A @ Z4 )
% 0.30/0.68                   => ( ( ord_less_real @ Z4 @ B2 )
% 0.30/0.68                     => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.30/0.68               => ? [C2: real] :
% 0.30/0.68                    ( ( ord_less_real @ A @ C2 )
% 0.30/0.68                    & ( ord_less_real @ C2 @ B2 )
% 0.30/0.68                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
% 0.30/0.68                      = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ ( F4 @ C2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % GMVT'
% 0.30/0.68  thf(fact_9981_summable__Leibniz_I3_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.68         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.30/0.68           => ! [N9: nat] :
% 0.30/0.68                ( member_real
% 0.30/0.68                @ ( suminf_real
% 0.30/0.68                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.30/0.68                @ ( set_or1222579329274155063t_real
% 0.30/0.68                  @ ( groups6591440286371151544t_real
% 0.30/0.68                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
% 0.30/0.68                  @ ( groups6591440286371151544t_real
% 0.30/0.68                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz(3)
% 0.30/0.68  thf(fact_9982_summable__Leibniz_I2_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.68         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.30/0.68           => ! [N9: nat] :
% 0.30/0.68                ( member_real
% 0.30/0.68                @ ( suminf_real
% 0.30/0.68                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.30/0.68                @ ( set_or1222579329274155063t_real
% 0.30/0.68                  @ ( groups6591440286371151544t_real
% 0.30/0.68                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 0.30/0.68                  @ ( groups6591440286371151544t_real
% 0.30/0.68                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz(2)
% 0.30/0.68  thf(fact_9983_mult__nat__left__at__top,axiom,
% 0.30/0.68      ! [C: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.68       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % mult_nat_left_at_top
% 0.30/0.68  thf(fact_9984_mult__nat__right__at__top,axiom,
% 0.30/0.68      ! [C: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.68       => ( filterlim_nat_nat
% 0.30/0.68          @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 0.30/0.68          @ at_top_nat
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % mult_nat_right_at_top
% 0.30/0.68  thf(fact_9985_monoseq__convergent,axiom,
% 0.30/0.68      ! [X9: nat > real,B5: real] :
% 0.30/0.68        ( ( topolo6980174941875973593q_real @ X9 )
% 0.30/0.68       => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X9 @ I3 ) ) @ B5 )
% 0.30/0.68         => ~ ! [L6: real] :
% 0.30/0.68                ~ ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % monoseq_convergent
% 0.30/0.68  thf(fact_9986_LIMSEQ__root,axiom,
% 0.30/0.68      ( filterlim_nat_real
% 0.30/0.68      @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.68      @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.68      @ at_top_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_root
% 0.30/0.68  thf(fact_9987_nested__sequence__unique,axiom,
% 0.30/0.68      ! [F: nat > real,G: nat > real] :
% 0.30/0.68        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
% 0.30/0.68         => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.68           => ( ( filterlim_nat_real
% 0.30/0.68                @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.30/0.68                @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68                @ at_top_nat )
% 0.30/0.68             => ? [L4: real] :
% 0.30/0.68                  ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
% 0.30/0.68                  & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 0.30/0.68                  & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
% 0.30/0.68                  & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nested_sequence_unique
% 0.30/0.68  thf(fact_9988_LIMSEQ__inverse__zero,axiom,
% 0.30/0.68      ! [X9: nat > real] :
% 0.30/0.68        ( ! [R3: real] :
% 0.30/0.68          ? [N7: nat] :
% 0.30/0.68          ! [N3: nat] :
% 0.30/0.68            ( ( ord_less_eq_nat @ N7 @ N3 )
% 0.30/0.68           => ( ord_less_real @ R3 @ ( X9 @ N3 ) ) )
% 0.30/0.68       => ( filterlim_nat_real
% 0.30/0.68          @ ^ [N2: nat] : ( inverse_inverse_real @ ( X9 @ N2 ) )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_inverse_zero
% 0.30/0.68  thf(fact_9989_LIMSEQ__root__const,axiom,
% 0.30/0.68      ! [C: real] :
% 0.30/0.68        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.68       => ( filterlim_nat_real
% 0.30/0.68          @ ^ [N2: nat] : ( root @ N2 @ C )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_root_const
% 0.30/0.68  thf(fact_9990_increasing__LIMSEQ,axiom,
% 0.30/0.68      ! [F: nat > real,L: real] :
% 0.30/0.68        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 0.30/0.68         => ( ! [E2: real] :
% 0.30/0.68                ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.68               => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
% 0.30/0.68           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % increasing_LIMSEQ
% 0.30/0.68  thf(fact_9991_LIMSEQ__realpow__zero,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.68       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.68         => ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_realpow_zero
% 0.30/0.68  thf(fact_9992_LIMSEQ__divide__realpow__zero,axiom,
% 0.30/0.68      ! [X3: real,A: real] :
% 0.30/0.68        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.68       => ( filterlim_nat_real
% 0.30/0.68          @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N2 ) )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_divide_realpow_zero
% 0.30/0.68  thf(fact_9993_LIMSEQ__abs__realpow__zero,axiom,
% 0.30/0.68      ! [C: real] :
% 0.30/0.68        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.30/0.68       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_abs_realpow_zero
% 0.30/0.68  thf(fact_9994_LIMSEQ__abs__realpow__zero2,axiom,
% 0.30/0.68      ! [C: real] :
% 0.30/0.68        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.30/0.68       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_abs_realpow_zero2
% 0.30/0.68  thf(fact_9995_LIMSEQ__inverse__realpow__zero,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.68       => ( filterlim_nat_real
% 0.30/0.68          @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N2 ) )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_inverse_realpow_zero
% 0.30/0.68  thf(fact_9996_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.30/0.68      ! [R: real] :
% 0.30/0.68        ( filterlim_nat_real
% 0.30/0.68        @ ^ [N2: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
% 0.30/0.68        @ ( topolo2815343760600316023s_real @ R )
% 0.30/0.68        @ at_top_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.30/0.68  thf(fact_9997_summable__Leibniz_I1_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.68         => ( summable_real
% 0.30/0.68            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz(1)
% 0.30/0.68  thf(fact_9998_summable,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.68         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.68           => ( summable_real
% 0.30/0.68              @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable
% 0.30/0.68  thf(fact_9999_cos__diff__limit__1,axiom,
% 0.30/0.68      ! [Theta: nat > real,Theta2: real] :
% 0.30/0.68        ( ( filterlim_nat_real
% 0.30/0.68          @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.68          @ at_top_nat )
% 0.30/0.68       => ~ ! [K3: nat > int] :
% 0.30/0.68              ~ ( filterlim_nat_real
% 0.30/0.68                @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.68                @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.30/0.68                @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % cos_diff_limit_1
% 0.30/0.68  thf(fact_10000_cos__limit__1,axiom,
% 0.30/0.68      ! [Theta: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real
% 0.30/0.68          @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.68          @ at_top_nat )
% 0.30/0.68       => ? [K3: nat > int] :
% 0.30/0.68            ( filterlim_nat_real
% 0.30/0.68            @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.30/0.68            @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68            @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % cos_limit_1
% 0.30/0.68  thf(fact_10001_summable__Leibniz_I4_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.68         => ( filterlim_nat_real
% 0.30/0.68            @ ^ [N2: nat] :
% 0.30/0.68                ( groups6591440286371151544t_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.68            @ ( topolo2815343760600316023s_real
% 0.30/0.68              @ ( suminf_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.68            @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz(4)
% 0.30/0.68  thf(fact_10002_zeroseq__arctan__series,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.68       => ( filterlim_nat_real
% 0.30/0.68          @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.30/0.68          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % zeroseq_arctan_series
% 0.30/0.68  thf(fact_10003_summable__Leibniz_H_I2_J,axiom,
% 0.30/0.68      ! [A: nat > real,N: nat] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.68         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.68           => ( ord_less_eq_real
% 0.30/0.68              @ ( groups6591440286371151544t_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.68              @ ( suminf_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz'(2)
% 0.30/0.68  thf(fact_10004_summable__Leibniz_H_I3_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.68         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.68           => ( filterlim_nat_real
% 0.30/0.68              @ ^ [N2: nat] :
% 0.30/0.68                  ( groups6591440286371151544t_real
% 0.30/0.68                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.68              @ ( topolo2815343760600316023s_real
% 0.30/0.68                @ ( suminf_real
% 0.30/0.68                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.68              @ at_top_nat ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz'(3)
% 0.30/0.68  thf(fact_10005_sums__alternating__upper__lower,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.68         => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68           => ? [L4: real] :
% 0.30/0.68                ( ! [N9: nat] :
% 0.30/0.68                    ( ord_less_eq_real
% 0.30/0.68                    @ ( groups6591440286371151544t_real
% 0.30/0.68                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 0.30/0.68                    @ L4 )
% 0.30/0.68                & ( filterlim_nat_real
% 0.30/0.68                  @ ^ [N2: nat] :
% 0.30/0.68                      ( groups6591440286371151544t_real
% 0.30/0.68                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.68                  @ ( topolo2815343760600316023s_real @ L4 )
% 0.30/0.68                  @ at_top_nat )
% 0.30/0.68                & ! [N9: nat] :
% 0.30/0.68                    ( ord_less_eq_real @ L4
% 0.30/0.68                    @ ( groups6591440286371151544t_real
% 0.30/0.68                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
% 0.30/0.68                & ( filterlim_nat_real
% 0.30/0.68                  @ ^ [N2: nat] :
% 0.30/0.68                      ( groups6591440286371151544t_real
% 0.30/0.68                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.30/0.68                  @ ( topolo2815343760600316023s_real @ L4 )
% 0.30/0.68                  @ at_top_nat ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sums_alternating_upper_lower
% 0.30/0.68  thf(fact_10006_summable__Leibniz_I5_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.68         => ( filterlim_nat_real
% 0.30/0.68            @ ^ [N2: nat] :
% 0.30/0.68                ( groups6591440286371151544t_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.30/0.68            @ ( topolo2815343760600316023s_real
% 0.30/0.68              @ ( suminf_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.68            @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz(5)
% 0.30/0.68  thf(fact_10007_summable__Leibniz_H_I5_J,axiom,
% 0.30/0.68      ! [A: nat > real] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.68         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.68           => ( filterlim_nat_real
% 0.30/0.68              @ ^ [N2: nat] :
% 0.30/0.68                  ( groups6591440286371151544t_real
% 0.30/0.68                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.30/0.68              @ ( topolo2815343760600316023s_real
% 0.30/0.68                @ ( suminf_real
% 0.30/0.68                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.30/0.68              @ at_top_nat ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz'(5)
% 0.30/0.68  thf(fact_10008_summable__Leibniz_H_I4_J,axiom,
% 0.30/0.68      ! [A: nat > real,N: nat] :
% 0.30/0.68        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.68       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.68         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.68           => ( ord_less_eq_real
% 0.30/0.68              @ ( suminf_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.30/0.68              @ ( groups6591440286371151544t_real
% 0.30/0.68                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.30/0.68                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_Leibniz'(4)
% 0.30/0.68  thf(fact_10009_eventually__sequentially__seg,axiom,
% 0.30/0.68      ! [P: nat > $o,K: nat] :
% 0.30/0.68        ( ( eventually_nat
% 0.30/0.68          @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.68          @ at_top_nat )
% 0.30/0.68        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % eventually_sequentially_seg
% 0.30/0.68  thf(fact_10010_sequentially__offset,axiom,
% 0.30/0.68      ! [P: nat > $o,K: nat] :
% 0.30/0.68        ( ( eventually_nat @ P @ at_top_nat )
% 0.30/0.68       => ( eventually_nat
% 0.30/0.68          @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 0.30/0.68          @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sequentially_offset
% 0.30/0.68  thf(fact_10011_eventually__sequentially,axiom,
% 0.30/0.68      ! [P: nat > $o] :
% 0.30/0.68        ( ( eventually_nat @ P @ at_top_nat )
% 0.30/0.68        = ( ? [N6: nat] :
% 0.30/0.68            ! [N2: nat] :
% 0.30/0.68              ( ( ord_less_eq_nat @ N6 @ N2 )
% 0.30/0.68             => ( P @ N2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % eventually_sequentially
% 0.30/0.68  thf(fact_10012_eventually__sequentiallyI,axiom,
% 0.30/0.68      ! [C: nat,P: nat > $o] :
% 0.30/0.68        ( ! [X5: nat] :
% 0.30/0.68            ( ( ord_less_eq_nat @ C @ X5 )
% 0.30/0.68           => ( P @ X5 ) )
% 0.30/0.68       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % eventually_sequentiallyI
% 0.30/0.68  thf(fact_10013_le__sequentially,axiom,
% 0.30/0.68      ! [F5: filter_nat] :
% 0.30/0.68        ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 0.30/0.68        = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % le_sequentially
% 0.30/0.68  thf(fact_10014_real__bounded__linear,axiom,
% 0.30/0.68      ( real_V5970128139526366754l_real
% 0.30/0.68      = ( ^ [F3: real > real] :
% 0.30/0.68          ? [C3: real] :
% 0.30/0.68            ( F3
% 0.30/0.68            = ( ^ [X: real] : ( times_times_real @ X @ C3 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % real_bounded_linear
% 0.30/0.68  thf(fact_10015_card__greaterThanAtMost,axiom,
% 0.30/0.68      ! [L: nat,U: nat] :
% 0.30/0.68        ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 0.30/0.68        = ( minus_minus_nat @ U @ L ) ) ).
% 0.30/0.68  
% 0.30/0.68  % card_greaterThanAtMost
% 0.30/0.68  thf(fact_10016_sqrt__at__top,axiom,
% 0.30/0.68      filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.30/0.68  
% 0.30/0.68  % sqrt_at_top
% 0.30/0.68  thf(fact_10017_eventually__at__left__real,axiom,
% 0.30/0.68      ! [B2: real,A: real] :
% 0.30/0.68        ( ( ord_less_real @ B2 @ A )
% 0.30/0.68       => ( eventually_real
% 0.30/0.68          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B2 @ A ) )
% 0.30/0.68          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % eventually_at_left_real
% 0.30/0.68  thf(fact_10018_filterlim__tan__at__left,axiom,
% 0.30/0.68      filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % filterlim_tan_at_left
% 0.30/0.68  thf(fact_10019_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.30/0.68      ! [B2: real,F: real > real,Flim: real] :
% 0.30/0.68        ( ! [X5: real] :
% 0.30/0.68            ( ( ord_less_eq_real @ B2 @ X5 )
% 0.30/0.68           => ? [Y5: real] :
% 0.30/0.68                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68                & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.30/0.68       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.30/0.68         => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_neg_imp_decreasing_at_top
% 0.30/0.68  thf(fact_10020_tendsto__arctan__at__top,axiom,
% 0.30/0.68      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.30/0.68  
% 0.30/0.68  % tendsto_arctan_at_top
% 0.30/0.68  thf(fact_10021_at__top__le__at__infinity,axiom,
% 0.30/0.68      ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 0.30/0.68  
% 0.30/0.68  % at_top_le_at_infinity
% 0.30/0.68  thf(fact_10022_greaterThanAtMost__upto,axiom,
% 0.30/0.68      ( set_or6656581121297822940st_int
% 0.30/0.68      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % greaterThanAtMost_upto
% 0.30/0.68  thf(fact_10023_filterlim__pow__at__bot__even,axiom,
% 0.30/0.68      ! [N: nat,F: real > real,F5: filter_real] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.30/0.68         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.68           => ( filterlim_real_real
% 0.30/0.68              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.30/0.68              @ at_top_real
% 0.30/0.68              @ F5 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % filterlim_pow_at_bot_even
% 0.30/0.68  thf(fact_10024_at__bot__le__at__infinity,axiom,
% 0.30/0.68      ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 0.30/0.68  
% 0.30/0.68  % at_bot_le_at_infinity
% 0.30/0.68  thf(fact_10025_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.30/0.68      ! [B2: real,F: real > real,Flim: real] :
% 0.30/0.68        ( ! [X5: real] :
% 0.30/0.68            ( ( ord_less_eq_real @ X5 @ B2 )
% 0.30/0.68           => ? [Y5: real] :
% 0.30/0.68                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68                & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.30/0.68       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.30/0.68         => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_pos_imp_increasing_at_bot
% 0.30/0.68  thf(fact_10026_filterlim__pow__at__bot__odd,axiom,
% 0.30/0.68      ! [N: nat,F: real > real,F5: filter_real] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.30/0.68         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.30/0.68           => ( filterlim_real_real
% 0.30/0.68              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.30/0.68              @ at_bot_real
% 0.30/0.68              @ F5 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % filterlim_pow_at_bot_odd
% 0.30/0.68  thf(fact_10027_tendsto__arctan__at__bot,axiom,
% 0.30/0.68      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.30/0.68  
% 0.30/0.68  % tendsto_arctan_at_bot
% 0.30/0.68  thf(fact_10028_Bseq__eq__bounded,axiom,
% 0.30/0.68      ! [F: nat > real,A: real,B2: real] :
% 0.30/0.68        ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.30/0.68       => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Bseq_eq_bounded
% 0.30/0.68  thf(fact_10029_Bseq__realpow,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.68       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.68         => ( bfun_nat_real @ ( power_power_real @ X3 ) @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Bseq_realpow
% 0.30/0.68  thf(fact_10030_tendsto__exp__limit__at__right,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( filterlim_real_real
% 0.30/0.68        @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X3 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 0.30/0.68        @ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
% 0.30/0.68        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % tendsto_exp_limit_at_right
% 0.30/0.68  thf(fact_10031_filterlim__tan__at__right,axiom,
% 0.30/0.68      filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % filterlim_tan_at_right
% 0.30/0.68  thf(fact_10032_eventually__at__right__real,axiom,
% 0.30/0.68      ! [A: real,B2: real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( eventually_real
% 0.30/0.68          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 0.30/0.68          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % eventually_at_right_real
% 0.30/0.68  thf(fact_10033_decseq__bounded,axiom,
% 0.30/0.68      ! [X9: nat > real,B5: real] :
% 0.30/0.68        ( ( order_9091379641038594480t_real @ X9 )
% 0.30/0.68       => ( ! [I3: nat] : ( ord_less_eq_real @ B5 @ ( X9 @ I3 ) )
% 0.30/0.68         => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % decseq_bounded
% 0.30/0.68  thf(fact_10034_decseq__convergent,axiom,
% 0.30/0.68      ! [X9: nat > real,B5: real] :
% 0.30/0.68        ( ( order_9091379641038594480t_real @ X9 )
% 0.30/0.68       => ( ! [I3: nat] : ( ord_less_eq_real @ B5 @ ( X9 @ I3 ) )
% 0.30/0.68         => ~ ! [L6: real] :
% 0.30/0.68                ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.30/0.68               => ~ ! [I2: nat] : ( ord_less_eq_real @ L6 @ ( X9 @ I2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % decseq_convergent
% 0.30/0.68  thf(fact_10035_GMVT,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real,G: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.68                & ( ord_less_eq_real @ X5 @ B2 ) )
% 0.30/0.68             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 0.30/0.68         => ( ! [X5: real] :
% 0.30/0.68                ( ( ( ord_less_real @ A @ X5 )
% 0.30/0.68                  & ( ord_less_real @ X5 @ B2 ) )
% 0.30/0.68               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.68           => ( ! [X5: real] :
% 0.30/0.68                  ( ( ( ord_less_eq_real @ A @ X5 )
% 0.30/0.68                    & ( ord_less_eq_real @ X5 @ B2 ) )
% 0.30/0.68                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
% 0.30/0.68             => ( ! [X5: real] :
% 0.30/0.68                    ( ( ( ord_less_real @ A @ X5 )
% 0.30/0.68                      & ( ord_less_real @ X5 @ B2 ) )
% 0.30/0.68                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.30/0.68               => ? [G_c: real,F_c: real,C2: real] :
% 0.30/0.68                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 0.30/0.68                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 0.30/0.68                    & ( ord_less_real @ A @ C2 )
% 0.30/0.68                    & ( ord_less_real @ C2 @ B2 )
% 0.30/0.68                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ G_c )
% 0.30/0.68                      = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % GMVT
% 0.30/0.68  thf(fact_10036_MVT,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68         => ( ! [X5: real] :
% 0.30/0.68                ( ( ord_less_real @ A @ X5 )
% 0.30/0.68               => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.68           => ? [L4: real,Z4: real] :
% 0.30/0.68                ( ( ord_less_real @ A @ Z4 )
% 0.30/0.68                & ( ord_less_real @ Z4 @ B2 )
% 0.30/0.68                & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 0.30/0.68                & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.30/0.68                  = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ L4 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % MVT
% 0.30/0.68  thf(fact_10037_continuous__on__arcosh_H,axiom,
% 0.30/0.68      ! [A4: set_real,F: real > real] :
% 0.30/0.68        ( ( topolo5044208981011980120l_real @ A4 @ F )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( member_real @ X5 @ A4 )
% 0.30/0.68             => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.30/0.68         => ( topolo5044208981011980120l_real @ A4
% 0.30/0.68            @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % continuous_on_arcosh'
% 0.30/0.68  thf(fact_10038_continuous__image__closed__interval,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_eq_real @ A @ B2 )
% 0.30/0.68       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68         => ? [C2: real,D3: real] :
% 0.30/0.68              ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 0.30/0.68                = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
% 0.30/0.68              & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % continuous_image_closed_interval
% 0.30/0.68  thf(fact_10039_continuous__on__arcosh,axiom,
% 0.30/0.68      ! [A4: set_real] :
% 0.30/0.68        ( ( ord_less_eq_set_real @ A4 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.30/0.68       => ( topolo5044208981011980120l_real @ A4 @ arcosh_real ) ) ).
% 0.30/0.68  
% 0.30/0.68  % continuous_on_arcosh
% 0.30/0.68  thf(fact_10040_Rolle__deriv,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( ( F @ A )
% 0.30/0.68            = ( F @ B2 ) )
% 0.30/0.68         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68           => ( ! [X5: real] :
% 0.30/0.68                  ( ( ord_less_real @ A @ X5 )
% 0.30/0.68                 => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68                   => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.68             => ? [Z4: real] :
% 0.30/0.68                  ( ( ord_less_real @ A @ Z4 )
% 0.30/0.68                  & ( ord_less_real @ Z4 @ B2 )
% 0.30/0.68                  & ( ( F4 @ Z4 )
% 0.30/0.68                    = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rolle_deriv
% 0.30/0.68  thf(fact_10041_mvt,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68         => ( ! [X5: real] :
% 0.30/0.68                ( ( ord_less_real @ A @ X5 )
% 0.30/0.68               => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68                 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.68           => ~ ! [Xi: real] :
% 0.30/0.68                  ( ( ord_less_real @ A @ Xi )
% 0.30/0.68                 => ( ( ord_less_real @ Xi @ B2 )
% 0.30/0.68                   => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 0.30/0.68                     != ( F4 @ Xi @ ( minus_minus_real @ B2 @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % mvt
% 0.30/0.68  thf(fact_10042_DERIV__pos__imp__increasing__open,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( ord_less_real @ A @ X5 )
% 0.30/0.68             => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68               => ? [Y5: real] :
% 0.30/0.68                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.30/0.68         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68           => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_pos_imp_increasing_open
% 0.30/0.68  thf(fact_10043_DERIV__neg__imp__decreasing__open,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ! [X5: real] :
% 0.30/0.68              ( ( ord_less_real @ A @ X5 )
% 0.30/0.68             => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68               => ? [Y5: real] :
% 0.30/0.68                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.30/0.68         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68           => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_neg_imp_decreasing_open
% 0.30/0.68  thf(fact_10044_DERIV__isconst__end,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68         => ( ! [X5: real] :
% 0.30/0.68                ( ( ord_less_real @ A @ X5 )
% 0.30/0.68               => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.68           => ( ( F @ B2 )
% 0.30/0.68              = ( F @ A ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_isconst_end
% 0.30/0.68  thf(fact_10045_continuous__on__artanh,axiom,
% 0.30/0.68      ! [A4: set_real] :
% 0.30/0.68        ( ( ord_less_eq_set_real @ A4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.30/0.68       => ( topolo5044208981011980120l_real @ A4 @ artanh_real ) ) ).
% 0.30/0.68  
% 0.30/0.68  % continuous_on_artanh
% 0.30/0.68  thf(fact_10046_DERIV__isconst2,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real,X3: real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68         => ( ! [X5: real] :
% 0.30/0.68                ( ( ord_less_real @ A @ X5 )
% 0.30/0.68               => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.68           => ( ( ord_less_eq_real @ A @ X3 )
% 0.30/0.68             => ( ( ord_less_eq_real @ X3 @ B2 )
% 0.30/0.68               => ( ( F @ X3 )
% 0.30/0.68                  = ( F @ A ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % DERIV_isconst2
% 0.30/0.68  thf(fact_10047_Rolle,axiom,
% 0.30/0.68      ! [A: real,B2: real,F: real > real] :
% 0.30/0.68        ( ( ord_less_real @ A @ B2 )
% 0.30/0.68       => ( ( ( F @ A )
% 0.30/0.68            = ( F @ B2 ) )
% 0.30/0.68         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 0.30/0.68           => ( ! [X5: real] :
% 0.30/0.68                  ( ( ord_less_real @ A @ X5 )
% 0.30/0.68                 => ( ( ord_less_real @ X5 @ B2 )
% 0.30/0.68                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.30/0.68             => ? [Z4: real] :
% 0.30/0.68                  ( ( ord_less_real @ A @ Z4 )
% 0.30/0.68                  & ( ord_less_real @ Z4 @ B2 )
% 0.30/0.68                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rolle
% 0.30/0.68  thf(fact_10048_mono__Suc,axiom,
% 0.30/0.68      order_mono_nat_nat @ suc ).
% 0.30/0.68  
% 0.30/0.68  % mono_Suc
% 0.30/0.68  thf(fact_10049_mono__times__nat,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % mono_times_nat
% 0.30/0.68  thf(fact_10050_incseq__bounded,axiom,
% 0.30/0.68      ! [X9: nat > real,B5: real] :
% 0.30/0.68        ( ( order_mono_nat_real @ X9 )
% 0.30/0.68       => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B5 )
% 0.30/0.68         => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % incseq_bounded
% 0.30/0.68  thf(fact_10051_incseq__convergent,axiom,
% 0.30/0.68      ! [X9: nat > real,B5: real] :
% 0.30/0.68        ( ( order_mono_nat_real @ X9 )
% 0.30/0.68       => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B5 )
% 0.30/0.68         => ~ ! [L6: real] :
% 0.30/0.68                ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.30/0.68               => ~ ! [I2: nat] : ( ord_less_eq_real @ ( X9 @ I2 ) @ L6 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % incseq_convergent
% 0.30/0.68  thf(fact_10052_mono__ge2__power__minus__self,axiom,
% 0.30/0.68      ! [K: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.68       => ( order_mono_nat_nat
% 0.30/0.68          @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % mono_ge2_power_minus_self
% 0.30/0.68  thf(fact_10053_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.30/0.68      ! [F: nat > real,G: nat > nat] :
% 0.30/0.68        ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.68       => ( ( order_mono_nat_real @ F )
% 0.30/0.68         => ( ( order_5726023648592871131at_nat @ G )
% 0.30/0.68           => ( ( bfun_nat_real
% 0.30/0.68                @ ^ [X: nat] : ( F @ ( G @ X ) )
% 0.30/0.68                @ at_top_nat )
% 0.30/0.68              = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nonneg_incseq_Bseq_subseq_iff
% 0.30/0.68  thf(fact_10054_strict__mono__imp__increasing,axiom,
% 0.30/0.68      ! [F: nat > nat,N: nat] :
% 0.30/0.68        ( ( order_5726023648592871131at_nat @ F )
% 0.30/0.68       => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % strict_mono_imp_increasing
% 0.30/0.68  thf(fact_10055_inj__sgn__power,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( inj_on_real_real
% 0.30/0.68          @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.30/0.68          @ top_top_set_real ) ) ).
% 0.30/0.68  
% 0.30/0.68  % inj_sgn_power
% 0.30/0.68  thf(fact_10056_list__encode_Oelims,axiom,
% 0.30/0.68      ! [X3: list_nat,Y3: nat] :
% 0.30/0.68        ( ( ( nat_list_encode @ X3 )
% 0.30/0.68          = Y3 )
% 0.30/0.68       => ( ( ( X3 = nil_nat )
% 0.30/0.68           => ( Y3 != zero_zero_nat ) )
% 0.30/0.68         => ~ ! [X5: nat,Xs3: list_nat] :
% 0.30/0.68                ( ( X3
% 0.30/0.68                  = ( cons_nat @ X5 @ Xs3 ) )
% 0.30/0.68               => ( Y3
% 0.30/0.68                 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % list_encode.elims
% 0.30/0.68  thf(fact_10057_log__inj,axiom,
% 0.30/0.68      ! [B2: real] :
% 0.30/0.68        ( ( ord_less_real @ one_one_real @ B2 )
% 0.30/0.68       => ( inj_on_real_real @ ( log @ B2 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % log_inj
% 0.30/0.68  thf(fact_10058_list__encode_Osimps_I2_J,axiom,
% 0.30/0.68      ! [X3: nat,Xs2: list_nat] :
% 0.30/0.68        ( ( nat_list_encode @ ( cons_nat @ X3 @ Xs2 ) )
% 0.30/0.68        = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % list_encode.simps(2)
% 0.30/0.68  thf(fact_10059_pos__deriv__imp__strict__mono,axiom,
% 0.30/0.68      ! [F: real > real,F4: real > real] :
% 0.30/0.68        ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.30/0.68       => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X5 ) )
% 0.30/0.68         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % pos_deriv_imp_strict_mono
% 0.30/0.68  thf(fact_10060_list__encode_Opelims,axiom,
% 0.30/0.68      ! [X3: list_nat,Y3: nat] :
% 0.30/0.68        ( ( ( nat_list_encode @ X3 )
% 0.30/0.68          = Y3 )
% 0.30/0.68       => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
% 0.30/0.68         => ( ( ( X3 = nil_nat )
% 0.30/0.68             => ( ( Y3 = zero_zero_nat )
% 0.30/0.68               => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.30/0.68           => ~ ! [X5: nat,Xs3: list_nat] :
% 0.30/0.68                  ( ( X3
% 0.30/0.68                    = ( cons_nat @ X5 @ Xs3 ) )
% 0.30/0.68                 => ( ( Y3
% 0.30/0.68                      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 0.30/0.68                   => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % list_encode.pelims
% 0.30/0.68  thf(fact_10061_inj__on__diff__nat,axiom,
% 0.30/0.68      ! [N4: set_nat,K: nat] :
% 0.30/0.68        ( ! [N3: nat] :
% 0.30/0.68            ( ( member_nat @ N3 @ N4 )
% 0.30/0.68           => ( ord_less_eq_nat @ K @ N3 ) )
% 0.30/0.68       => ( inj_on_nat_nat
% 0.30/0.68          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 0.30/0.68          @ N4 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % inj_on_diff_nat
% 0.30/0.68  thf(fact_10062_inj__Suc,axiom,
% 0.30/0.68      ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% 0.30/0.68  
% 0.30/0.68  % inj_Suc
% 0.30/0.68  thf(fact_10063_summable__reindex,axiom,
% 0.30/0.68      ! [F: nat > real,G: nat > nat] :
% 0.30/0.68        ( ( summable_real @ F )
% 0.30/0.68       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.30/0.68         => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.68           => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % summable_reindex
% 0.30/0.68  thf(fact_10064_suminf__reindex__mono,axiom,
% 0.30/0.68      ! [F: nat > real,G: nat > nat] :
% 0.30/0.68        ( ( summable_real @ F )
% 0.30/0.68       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.30/0.68         => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.68           => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % suminf_reindex_mono
% 0.30/0.68  thf(fact_10065_inj__on__char__of__nat,axiom,
% 0.30/0.68      inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % inj_on_char_of_nat
% 0.30/0.68  thf(fact_10066_suminf__reindex,axiom,
% 0.30/0.68      ! [F: nat > real,G: nat > nat] :
% 0.30/0.68        ( ( summable_real @ F )
% 0.30/0.68       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.30/0.68         => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.30/0.68           => ( ! [X5: nat] :
% 0.30/0.68                  ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.30/0.68                 => ( ( F @ X5 )
% 0.30/0.68                    = zero_zero_real ) )
% 0.30/0.68             => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.30/0.68                = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % suminf_reindex
% 0.30/0.68  thf(fact_10067_powr__real__of__int_H,axiom,
% 0.30/0.68      ! [X3: real,N: int] :
% 0.30/0.68        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.68       => ( ( ( X3 != zero_zero_real )
% 0.30/0.68            | ( ord_less_int @ zero_zero_int @ N ) )
% 0.30/0.68         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
% 0.30/0.68            = ( power_int_real @ X3 @ N ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % powr_real_of_int'
% 0.30/0.68  thf(fact_10068_uniformity__real__def,axiom,
% 0.30/0.68      ( topolo1511823702728130853y_real
% 0.30/0.68      = ( comple2936214249959783750l_real
% 0.30/0.68        @ ( image_2178119161166701260l_real
% 0.30/0.68          @ ^ [E3: real] :
% 0.30/0.68              ( princi6114159922880469582l_real
% 0.30/0.68              @ ( collec3799799289383736868l_real
% 0.30/0.68                @ ( produc5414030515140494994real_o
% 0.30/0.68                  @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 0.30/0.68          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % uniformity_real_def
% 0.30/0.68  thf(fact_10069_uniformity__complex__def,axiom,
% 0.30/0.68      ( topolo896644834953643431omplex
% 0.30/0.68      = ( comple8358262395181532106omplex
% 0.30/0.68        @ ( image_5971271580939081552omplex
% 0.30/0.68          @ ^ [E3: real] :
% 0.30/0.68              ( princi3496590319149328850omplex
% 0.30/0.68              @ ( collec8663557070575231912omplex
% 0.30/0.68                @ ( produc6771430404735790350plex_o
% 0.30/0.68                  @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 0.30/0.68          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % uniformity_complex_def
% 0.30/0.68  thf(fact_10070_eventually__prod__sequentially,axiom,
% 0.30/0.68      ! [P: product_prod_nat_nat > $o] :
% 0.30/0.68        ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.30/0.68        = ( ? [N6: nat] :
% 0.30/0.68            ! [M3: nat] :
% 0.30/0.68              ( ( ord_less_eq_nat @ N6 @ M3 )
% 0.30/0.68             => ! [N2: nat] :
% 0.30/0.68                  ( ( ord_less_eq_nat @ N6 @ N2 )
% 0.30/0.68                 => ( P @ ( product_Pair_nat_nat @ N2 @ M3 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % eventually_prod_sequentially
% 0.30/0.68  thf(fact_10071_min__Suc__Suc,axiom,
% 0.30/0.68      ! [M: nat,N: nat] :
% 0.30/0.68        ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.30/0.68        = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % min_Suc_Suc
% 0.30/0.68  thf(fact_10072_min__0L,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.30/0.68        = zero_zero_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % min_0L
% 0.30/0.68  thf(fact_10073_min__0R,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.30/0.68        = zero_zero_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % min_0R
% 0.30/0.68  thf(fact_10074_min__numeral__Suc,axiom,
% 0.30/0.68      ! [K: num,N: nat] :
% 0.30/0.68        ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.30/0.68        = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % min_numeral_Suc
% 0.30/0.68  thf(fact_10075_min__Suc__numeral,axiom,
% 0.30/0.68      ! [N: nat,K: num] :
% 0.30/0.68        ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.68        = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % min_Suc_numeral
% 0.30/0.68  thf(fact_10076_inf__nat__def,axiom,
% 0.30/0.68      inf_inf_nat = ord_min_nat ).
% 0.30/0.68  
% 0.30/0.68  % inf_nat_def
% 0.30/0.68  thf(fact_10077_min__diff,axiom,
% 0.30/0.68      ! [M: nat,I: nat,N: nat] :
% 0.30/0.68        ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.30/0.68        = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.30/0.68  
% 0.30/0.68  % min_diff
% 0.30/0.68  thf(fact_10078_nat__mult__min__right,axiom,
% 0.30/0.68      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.68        ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 0.30/0.68        = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nat_mult_min_right
% 0.30/0.68  thf(fact_10079_nat__mult__min__left,axiom,
% 0.30/0.68      ! [M: nat,N: nat,Q2: nat] :
% 0.30/0.68        ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 0.30/0.68        = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nat_mult_min_left
% 0.30/0.68  thf(fact_10080_concat__bit__assoc__sym,axiom,
% 0.30/0.68      ! [M: nat,N: nat,K: int,L: int,R: int] :
% 0.30/0.68        ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R )
% 0.30/0.68        = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % concat_bit_assoc_sym
% 0.30/0.68  thf(fact_10081_take__bit__concat__bit__eq,axiom,
% 0.30/0.68      ! [M: nat,N: nat,K: int,L: int] :
% 0.30/0.68        ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 0.30/0.68        = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % take_bit_concat_bit_eq
% 0.30/0.68  thf(fact_10082_min__Suc1,axiom,
% 0.30/0.68      ! [N: nat,M: nat] :
% 0.30/0.68        ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.30/0.68        = ( case_nat_nat @ zero_zero_nat
% 0.30/0.68          @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
% 0.30/0.68          @ M ) ) ).
% 0.30/0.68  
% 0.30/0.68  % min_Suc1
% 0.30/0.68  thf(fact_10083_min__Suc2,axiom,
% 0.30/0.68      ! [M: nat,N: nat] :
% 0.30/0.68        ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.30/0.68        = ( case_nat_nat @ zero_zero_nat
% 0.30/0.68          @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
% 0.30/0.68          @ M ) ) ).
% 0.30/0.68  
% 0.30/0.68  % min_Suc2
% 0.30/0.68  thf(fact_10084_min__enat__simps_I3_J,axiom,
% 0.30/0.68      ! [Q2: extended_enat] :
% 0.30/0.68        ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.30/0.68        = zero_z5237406670263579293d_enat ) ).
% 0.30/0.68  
% 0.30/0.68  % min_enat_simps(3)
% 0.30/0.68  thf(fact_10085_min__enat__simps_I2_J,axiom,
% 0.30/0.68      ! [Q2: extended_enat] :
% 0.30/0.68        ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.30/0.68        = zero_z5237406670263579293d_enat ) ).
% 0.30/0.68  
% 0.30/0.68  % min_enat_simps(2)
% 0.30/0.68  thf(fact_10086_inf__enat__def,axiom,
% 0.30/0.68      inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 0.30/0.68  
% 0.30/0.68  % inf_enat_def
% 0.30/0.68  thf(fact_10087_hd__upt,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ord_less_nat @ I @ J )
% 0.30/0.68       => ( ( hd_nat @ ( upt @ I @ J ) )
% 0.30/0.68          = I ) ) ).
% 0.30/0.68  
% 0.30/0.68  % hd_upt
% 0.30/0.68  thf(fact_10088_drop__upt,axiom,
% 0.30/0.68      ! [M: nat,I: nat,J: nat] :
% 0.30/0.68        ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 0.30/0.68        = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 0.30/0.68  
% 0.30/0.68  % drop_upt
% 0.30/0.68  thf(fact_10089_length__upt,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.30/0.68        = ( minus_minus_nat @ J @ I ) ) ).
% 0.30/0.68  
% 0.30/0.68  % length_upt
% 0.30/0.68  thf(fact_10090_nth__upt,axiom,
% 0.30/0.68      ! [I: nat,K: nat,J: nat] :
% 0.30/0.68        ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.30/0.68       => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.30/0.68          = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nth_upt
% 0.30/0.68  thf(fact_10091_take__upt,axiom,
% 0.30/0.68      ! [I: nat,M: nat,N: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.30/0.68       => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.30/0.68          = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % take_upt
% 0.30/0.68  thf(fact_10092_upt__conv__Nil,axiom,
% 0.30/0.68      ! [J: nat,I: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.68       => ( ( upt @ I @ J )
% 0.30/0.68          = nil_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_conv_Nil
% 0.30/0.68  thf(fact_10093_upt__eq__Nil__conv,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ( upt @ I @ J )
% 0.30/0.68          = nil_nat )
% 0.30/0.68        = ( ( J = zero_zero_nat )
% 0.30/0.68          | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_eq_Nil_conv
% 0.30/0.68  thf(fact_10094_upt__rec__numeral,axiom,
% 0.30/0.68      ! [M: num,N: num] :
% 0.30/0.68        ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.68         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.68            = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.30/0.68        & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.68         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.30/0.68            = nil_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_rec_numeral
% 0.30/0.68  thf(fact_10095_map__add__upt,axiom,
% 0.30/0.68      ! [N: nat,M: nat] :
% 0.30/0.68        ( ( map_nat_nat
% 0.30/0.68          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 0.30/0.68          @ ( upt @ zero_zero_nat @ M ) )
% 0.30/0.68        = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % map_add_upt
% 0.30/0.68  thf(fact_10096_upt__add__eq__append,axiom,
% 0.30/0.68      ! [I: nat,J: nat,K: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.68       => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.68          = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_add_eq_append
% 0.30/0.68  thf(fact_10097_greaterThanAtMost__upt,axiom,
% 0.30/0.68      ( set_or6659071591806873216st_nat
% 0.30/0.68      = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M3 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % greaterThanAtMost_upt
% 0.30/0.68  thf(fact_10098_greaterThanLessThan__upt,axiom,
% 0.30/0.68      ( set_or5834768355832116004an_nat
% 0.30/0.68      = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M3 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % greaterThanLessThan_upt
% 0.30/0.68  thf(fact_10099_atLeastAtMost__upt,axiom,
% 0.30/0.68      ( set_or1269000886237332187st_nat
% 0.30/0.68      = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M3 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % atLeastAtMost_upt
% 0.30/0.68  thf(fact_10100_atLeastLessThan__upt,axiom,
% 0.30/0.68      ( set_or4665077453230672383an_nat
% 0.30/0.68      = ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % atLeastLessThan_upt
% 0.30/0.68  thf(fact_10101_atLeast__upt,axiom,
% 0.30/0.68      ( set_ord_lessThan_nat
% 0.30/0.68      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % atLeast_upt
% 0.30/0.68  thf(fact_10102_atMost__upto,axiom,
% 0.30/0.68      ( set_ord_atMost_nat
% 0.30/0.68      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % atMost_upto
% 0.30/0.68  thf(fact_10103_upt__conv__Cons,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ord_less_nat @ I @ J )
% 0.30/0.68       => ( ( upt @ I @ J )
% 0.30/0.68          = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_conv_Cons
% 0.30/0.68  thf(fact_10104_map__decr__upt,axiom,
% 0.30/0.68      ! [M: nat,N: nat] :
% 0.30/0.68        ( ( map_nat_nat
% 0.30/0.68          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.68          @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.30/0.68        = ( upt @ M @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % map_decr_upt
% 0.30/0.68  thf(fact_10105_upt__eq__Cons__conv,axiom,
% 0.30/0.68      ! [I: nat,J: nat,X3: nat,Xs2: list_nat] :
% 0.30/0.68        ( ( ( upt @ I @ J )
% 0.30/0.68          = ( cons_nat @ X3 @ Xs2 ) )
% 0.30/0.68        = ( ( ord_less_nat @ I @ J )
% 0.30/0.68          & ( I = X3 )
% 0.30/0.68          & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.30/0.68            = Xs2 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_eq_Cons_conv
% 0.30/0.68  thf(fact_10106_upt__rec,axiom,
% 0.30/0.68      ( upt
% 0.30/0.68      = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_rec
% 0.30/0.68  thf(fact_10107_upt__Suc__append,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.68       => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.68          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_Suc_append
% 0.30/0.68  thf(fact_10108_upt__Suc,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.68         => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.68            = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.30/0.68        & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.30/0.68         => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.68            = nil_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % upt_Suc
% 0.30/0.68  thf(fact_10109_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.68      ! [N: nat,J: nat,I: nat] :
% 0.30/0.68        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.30/0.68       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.30/0.68          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nth_sorted_list_of_set_greaterThanLessThan
% 0.30/0.68  thf(fact_10110_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.68      ! [N: nat,J: nat,I: nat] :
% 0.30/0.68        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.30/0.68       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.30/0.68          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nth_sorted_list_of_set_greaterThanAtMost
% 0.30/0.68  thf(fact_10111_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.30/0.68       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.30/0.68          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_list_of_set_greaterThanAtMost
% 0.30/0.68  thf(fact_10112_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.68      ! [I: nat,J: nat] :
% 0.30/0.68        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.30/0.68       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.30/0.68          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_list_of_set_greaterThanLessThan
% 0.30/0.68  thf(fact_10113_sum__list__upt,axiom,
% 0.30/0.68      ! [M: nat,N: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ M @ N )
% 0.30/0.68       => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.30/0.68          = ( groups3542108847815614940at_nat
% 0.30/0.68            @ ^ [X: nat] : X
% 0.30/0.68            @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sum_list_upt
% 0.30/0.68  thf(fact_10114_card__length__sum__list__rec,axiom,
% 0.30/0.68      ! [M: nat,N4: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.30/0.68       => ( ( finite_card_list_nat
% 0.30/0.68            @ ( collect_list_nat
% 0.30/0.68              @ ^ [L2: list_nat] :
% 0.30/0.68                  ( ( ( size_size_list_nat @ L2 )
% 0.30/0.68                    = M )
% 0.30/0.68                  & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.68                    = N4 ) ) ) )
% 0.30/0.68          = ( plus_plus_nat
% 0.30/0.68            @ ( finite_card_list_nat
% 0.30/0.68              @ ( collect_list_nat
% 0.30/0.68                @ ^ [L2: list_nat] :
% 0.30/0.68                    ( ( ( size_size_list_nat @ L2 )
% 0.30/0.68                      = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.30/0.68                    & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.68                      = N4 ) ) ) )
% 0.30/0.68            @ ( finite_card_list_nat
% 0.30/0.68              @ ( collect_list_nat
% 0.30/0.68                @ ^ [L2: list_nat] :
% 0.30/0.68                    ( ( ( size_size_list_nat @ L2 )
% 0.30/0.68                      = M )
% 0.30/0.68                    & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 0.30/0.68                      = N4 ) ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % card_length_sum_list_rec
% 0.30/0.68  thf(fact_10115_card__length__sum__list,axiom,
% 0.30/0.68      ! [M: nat,N4: nat] :
% 0.30/0.68        ( ( finite_card_list_nat
% 0.30/0.68          @ ( collect_list_nat
% 0.30/0.68            @ ^ [L2: list_nat] :
% 0.30/0.68                ( ( ( size_size_list_nat @ L2 )
% 0.30/0.68                  = M )
% 0.30/0.68                & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.68                  = N4 ) ) ) )
% 0.30/0.68        = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ one_one_nat ) @ N4 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % card_length_sum_list
% 0.30/0.68  thf(fact_10116_sorted__upt,axiom,
% 0.30/0.68      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_upt
% 0.30/0.68  thf(fact_10117_sorted__wrt__upt,axiom,
% 0.30/0.68      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_wrt_upt
% 0.30/0.68  thf(fact_10118_sorted__wrt__less__idx,axiom,
% 0.30/0.68      ! [Ns: list_nat,I: nat] :
% 0.30/0.68        ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.30/0.68       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.30/0.68         => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_wrt_less_idx
% 0.30/0.68  thf(fact_10119_sorted__wrt__upto,axiom,
% 0.30/0.68      ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_wrt_upto
% 0.30/0.68  thf(fact_10120_sorted__upto,axiom,
% 0.30/0.68      ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sorted_upto
% 0.30/0.68  thf(fact_10121_VEBT_Osize_I3_J,axiom,
% 0.30/0.68      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.30/0.68        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.30/0.68        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % VEBT.size(3)
% 0.30/0.68  thf(fact_10122_VEBT_Osize__gen_I1_J,axiom,
% 0.30/0.68      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.30/0.68        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.30/0.68        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % VEBT.size_gen(1)
% 0.30/0.68  thf(fact_10123_card__le__Suc__Max,axiom,
% 0.30/0.68      ! [S2: set_nat] :
% 0.30/0.68        ( ( finite_finite_nat @ S2 )
% 0.30/0.68       => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % card_le_Suc_Max
% 0.30/0.68  thf(fact_10124_divide__nat__def,axiom,
% 0.30/0.68      ( divide_divide_nat
% 0.30/0.68      = ( ^ [M3: nat,N2: nat] :
% 0.30/0.68            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.30/0.68            @ ( lattic8265883725875713057ax_nat
% 0.30/0.68              @ ( collect_nat
% 0.30/0.68                @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M3 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % divide_nat_def
% 0.30/0.68  thf(fact_10125_gcd__is__Max__divisors__nat,axiom,
% 0.30/0.68      ! [N: nat,M: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( ( gcd_gcd_nat @ M @ N )
% 0.30/0.68          = ( lattic8265883725875713057ax_nat
% 0.30/0.68            @ ( collect_nat
% 0.30/0.68              @ ^ [D2: nat] :
% 0.30/0.68                  ( ( dvd_dvd_nat @ D2 @ M )
% 0.30/0.68                  & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % gcd_is_Max_divisors_nat
% 0.30/0.68  thf(fact_10126_atLeastLessThan__add__Un,axiom,
% 0.30/0.68      ! [I: nat,J: nat,K: nat] :
% 0.30/0.68        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.68       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.68          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % atLeastLessThan_add_Un
% 0.30/0.68  thf(fact_10127_sup__nat__def,axiom,
% 0.30/0.68      sup_sup_nat = ord_max_nat ).
% 0.30/0.68  
% 0.30/0.68  % sup_nat_def
% 0.30/0.68  thf(fact_10128_sup__enat__def,axiom,
% 0.30/0.68      sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 0.30/0.68  
% 0.30/0.68  % sup_enat_def
% 0.30/0.68  thf(fact_10129_quotient__of__def,axiom,
% 0.30/0.68      ( quotient_of
% 0.30/0.68      = ( ^ [X: rat] :
% 0.30/0.68            ( the_Pr4378521158711661632nt_int
% 0.30/0.68            @ ^ [Pair: product_prod_int_int] :
% 0.30/0.68                ( ( X
% 0.30/0.68                  = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.30/0.68                & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.30/0.68                & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % quotient_of_def
% 0.30/0.68  thf(fact_10130_coprime__abs__left__iff,axiom,
% 0.30/0.68      ! [K: int,L: int] :
% 0.30/0.68        ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
% 0.30/0.68        = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_abs_left_iff
% 0.30/0.68  thf(fact_10131_coprime__abs__right__iff,axiom,
% 0.30/0.68      ! [K: int,L: int] :
% 0.30/0.68        ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
% 0.30/0.68        = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_abs_right_iff
% 0.30/0.68  thf(fact_10132_normalize__stable,axiom,
% 0.30/0.68      ! [Q2: int,P4: int] :
% 0.30/0.68        ( ( ord_less_int @ zero_zero_int @ Q2 )
% 0.30/0.68       => ( ( algebr932160517623751201me_int @ P4 @ Q2 )
% 0.30/0.68         => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.68            = ( product_Pair_int_int @ P4 @ Q2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % normalize_stable
% 0.30/0.68  thf(fact_10133_Rat__cases,axiom,
% 0.30/0.68      ! [Q2: rat] :
% 0.30/0.68        ~ ! [A3: int,B3: int] :
% 0.30/0.68            ( ( Q2
% 0.30/0.68              = ( fract @ A3 @ B3 ) )
% 0.30/0.68           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.30/0.68             => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rat_cases
% 0.30/0.68  thf(fact_10134_Rat__induct,axiom,
% 0.30/0.68      ! [P: rat > $o,Q2: rat] :
% 0.30/0.68        ( ! [A3: int,B3: int] :
% 0.30/0.68            ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.30/0.68           => ( ( algebr932160517623751201me_int @ A3 @ B3 )
% 0.30/0.68             => ( P @ ( fract @ A3 @ B3 ) ) ) )
% 0.30/0.68       => ( P @ Q2 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rat_induct
% 0.30/0.68  thf(fact_10135_quotient__of__coprime,axiom,
% 0.30/0.68      ! [R: rat,P4: int,Q2: int] :
% 0.30/0.68        ( ( ( quotient_of @ R )
% 0.30/0.68          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.68       => ( algebr932160517623751201me_int @ P4 @ Q2 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % quotient_of_coprime
% 0.30/0.68  thf(fact_10136_normalize__coprime,axiom,
% 0.30/0.68      ! [R: product_prod_int_int,P4: int,Q2: int] :
% 0.30/0.68        ( ( ( normalize @ R )
% 0.30/0.68          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.30/0.68       => ( algebr932160517623751201me_int @ P4 @ Q2 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % normalize_coprime
% 0.30/0.68  thf(fact_10137_coprime__crossproduct__int,axiom,
% 0.30/0.68      ! [A: int,D: int,B2: int,C: int] :
% 0.30/0.68        ( ( algebr932160517623751201me_int @ A @ D )
% 0.30/0.68       => ( ( algebr932160517623751201me_int @ B2 @ C )
% 0.30/0.68         => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 0.30/0.68              = ( times_times_int @ ( abs_abs_int @ B2 ) @ ( abs_abs_int @ D ) ) )
% 0.30/0.68            = ( ( ( abs_abs_int @ A )
% 0.30/0.68                = ( abs_abs_int @ B2 ) )
% 0.30/0.68              & ( ( abs_abs_int @ C )
% 0.30/0.68                = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_crossproduct_int
% 0.30/0.68  thf(fact_10138_coprime__common__divisor__int,axiom,
% 0.30/0.68      ! [A: int,B2: int,X3: int] :
% 0.30/0.68        ( ( algebr932160517623751201me_int @ A @ B2 )
% 0.30/0.68       => ( ( dvd_dvd_int @ X3 @ A )
% 0.30/0.68         => ( ( dvd_dvd_int @ X3 @ B2 )
% 0.30/0.68           => ( ( abs_abs_int @ X3 )
% 0.30/0.68              = one_one_int ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_common_divisor_int
% 0.30/0.68  thf(fact_10139_Rat__cases__nonzero,axiom,
% 0.30/0.68      ! [Q2: rat] :
% 0.30/0.68        ( ! [A3: int,B3: int] :
% 0.30/0.68            ( ( Q2
% 0.30/0.68              = ( fract @ A3 @ B3 ) )
% 0.30/0.68           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.30/0.68             => ( ( A3 != zero_zero_int )
% 0.30/0.68               => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
% 0.30/0.68       => ( Q2 = zero_zero_rat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rat_cases_nonzero
% 0.30/0.68  thf(fact_10140_quotient__of__unique,axiom,
% 0.30/0.68      ! [R: rat] :
% 0.30/0.68      ? [X5: product_prod_int_int] :
% 0.30/0.68        ( ( R
% 0.30/0.68          = ( fract @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) )
% 0.30/0.68        & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X5 ) )
% 0.30/0.68        & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) )
% 0.30/0.68        & ! [Y5: product_prod_int_int] :
% 0.30/0.68            ( ( ( R
% 0.30/0.68                = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.30/0.68              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 0.30/0.68              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.30/0.68           => ( Y5 = X5 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % quotient_of_unique
% 0.30/0.68  thf(fact_10141_coprime__int__iff,axiom,
% 0.30/0.68      ! [M: nat,N: nat] :
% 0.30/0.68        ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.68        = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_int_iff
% 0.30/0.68  thf(fact_10142_coprime__nat__abs__right__iff,axiom,
% 0.30/0.68      ! [N: nat,K: int] :
% 0.30/0.68        ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.30/0.68        = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_nat_abs_right_iff
% 0.30/0.68  thf(fact_10143_coprime__nat__abs__left__iff,axiom,
% 0.30/0.68      ! [K: int,N: nat] :
% 0.30/0.68        ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.30/0.68        = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_nat_abs_left_iff
% 0.30/0.68  thf(fact_10144_coprime__Suc__0__left,axiom,
% 0.30/0.68      ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_Suc_0_left
% 0.30/0.68  thf(fact_10145_coprime__Suc__0__right,axiom,
% 0.30/0.68      ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_Suc_0_right
% 0.30/0.68  thf(fact_10146_coprime__crossproduct__nat,axiom,
% 0.30/0.68      ! [A: nat,D: nat,B2: nat,C: nat] :
% 0.30/0.68        ( ( algebr934650988132801477me_nat @ A @ D )
% 0.30/0.68       => ( ( algebr934650988132801477me_nat @ B2 @ C )
% 0.30/0.68         => ( ( ( times_times_nat @ A @ C )
% 0.30/0.68              = ( times_times_nat @ B2 @ D ) )
% 0.30/0.68            = ( ( A = B2 )
% 0.30/0.68              & ( C = D ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_crossproduct_nat
% 0.30/0.68  thf(fact_10147_coprime__common__divisor__nat,axiom,
% 0.30/0.68      ! [A: nat,B2: nat,X3: nat] :
% 0.30/0.68        ( ( algebr934650988132801477me_nat @ A @ B2 )
% 0.30/0.68       => ( ( dvd_dvd_nat @ X3 @ A )
% 0.30/0.68         => ( ( dvd_dvd_nat @ X3 @ B2 )
% 0.30/0.68           => ( X3 = one_one_nat ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_common_divisor_nat
% 0.30/0.68  thf(fact_10148_Gcd__int__set__eq__fold,axiom,
% 0.30/0.68      ! [Xs2: list_int] :
% 0.30/0.68        ( ( gcd_Gcd_int @ ( set_int2 @ Xs2 ) )
% 0.30/0.68        = ( fold_int_int @ gcd_gcd_int @ Xs2 @ zero_zero_int ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Gcd_int_set_eq_fold
% 0.30/0.68  thf(fact_10149_Gcd__nat__set__eq__fold,axiom,
% 0.30/0.68      ! [Xs2: list_nat] :
% 0.30/0.68        ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
% 0.30/0.68        = ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Gcd_nat_set_eq_fold
% 0.30/0.68  thf(fact_10150_Gcd__int__greater__eq__0,axiom,
% 0.30/0.68      ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Gcd_int_greater_eq_0
% 0.30/0.68  thf(fact_10151_coprime__diff__one__right__nat,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_diff_one_right_nat
% 0.30/0.68  thf(fact_10152_coprime__diff__one__left__nat,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.30/0.68       => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.30/0.68  
% 0.30/0.68  % coprime_diff_one_left_nat
% 0.30/0.68  thf(fact_10153_Rats__abs__nat__div__natE,axiom,
% 0.30/0.68      ! [X3: real] :
% 0.30/0.68        ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.30/0.68       => ~ ! [M4: nat,N3: nat] :
% 0.30/0.68              ( ( N3 != zero_zero_nat )
% 0.30/0.68             => ( ( ( abs_abs_real @ X3 )
% 0.30/0.68                  = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.30/0.68               => ~ ( algebr934650988132801477me_nat @ M4 @ N3 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rats_abs_nat_div_natE
% 0.30/0.68  thf(fact_10154_Rat_Opositive_Orsp,axiom,
% 0.30/0.68      ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.30/0.68      @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 )
% 0.30/0.68      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 0.30/0.68      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Rat.positive.rsp
% 0.30/0.68  thf(fact_10155_less__eq__integer_Orsp,axiom,
% 0.30/0.68      ( bNF_re3403563459893282935_int_o
% 0.30/0.68      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re5089333283451836215nt_o_o
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ord_less_eq_int
% 0.30/0.68      @ ord_less_eq_int ) ).
% 0.30/0.68  
% 0.30/0.68  % less_eq_integer.rsp
% 0.30/0.68  thf(fact_10156_less__eq__natural_Orsp,axiom,
% 0.30/0.68      ( bNF_re578469030762574527_nat_o
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re4705727531993890431at_o_o
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ord_less_eq_nat
% 0.30/0.68      @ ord_less_eq_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % less_eq_natural.rsp
% 0.30/0.68  thf(fact_10157_plus__rat_Orsp,axiom,
% 0.30/0.68      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.30/0.68      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.30/0.68      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % plus_rat.rsp
% 0.30/0.68  thf(fact_10158_times__rat_Orsp,axiom,
% 0.30/0.68      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.30/0.68      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.30/0.68      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % times_rat.rsp
% 0.30/0.68  thf(fact_10159_Fract_Orsp,axiom,
% 0.30/0.68      ( bNF_re157797125943740599nt_int
% 0.30/0.68      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re6250860962936578807nt_int
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68        @ ratrel )
% 0.30/0.68      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
% 0.30/0.68      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Fract.rsp
% 0.30/0.68  thf(fact_10160_plus__natural_Orsp,axiom,
% 0.30/0.68      ( bNF_re1345281282404953727at_nat
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re5653821019739307937at_nat
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.68      @ plus_plus_nat
% 0.30/0.68      @ plus_plus_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % plus_natural.rsp
% 0.30/0.68  thf(fact_10161_minus__natural_Orsp,axiom,
% 0.30/0.68      ( bNF_re1345281282404953727at_nat
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re5653821019739307937at_nat
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.68      @ minus_minus_nat
% 0.30/0.68      @ minus_minus_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % minus_natural.rsp
% 0.30/0.68  thf(fact_10162_times__natural_Orsp,axiom,
% 0.30/0.68      ( bNF_re1345281282404953727at_nat
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re5653821019739307937at_nat
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.68      @ times_times_nat
% 0.30/0.68      @ times_times_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % times_natural.rsp
% 0.30/0.68  thf(fact_10163_times__integer_Orsp,axiom,
% 0.30/0.68      ( bNF_re711492959462206631nt_int
% 0.30/0.68      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re4712519889275205905nt_int
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.68      @ times_times_int
% 0.30/0.68      @ times_times_int ) ).
% 0.30/0.68  
% 0.30/0.68  % times_integer.rsp
% 0.30/0.68  thf(fact_10164_divide__natural_Orsp,axiom,
% 0.30/0.68      ( bNF_re1345281282404953727at_nat
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re5653821019739307937at_nat
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 0.30/0.68      @ divide_divide_nat
% 0.30/0.68      @ divide_divide_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % divide_natural.rsp
% 0.30/0.68  thf(fact_10165_sub_Orsp,axiom,
% 0.30/0.68      ( bNF_re8402795839162346335um_int
% 0.30/0.68      @ ^ [Y6: num,Z5: num] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re1822329894187522285nt_int
% 0.30/0.68        @ ^ [Y6: num,Z5: num] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.68      @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % sub.rsp
% 0.30/0.68  thf(fact_10166_less__natural_Orsp,axiom,
% 0.30/0.68      ( bNF_re578469030762574527_nat_o
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re4705727531993890431at_o_o
% 0.30/0.68        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ord_less_nat
% 0.30/0.68      @ ord_less_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % less_natural.rsp
% 0.30/0.68  thf(fact_10167_less__integer_Orsp,axiom,
% 0.30/0.68      ( bNF_re3403563459893282935_int_o
% 0.30/0.68      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re5089333283451836215nt_o_o
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ord_less_int
% 0.30/0.68      @ ord_less_int ) ).
% 0.30/0.68  
% 0.30/0.68  % less_integer.rsp
% 0.30/0.68  thf(fact_10168_uminus__rat_Orsp,axiom,
% 0.30/0.68      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 0.30/0.68      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) )
% 0.30/0.68      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % uminus_rat.rsp
% 0.30/0.68  thf(fact_10169_inverse__rat_Orsp,axiom,
% 0.30/0.68      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 0.30/0.68      @ ^ [X: product_prod_int_int] :
% 0.30/0.68          ( if_Pro3027730157355071871nt_int
% 0.30/0.68          @ ( ( product_fst_int_int @ X )
% 0.30/0.68            = zero_zero_int )
% 0.30/0.68          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.68          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) )
% 0.30/0.68      @ ^ [X: product_prod_int_int] :
% 0.30/0.68          ( if_Pro3027730157355071871nt_int
% 0.30/0.68          @ ( ( product_fst_int_int @ X )
% 0.30/0.68            = zero_zero_int )
% 0.30/0.68          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.68          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % inverse_rat.rsp
% 0.30/0.68  thf(fact_10170_plus__rat_Otransfer,axiom,
% 0.30/0.68      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.30/0.68      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.30/0.68      @ plus_plus_rat ) ).
% 0.30/0.68  
% 0.30/0.68  % plus_rat.transfer
% 0.30/0.68  thf(fact_10171_one__rat_Otransfer,axiom,
% 0.30/0.68      pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% 0.30/0.68  
% 0.30/0.68  % one_rat.transfer
% 0.30/0.68  thf(fact_10172_zero__rat_Otransfer,axiom,
% 0.30/0.68      pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 0.30/0.68  
% 0.30/0.68  % zero_rat.transfer
% 0.30/0.68  thf(fact_10173_Fract_Otransfer,axiom,
% 0.30/0.68      ( bNF_re3461391660133120880nt_rat
% 0.30/0.68      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68      @ ( bNF_re2214769303045360666nt_rat
% 0.30/0.68        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 0.30/0.68        @ pcr_rat )
% 0.30/0.68      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
% 0.30/0.68      @ fract ) ).
% 0.30/0.68  
% 0.30/0.68  % Fract.transfer
% 0.30/0.68  thf(fact_10174_uminus__rat_Otransfer,axiom,
% 0.30/0.68      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 0.30/0.68      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) )
% 0.30/0.68      @ uminus_uminus_rat ) ).
% 0.30/0.68  
% 0.30/0.68  % uminus_rat.transfer
% 0.30/0.68  thf(fact_10175_times__rat_Otransfer,axiom,
% 0.30/0.68      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.30/0.68      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 0.30/0.68      @ times_times_rat ) ).
% 0.30/0.68  
% 0.30/0.68  % times_rat.transfer
% 0.30/0.68  thf(fact_10176_Rat_Opositive_Otransfer,axiom,
% 0.30/0.68      ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.30/0.68      @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 )
% 0.30/0.68      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 0.30/0.68      @ positive ) ).
% 0.30/0.68  
% 0.30/0.68  % Rat.positive.transfer
% 0.30/0.68  thf(fact_10177_inverse__rat_Otransfer,axiom,
% 0.30/0.68      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 0.30/0.68      @ ^ [X: product_prod_int_int] :
% 0.30/0.68          ( if_Pro3027730157355071871nt_int
% 0.30/0.68          @ ( ( product_fst_int_int @ X )
% 0.30/0.68            = zero_zero_int )
% 0.30/0.68          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.68          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) )
% 0.30/0.68      @ inverse_inverse_rat ) ).
% 0.30/0.68  
% 0.30/0.68  % inverse_rat.transfer
% 0.30/0.68  thf(fact_10178_times__int_Otransfer,axiom,
% 0.30/0.68      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [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 ) ) ) ) )
% 0.30/0.68      @ times_times_int ) ).
% 0.30/0.68  
% 0.30/0.68  % times_int.transfer
% 0.30/0.68  thf(fact_10179_zero__int_Otransfer,axiom,
% 0.30/0.68      pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 0.30/0.68  
% 0.30/0.68  % zero_int.transfer
% 0.30/0.68  thf(fact_10180_int__transfer,axiom,
% 0.30/0.68      ( bNF_re6830278522597306478at_int
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ pcr_int
% 0.30/0.68      @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
% 0.30/0.68      @ semiri1314217659103216013at_int ) ).
% 0.30/0.68  
% 0.30/0.68  % int_transfer
% 0.30/0.68  thf(fact_10181_uminus__int_Otransfer,axiom,
% 0.30/0.68      ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 0.30/0.68      @ ( produc2626176000494625587at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
% 0.30/0.68      @ uminus_uminus_int ) ).
% 0.30/0.68  
% 0.30/0.68  % uminus_int.transfer
% 0.30/0.68  thf(fact_10182_nat_Otransfer,axiom,
% 0.30/0.68      ( bNF_re4555766996558763186at_nat @ pcr_int
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.30/0.68      @ nat2 ) ).
% 0.30/0.68  
% 0.30/0.68  % nat.transfer
% 0.30/0.68  thf(fact_10183_one__int_Otransfer,axiom,
% 0.30/0.68      pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.30/0.68  
% 0.30/0.68  % one_int.transfer
% 0.30/0.68  thf(fact_10184_less__int_Otransfer,axiom,
% 0.30/0.68      ( bNF_re717283939379294677_int_o @ pcr_int
% 0.30/0.68      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.30/0.68      @ ord_less_int ) ).
% 0.30/0.68  
% 0.30/0.68  % less_int.transfer
% 0.30/0.68  thf(fact_10185_less__eq__int_Otransfer,axiom,
% 0.30/0.68      ( bNF_re717283939379294677_int_o @ pcr_int
% 0.30/0.68      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.30/0.68      @ ord_less_eq_int ) ).
% 0.30/0.68  
% 0.30/0.68  % less_eq_int.transfer
% 0.30/0.68  thf(fact_10186_plus__int_Otransfer,axiom,
% 0.30/0.68      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 0.30/0.68      @ plus_plus_int ) ).
% 0.30/0.68  
% 0.30/0.68  % plus_int.transfer
% 0.30/0.68  thf(fact_10187_minus__int_Otransfer,axiom,
% 0.30/0.68      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
% 0.30/0.68      @ minus_minus_int ) ).
% 0.30/0.68  
% 0.30/0.68  % minus_int.transfer
% 0.30/0.68  thf(fact_10188_times__int_Orsp,axiom,
% 0.30/0.68      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [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 ) ) ) ) )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [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 ) ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % times_int.rsp
% 0.30/0.68  thf(fact_10189_intrel__iff,axiom,
% 0.30/0.68      ! [X3: nat,Y3: nat,U: nat,V: nat] :
% 0.30/0.68        ( ( intrel @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 0.30/0.68        = ( ( plus_plus_nat @ X3 @ V )
% 0.30/0.68          = ( plus_plus_nat @ U @ Y3 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % intrel_iff
% 0.30/0.68  thf(fact_10190_nat_Orsp,axiom,
% 0.30/0.68      ( bNF_re8246922863344978751at_nat @ intrel
% 0.30/0.68      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 0.30/0.68      @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.30/0.68      @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % nat.rsp
% 0.30/0.68  thf(fact_10191_uminus__int_Orsp,axiom,
% 0.30/0.68      ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 0.30/0.68      @ ( produc2626176000494625587at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
% 0.30/0.68      @ ( produc2626176000494625587at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % uminus_int.rsp
% 0.30/0.68  thf(fact_10192_one__int_Orsp,axiom,
% 0.30/0.68      intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % one_int.rsp
% 0.30/0.68  thf(fact_10193_zero__int_Orsp,axiom,
% 0.30/0.68      intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.30/0.68  
% 0.30/0.68  % zero_int.rsp
% 0.30/0.68  thf(fact_10194_intrel__def,axiom,
% 0.30/0.68      ( intrel
% 0.30/0.68      = ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] :
% 0.30/0.68                ( ( plus_plus_nat @ X @ V4 )
% 0.30/0.68                = ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % intrel_def
% 0.30/0.68  thf(fact_10195_less__int_Orsp,axiom,
% 0.30/0.68      ( bNF_re4202695980764964119_nat_o @ intrel
% 0.30/0.68      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.30/0.68      @ ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % less_int.rsp
% 0.30/0.68  thf(fact_10196_less__eq__int_Orsp,axiom,
% 0.30/0.68      ( bNF_re4202695980764964119_nat_o @ intrel
% 0.30/0.68      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.30/0.68        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 0.30/0.68      @ ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.30/0.68      @ ( produc8739625826339149834_nat_o
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc6081775807080527818_nat_o
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % less_eq_int.rsp
% 0.30/0.68  thf(fact_10197_minus__int_Orsp,axiom,
% 0.30/0.68      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % minus_int.rsp
% 0.30/0.68  thf(fact_10198_plus__int_Orsp,axiom,
% 0.30/0.68      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 0.30/0.68      @ ( produc27273713700761075at_nat
% 0.30/0.68        @ ^ [X: nat,Y: nat] :
% 0.30/0.68            ( produc2626176000494625587at_nat
% 0.30/0.68            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % plus_int.rsp
% 0.30/0.68  thf(fact_10199_vanishes__mult__bounded,axiom,
% 0.30/0.68      ! [X9: nat > rat,Y7: nat > rat] :
% 0.30/0.68        ( ? [A8: rat] :
% 0.30/0.68            ( ( ord_less_rat @ zero_zero_rat @ A8 )
% 0.30/0.68            & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N3 ) ) @ A8 ) )
% 0.30/0.68       => ( ( vanishes @ Y7 )
% 0.30/0.68         => ( vanishes
% 0.30/0.68            @ ^ [N2: nat] : ( times_times_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishes_mult_bounded
% 0.30/0.68  thf(fact_10200_vanishes__const,axiom,
% 0.30/0.68      ! [C: rat] :
% 0.30/0.68        ( ( vanishes
% 0.30/0.68          @ ^ [N2: nat] : C )
% 0.30/0.68        = ( C = zero_zero_rat ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishes_const
% 0.30/0.68  thf(fact_10201_vanishes__minus,axiom,
% 0.30/0.68      ! [X9: nat > rat] :
% 0.30/0.68        ( ( vanishes @ X9 )
% 0.30/0.68       => ( vanishes
% 0.30/0.68          @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X9 @ N2 ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishes_minus
% 0.30/0.68  thf(fact_10202_vanishes__add,axiom,
% 0.30/0.68      ! [X9: nat > rat,Y7: nat > rat] :
% 0.30/0.68        ( ( vanishes @ X9 )
% 0.30/0.68       => ( ( vanishes @ Y7 )
% 0.30/0.68         => ( vanishes
% 0.30/0.68            @ ^ [N2: nat] : ( plus_plus_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishes_add
% 0.30/0.68  thf(fact_10203_vanishes__diff,axiom,
% 0.30/0.68      ! [X9: nat > rat,Y7: nat > rat] :
% 0.30/0.68        ( ( vanishes @ X9 )
% 0.30/0.68       => ( ( vanishes @ Y7 )
% 0.30/0.68         => ( vanishes
% 0.30/0.68            @ ^ [N2: nat] : ( minus_minus_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishes_diff
% 0.30/0.68  thf(fact_10204_vanishes__def,axiom,
% 0.30/0.68      ( vanishes
% 0.30/0.68      = ( ^ [X7: nat > rat] :
% 0.30/0.68          ! [R5: rat] :
% 0.30/0.68            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.30/0.68           => ? [K2: nat] :
% 0.30/0.68              ! [N2: nat] :
% 0.30/0.68                ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.30/0.68               => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishes_def
% 0.30/0.68  thf(fact_10205_vanishesI,axiom,
% 0.30/0.68      ! [X9: nat > rat] :
% 0.30/0.68        ( ! [R3: rat] :
% 0.30/0.68            ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.30/0.68           => ? [K4: nat] :
% 0.30/0.68              ! [N3: nat] :
% 0.30/0.68                ( ( ord_less_eq_nat @ K4 @ N3 )
% 0.30/0.68               => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N3 ) ) @ R3 ) ) )
% 0.30/0.68       => ( vanishes @ X9 ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishesI
% 0.30/0.68  thf(fact_10206_vanishesD,axiom,
% 0.30/0.68      ! [X9: nat > rat,R: rat] :
% 0.30/0.68        ( ( vanishes @ X9 )
% 0.30/0.68       => ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.30/0.68         => ? [K3: nat] :
% 0.30/0.68            ! [N9: nat] :
% 0.30/0.68              ( ( ord_less_eq_nat @ K3 @ N9 )
% 0.30/0.68             => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N9 ) ) @ R ) ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % vanishesD
% 0.30/0.68  thf(fact_10207_Field__natLeq__on,axiom,
% 0.30/0.68      ! [N: nat] :
% 0.30/0.68        ( ( field_nat
% 0.30/0.68          @ ( collec3392354462482085612at_nat
% 0.30/0.68            @ ( produc6081775807080527818_nat_o
% 0.30/0.68              @ ^ [X: nat,Y: nat] :
% 0.30/0.68                  ( ( ord_less_nat @ X @ N )
% 0.30/0.68                  & ( ord_less_nat @ Y @ N )
% 0.30/0.68                  & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 0.30/0.68        = ( collect_nat
% 0.30/0.68          @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % Field_natLeq_on
% 0.30/0.68  thf(fact_10208_natLess__def,axiom,
% 0.30/0.68      ( bNF_Ca8459412986667044542atLess
% 0.30/0.68      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  % natLess_def
% 0.30/0.68  
% 0.30/0.68  % Helper facts (38)
% 0.30/0.68  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.30/0.68      ! [X3: int,Y3: int] :
% 0.30/0.68        ( ( if_int @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.30/0.68      ! [X3: int,Y3: int] :
% 0.30/0.68        ( ( if_int @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.68      ! [X3: nat,Y3: nat] :
% 0.30/0.68        ( ( if_nat @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.68      ! [X3: nat,Y3: nat] :
% 0.30/0.68        ( ( if_nat @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.30/0.68      ! [X3: num,Y3: num] :
% 0.30/0.68        ( ( if_num @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.30/0.68      ! [X3: num,Y3: num] :
% 0.30/0.68        ( ( if_num @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.30/0.68      ! [X3: rat,Y3: rat] :
% 0.30/0.68        ( ( if_rat @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.30/0.68      ! [X3: rat,Y3: rat] :
% 0.30/0.68        ( ( if_rat @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.30/0.68      ! [X3: real,Y3: real] :
% 0.30/0.68        ( ( if_real @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.30/0.68      ! [X3: real,Y3: real] :
% 0.30/0.68        ( ( if_real @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.30/0.68      ! [P: real > $o] :
% 0.30/0.68        ( ( P @ ( fChoice_real @ P ) )
% 0.30/0.68        = ( ? [X7: real] : ( P @ X7 ) ) ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.30/0.68      ! [X3: complex,Y3: complex] :
% 0.30/0.68        ( ( if_complex @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.30/0.68      ! [X3: complex,Y3: complex] :
% 0.30/0.68        ( ( if_complex @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.30/0.68      ! [X3: extended_enat,Y3: extended_enat] :
% 0.30/0.68        ( ( if_Extended_enat @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.30/0.68      ! [X3: extended_enat,Y3: extended_enat] :
% 0.30/0.68        ( ( if_Extended_enat @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.30/0.68      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.68        ( ( if_Code_integer @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.30/0.68      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.68        ( ( if_Code_integer @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.30/0.68      ! [X3: set_int,Y3: set_int] :
% 0.30/0.68        ( ( if_set_int @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.30/0.68      ! [X3: set_int,Y3: set_int] :
% 0.30/0.68        ( ( if_set_int @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.30/0.68      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.30/0.68        ( ( if_VEBT_VEBT @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.30/0.68      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.30/0.68        ( ( if_VEBT_VEBT @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.30/0.68      ! [X3: list_int,Y3: list_int] :
% 0.30/0.68        ( ( if_list_int @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.30/0.68      ! [X3: list_int,Y3: list_int] :
% 0.30/0.68        ( ( if_list_int @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.30/0.68      ! [X3: list_nat,Y3: list_nat] :
% 0.30/0.68        ( ( if_list_nat @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.30/0.68      ! [X3: list_nat,Y3: list_nat] :
% 0.30/0.68        ( ( if_list_nat @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.30/0.68      ! [X3: option_nat,Y3: option_nat] :
% 0.30/0.68        ( ( if_option_nat @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.30/0.68      ! [X3: option_nat,Y3: option_nat] :
% 0.30/0.68        ( ( if_option_nat @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.30/0.68      ! [X3: option_num,Y3: option_num] :
% 0.30/0.68        ( ( if_option_num @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.30/0.68      ! [X3: option_num,Y3: option_num] :
% 0.30/0.68        ( ( if_option_num @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.68      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 0.30/0.68        ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.68      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 0.30/0.68        ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.30/0.68      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.30/0.68        ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.30/0.68      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.30/0.68        ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.30/0.68      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.30/0.68        ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.30/0.68      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.30/0.68        ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.68      ! [P: $o] :
% 0.30/0.68        ( ( P = $true )
% 0.30/0.68        | ( P = $false ) ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.68      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.30/0.68        ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y3 )
% 0.30/0.68        = Y3 ) ).
% 0.30/0.68  
% 0.30/0.68  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.68      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.30/0.68        ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y3 )
% 0.30/0.68        = X3 ) ).
% 0.30/0.68  
% 0.30/0.68  % Conjectures (2)
% 0.30/0.68  thf(conj_0,hypothesis,
% 0.30/0.68      ! [Minilow: nat] :
% 0.30/0.68        ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.68          = ( some_nat @ Minilow ) )
% 0.30/0.68       => thesis ) ).
% 0.30/0.68  
% 0.30/0.68  thf(conj_1,conjecture,
% 0.30/0.68      thesis ).
% 0.30/0.68  
% 0.30/0.68  %------------------------------------------------------------------------------
% 0.30/0.68  ------------------------
% 6.44/2.65  ---- Embedded in THF ---
% 6.44/2.65  %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.44/2.65  %%% Generated on Thu Feb 26 16:58:12 EST 2026
% 6.44/2.65  %%% using '$$dhol' embedding, version 1.3.0.
% 6.44/2.65  %%% Logic specification used:
% 6.44/2.65  %%% thf(spec, logic, $$dhol).
% 6.44/2.65  
% 6.44/2.65  % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2DTF22586.p
% See solution above
% 6.44/2.67  ------------------------
% 9.07/5.14  ---- Cleaned THF ---
% 9.07/5.14  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,
% 9.07/5.14      produc5542196010084753463at_nat: $tType ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      set_Pr1281608226676607948nteger: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.07/5.14      produc1908205239877642774nteger: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 9.07/5.14      produc5491161045314408544at_nat: $tType ).
% 9.07/5.14  
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% 9.07/5.14  thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 9.07/5.14      filter_real: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      option_num: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
% 9.07/5.14      option_nat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 9.07/5.14      filter_nat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
% 9.07/5.14      set_char: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 9.07/5.14      list_real: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 9.07/5.14      set_real: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      list_nat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 9.07/5.14      list_int: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      vEBT_VEBT: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 9.07/5.14      set_rat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
% 9.07/5.14      set_num: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 9.07/5.14      set_nat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.14      set_int: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Code____Numeral__Ointeger,type,
% 9.07/5.14      code_integer: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Extended____Nat__Oenat,type,
% 9.07/5.14      extended_enat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__List__Olist_I_Eo_J,type,
% 9.07/5.14      list_o: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Complex__Ocomplex,type,
% 9.07/5.14      complex: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 9.07/5.14      set_o: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__String__Ochar,type,
% 9.07/5.14      char: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Real__Oreal,type,
% 9.07/5.14      real: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Rat__Orat,type,
% 9.07/5.14      rat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Num__Onum,type,
% 9.07/5.14      num: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Nat__Onat,type,
% 9.07/5.14      nat: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(ty_n_t__Int__Oint,type,
% 9.07/5.14      int: $tType ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 9.07/5.14      archim7802044766580827645g_real: real > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 9.07/5.14      archim3151403230148437115or_rat: rat > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 9.07/5.14      archim6058952711729229775r_real: real > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
% 9.07/5.14      bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 9.07/5.14      bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.07/5.14      bNF_re157797125943740599nt_int: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > product_prod_int_int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_062_It__Int__Oint_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Int__Oint_Mt__Rat__Orat_J,type,
% 9.07/5.14      bNF_re3461391660133120880nt_rat: ( int > int > $o ) > ( ( int > product_prod_int_int ) > ( int > rat ) > $o ) > ( int > int > product_prod_int_int ) > ( int > int > rat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 9.07/5.14      bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.14      bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      bNF_re6250860962936578807nt_int: ( int > int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( int > product_prod_int_int ) > ( int > product_prod_int_int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 9.07/5.14      bNF_re2214769303045360666nt_rat: ( int > int > $o ) > ( product_prod_int_int > rat > $o ) > ( int > product_prod_int_int ) > ( int > rat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_M_Eo_J_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.07/5.14      bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 9.07/5.14      bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 9.07/5.14      bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001_062_It__Num__Onum_Mt__Int__Oint_J_001_062_It__Num__Onum_Mt__Int__Oint_J,type,
% 9.07/5.14      bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.14      bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.07/5.14      bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 9.07/5.14      bNF_re8699439704749558557nt_o_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      bNF_re7145576690424134365nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 9.07/5.14      bNF_re7627151682743391978at_rat: ( product_prod_int_int > rat > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( rat > rat > rat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001_Eo_001_Eo,type,
% 9.07/5.14      bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 9.07/5.14      bNF_re8279943556446156061nt_rat: ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > rat > $o ) > ( product_prod_int_int > product_prod_int_int ) > ( rat > rat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 9.07/5.14      bNF_re717283939379294677_int_o: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( int > int > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      bNF_re7408651293131936558nt_int: ( product_prod_nat_nat > int > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( int > int > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001_Eo_001_Eo,type,
% 9.07/5.14      bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 9.07/5.14      bNF_re7400052026677387805at_int: ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( int > int ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.07/5.14      bNF_re4202695980764964119_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.07/5.14      bNF_re3099431351363272937at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 9.07/5.14      bNF_re3666534408544137501at_o_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      bNF_re8246922863344978751at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( product_prod_nat_nat > nat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      bNF_re2241393799969408733at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat ) > ( product_prod_nat_nat > product_prod_nat_nat ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Binomial_Obinomial,type,
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% 9.07/5.14  thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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% 9.07/5.14  thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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% 9.07/5.14  thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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% 9.07/5.14  thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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% 9.07/5.14  thf(sy_c_Bit__Operations_Oconcat__bit,type,
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% 9.07/5.14      semiri3624122377584611663nteger: nat > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 9.07/5.14      semiri5044797733671781792omplex: nat > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 9.07/5.14      semiri1406184849735516958ct_int: nat > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 9.07/5.14      semiri1408675320244567234ct_nat: nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 9.07/5.14      semiri773545260158071498ct_rat: nat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 9.07/5.14      semiri2265585572941072030t_real: nat > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 9.07/5.14      invers8013647133539491842omplex: complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 9.07/5.14      inverse_inverse_rat: rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 9.07/5.14      inverse_inverse_real: real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 9.07/5.14      at_bot_real: filter_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 9.07/5.14      at_top_nat: filter_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 9.07/5.14      at_top_real: filter_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 9.07/5.14      eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 9.07/5.14      eventually_real: ( real > $o ) > filter_real > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.07/5.14      filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.14      filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.07/5.14      princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.07/5.14      princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 9.07/5.14      finite_card_o: set_o > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 9.07/5.14      finite_card_complex: set_complex > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      finite_card_list_nat: set_list_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 9.07/5.14      finite_card_nat: set_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 9.07/5.14      finite_card_char: set_char > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
% 9.07/5.14      finite_finite_o: set_o > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 9.07/5.14      finite3207457112153483333omplex: set_complex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 9.07/5.14      finite_finite_int: set_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
% 9.07/5.14      finite_finite_list_o: set_list_o > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.07/5.14      finite8712137658972009173omplex: set_list_complex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
% 9.07/5.14      finite3922522038869484883st_int: set_list_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      finite8100373058378681591st_nat: set_list_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.14      finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 9.07/5.14      finite_finite_nat: set_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
% 9.07/5.14      finite_finite_num: set_num > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
% 9.07/5.14      finite_finite_rat: set_rat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 9.07/5.14      finite_finite_real: set_real > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.07/5.14      finite6551019134538273531omplex: set_set_complex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.14      finite6197958912794628473et_int: set_set_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.07/5.14      finite1152437895449049373et_nat: set_set_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 9.07/5.14      bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.07/5.14      bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 9.07/5.14      comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
% 9.07/5.14      comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.07/5.14      comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Oid_001_Eo,type,
% 9.07/5.14      id_o: $o > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
% 9.07/5.14      id_nat: nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 9.07/5.14      inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.14      inj_on_real_real: ( real > real ) > set_real > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001_062_It__Int__Oint_M_Eo_J,type,
% 9.07/5.14      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 ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      map_fu4960017516451851995nt_int: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ) > ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > int > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001_Eo_001_Eo,type,
% 9.07/5.14      map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Int__Oint_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint,type,
% 9.07/5.14      map_fu3667384564859982768at_int: ( int > product_prod_nat_nat ) > ( product_prod_nat_nat > int ) > ( product_prod_nat_nat > product_prod_nat_nat ) > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Rat__Orat_Mt__Rat__Orat_J,type,
% 9.07/5.14      map_fu4333342158222067775at_rat: ( rat > product_prod_int_int ) > ( ( product_prod_int_int > product_prod_int_int ) > rat > rat ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > rat > rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_Eo_001_Eo,type,
% 9.07/5.14      map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Omap__fun_001t__Rat__Orat_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Rat__Orat,type,
% 9.07/5.14      map_fu5673905371560938248nt_rat: ( rat > product_prod_int_int ) > ( product_prod_int_int > rat ) > ( product_prod_int_int > product_prod_int_int ) > rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.14      the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
% 9.07/5.14      gcd_Gcd_int: set_int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 9.07/5.14      gcd_Gcd_nat: set_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_Obezw,type,
% 9.07/5.14      bezw: nat > nat > product_prod_int_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_Obezw__rel,type,
% 9.07/5.14      bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 9.07/5.14      gcd_gcd_int: int > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 9.07/5.14      gcd_gcd_nat: nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_GCD_Ogcd__nat__rel,type,
% 9.07/5.14      gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      abs_abs_Code_integer: code_integer > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 9.07/5.14      abs_abs_complex: complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 9.07/5.14      abs_abs_int: int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 9.07/5.14      abs_abs_rat: rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 9.07/5.14      abs_abs_real: real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
% 9.07/5.14      minus_minus_complex: complex > complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
% 9.07/5.14      minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 9.07/5.14      minus_minus_int: int > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 9.07/5.14      minus_minus_nat: nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
% 9.07/5.14      minus_minus_rat: rat > rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 9.07/5.14      minus_minus_real: real > real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.07/5.14      minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 9.07/5.14      groups4077766827762148844at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Complex__Ocomplex,type,
% 9.07/5.14      groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Int__Oint,type,
% 9.07/5.14      groups4694064378042380927al_int: ( real > int ) > set_real > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.07/5.14      groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Rat__Orat,type,
% 9.07/5.14      groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.14      groups1681761925125756287l_real: ( real > real ) > set_real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
% 9.07/5.14      groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
% 9.07/5.14      groups4561878855575611511st_nat: list_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_HOL_OThe_001t__Int__Oint,type,
% 9.07/5.14      the_int: ( int > $o ) > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
% 9.07/5.14      the_real: ( real > $o ) > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      if_Code_integer: $o > code_integer > code_integer > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Complex__Ocomplex,type,
% 9.07/5.14      if_complex: $o > complex > complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Extended____Nat__Oenat,type,
% 9.07/5.14      if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Int__Oint,type,
% 9.07/5.14      if_int: $o > int > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
% 9.07/5.14      if_list_int: $o > list_int > list_int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      if_list_nat: $o > list_nat > list_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Nat__Onat,type,
% 9.07/5.14      if_nat: $o > nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Num__Onum,type,
% 9.07/5.14      if_num: $o > num > num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.07/5.14      if_option_nat: $o > option_nat > option_nat > option_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      if_option_num: $o > option_num > option_num > option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 9.07/5.14      if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.07/5.14      if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Rat__Orat,type,
% 9.07/5.14      if_rat: $o > rat > rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Real__Oreal,type,
% 9.07/5.14      if_real: $o > real > real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.14      if_set_int: $o > set_int > set_int > set_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_OAbs__Integ,type,
% 9.07/5.14      abs_Integ: product_prod_nat_nat > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_ORep__Integ,type,
% 9.07/5.14      rep_Integ: int > product_prod_nat_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oint__ge__less__than,type,
% 9.07/5.14      int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oint__ge__less__than2,type,
% 9.07/5.14      int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Ointrel,type,
% 9.07/5.14      intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Onat,type,
% 9.07/5.14      nat2: int > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Opcr__int,type,
% 9.07/5.14      pcr_int: product_prod_nat_nat > int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Opower__int_001t__Real__Oreal,type,
% 9.07/5.14      power_int_real: real > int > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 9.07/5.14      ring_1_Ints_real: set_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 9.07/5.14      ring_17405671764205052669omplex: int > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 9.07/5.14      ring_1_of_int_int: int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 9.07/5.14      ring_1_of_int_rat: int > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 9.07/5.14      ring_1_of_int_real: int > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
% 9.07/5.14      inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
% 9.07/5.14      inf_inf_nat: nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 9.07/5.14      semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
% 9.07/5.14      sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
% 9.07/5.14      sup_sup_nat: nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.07/5.14      sup_sup_set_nat: set_nat > set_nat > set_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 9.07/5.14      lattic8265883725875713057ax_nat: set_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.07/5.14      bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
% 9.07/5.14      at_infinity_real: filter_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 9.07/5.14      append_int: list_int > list_int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 9.07/5.14      append_nat: list_nat > list_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 9.07/5.14      drop_nat: nat > list_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Ofold_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.14      fold_int_int: ( int > int > int ) > list_int > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Ofold_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 9.07/5.14      linord2614967742042102400et_nat: set_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 9.07/5.14      cons_int: int > list_int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 9.07/5.14      cons_nat: nat > list_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 9.07/5.14      nil_int: list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 9.07/5.14      nil_nat: list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 9.07/5.14      hd_nat: list_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 9.07/5.14      set_o2: list_o > set_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 9.07/5.14      set_complex2: list_complex > set_complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 9.07/5.14      set_int2: list_int > set_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      set_list_nat2: list_list_nat > set_list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 9.07/5.14      set_nat2: list_nat > set_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 9.07/5.14      set_real2: list_real > set_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist__update_001_Eo,type,
% 9.07/5.14      list_update_o: list_o > nat > $o > list_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
% 9.07/5.14      list_update_complex: list_complex > nat > complex > list_complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 9.07/5.14      list_update_int: list_int > nat > int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 9.07/5.14      list_update_nat: list_nat > nat > nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 9.07/5.14      list_update_real: list_real > nat > real > list_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001_Eo,type,
% 9.07/5.14      nth_o: list_o > nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 9.07/5.14      nth_complex: list_complex > nat > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 9.07/5.14      nth_int: list_int > nat > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      nth_list_nat: list_list_nat > nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 9.07/5.14      nth_nat: list_nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 9.07/5.14      nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.14      nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
% 9.07/5.14      nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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% 9.07/5.14  
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% 9.07/5.14      nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.14      nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).
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% 9.07/5.14      nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.07/5.14      nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 9.07/5.14  
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% 9.07/5.14      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.14      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 9.07/5.14      nth_real: list_real > nat > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 9.07/5.14      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 9.07/5.14      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 9.07/5.14      product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 9.07/5.14      product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.14      product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
% 9.07/5.14      product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 9.07/5.14      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.07/5.14      product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.07/5.14      produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.07/5.14      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oreplicate_001_Eo,type,
% 9.07/5.14      replicate_o: nat > $o > list_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 9.07/5.14      replicate_complex: nat > complex > list_complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 9.07/5.14      replicate_int: nat > int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 9.07/5.14      replicate_nat: nat > nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 9.07/5.14      replicate_real: nat > real > list_real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 9.07/5.14      sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 9.07/5.14      sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 9.07/5.14      take_nat: nat > list_nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oupt,type,
% 9.07/5.14      upt: nat > nat > list_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oupto,type,
% 9.07/5.14      upto: int > int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oupto__aux,type,
% 9.07/5.14      upto_aux: int > int > list_int > list_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_List_Oupto__rel,type,
% 9.07/5.14      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_OSuc,type,
% 9.07/5.14      suc: nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 9.07/5.14      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 9.07/5.14      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Onat_Opred,type,
% 9.07/5.14      pred: nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      semiri4939895301339042750nteger: nat > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 9.07/5.14      semiri8010041392384452111omplex: nat > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 9.07/5.14      semiri4216267220026989637d_enat: nat > extended_enat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 9.07/5.14      semiri1314217659103216013at_int: nat > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 9.07/5.14      semiri1316708129612266289at_nat: nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 9.07/5.14      semiri681578069525770553at_rat: nat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 9.07/5.14      semiri5074537144036343181t_real: nat > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 9.07/5.14      semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
% 9.07/5.14      semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
% 9.07/5.14      semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 9.07/5.14      semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 9.07/5.14      semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 9.07/5.14      size_size_list_o: list_o > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.07/5.14      size_s3451745648224563538omplex: list_complex > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 9.07/5.14      size_size_list_int: list_int > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.07/5.14      size_s3023201423986296836st_nat: list_list_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.14      size_size_list_nat: list_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.07/5.14      size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 9.07/5.14      size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.07/5.14      size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 9.07/5.14      size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.07/5.14      size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.07/5.14      size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.07/5.14      size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 9.07/5.14      size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 9.07/5.14      size_size_list_real: list_real > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.14      size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 9.07/5.14      size_size_num: num > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.07/5.14      size_size_option_nat: option_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      size_size_option_num: option_num > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.07/5.14      size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Olist__encode,type,
% 9.07/5.14      nat_list_encode: list_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 9.07/5.14      nat_list_encode_rel: list_nat > list_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 9.07/5.14      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 9.07/5.14      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 9.07/5.14      nat_prod_encode: product_prod_nat_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 9.07/5.14      nat_set_decode: nat > set_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 9.07/5.14      nat_set_encode: set_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Nat__Bijection_Otriangle,type,
% 9.07/5.14      nat_triangle: nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_NthRoot_Oroot,type,
% 9.07/5.14      root: nat > real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_NthRoot_Osqrt,type,
% 9.07/5.14      sqrt: real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_OBitM,type,
% 9.07/5.14      bitM: num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oinc,type,
% 9.07/5.14      inc: num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 9.07/5.14      neg_nu7009210354673126013omplex: complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 9.07/5.14      neg_numeral_dbl_int: int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 9.07/5.14      neg_numeral_dbl_rat: rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 9.07/5.14      neg_numeral_dbl_real: real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 9.07/5.14      neg_nu6511756317524482435omplex: complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 9.07/5.14      neg_nu3811975205180677377ec_int: int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 9.07/5.14      neg_nu3179335615603231917ec_rat: rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 9.07/5.14      neg_nu6075765906172075777c_real: real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 9.07/5.14      neg_nu8557863876264182079omplex: complex > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 9.07/5.14      neg_nu5851722552734809277nc_int: int > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 9.07/5.14      neg_nu5219082963157363817nc_rat: rat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 9.07/5.14      neg_nu8295874005876285629c_real: real > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 9.07/5.14      neg_numeral_sub_int: num > num > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onum_OBit0,type,
% 9.07/5.14      bit0: num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onum_OBit1,type,
% 9.07/5.14      bit1: num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onum_OOne,type,
% 9.07/5.14      one: num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onum_Osize__num,type,
% 9.07/5.14      size_num: num > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onum__of__nat,type,
% 9.07/5.14      num_of_nat: nat > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      numera6620942414471956472nteger: num > code_integer ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 9.07/5.14      numera6690914467698888265omplex: num > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 9.07/5.14      numera1916890842035813515d_enat: num > extended_enat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 9.07/5.14      numeral_numeral_int: num > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 9.07/5.14      numeral_numeral_nat: num > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 9.07/5.14      numeral_numeral_rat: num > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 9.07/5.14      numeral_numeral_real: num > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Opow,type,
% 9.07/5.14      pow: num > num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Opred__numeral,type,
% 9.07/5.14      pred_numeral: num > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Num_Osqr,type,
% 9.07/5.14      sqr: num > num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 9.07/5.14      none_nat: option_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 9.07/5.14      none_num: option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 9.07/5.14      some_nat: nat > option_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 9.07/5.14      some_num: num > option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 9.07/5.14      case_option_int_num: int > ( num > int ) > option_num > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.07/5.14      map_option_num_num: ( num > num ) > option_num > option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
% 9.07/5.14      size_option_nat: ( nat > nat ) > option_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 9.07/5.14      size_option_num: ( num > nat ) > option_num > nat ).
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% 9.07/5.14  thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 9.07/5.14      the_nat: option_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
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% 9.07/5.14  
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
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% 9.07/5.14  
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% 9.07/5.14  
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% 9.07/5.14  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 9.07/5.14  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.07/5.14  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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% 9.07/5.14  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
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% 9.07/5.14  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
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% 9.07/5.14  thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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% 9.07/5.14  
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% 9.07/5.14      produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 9.07/5.14      produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 9.07/5.14      produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.07/5.14      produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.14      produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.07/5.14      produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 9.07/5.14      product_Pair_o_int: $o > int > product_prod_o_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 9.07/5.14      produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
% 9.07/5.14      product_Pair_int_o: int > $o > product_prod_int_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.14      product_Pair_int_int: int > int > product_prod_int_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
% 9.07/5.14      product_Pair_int_nat: int > nat > product_prod_int_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 9.07/5.14      product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.14      produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 9.07/5.14      product_Pair_num_num: num > num > product_prod_num_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.07/5.14      produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.07/5.14      produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.07/5.14      produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.14      produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.07/5.14      produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.07/5.14      produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.07/5.14      produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.07/5.14      produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.14      produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 9.07/5.14      produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_Eo,type,
% 9.07/5.14      produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).
% 9.07/5.14  
% 9.07/5.14  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__Complex__Ocomplex_J,type,
% 9.07/5.14      produc2592262431452330817omplex: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex ) > produc8763457246119570046nteger > set_complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.14      produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 9.07/5.14      produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 9.07/5.14      produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 9.07/5.14      produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 9.07/5.14      produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 9.07/5.14      produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.07/5.14      produc8580519160106071146omplex: ( int > int > set_complex ) > product_prod_int_int > set_complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.14      produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.07/5.14      produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.07/5.14      produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 9.07/5.14      produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.07/5.14      produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 9.07/5.14      produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.14      produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 9.07/5.14  
% 9.07/5.14  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,
% 9.07/5.14      produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.07/5.14      produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.07/5.14      produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.07/5.14      produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 9.07/5.14      produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J_001_Eo,type,
% 9.07/5.14      produc3914248068834153634plex_o: ( set_complex > set_complex > $o ) > produc8064648209034914857omplex > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J_001_Eo,type,
% 9.07/5.14      produc4109468873575309990_int_o: ( set_int > set_int > $o ) > produc2115011035271226405et_int > $o ).
% 9.07/5.14  
% 9.07/5.14  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
% 9.07/5.15      produc6247414631856714078_nat_o: ( set_nat > set_nat > $o ) > produc7819656566062154093et_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.15      product_fst_int_int: product_prod_int_int > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.15      product_fst_nat_nat: product_prod_nat_nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.15      product_snd_int_int: product_prod_int_int > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.15      product_snd_nat_nat: product_prod_nat_nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_OAbs__Rat,type,
% 9.07/5.15      abs_Rat: product_prod_int_int > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_OFract,type,
% 9.07/5.15      fract: int > int > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_OFrct,type,
% 9.07/5.15      frct: product_prod_int_int > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_ORep__Rat,type,
% 9.07/5.15      rep_Rat: rat > product_prod_int_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 9.07/5.15      field_5140801741446780682s_real: set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 9.07/5.15      field_7254667332652039916t_real: rat > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Onormalize,type,
% 9.07/5.15      normalize: product_prod_int_int > product_prod_int_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Oof__int,type,
% 9.07/5.15      of_int: int > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Opcr__rat,type,
% 9.07/5.15      pcr_rat: product_prod_int_int > rat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Opositive,type,
% 9.07/5.15      positive: rat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Oquotient__of,type,
% 9.07/5.15      quotient_of: rat > product_prod_int_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rat_Oratrel,type,
% 9.07/5.15      ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real_Ovanishes,type,
% 9.07/5.15      vanishes: ( nat > rat ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 9.07/5.15      real_V2521375963428798218omplex: set_complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.15      real_V5970128139526366754l_real: ( real > real ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 9.07/5.15      real_V3694042436643373181omplex: complex > complex > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 9.07/5.15      real_V975177566351809787t_real: real > real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 9.07/5.15      real_V1022390504157884413omplex: complex > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 9.07/5.15      real_V7735802525324610683m_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 9.07/5.15      real_V4546457046886955230omplex: real > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 9.07/5.15      real_V2046097035970521341omplex: real > complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 9.07/5.15      real_V1485227260804924795R_real: real > real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Relation_OField_001t__Nat__Onat,type,
% 9.07/5.15      field_nat: set_Pr1261947904930325089at_nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 9.07/5.15      algebr932160517623751201me_int: int > int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 9.07/5.15      algebr934650988132801477me_nat: nat > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 9.07/5.15      divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 9.07/5.15      divide1717551699836669952omplex: complex > complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 9.07/5.15      divide_divide_int: int > int > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 9.07/5.15      divide_divide_nat: nat > nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 9.07/5.15      divide_divide_rat: rat > rat > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 9.07/5.15      divide_divide_real: real > real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 9.07/5.15      dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 9.07/5.15      dvd_dvd_complex: complex > complex > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 9.07/5.15      dvd_dvd_int: int > int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 9.07/5.15      dvd_dvd_nat: nat > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 9.07/5.15      dvd_dvd_rat: rat > rat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 9.07/5.15      dvd_dvd_real: real > real > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 9.07/5.15      modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 9.07/5.15      modulo_modulo_int: int > int > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 9.07/5.15      modulo_modulo_nat: nat > nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 9.07/5.15      zero_n356916108424825756nteger: $o > code_integer ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 9.07/5.15      zero_n1201886186963655149omplex: $o > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 9.07/5.15      zero_n2684676970156552555ol_int: $o > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 9.07/5.15      zero_n2687167440665602831ol_nat: $o > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 9.07/5.15      zero_n2052037380579107095ol_rat: $o > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 9.07/5.15      zero_n3304061248610475627l_real: $o > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 9.07/5.15      suminf_complex: ( nat > complex ) > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
% 9.07/5.15      suminf_int: ( nat > int ) > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
% 9.07/5.15      suminf_nat: ( nat > nat ) > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 9.07/5.15      suminf_real: ( nat > real ) > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 9.07/5.15      summable_complex: ( nat > complex ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osummable_001t__Int__Oint,type,
% 9.07/5.15      summable_int: ( nat > int ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
% 9.07/5.15      summable_nat: ( nat > nat ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 9.07/5.15      summable_real: ( nat > real ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
% 9.07/5.15      sums_complex: ( nat > complex ) > complex > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osums_001t__Int__Oint,type,
% 9.07/5.15      sums_int: ( nat > int ) > int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osums_001t__Nat__Onat,type,
% 9.07/5.15      sums_nat: ( nat > nat ) > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 9.07/5.15      sums_real: ( nat > real ) > real > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001_Eo,type,
% 9.07/5.15      collect_o: ( $o > $o ) > set_o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 9.07/5.15      collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 9.07/5.15      collect_complex: ( complex > $o ) > set_complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 9.07/5.15      collect_int: ( int > $o ) > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
% 9.07/5.15      collect_list_o: ( list_o > $o ) > set_list_o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.07/5.15      collect_list_complex: ( list_complex > $o ) > set_list_complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
% 9.07/5.15      collect_list_int: ( list_int > $o ) > set_list_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.15      collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.15      collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 9.07/5.15      collect_nat: ( nat > $o ) > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Num__Onum,type,
% 9.07/5.15      collect_num: ( num > $o ) > set_num ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 9.07/5.15      collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.15      collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.15      collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
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% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 9.07/5.15      collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 9.07/5.15      collec5108298041176329748omplex: ( produc8064648209034914857omplex > $o ) > set_Pr6308028481084910985omplex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
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% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
% 9.07/5.15      collect_rat: ( rat > $o ) > set_rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 9.07/5.15      collect_real: ( real > $o ) > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.07/5.15      collect_set_complex: ( set_complex > $o ) > set_set_complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.15      collect_set_int: ( set_int > $o ) > set_set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.07/5.15      collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.15      collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 9.07/5.15      image_int_int: ( int > int ) > set_int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 9.07/5.15      image_nat_int: ( nat > int ) > set_nat > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.07/5.15      image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.07/5.15      image_nat_real: ( nat > real ) > set_nat > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
% 9.07/5.15      image_nat_char: ( nat > char ) > set_nat > set_char ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 9.07/5.15      image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 9.07/5.15      image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.15      image_real_real: ( real > real ) > set_real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 9.07/5.15      image_char_nat: ( char > nat ) > set_char > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 9.07/5.15      insert_int: int > set_int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 9.07/5.15      insert_nat: nat > set_nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 9.07/5.15      insert_real: real > set_real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 9.07/5.15      set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 9.07/5.15      set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 9.07/5.15      set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 9.07/5.15      set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 9.07/5.15      set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel_001t__Nat__Onat,type,
% 9.07/5.15      set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 9.07/5.15      set_or1266510415728281911st_int: int > int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 9.07/5.15      set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 9.07/5.15      set_or7049704709247886629st_num: num > num > set_num ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 9.07/5.15      set_or633870826150836451st_rat: rat > rat > set_rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 9.07/5.15      set_or1222579329274155063t_real: real > real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.15      set_or370866239135849197et_int: set_int > set_int > set_set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 9.07/5.15      set_or4662586982721622107an_int: int > int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 9.07/5.15      set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 9.07/5.15      set_ord_atLeast_real: real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
% 9.07/5.15      set_ord_atMost_int: int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 9.07/5.15      set_ord_atMost_nat: nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
% 9.07/5.15      set_ord_atMost_num: num > set_num ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
% 9.07/5.15      set_ord_atMost_rat: rat > set_rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
% 9.07/5.15      set_ord_atMost_real: real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.15      set_or58775011639299419et_int: set_int > set_set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 9.07/5.15      set_or6656581121297822940st_int: int > int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 9.07/5.15      set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 9.07/5.15      set_or5832277885323065728an_int: int > int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 9.07/5.15      set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 9.07/5.15      set_or1633881224788618240n_real: real > real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 9.07/5.15      set_or5849166863359141190n_real: real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 9.07/5.15      set_ord_lessThan_int: int > set_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 9.07/5.15      set_ord_lessThan_nat: nat > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 9.07/5.15      set_ord_lessThan_num: num > set_num ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 9.07/5.15      set_ord_lessThan_rat: rat > set_rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 9.07/5.15      set_or5984915006950818249n_real: real > set_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_String_Oascii__of,type,
% 9.07/5.15      ascii_of: char > char ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_String_Ochar_OChar,type,
% 9.07/5.15      char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 9.07/5.15      comm_s629917340098488124ar_nat: char > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_String_Ointeger__of__char,type,
% 9.07/5.15      integer_of_char: char > code_integer ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 9.07/5.15      unique3096191561947761185of_nat: nat > char ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.15      topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.07/5.15      topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 9.07/5.15      topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 9.07/5.15      topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 9.07/5.15      topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 9.07/5.15      topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 9.07/5.15      topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.15      topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 9.07/5.15      topolo2177554685111907308n_real: real > set_real > filter_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 9.07/5.15      topolo2815343760600316023s_real: real > filter_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 9.07/5.15      topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 9.07/5.15      topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 9.07/5.15      topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oarccos,type,
% 9.07/5.15      arccos: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 9.07/5.15      arcosh_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oarcsin,type,
% 9.07/5.15      arcsin: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oarctan,type,
% 9.07/5.15      arctan: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 9.07/5.15      arsinh_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 9.07/5.15      artanh_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 9.07/5.15      cos_complex: complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 9.07/5.15      cos_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Ocos__coeff,type,
% 9.07/5.15      cos_coeff: nat > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 9.07/5.15      cosh_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 9.07/5.15      cot_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 9.07/5.15      diffs_complex: ( nat > complex ) > nat > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 9.07/5.15      diffs_int: ( nat > int ) > nat > int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 9.07/5.15      diffs_rat: ( nat > rat ) > nat > rat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 9.07/5.15      diffs_real: ( nat > real ) > nat > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 9.07/5.15      exp_complex: complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 9.07/5.15      exp_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 9.07/5.15      ln_ln_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Olog,type,
% 9.07/5.15      log: real > real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Opi,type,
% 9.07/5.15      pi: real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 9.07/5.15      powr_real: real > real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 9.07/5.15      sin_complex: complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 9.07/5.15      sin_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Osin__coeff,type,
% 9.07/5.15      sin_coeff: nat > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 9.07/5.15      sinh_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 9.07/5.15      tan_complex: complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 9.07/5.15      tan_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 9.07/5.15      tanh_complex: complex > complex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 9.07/5.15      tanh_real: real > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 9.07/5.15      vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 9.07/5.15      vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 9.07/5.15      vEBT_size_VEBT: vEBT_VEBT > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 9.07/5.15      vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 9.07/5.15      vEBT_VEBT_high: nat > nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 9.07/5.15      vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 9.07/5.15      vEBT_VEBT_low: nat > nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 9.07/5.15      vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 9.07/5.15      vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 9.07/5.15      vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 9.07/5.15      vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 9.07/5.15      vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 9.07/5.15      vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 9.07/5.15      vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 9.07/5.15      vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 9.07/5.15      vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 9.07/5.15      vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 9.07/5.15      vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 9.07/5.15      vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 9.07/5.15      vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 9.07/5.15      vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 9.07/5.15      vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 9.07/5.15      vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Member_Ovebt__member,type,
% 9.07/5.15      vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 9.07/5.15      vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 9.07/5.15      vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 9.07/5.15      vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 9.07/5.15      vEBT_VEBT_less: option_nat > option_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 9.07/5.15      vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 9.07/5.15      vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 9.07/5.15      vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 9.07/5.15      vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 9.07/5.15      vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 9.07/5.15      vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.15      vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Nat__Onat,type,
% 9.07/5.15      vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Num__Onum,type,
% 9.07/5.15      vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.15      vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 9.07/5.15      vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 9.07/5.15      vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 9.07/5.15      vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 9.07/5.15      vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 9.07/5.15      vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 9.07/5.15      vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 9.07/5.15      vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 9.07/5.15      vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.15      accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 9.07/5.15      accp_nat: ( nat > nat > $o ) > nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 9.07/5.15      accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 9.07/5.15      accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 9.07/5.15      accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__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,
% 9.07/5.15      accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.15      accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.07/5.15      accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 9.07/5.15      accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.07/5.15      accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.15      accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Ofinite__psubset_001t__Complex__Ocomplex,type,
% 9.07/5.15      finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Ofinite__psubset_001t__Int__Oint,type,
% 9.07/5.15      finite_psubset_int: set_Pr2522554150109002629et_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
% 9.07/5.15      finite_psubset_nat: set_Pr5488025237498180813et_nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
% 9.07/5.15      measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_fChoice_001t__Real__Oreal,type,
% 9.07/5.15      fChoice_real: ( real > $o ) > real ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001_Eo,type,
% 9.07/5.15      member_o: $o > set_o > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Complex__Ocomplex,type,
% 9.07/5.15      member_complex: complex > set_complex > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Int__Oint,type,
% 9.07/5.15      member_int: int > set_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 9.07/5.15      member_list_o: list_o > set_list_o > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 9.07/5.15      member_list_nat: list_nat > set_list_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.07/5.15      member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Nat__Onat,type,
% 9.07/5.15      member_nat: nat > set_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Num__Onum,type,
% 9.07/5.15      member_num: num > set_num > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.07/5.15      member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.07/5.15      member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
% 9.07/5.15      member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 9.07/5.15      member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.07/5.15      member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 9.07/5.15      member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
% 9.07/5.15      member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 9.07/5.15      member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Rat__Orat,type,
% 9.07/5.15      member_rat: rat > set_rat > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Real__Oreal,type,
% 9.07/5.15      member_real: real > set_real > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 9.07/5.15      member_set_int: set_int > set_set_int > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 9.07/5.15      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_deg____,type,
% 9.07/5.15      deg: nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_m____,type,
% 9.07/5.15      m: nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_ma____,type,
% 9.07/5.15      ma: nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_mi____,type,
% 9.07/5.15      mi: nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_na____,type,
% 9.07/5.15      na: nat ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_summary____,type,
% 9.07/5.15      summary: vEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_thesis____,type,
% 9.07/5.15      thesis: $o ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_treeList____,type,
% 9.07/5.15      treeList: list_VEBT_VEBT ).
% 9.07/5.15  
% 9.07/5.15  thf(sy_v_xa____,type,
% 9.07/5.15      xa: nat ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_0_False,axiom,
% 9.07/5.15      ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.15     != none_nat ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_1__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 9.07/5.15      ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15      = na ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_2_semiring__norm_I85_J,axiom,
% 9.07/5.15      ! [M: num] :
% 9.07/5.15        ( ( bit0 @ M )
% 9.07/5.15       != one ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_3_semiring__norm_I83_J,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( one
% 9.07/5.15       != ( bit0 @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_4__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 9.07/5.15      ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_5_numeral__Bit0__div__2,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15        = ( numeral_numeral_nat @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_6_numeral__Bit0__div__2,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.15        = ( numeral_numeral_int @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_7_high__def,axiom,
% 9.07/5.15      ( vEBT_VEBT_high
% 9.07/5.15      = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_8_True,axiom,
% 9.07/5.15      ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_9_i1,axiom,
% 9.07/5.15      ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.15        = none_nat )
% 9.07/5.15      | ~ ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_10_divide__numeral__1,axiom,
% 9.07/5.15      ! [A: complex] :
% 9.07/5.15        ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 9.07/5.15        = A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_11_divide__numeral__1,axiom,
% 9.07/5.15      ! [A: real] :
% 9.07/5.15        ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 9.07/5.15        = A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_12_divide__numeral__1,axiom,
% 9.07/5.15      ! [A: rat] :
% 9.07/5.15        ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 9.07/5.15        = A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_13_option_Oinject,axiom,
% 9.07/5.15      ! [X2: nat,Y2: nat] :
% 9.07/5.15        ( ( ( some_nat @ X2 )
% 9.07/5.15          = ( some_nat @ Y2 ) )
% 9.07/5.15        = ( X2 = Y2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_14_option_Oinject,axiom,
% 9.07/5.15      ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 9.07/5.15        ( ( ( some_P7363390416028606310at_nat @ X2 )
% 9.07/5.15          = ( some_P7363390416028606310at_nat @ Y2 ) )
% 9.07/5.15        = ( X2 = Y2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_15_option_Oinject,axiom,
% 9.07/5.15      ! [X2: num,Y2: num] :
% 9.07/5.15        ( ( ( some_num @ X2 )
% 9.07/5.15          = ( some_num @ Y2 ) )
% 9.07/5.15        = ( X2 = Y2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_16_verit__eq__simplify_I8_J,axiom,
% 9.07/5.15      ! [X2: num,Y2: num] :
% 9.07/5.15        ( ( ( bit0 @ X2 )
% 9.07/5.15          = ( bit0 @ Y2 ) )
% 9.07/5.15        = ( X2 = Y2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_17_semiring__norm_I87_J,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ( bit0 @ M )
% 9.07/5.15          = ( bit0 @ N ) )
% 9.07/5.15        = ( M = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_18__C4_Ohyps_C_I5_J,axiom,
% 9.07/5.15      m = na ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_19_max__in__set__def,axiom,
% 9.07/5.15      ( vEBT_VEBT_max_in_set
% 9.07/5.15      = ( ^ [Xs: set_nat,X: nat] :
% 9.07/5.15            ( ( member_nat @ X @ Xs )
% 9.07/5.15            & ! [Y: nat] :
% 9.07/5.15                ( ( member_nat @ Y @ Xs )
% 9.07/5.15               => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_20_greater__shift,axiom,
% 9.07/5.15      ( ord_less_nat
% 9.07/5.15      = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_21_min__in__set__def,axiom,
% 9.07/5.15      ( vEBT_VEBT_min_in_set
% 9.07/5.15      = ( ^ [Xs: set_nat,X: nat] :
% 9.07/5.15            ( ( member_nat @ X @ Xs )
% 9.07/5.15            & ! [Y: nat] :
% 9.07/5.15                ( ( member_nat @ Y @ Xs )
% 9.07/5.15               => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_22_power__shift,axiom,
% 9.07/5.15      ! [X3: nat,Y3: nat,Z: nat] :
% 9.07/5.15        ( ( ( power_power_nat @ X3 @ Y3 )
% 9.07/5.15          = Z )
% 9.07/5.15        = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 9.07/5.15          = ( some_nat @ Z ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_23_bit__split__inv,axiom,
% 9.07/5.15      ! [X3: nat,D: nat] :
% 9.07/5.15        ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
% 9.07/5.15        = X3 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_24_numeral__eq__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ( numera6690914467698888265omplex @ M )
% 9.07/5.15          = ( numera6690914467698888265omplex @ N ) )
% 9.07/5.15        = ( M = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_25_numeral__eq__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ( numeral_numeral_real @ M )
% 9.07/5.15          = ( numeral_numeral_real @ N ) )
% 9.07/5.15        = ( M = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_26_numeral__eq__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ( numeral_numeral_rat @ M )
% 9.07/5.15          = ( numeral_numeral_rat @ N ) )
% 9.07/5.15        = ( M = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_27_numeral__eq__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ( numeral_numeral_nat @ M )
% 9.07/5.15          = ( numeral_numeral_nat @ N ) )
% 9.07/5.15        = ( M = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_28_numeral__eq__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ( numeral_numeral_int @ M )
% 9.07/5.15          = ( numeral_numeral_int @ N ) )
% 9.07/5.15        = ( M = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_29_numeral__le__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.07/5.15        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_30_numeral__le__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.07/5.15        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_31_numeral__le__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.15        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_32_numeral__le__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.07/5.15        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_33_numeral__less__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.07/5.15        = ( ord_less_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_34_numeral__less__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.07/5.15        = ( ord_less_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_35_numeral__less__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.15        = ( ord_less_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_36_numeral__less__iff,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.07/5.15        = ( ord_less_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_37_not__Some__eq,axiom,
% 9.07/5.15      ! [X3: option_nat] :
% 9.07/5.15        ( ( ! [Y: nat] :
% 9.07/5.15              ( X3
% 9.07/5.15             != ( some_nat @ Y ) ) )
% 9.07/5.15        = ( X3 = none_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_38_not__Some__eq,axiom,
% 9.07/5.15      ! [X3: option4927543243414619207at_nat] :
% 9.07/5.15        ( ( ! [Y: product_prod_nat_nat] :
% 9.07/5.15              ( X3
% 9.07/5.15             != ( some_P7363390416028606310at_nat @ Y ) ) )
% 9.07/5.15        = ( X3 = none_P5556105721700978146at_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_39_not__Some__eq,axiom,
% 9.07/5.15      ! [X3: option_num] :
% 9.07/5.15        ( ( ! [Y: num] :
% 9.07/5.15              ( X3
% 9.07/5.15             != ( some_num @ Y ) ) )
% 9.07/5.15        = ( X3 = none_num ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_40_not__None__eq,axiom,
% 9.07/5.15      ! [X3: option_nat] :
% 9.07/5.15        ( ( X3 != none_nat )
% 9.07/5.15        = ( ? [Y: nat] :
% 9.07/5.15              ( X3
% 9.07/5.15              = ( some_nat @ Y ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_41_not__None__eq,axiom,
% 9.07/5.15      ! [X3: option4927543243414619207at_nat] :
% 9.07/5.15        ( ( X3 != none_P5556105721700978146at_nat )
% 9.07/5.15        = ( ? [Y: product_prod_nat_nat] :
% 9.07/5.15              ( X3
% 9.07/5.15              = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_42_not__None__eq,axiom,
% 9.07/5.15      ! [X3: option_num] :
% 9.07/5.15        ( ( X3 != none_num )
% 9.07/5.15        = ( ? [Y: num] :
% 9.07/5.15              ( X3
% 9.07/5.15              = ( some_num @ Y ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_43__C4_Ohyps_C_I4_J,axiom,
% 9.07/5.15      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.07/5.15      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_44__092_060open_062length_AtreeList_A_061_A2_A_094_An_092_060close_062,axiom,
% 9.07/5.15      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.07/5.15      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_45__092_060open_062high_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
% 9.07/5.15      ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
% 9.07/5.15      & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_46__C4_Ohyps_C_I10_J,axiom,
% 9.07/5.15      ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_47_less__shift,axiom,
% 9.07/5.15      ( ord_less_nat
% 9.07/5.15      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_48_lesseq__shift,axiom,
% 9.07/5.15      ( ord_less_eq_nat
% 9.07/5.15      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_49_verit__comp__simplify1_I3_J,axiom,
% 9.07/5.15      ! [B: real,A2: real] :
% 9.07/5.15        ( ( ~ ( ord_less_eq_real @ B @ A2 ) )
% 9.07/5.15        = ( ord_less_real @ A2 @ B ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_50_verit__comp__simplify1_I3_J,axiom,
% 9.07/5.15      ! [B: rat,A2: rat] :
% 9.07/5.15        ( ( ~ ( ord_less_eq_rat @ B @ A2 ) )
% 9.07/5.15        = ( ord_less_rat @ A2 @ B ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_51_verit__comp__simplify1_I3_J,axiom,
% 9.07/5.15      ! [B: num,A2: num] :
% 9.07/5.15        ( ( ~ ( ord_less_eq_num @ B @ A2 ) )
% 9.07/5.15        = ( ord_less_num @ A2 @ B ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_52_verit__comp__simplify1_I3_J,axiom,
% 9.07/5.15      ! [B: nat,A2: nat] :
% 9.07/5.15        ( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
% 9.07/5.15        = ( ord_less_nat @ A2 @ B ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_53_verit__comp__simplify1_I3_J,axiom,
% 9.07/5.15      ! [B: int,A2: int] :
% 9.07/5.15        ( ( ~ ( ord_less_eq_int @ B @ A2 ) )
% 9.07/5.15        = ( ord_less_int @ A2 @ B ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_54_verit__comp__simplify1_I2_J,axiom,
% 9.07/5.15      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_55_verit__comp__simplify1_I2_J,axiom,
% 9.07/5.15      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_56_verit__comp__simplify1_I2_J,axiom,
% 9.07/5.15      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_57_verit__comp__simplify1_I2_J,axiom,
% 9.07/5.15      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_58_verit__comp__simplify1_I2_J,axiom,
% 9.07/5.15      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_59_verit__comp__simplify1_I1_J,axiom,
% 9.07/5.15      ! [A: real] :
% 9.07/5.15        ~ ( ord_less_real @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_60_verit__comp__simplify1_I1_J,axiom,
% 9.07/5.15      ! [A: rat] :
% 9.07/5.15        ~ ( ord_less_rat @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_61_verit__comp__simplify1_I1_J,axiom,
% 9.07/5.15      ! [A: num] :
% 9.07/5.15        ~ ( ord_less_num @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_62_verit__comp__simplify1_I1_J,axiom,
% 9.07/5.15      ! [A: nat] :
% 9.07/5.15        ~ ( ord_less_nat @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_63_verit__comp__simplify1_I1_J,axiom,
% 9.07/5.15      ! [A: int] :
% 9.07/5.15        ~ ( ord_less_int @ A @ A ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_64_verit__la__disequality,axiom,
% 9.07/5.15      ! [A: rat,B2: rat] :
% 9.07/5.15        ( ( A = B2 )
% 9.07/5.15        | ~ ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.15        | ~ ( ord_less_eq_rat @ B2 @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_65_verit__la__disequality,axiom,
% 9.07/5.15      ! [A: num,B2: num] :
% 9.07/5.15        ( ( A = B2 )
% 9.07/5.15        | ~ ( ord_less_eq_num @ A @ B2 )
% 9.07/5.15        | ~ ( ord_less_eq_num @ B2 @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_66_verit__la__disequality,axiom,
% 9.07/5.15      ! [A: nat,B2: nat] :
% 9.07/5.15        ( ( A = B2 )
% 9.07/5.15        | ~ ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.15        | ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_67_verit__la__disequality,axiom,
% 9.07/5.15      ! [A: int,B2: int] :
% 9.07/5.15        ( ( A = B2 )
% 9.07/5.15        | ~ ( ord_less_eq_int @ A @ B2 )
% 9.07/5.15        | ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_68_div__le__dividend,axiom,
% 9.07/5.15      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_69_div__le__mono,axiom,
% 9.07/5.15      ! [M: nat,N: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_70_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 9.07/5.15        ( ( ( X3 = none_nat )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_nat )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: nat,B3: nat] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_nat @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_nat @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_71_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 9.07/5.15        ( ( ( X3 = none_nat )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: nat,B3: product_prod_nat_nat] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_nat @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_72_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 9.07/5.15        ( ( ( X3 = none_nat )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_num )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: nat,B3: num] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_nat @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_num @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_73_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 9.07/5.15        ( ( ( X3 = none_P5556105721700978146at_nat )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_nat )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: product_prod_nat_nat,B3: nat] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_nat @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_74_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 9.07/5.15        ( ( ( X3 = none_P5556105721700978146at_nat )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_75_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 9.07/5.15        ( ( ( X3 = none_P5556105721700978146at_nat )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_num )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: product_prod_nat_nat,B3: num] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_num @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_76_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 9.07/5.15        ( ( ( X3 = none_num )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_nat )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: num,B3: nat] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_num @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_nat @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_77_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 9.07/5.15        ( ( ( X3 = none_num )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: num,B3: product_prod_nat_nat] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_num @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_78_combine__options__cases,axiom,
% 9.07/5.15      ! [X3: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 9.07/5.15        ( ( ( X3 = none_num )
% 9.07/5.15         => ( P @ X3 @ Y3 ) )
% 9.07/5.15       => ( ( ( Y3 = none_num )
% 9.07/5.15           => ( P @ X3 @ Y3 ) )
% 9.07/5.15         => ( ! [A3: num,B3: num] :
% 9.07/5.15                ( ( X3
% 9.07/5.15                  = ( some_num @ A3 ) )
% 9.07/5.15               => ( ( Y3
% 9.07/5.15                    = ( some_num @ B3 ) )
% 9.07/5.15                 => ( P @ X3 @ Y3 ) ) )
% 9.07/5.15           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_79_split__option__all,axiom,
% 9.07/5.15      ( ( ^ [P2: option_nat > $o] :
% 9.07/5.15          ! [X4: option_nat] : ( P2 @ X4 ) )
% 9.07/5.15      = ( ^ [P3: option_nat > $o] :
% 9.07/5.15            ( ( P3 @ none_nat )
% 9.07/5.15            & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_80_split__option__all,axiom,
% 9.07/5.15      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.07/5.15          ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 9.07/5.15      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.07/5.15            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.07/5.15            & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_81_split__option__all,axiom,
% 9.07/5.15      ( ( ^ [P2: option_num > $o] :
% 9.07/5.15          ! [X4: option_num] : ( P2 @ X4 ) )
% 9.07/5.15      = ( ^ [P3: option_num > $o] :
% 9.07/5.15            ( ( P3 @ none_num )
% 9.07/5.15            & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_82_split__option__ex,axiom,
% 9.07/5.15      ( ( ^ [P2: option_nat > $o] :
% 9.07/5.15          ? [X4: option_nat] : ( P2 @ X4 ) )
% 9.07/5.15      = ( ^ [P3: option_nat > $o] :
% 9.07/5.15            ( ( P3 @ none_nat )
% 9.07/5.15            | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_83_split__option__ex,axiom,
% 9.07/5.15      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.07/5.15          ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 9.07/5.15      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.07/5.15            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.07/5.15            | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_84_split__option__ex,axiom,
% 9.07/5.15      ( ( ^ [P2: option_num > $o] :
% 9.07/5.15          ? [X4: option_num] : ( P2 @ X4 ) )
% 9.07/5.15      = ( ^ [P3: option_num > $o] :
% 9.07/5.15            ( ( P3 @ none_num )
% 9.07/5.15            | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_85_option_Oexhaust,axiom,
% 9.07/5.15      ! [Y3: option_nat] :
% 9.07/5.15        ( ( Y3 != none_nat )
% 9.07/5.15       => ~ ! [X22: nat] :
% 9.07/5.15              ( Y3
% 9.07/5.15             != ( some_nat @ X22 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_86_option_Oexhaust,axiom,
% 9.07/5.15      ! [Y3: option4927543243414619207at_nat] :
% 9.07/5.15        ( ( Y3 != none_P5556105721700978146at_nat )
% 9.07/5.15       => ~ ! [X22: product_prod_nat_nat] :
% 9.07/5.15              ( Y3
% 9.07/5.15             != ( some_P7363390416028606310at_nat @ X22 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_87_option_Oexhaust,axiom,
% 9.07/5.15      ! [Y3: option_num] :
% 9.07/5.15        ( ( Y3 != none_num )
% 9.07/5.15       => ~ ! [X22: num] :
% 9.07/5.15              ( Y3
% 9.07/5.15             != ( some_num @ X22 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_88_option_OdiscI,axiom,
% 9.07/5.15      ! [Option: option_nat,X2: nat] :
% 9.07/5.15        ( ( Option
% 9.07/5.15          = ( some_nat @ X2 ) )
% 9.07/5.15       => ( Option != none_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_89_option_OdiscI,axiom,
% 9.07/5.15      ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
% 9.07/5.15        ( ( Option
% 9.07/5.15          = ( some_P7363390416028606310at_nat @ X2 ) )
% 9.07/5.15       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_90_option_OdiscI,axiom,
% 9.07/5.15      ! [Option: option_num,X2: num] :
% 9.07/5.15        ( ( Option
% 9.07/5.15          = ( some_num @ X2 ) )
% 9.07/5.15       => ( Option != none_num ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_91_option_Odistinct_I1_J,axiom,
% 9.07/5.15      ! [X2: nat] :
% 9.07/5.15        ( none_nat
% 9.07/5.15       != ( some_nat @ X2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_92_option_Odistinct_I1_J,axiom,
% 9.07/5.15      ! [X2: product_prod_nat_nat] :
% 9.07/5.15        ( none_P5556105721700978146at_nat
% 9.07/5.15       != ( some_P7363390416028606310at_nat @ X2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_93_option_Odistinct_I1_J,axiom,
% 9.07/5.15      ! [X2: num] :
% 9.07/5.15        ( none_num
% 9.07/5.15       != ( some_num @ X2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_94_mem__Collect__eq,axiom,
% 9.07/5.15      ! [A: complex,P: complex > $o] :
% 9.07/5.15        ( ( member_complex @ A @ ( collect_complex @ P ) )
% 9.07/5.15        = ( P @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_95_mem__Collect__eq,axiom,
% 9.07/5.15      ! [A: real,P: real > $o] :
% 9.07/5.15        ( ( member_real @ A @ ( collect_real @ P ) )
% 9.07/5.15        = ( P @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_96_mem__Collect__eq,axiom,
% 9.07/5.15      ! [A: list_nat,P: list_nat > $o] :
% 9.07/5.15        ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 9.07/5.15        = ( P @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_97_mem__Collect__eq,axiom,
% 9.07/5.15      ! [A: nat,P: nat > $o] :
% 9.07/5.15        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 9.07/5.15        = ( P @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_98_mem__Collect__eq,axiom,
% 9.07/5.15      ! [A: int,P: int > $o] :
% 9.07/5.15        ( ( member_int @ A @ ( collect_int @ P ) )
% 9.07/5.15        = ( P @ A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_99_Collect__mem__eq,axiom,
% 9.07/5.15      ! [A4: set_complex] :
% 9.07/5.15        ( ( collect_complex
% 9.07/5.15          @ ^ [X: complex] : ( member_complex @ X @ A4 ) )
% 9.07/5.15        = A4 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_100_Collect__mem__eq,axiom,
% 9.07/5.15      ! [A4: set_real] :
% 9.07/5.15        ( ( collect_real
% 9.07/5.15          @ ^ [X: real] : ( member_real @ X @ A4 ) )
% 9.07/5.15        = A4 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_101_Collect__mem__eq,axiom,
% 9.07/5.15      ! [A4: set_list_nat] :
% 9.07/5.15        ( ( collect_list_nat
% 9.07/5.15          @ ^ [X: list_nat] : ( member_list_nat @ X @ A4 ) )
% 9.07/5.15        = A4 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_102_Collect__mem__eq,axiom,
% 9.07/5.15      ! [A4: set_nat] :
% 9.07/5.15        ( ( collect_nat
% 9.07/5.15          @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
% 9.07/5.15        = A4 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_103_Collect__mem__eq,axiom,
% 9.07/5.15      ! [A4: set_int] :
% 9.07/5.15        ( ( collect_int
% 9.07/5.15          @ ^ [X: int] : ( member_int @ X @ A4 ) )
% 9.07/5.15        = A4 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_104_Collect__cong,axiom,
% 9.07/5.15      ! [P: complex > $o,Q: complex > $o] :
% 9.07/5.15        ( ! [X5: complex] :
% 9.07/5.15            ( ( P @ X5 )
% 9.07/5.15            = ( Q @ X5 ) )
% 9.07/5.15       => ( ( collect_complex @ P )
% 9.07/5.15          = ( collect_complex @ Q ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_105_Collect__cong,axiom,
% 9.07/5.15      ! [P: real > $o,Q: real > $o] :
% 9.07/5.15        ( ! [X5: real] :
% 9.07/5.15            ( ( P @ X5 )
% 9.07/5.15            = ( Q @ X5 ) )
% 9.07/5.15       => ( ( collect_real @ P )
% 9.07/5.15          = ( collect_real @ Q ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_106_Collect__cong,axiom,
% 9.07/5.15      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.07/5.15        ( ! [X5: list_nat] :
% 9.07/5.15            ( ( P @ X5 )
% 9.07/5.15            = ( Q @ X5 ) )
% 9.07/5.15       => ( ( collect_list_nat @ P )
% 9.07/5.15          = ( collect_list_nat @ Q ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_107_Collect__cong,axiom,
% 9.07/5.15      ! [P: nat > $o,Q: nat > $o] :
% 9.07/5.15        ( ! [X5: nat] :
% 9.07/5.15            ( ( P @ X5 )
% 9.07/5.15            = ( Q @ X5 ) )
% 9.07/5.15       => ( ( collect_nat @ P )
% 9.07/5.15          = ( collect_nat @ Q ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_108_Collect__cong,axiom,
% 9.07/5.15      ! [P: int > $o,Q: int > $o] :
% 9.07/5.15        ( ! [X5: int] :
% 9.07/5.15            ( ( P @ X5 )
% 9.07/5.15            = ( Q @ X5 ) )
% 9.07/5.15       => ( ( collect_int @ P )
% 9.07/5.15          = ( collect_int @ Q ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_109_verit__eq__simplify_I10_J,axiom,
% 9.07/5.15      ! [X2: num] :
% 9.07/5.15        ( one
% 9.07/5.15       != ( bit0 @ X2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_110__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
% 9.07/5.15      vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_111_minminNull,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT] :
% 9.07/5.15        ( ( ( vEBT_vebt_mint @ T )
% 9.07/5.15          = none_nat )
% 9.07/5.15       => ( vEBT_VEBT_minNull @ T ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_112_minNullmin,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT] :
% 9.07/5.15        ( ( vEBT_VEBT_minNull @ T )
% 9.07/5.15       => ( ( vEBT_vebt_mint @ T )
% 9.07/5.15          = none_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_113_power2__nat__le__imp__le,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_114_power2__nat__le__eq__le,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.15        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_115_self__le__ge2__pow,axiom,
% 9.07/5.15      ! [K: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.07/5.15       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_116_less__exp,axiom,
% 9.07/5.15      ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_117_high__bound__aux,axiom,
% 9.07/5.15      ! [Ma: nat,N: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.07/5.15       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_118_power__minus__is__div,axiom,
% 9.07/5.15      ! [B2: nat,A: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ B2 @ A )
% 9.07/5.15       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B2 ) )
% 9.07/5.15          = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_119_misiz,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,M: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( some_nat @ M )
% 9.07/5.15            = ( vEBT_vebt_mint @ T ) )
% 9.07/5.15         => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_120_pow__sum,axiom,
% 9.07/5.15      ! [A: nat,B2: nat] :
% 9.07/5.15        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.07/5.15        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_121__092_060open_062x_A_092_060le_062_Ama_092_060close_062,axiom,
% 9.07/5.15      ord_less_eq_nat @ xa @ ma ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_122__C4_Ohyps_C_I6_J,axiom,
% 9.07/5.15      ( deg
% 9.07/5.15      = ( plus_plus_nat @ na @ m ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_123__C4_Ohyps_C_I9_J,axiom,
% 9.07/5.15      ord_less_eq_nat @ mi @ ma ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_124__C4_Ohyps_C_I2_J,axiom,
% 9.07/5.15      vEBT_invar_vebt @ summary @ m ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_125_add__numeral__left,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: complex] :
% 9.07/5.15        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 9.07/5.15        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_126_add__numeral__left,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: real] :
% 9.07/5.15        ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 9.07/5.15        = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_127_add__numeral__left,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: rat] :
% 9.07/5.15        ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 9.07/5.15        = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_128_add__numeral__left,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: nat] :
% 9.07/5.15        ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 9.07/5.15        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_129_add__numeral__left,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: int] :
% 9.07/5.15        ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 9.07/5.15        = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_130_numeral__plus__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 9.07/5.15        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_131_numeral__plus__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.07/5.15        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_132_numeral__plus__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.07/5.15        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_133_numeral__plus__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.15        = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_134_numeral__plus__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.07/5.15        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_135_semiring__norm_I78_J,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.07/5.15        = ( ord_less_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_136_semiring__norm_I71_J,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.07/5.15        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_137_semiring__norm_I75_J,axiom,
% 9.07/5.15      ! [M: num] :
% 9.07/5.15        ~ ( ord_less_num @ M @ one ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_138_semiring__norm_I68_J,axiom,
% 9.07/5.15      ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_139_semiring__norm_I76_J,axiom,
% 9.07/5.15      ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_140_semiring__norm_I69_J,axiom,
% 9.07/5.15      ! [M: num] :
% 9.07/5.15        ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_141_add__self__div__2,axiom,
% 9.07/5.15      ! [M: nat] :
% 9.07/5.15        ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15        = M ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_142_local_Opower__def,axiom,
% 9.07/5.15      ( vEBT_VEBT_power
% 9.07/5.15      = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_143_is__num__normalize_I1_J,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_144_is__num__normalize_I1_J,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_145_is__num__normalize_I1_J,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_146_le__num__One__iff,axiom,
% 9.07/5.15      ! [X3: num] :
% 9.07/5.15        ( ( ord_less_eq_num @ X3 @ one )
% 9.07/5.15        = ( X3 = one ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_147_numeral__Bit0,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 9.07/5.15        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_148_numeral__Bit0,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 9.07/5.15        = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_149_numeral__Bit0,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 9.07/5.15        = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_150_numeral__Bit0,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 9.07/5.15        = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_151_numeral__Bit0,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 9.07/5.15        = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_152_power2__commute,axiom,
% 9.07/5.15      ! [X3: complex,Y3: complex] :
% 9.07/5.15        ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15        = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_153_power2__commute,axiom,
% 9.07/5.15      ! [X3: real,Y3: real] :
% 9.07/5.15        ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15        = ( power_power_real @ ( minus_minus_real @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_154_power2__commute,axiom,
% 9.07/5.15      ! [X3: rat,Y3: rat] :
% 9.07/5.15        ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15        = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_155_power2__commute,axiom,
% 9.07/5.15      ! [X3: int,Y3: int] :
% 9.07/5.15        ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.15        = ( power_power_int @ ( minus_minus_int @ Y3 @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_156_diff__le__diff__pow,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.07/5.15       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_157_power__divide,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,N: nat] :
% 9.07/5.15        ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B2 ) @ N )
% 9.07/5.15        = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_158_power__divide,axiom,
% 9.07/5.15      ! [A: real,B2: real,N: nat] :
% 9.07/5.15        ( ( power_power_real @ ( divide_divide_real @ A @ B2 ) @ N )
% 9.07/5.15        = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_159_power__divide,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,N: nat] :
% 9.07/5.15        ( ( power_power_rat @ ( divide_divide_rat @ A @ B2 ) @ N )
% 9.07/5.15        = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_160_mint__corr,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( vEBT_vebt_mint @ T )
% 9.07/5.15            = ( some_nat @ X3 ) )
% 9.07/5.15         => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_161_mint__sound,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 9.07/5.15         => ( ( vEBT_vebt_mint @ T )
% 9.07/5.15            = ( some_nat @ X3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_162_add__shift,axiom,
% 9.07/5.15      ! [X3: nat,Y3: nat,Z: nat] :
% 9.07/5.15        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.07/5.15          = Z )
% 9.07/5.15        = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 9.07/5.15          = ( some_nat @ Z ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_163_Nat_Oadd__diff__assoc,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.07/5.15          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_164_Nat_Oadd__diff__assoc2,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.07/5.15          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_165_Nat_Odiff__diff__right,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.07/5.15          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_166_le__add__diff__inverse,axiom,
% 9.07/5.15      ! [B2: real,A: real] :
% 9.07/5.15        ( ( ord_less_eq_real @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_167_le__add__diff__inverse,axiom,
% 9.07/5.15      ! [B2: rat,A: rat] :
% 9.07/5.15        ( ( ord_less_eq_rat @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_168_le__add__diff__inverse,axiom,
% 9.07/5.15      ! [B2: nat,A: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_169_le__add__diff__inverse,axiom,
% 9.07/5.15      ! [B2: int,A: int] :
% 9.07/5.15        ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_170_le__add__diff__inverse2,axiom,
% 9.07/5.15      ! [B2: real,A: real] :
% 9.07/5.15        ( ( ord_less_eq_real @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_171_le__add__diff__inverse2,axiom,
% 9.07/5.15      ! [B2: rat,A: rat] :
% 9.07/5.15        ( ( ord_less_eq_rat @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_172_le__add__diff__inverse2,axiom,
% 9.07/5.15      ! [B2: nat,A: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_173_le__add__diff__inverse2,axiom,
% 9.07/5.15      ! [B2: int,A: int] :
% 9.07/5.15        ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.15       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_174_member__bound,axiom,
% 9.07/5.15      ! [Tree: vEBT_VEBT,X3: nat,N: nat] :
% 9.07/5.15        ( ( vEBT_vebt_member @ Tree @ X3 )
% 9.07/5.15       => ( ( vEBT_invar_vebt @ Tree @ N )
% 9.07/5.15         => ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_175__C4_Ohyps_C_I11_J,axiom,
% 9.07/5.15      ( ( mi != ma )
% 9.07/5.15     => ! [I2: nat] :
% 9.07/5.15          ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.07/5.15         => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 9.07/5.15                = I2 )
% 9.07/5.15             => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 9.07/5.15            & ! [X6: nat] :
% 9.07/5.15                ( ( ( ( vEBT_VEBT_high @ X6 @ na )
% 9.07/5.15                    = I2 )
% 9.07/5.15                  & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ X6 @ na ) ) )
% 9.07/5.15               => ( ( ord_less_nat @ mi @ X6 )
% 9.07/5.15                  & ( ord_less_eq_nat @ X6 @ ma ) ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_176_div__exp__eq,axiom,
% 9.07/5.15      ! [A: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15        = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_177_div__exp__eq,axiom,
% 9.07/5.15      ! [A: int,M: nat,N: nat] :
% 9.07/5.15        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15        = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_178_field__less__half__sum,axiom,
% 9.07/5.15      ! [X3: real,Y3: real] :
% 9.07/5.15        ( ( ord_less_real @ X3 @ Y3 )
% 9.07/5.15       => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_179_field__less__half__sum,axiom,
% 9.07/5.15      ! [X3: rat,Y3: rat] :
% 9.07/5.15        ( ( ord_less_rat @ X3 @ Y3 )
% 9.07/5.15       => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_180_not__min__Null__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT] :
% 9.07/5.15        ( ~ ( vEBT_VEBT_minNull @ T )
% 9.07/5.15       => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_181_min__Null__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,X3: nat] :
% 9.07/5.15        ( ( vEBT_VEBT_minNull @ T )
% 9.07/5.15       => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_182_both__member__options__equiv__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 9.07/5.15          = ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_183_valid__member__both__member__options,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 9.07/5.15         => ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_184_add__def,axiom,
% 9.07/5.15      ( vEBT_VEBT_add
% 9.07/5.15      = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_185_mint__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( vEBT_vebt_mint @ T )
% 9.07/5.15            = ( some_nat @ Maxi ) )
% 9.07/5.15         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_186_member__correct,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( vEBT_vebt_member @ T @ X3 )
% 9.07/5.15          = ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_187_mint__corr__help,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,Mini: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( vEBT_vebt_mint @ T )
% 9.07/5.15            = ( some_nat @ Mini ) )
% 9.07/5.15         => ( ( vEBT_vebt_member @ T @ X3 )
% 9.07/5.15           => ( ord_less_eq_nat @ Mini @ X3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_188_set__vebt__set__vebt_H__valid,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( vEBT_set_vebt @ T )
% 9.07/5.15          = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_189_nat__add__left__cancel__less,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.07/5.15        = ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_190_nat__add__left__cancel__le,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.07/5.15        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_191_diff__diff__cancel,axiom,
% 9.07/5.15      ! [I: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ N )
% 9.07/5.15       => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 9.07/5.15          = I ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_192_diff__diff__left,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.07/5.15        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_193_semiring__norm_I6_J,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.07/5.15        = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_194_semiring__norm_I2_J,axiom,
% 9.07/5.15      ( ( plus_plus_num @ one @ one )
% 9.07/5.15      = ( bit0 @ one ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_195_pred__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,X3: nat,Y3: nat] :
% 9.07/5.15        ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Y3 )
% 9.07/5.15        = ( ( vEBT_vebt_member @ T @ Y3 )
% 9.07/5.15          & ( ord_less_nat @ Y3 @ X3 )
% 9.07/5.15          & ! [Z2: nat] :
% 9.07/5.15              ( ( ( vEBT_vebt_member @ T @ Z2 )
% 9.07/5.15                & ( ord_less_nat @ Z2 @ X3 ) )
% 9.07/5.15             => ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_196__C4_Ohyps_C_I7_J,axiom,
% 9.07/5.15      ! [I2: nat] :
% 9.07/5.15        ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.07/5.15       => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ X7 ) )
% 9.07/5.15          = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_197_add__One__commute,axiom,
% 9.07/5.15      ! [N: num] :
% 9.07/5.15        ( ( plus_plus_num @ one @ N )
% 9.07/5.15        = ( plus_plus_num @ N @ one ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_198_linorder__neqE__linordered__idom,axiom,
% 9.07/5.15      ! [X3: real,Y3: real] :
% 9.07/5.15        ( ( X3 != Y3 )
% 9.07/5.15       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.07/5.15         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_199_linorder__neqE__linordered__idom,axiom,
% 9.07/5.15      ! [X3: rat,Y3: rat] :
% 9.07/5.15        ( ( X3 != Y3 )
% 9.07/5.15       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.07/5.15         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_200_linorder__neqE__linordered__idom,axiom,
% 9.07/5.15      ! [X3: int,Y3: int] :
% 9.07/5.15        ( ( X3 != Y3 )
% 9.07/5.15       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.07/5.15         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_201_nat__neq__iff,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( M != N )
% 9.07/5.15        = ( ( ord_less_nat @ M @ N )
% 9.07/5.15          | ( ord_less_nat @ N @ M ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_202_less__not__refl,axiom,
% 9.07/5.15      ! [N: nat] :
% 9.07/5.15        ~ ( ord_less_nat @ N @ N ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_203_less__not__refl2,axiom,
% 9.07/5.15      ! [N: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_nat @ N @ M )
% 9.07/5.15       => ( M != N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_204_less__not__refl3,axiom,
% 9.07/5.15      ! [S: nat,T: nat] :
% 9.07/5.15        ( ( ord_less_nat @ S @ T )
% 9.07/5.15       => ( S != T ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_205_less__irrefl__nat,axiom,
% 9.07/5.15      ! [N: nat] :
% 9.07/5.15        ~ ( ord_less_nat @ N @ N ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_206_nat__less__induct,axiom,
% 9.07/5.15      ! [P: nat > $o,N: nat] :
% 9.07/5.15        ( ! [N3: nat] :
% 9.07/5.15            ( ! [M2: nat] :
% 9.07/5.15                ( ( ord_less_nat @ M2 @ N3 )
% 9.07/5.15               => ( P @ M2 ) )
% 9.07/5.15           => ( P @ N3 ) )
% 9.07/5.15       => ( P @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_207_infinite__descent,axiom,
% 9.07/5.15      ! [P: nat > $o,N: nat] :
% 9.07/5.15        ( ! [N3: nat] :
% 9.07/5.15            ( ~ ( P @ N3 )
% 9.07/5.15           => ? [M2: nat] :
% 9.07/5.15                ( ( ord_less_nat @ M2 @ N3 )
% 9.07/5.15                & ~ ( P @ M2 ) ) )
% 9.07/5.15       => ( P @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_208_linorder__neqE__nat,axiom,
% 9.07/5.15      ! [X3: nat,Y3: nat] :
% 9.07/5.15        ( ( X3 != Y3 )
% 9.07/5.15       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.07/5.15         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_209_le__refl,axiom,
% 9.07/5.15      ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_210_le__trans,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ( ord_less_eq_nat @ J @ K )
% 9.07/5.15         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_211_eq__imp__le,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( M = N )
% 9.07/5.15       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_212_le__antisym,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15       => ( ( ord_less_eq_nat @ N @ M )
% 9.07/5.15         => ( M = N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_213_nat__le__linear,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15        | ( ord_less_eq_nat @ N @ M ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_214_Nat_Oex__has__greatest__nat,axiom,
% 9.07/5.15      ! [P: nat > $o,K: nat,B2: nat] :
% 9.07/5.15        ( ( P @ K )
% 9.07/5.15       => ( ! [Y4: nat] :
% 9.07/5.15              ( ( P @ Y4 )
% 9.07/5.15             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 9.07/5.15         => ? [X5: nat] :
% 9.07/5.15              ( ( P @ X5 )
% 9.07/5.15              & ! [Y5: nat] :
% 9.07/5.15                  ( ( P @ Y5 )
% 9.07/5.15                 => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_215_size__neq__size__imp__neq,axiom,
% 9.07/5.15      ! [X3: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 9.07/5.15        ( ( ( size_s6755466524823107622T_VEBT @ X3 )
% 9.07/5.15         != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 9.07/5.15       => ( X3 != Y3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_216_size__neq__size__imp__neq,axiom,
% 9.07/5.15      ! [X3: num,Y3: num] :
% 9.07/5.15        ( ( ( size_size_num @ X3 )
% 9.07/5.15         != ( size_size_num @ Y3 ) )
% 9.07/5.15       => ( X3 != Y3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_217_size__neq__size__imp__neq,axiom,
% 9.07/5.15      ! [X3: list_o,Y3: list_o] :
% 9.07/5.15        ( ( ( size_size_list_o @ X3 )
% 9.07/5.15         != ( size_size_list_o @ Y3 ) )
% 9.07/5.15       => ( X3 != Y3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_218_size__neq__size__imp__neq,axiom,
% 9.07/5.15      ! [X3: list_nat,Y3: list_nat] :
% 9.07/5.15        ( ( ( size_size_list_nat @ X3 )
% 9.07/5.15         != ( size_size_list_nat @ Y3 ) )
% 9.07/5.15       => ( X3 != Y3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_219_size__neq__size__imp__neq,axiom,
% 9.07/5.15      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
% 9.07/5.15        ( ( ( size_size_VEBT_VEBT @ X3 )
% 9.07/5.15         != ( size_size_VEBT_VEBT @ Y3 ) )
% 9.07/5.15       => ( X3 != Y3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_220_diff__commute,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.07/5.15        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_221_add__diff__add,axiom,
% 9.07/5.15      ! [A: complex,C: complex,B2: complex,D: complex] :
% 9.07/5.15        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B2 @ D ) )
% 9.07/5.15        = ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ ( minus_minus_complex @ C @ D ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_222_add__diff__add,axiom,
% 9.07/5.15      ! [A: real,C: real,B2: real,D: real] :
% 9.07/5.15        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
% 9.07/5.15        = ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_223_add__diff__add,axiom,
% 9.07/5.15      ! [A: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.15        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) )
% 9.07/5.15        = ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_224_add__diff__add,axiom,
% 9.07/5.15      ! [A: int,C: int,B2: int,D: int] :
% 9.07/5.15        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) )
% 9.07/5.15        = ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_225_nat__less__le,axiom,
% 9.07/5.15      ( ord_less_nat
% 9.07/5.15      = ( ^ [M3: nat,N2: nat] :
% 9.07/5.15            ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.07/5.15            & ( M3 != N2 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_226_less__imp__le__nat,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_nat @ M @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_227_le__eq__less__or__eq,axiom,
% 9.07/5.15      ( ord_less_eq_nat
% 9.07/5.15      = ( ^ [M3: nat,N2: nat] :
% 9.07/5.15            ( ( ord_less_nat @ M3 @ N2 )
% 9.07/5.15            | ( M3 = N2 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_228_less__or__eq__imp__le,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ( ord_less_nat @ M @ N )
% 9.07/5.15          | ( M = N ) )
% 9.07/5.15       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_229_le__neq__implies__less,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15       => ( ( M != N )
% 9.07/5.15         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_230_less__mono__imp__le__mono,axiom,
% 9.07/5.15      ! [F: nat > nat,I: nat,J: nat] :
% 9.07/5.15        ( ! [I3: nat,J2: nat] :
% 9.07/5.15            ( ( ord_less_nat @ I3 @ J2 )
% 9.07/5.15           => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 9.07/5.15       => ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_231_add__lessD1,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.07/5.15       => ( ord_less_nat @ I @ K ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_232_add__less__mono,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_nat @ I @ J )
% 9.07/5.15       => ( ( ord_less_nat @ K @ L )
% 9.07/5.15         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_233_not__add__less1,axiom,
% 9.07/5.15      ! [I: nat,J: nat] :
% 9.07/5.15        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_234_not__add__less2,axiom,
% 9.07/5.15      ! [J: nat,I: nat] :
% 9.07/5.15        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_235_add__less__mono1,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_nat @ I @ J )
% 9.07/5.15       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_236_trans__less__add1,axiom,
% 9.07/5.15      ! [I: nat,J: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_nat @ I @ J )
% 9.07/5.15       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_237_trans__less__add2,axiom,
% 9.07/5.15      ! [I: nat,J: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_nat @ I @ J )
% 9.07/5.15       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_238_less__add__eq__less,axiom,
% 9.07/5.15      ! [K: nat,L: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_nat @ K @ L )
% 9.07/5.15       => ( ( ( plus_plus_nat @ M @ L )
% 9.07/5.15            = ( plus_plus_nat @ K @ N ) )
% 9.07/5.15         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_239_add__leE,axiom,
% 9.07/5.15      ! [M: nat,K: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.07/5.15       => ~ ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15           => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_240_le__add1,axiom,
% 9.07/5.15      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_241_le__add2,axiom,
% 9.07/5.15      ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_242_add__leD1,axiom,
% 9.07/5.15      ! [M: nat,K: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_243_add__leD2,axiom,
% 9.07/5.15      ! [M: nat,K: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ K @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_244_le__Suc__ex,axiom,
% 9.07/5.15      ! [K: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ L )
% 9.07/5.15       => ? [N3: nat] :
% 9.07/5.15            ( L
% 9.07/5.15            = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_245_add__le__mono,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ( ord_less_eq_nat @ K @ L )
% 9.07/5.15         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_246_add__le__mono1,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_247_trans__le__add1,axiom,
% 9.07/5.15      ! [I: nat,J: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_248_trans__le__add2,axiom,
% 9.07/5.15      ! [I: nat,J: nat,M: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_249_nat__le__iff__add,axiom,
% 9.07/5.15      ( ord_less_eq_nat
% 9.07/5.15      = ( ^ [M3: nat,N2: nat] :
% 9.07/5.15          ? [K2: nat] :
% 9.07/5.15            ( N2
% 9.07/5.15            = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_250_diff__less__mono2,axiom,
% 9.07/5.15      ! [M: nat,N: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_nat @ M @ N )
% 9.07/5.15       => ( ( ord_less_nat @ M @ L )
% 9.07/5.15         => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_251_less__imp__diff__less,axiom,
% 9.07/5.15      ! [J: nat,K: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_nat @ J @ K )
% 9.07/5.15       => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_252_eq__diff__iff,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ M )
% 9.07/5.15       => ( ( ord_less_eq_nat @ K @ N )
% 9.07/5.15         => ( ( ( minus_minus_nat @ M @ K )
% 9.07/5.15              = ( minus_minus_nat @ N @ K ) )
% 9.07/5.15            = ( M = N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_253_le__diff__iff,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ M )
% 9.07/5.15       => ( ( ord_less_eq_nat @ K @ N )
% 9.07/5.15         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.07/5.15            = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_254_Nat_Odiff__diff__eq,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ M )
% 9.07/5.15       => ( ( ord_less_eq_nat @ K @ N )
% 9.07/5.15         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.07/5.15            = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_255_diff__le__mono,axiom,
% 9.07/5.15      ! [M: nat,N: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_256_diff__le__self,axiom,
% 9.07/5.15      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_257_le__diff__iff_H,axiom,
% 9.07/5.15      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ A @ C )
% 9.07/5.15       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.07/5.15         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
% 9.07/5.15            = ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_258_diff__le__mono2,axiom,
% 9.07/5.15      ! [M: nat,N: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_259_Nat_Odiff__cancel,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.07/5.15        = ( minus_minus_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_260_diff__cancel2,axiom,
% 9.07/5.15      ! [M: nat,K: nat,N: nat] :
% 9.07/5.15        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 9.07/5.15        = ( minus_minus_nat @ M @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_261_diff__add__inverse,axiom,
% 9.07/5.15      ! [N: nat,M: nat] :
% 9.07/5.15        ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 9.07/5.15        = M ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_262_diff__add__inverse2,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 9.07/5.15        = M ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_263_add__le__add__imp__diff__le,axiom,
% 9.07/5.15      ! [I: real,K: real,N: real,J: real] :
% 9.07/5.15        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.07/5.15       => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.07/5.15         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.07/5.15           => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.07/5.15             => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_264_add__le__add__imp__diff__le,axiom,
% 9.07/5.15      ! [I: rat,K: rat,N: rat,J: rat] :
% 9.07/5.15        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.07/5.15       => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.07/5.15         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.07/5.15           => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.07/5.15             => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_265_add__le__add__imp__diff__le,axiom,
% 9.07/5.15      ! [I: nat,K: nat,N: nat,J: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.07/5.15       => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.07/5.15         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.07/5.15           => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.07/5.15             => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_266_add__le__add__imp__diff__le,axiom,
% 9.07/5.15      ! [I: int,K: int,N: int,J: int] :
% 9.07/5.15        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.07/5.15       => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.07/5.15         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.07/5.15           => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.07/5.15             => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_267_add__le__imp__le__diff,axiom,
% 9.07/5.15      ! [I: real,K: real,N: real] :
% 9.07/5.15        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.07/5.15       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_268_add__le__imp__le__diff,axiom,
% 9.07/5.15      ! [I: rat,K: rat,N: rat] :
% 9.07/5.15        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.07/5.15       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_269_add__le__imp__le__diff,axiom,
% 9.07/5.15      ! [I: nat,K: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.07/5.15       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_270_add__le__imp__le__diff,axiom,
% 9.07/5.15      ! [I: int,K: int,N: int] :
% 9.07/5.15        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.07/5.15       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_271_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.07/5.15      ! [A: real,B2: real] :
% 9.07/5.15        ( ~ ( ord_less_real @ A @ B2 )
% 9.07/5.15       => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_272_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.07/5.15      ! [A: rat,B2: rat] :
% 9.07/5.15        ( ~ ( ord_less_rat @ A @ B2 )
% 9.07/5.15       => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_273_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.07/5.15      ! [A: nat,B2: nat] :
% 9.07/5.15        ( ~ ( ord_less_nat @ A @ B2 )
% 9.07/5.15       => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_274_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.07/5.15      ! [A: int,B2: int] :
% 9.07/5.15        ( ~ ( ord_less_int @ A @ B2 )
% 9.07/5.15       => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A @ B2 ) )
% 9.07/5.15          = A ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_275_mono__nat__linear__lb,axiom,
% 9.07/5.15      ! [F: nat > nat,M: nat,K: nat] :
% 9.07/5.15        ( ! [M4: nat,N3: nat] :
% 9.07/5.15            ( ( ord_less_nat @ M4 @ N3 )
% 9.07/5.15           => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 9.07/5.15       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_276_less__diff__iff,axiom,
% 9.07/5.15      ! [K: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ M )
% 9.07/5.15       => ( ( ord_less_eq_nat @ K @ N )
% 9.07/5.15         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.07/5.15            = ( ord_less_nat @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_277_diff__less__mono,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.15        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.15       => ( ( ord_less_eq_nat @ C @ A )
% 9.07/5.15         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_278_add__diff__inverse__nat,axiom,
% 9.07/5.15      ! [M: nat,N: nat] :
% 9.07/5.15        ( ~ ( ord_less_nat @ M @ N )
% 9.07/5.15       => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 9.07/5.15          = M ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_279_less__diff__conv,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.07/5.15        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_280_Nat_Ole__imp__diff__is__add,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ( ( minus_minus_nat @ J @ I )
% 9.07/5.15            = K )
% 9.07/5.15          = ( J
% 9.07/5.15            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_281_Nat_Odiff__add__assoc2,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 9.07/5.15          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_282_Nat_Odiff__add__assoc,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.07/5.15          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_283_Nat_Ole__diff__conv2,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.07/5.15          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_284_le__diff__conv,axiom,
% 9.07/5.15      ! [J: nat,K: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.07/5.15        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_285_less__diff__conv2,axiom,
% 9.07/5.15      ! [K: nat,J: nat,I: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.15       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.07/5.15          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_286_field__sum__of__halves,axiom,
% 9.07/5.15      ! [X3: real] :
% 9.07/5.15        ( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.07/5.15        = X3 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_287_field__sum__of__halves,axiom,
% 9.07/5.15      ! [X3: rat] :
% 9.07/5.15        ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.07/5.15        = X3 ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_288_in__children__def,axiom,
% 9.07/5.15      ( vEBT_V5917875025757280293ildren
% 9.07/5.15      = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_289_maxt__sound,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 9.07/5.15         => ( ( vEBT_vebt_maxt @ T )
% 9.07/5.15            = ( some_nat @ X3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_290_maxt__corr,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( vEBT_vebt_maxt @ T )
% 9.07/5.15            = ( some_nat @ X3 ) )
% 9.07/5.15         => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_291_post__member__pre__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15         => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15           => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X3 ) @ Y3 )
% 9.07/5.15             => ( ( vEBT_vebt_member @ T @ Y3 )
% 9.07/5.15                | ( X3 = Y3 ) ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_292_valid__insert__both__member__options__pres,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat,Y3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15         => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15           => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 9.07/5.15             => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X3 ) ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_293_valid__insert__both__member__options__add,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15         => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X3 ) @ X3 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_294_maxt__corr__help,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,Maxi: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( vEBT_vebt_maxt @ T )
% 9.07/5.15            = ( some_nat @ Maxi ) )
% 9.07/5.15         => ( ( vEBT_vebt_member @ T @ X3 )
% 9.07/5.15           => ( ord_less_eq_nat @ X3 @ Maxi ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_295_maxt__member,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.15       => ( ( ( vEBT_vebt_maxt @ T )
% 9.07/5.15            = ( some_nat @ Maxi ) )
% 9.07/5.15         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_296_both__member__options__ding,axiom,
% 9.07/5.15      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 9.07/5.15       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.07/5.15         => ( ( 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 ) ) ) ) )
% 9.07/5.15           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_297_maxbmo,axiom,
% 9.07/5.15      ! [T: vEBT_VEBT,X3: nat] :
% 9.07/5.15        ( ( ( vEBT_vebt_maxt @ T )
% 9.07/5.15          = ( some_nat @ X3 ) )
% 9.07/5.15       => ( vEBT_V8194947554948674370ptions @ T @ X3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_298_bit__concat__def,axiom,
% 9.07/5.15      ( vEBT_VEBT_bit_concat
% 9.07/5.15      = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_299_low__inv,axiom,
% 9.07/5.15      ! [X3: nat,N: nat,Y3: nat] :
% 9.07/5.15        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
% 9.07/5.15          = X3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_300_deg__deg__n,axiom,
% 9.07/5.15      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.07/5.15        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 9.07/5.15       => ( Deg = N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_301_mult__numeral__left__semiring__numeral,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: complex] :
% 9.07/5.15        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 9.07/5.15        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_302_mult__numeral__left__semiring__numeral,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: real] :
% 9.07/5.15        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 9.07/5.15        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_303_mult__numeral__left__semiring__numeral,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 9.07/5.15        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_304_mult__numeral__left__semiring__numeral,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 9.07/5.15        = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_305_mult__numeral__left__semiring__numeral,axiom,
% 9.07/5.15      ! [V: num,W: num,Z: int] :
% 9.07/5.15        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 9.07/5.15        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_306_numeral__times__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 9.07/5.15        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_307_numeral__times__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.07/5.15        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_308_numeral__times__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.07/5.15        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_309_numeral__times__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.15        = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_310_numeral__times__numeral,axiom,
% 9.07/5.15      ! [M: num,N: num] :
% 9.07/5.15        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.07/5.15        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_311_high__inv,axiom,
% 9.07/5.15      ! [X3: nat,N: nat,Y3: nat] :
% 9.07/5.15        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.15       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
% 9.07/5.15          = Y3 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_312_distrib__right__numeral,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,V: num] :
% 9.07/5.15        ( ( times_times_complex @ ( plus_plus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 9.07/5.15        = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_313_distrib__right__numeral,axiom,
% 9.07/5.15      ! [A: real,B2: real,V: num] :
% 9.07/5.15        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_314_distrib__right__numeral,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,V: num] :
% 9.07/5.15        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_315_distrib__right__numeral,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,V: num] :
% 9.07/5.15        ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ ( numeral_numeral_nat @ V ) )
% 9.07/5.15        = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_316_distrib__right__numeral,axiom,
% 9.07/5.15      ! [A: int,B2: int,V: num] :
% 9.07/5.15        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_317_distrib__left__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: complex,C: complex] :
% 9.07/5.15        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_318_distrib__left__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_319_distrib__left__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_320_distrib__left__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: nat,C: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_321_distrib__left__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_322_right__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: complex,C: complex] :
% 9.07/5.15        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_323_right__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_324_right__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_325_right__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [V: num,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_326_left__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,V: num] :
% 9.07/5.15        ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
% 9.07/5.15        = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_327_left__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [A: real,B2: real,V: num] :
% 9.07/5.15        ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ ( numeral_numeral_real @ V ) )
% 9.07/5.15        = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_328_left__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,V: num] :
% 9.07/5.15        ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ ( numeral_numeral_rat @ V ) )
% 9.07/5.15        = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_329_left__diff__distrib__numeral,axiom,
% 9.07/5.15      ! [A: int,B2: int,V: num] :
% 9.07/5.15        ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ ( numeral_numeral_int @ V ) )
% 9.07/5.15        = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_330_divide__le__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [B2: real,W: num,A: real] :
% 9.07/5.15        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 9.07/5.15        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_331_divide__le__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [B2: rat,W: num,A: rat] :
% 9.07/5.15        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 9.07/5.15        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_332_le__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [A: real,B2: real,W: num] :
% 9.07/5.15        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 9.07/5.15        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_333_le__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,W: num] :
% 9.07/5.15        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 9.07/5.15        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_334_divide__less__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [B2: real,W: num,A: real] :
% 9.07/5.15        ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A )
% 9.07/5.15        = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_335_divide__less__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [B2: rat,W: num,A: rat] :
% 9.07/5.15        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A )
% 9.07/5.15        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_336_less__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [A: real,B2: real,W: num] :
% 9.07/5.15        ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 9.07/5.15        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_337_less__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,W: num] :
% 9.07/5.15        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 9.07/5.15        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_338_power__add__numeral2,axiom,
% 9.07/5.15      ! [A: complex,M: num,N: num,B2: complex] :
% 9.07/5.15        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 9.07/5.15        = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_339_power__add__numeral2,axiom,
% 9.07/5.15      ! [A: real,M: num,N: num,B2: real] :
% 9.07/5.15        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 9.07/5.15        = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_340_power__add__numeral2,axiom,
% 9.07/5.15      ! [A: rat,M: num,N: num,B2: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 9.07/5.15        = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_341_power__add__numeral2,axiom,
% 9.07/5.15      ! [A: nat,M: num,N: num,B2: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 9.07/5.15        = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_342_power__add__numeral2,axiom,
% 9.07/5.15      ! [A: int,M: num,N: num,B2: int] :
% 9.07/5.15        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B2 ) )
% 9.07/5.15        = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B2 ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_343_power__add__numeral,axiom,
% 9.07/5.15      ! [A: complex,M: num,N: num] :
% 9.07/5.15        ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.07/5.15        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_344_power__add__numeral,axiom,
% 9.07/5.15      ! [A: real,M: num,N: num] :
% 9.07/5.15        ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.07/5.15        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_345_power__add__numeral,axiom,
% 9.07/5.15      ! [A: rat,M: num,N: num] :
% 9.07/5.15        ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.07/5.15        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_346_power__add__numeral,axiom,
% 9.07/5.15      ! [A: nat,M: num,N: num] :
% 9.07/5.15        ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.07/5.15        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_347_power__add__numeral,axiom,
% 9.07/5.15      ! [A: int,M: num,N: num] :
% 9.07/5.15        ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 9.07/5.15        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_348__C4_Ohyps_C_I8_J,axiom,
% 9.07/5.15      ( ( mi = ma )
% 9.07/5.15     => ! [X6: vEBT_VEBT] :
% 9.07/5.15          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.07/5.15         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_349_combine__common__factor,axiom,
% 9.07/5.15      ! [A: real,E: real,B2: real,C: real] :
% 9.07/5.15        ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C ) )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ E ) @ C ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_350_combine__common__factor,axiom,
% 9.07/5.15      ! [A: rat,E: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C ) )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ E ) @ C ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_351_combine__common__factor,axiom,
% 9.07/5.15      ! [A: nat,E: nat,B2: nat,C: nat] :
% 9.07/5.15        ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
% 9.07/5.15        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ E ) @ C ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_352_combine__common__factor,axiom,
% 9.07/5.15      ! [A: int,E: int,B2: int,C: int] :
% 9.07/5.15        ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ E ) @ C ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_353_distrib__right,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_354_distrib__right,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_355_distrib__right,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_356_distrib__right,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_357_distrib__left,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_358_distrib__left,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_359_distrib__left,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.15        ( ( times_times_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_360_distrib__left,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_361_comm__semiring__class_Odistrib,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_362_comm__semiring__class_Odistrib,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_363_comm__semiring__class_Odistrib,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_364_comm__semiring__class_Odistrib,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_365_ring__class_Oring__distribs_I1_J,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_366_ring__class_Oring__distribs_I1_J,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_367_ring__class_Oring__distribs_I1_J,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_368_ring__class_Oring__distribs_I2_J,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_369_ring__class_Oring__distribs_I2_J,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_370_ring__class_Oring__distribs_I2_J,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.15        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_371_right__diff__distrib_H,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.15        ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_372_right__diff__distrib_H,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_373_right__diff__distrib_H,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_374_right__diff__distrib_H,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.15        ( ( times_times_nat @ A @ ( minus_minus_nat @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_375_right__diff__distrib_H,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_376_left__diff__distrib_H,axiom,
% 9.07/5.15      ! [B2: complex,C: complex,A: complex] :
% 9.07/5.15        ( ( times_times_complex @ ( minus_minus_complex @ B2 @ C ) @ A )
% 9.07/5.15        = ( minus_minus_complex @ ( times_times_complex @ B2 @ A ) @ ( times_times_complex @ C @ A ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_377_left__diff__distrib_H,axiom,
% 9.07/5.15      ! [B2: real,C: real,A: real] :
% 9.07/5.15        ( ( times_times_real @ ( minus_minus_real @ B2 @ C ) @ A )
% 9.07/5.15        = ( minus_minus_real @ ( times_times_real @ B2 @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_378_left__diff__distrib_H,axiom,
% 9.07/5.15      ! [B2: rat,C: rat,A: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C ) @ A )
% 9.07/5.15        = ( minus_minus_rat @ ( times_times_rat @ B2 @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_379_left__diff__distrib_H,axiom,
% 9.07/5.15      ! [B2: nat,C: nat,A: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C ) @ A )
% 9.07/5.15        = ( minus_minus_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_380_left__diff__distrib_H,axiom,
% 9.07/5.15      ! [B2: int,C: int,A: int] :
% 9.07/5.15        ( ( times_times_int @ ( minus_minus_int @ B2 @ C ) @ A )
% 9.07/5.15        = ( minus_minus_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_381_right__diff__distrib,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.15        ( ( times_times_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_complex @ ( times_times_complex @ A @ B2 ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_382_right__diff__distrib,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_real @ ( times_times_real @ A @ B2 ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_383_right__diff__distrib,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_rat @ ( times_times_rat @ A @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_384_right__diff__distrib,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 9.07/5.15        = ( minus_minus_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_385_left__diff__distrib,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.15        ( ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 9.07/5.15        = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_386_left__diff__distrib,axiom,
% 9.07/5.15      ! [A: real,B2: real,C: real] :
% 9.07/5.15        ( ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 9.07/5.15        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_387_left__diff__distrib,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.15        ( ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 9.07/5.15        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_388_left__diff__distrib,axiom,
% 9.07/5.15      ! [A: int,B2: int,C: int] :
% 9.07/5.15        ( ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.07/5.15        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_389_power__commuting__commutes,axiom,
% 9.07/5.15      ! [X3: complex,Y3: complex,N: nat] :
% 9.07/5.15        ( ( ( times_times_complex @ X3 @ Y3 )
% 9.07/5.15          = ( times_times_complex @ Y3 @ X3 ) )
% 9.07/5.15       => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ Y3 )
% 9.07/5.15          = ( times_times_complex @ Y3 @ ( power_power_complex @ X3 @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_390_power__commuting__commutes,axiom,
% 9.07/5.15      ! [X3: real,Y3: real,N: nat] :
% 9.07/5.15        ( ( ( times_times_real @ X3 @ Y3 )
% 9.07/5.15          = ( times_times_real @ Y3 @ X3 ) )
% 9.07/5.15       => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ Y3 )
% 9.07/5.15          = ( times_times_real @ Y3 @ ( power_power_real @ X3 @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_391_power__commuting__commutes,axiom,
% 9.07/5.15      ! [X3: rat,Y3: rat,N: nat] :
% 9.07/5.15        ( ( ( times_times_rat @ X3 @ Y3 )
% 9.07/5.15          = ( times_times_rat @ Y3 @ X3 ) )
% 9.07/5.15       => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ Y3 )
% 9.07/5.15          = ( times_times_rat @ Y3 @ ( power_power_rat @ X3 @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_392_power__commuting__commutes,axiom,
% 9.07/5.15      ! [X3: nat,Y3: nat,N: nat] :
% 9.07/5.15        ( ( ( times_times_nat @ X3 @ Y3 )
% 9.07/5.15          = ( times_times_nat @ Y3 @ X3 ) )
% 9.07/5.15       => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ Y3 )
% 9.07/5.15          = ( times_times_nat @ Y3 @ ( power_power_nat @ X3 @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_393_power__commuting__commutes,axiom,
% 9.07/5.15      ! [X3: int,Y3: int,N: nat] :
% 9.07/5.15        ( ( ( times_times_int @ X3 @ Y3 )
% 9.07/5.15          = ( times_times_int @ Y3 @ X3 ) )
% 9.07/5.15       => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ Y3 )
% 9.07/5.15          = ( times_times_int @ Y3 @ ( power_power_int @ X3 @ N ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_394_power__mult__distrib,axiom,
% 9.07/5.15      ! [A: complex,B2: complex,N: nat] :
% 9.07/5.15        ( ( power_power_complex @ ( times_times_complex @ A @ B2 ) @ N )
% 9.07/5.15        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_395_power__mult__distrib,axiom,
% 9.07/5.15      ! [A: real,B2: real,N: nat] :
% 9.07/5.15        ( ( power_power_real @ ( times_times_real @ A @ B2 ) @ N )
% 9.07/5.15        = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_396_power__mult__distrib,axiom,
% 9.07/5.15      ! [A: rat,B2: rat,N: nat] :
% 9.07/5.15        ( ( power_power_rat @ ( times_times_rat @ A @ B2 ) @ N )
% 9.07/5.15        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_397_power__mult__distrib,axiom,
% 9.07/5.15      ! [A: nat,B2: nat,N: nat] :
% 9.07/5.15        ( ( power_power_nat @ ( times_times_nat @ A @ B2 ) @ N )
% 9.07/5.15        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_398_power__mult__distrib,axiom,
% 9.07/5.15      ! [A: int,B2: int,N: nat] :
% 9.07/5.15        ( ( power_power_int @ ( times_times_int @ A @ B2 ) @ N )
% 9.07/5.15        = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_399_power__commutes,axiom,
% 9.07/5.15      ! [A: complex,N: nat] :
% 9.07/5.15        ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 9.07/5.15        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_400_power__commutes,axiom,
% 9.07/5.15      ! [A: real,N: nat] :
% 9.07/5.15        ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 9.07/5.15        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_401_power__commutes,axiom,
% 9.07/5.15      ! [A: rat,N: nat] :
% 9.07/5.15        ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 9.07/5.15        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_402_power__commutes,axiom,
% 9.07/5.15      ! [A: nat,N: nat] :
% 9.07/5.15        ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 9.07/5.15        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_403_power__commutes,axiom,
% 9.07/5.15      ! [A: int,N: nat] :
% 9.07/5.15        ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 9.07/5.15        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_404_power__mult,axiom,
% 9.07/5.15      ! [A: nat,M: nat,N: nat] :
% 9.07/5.15        ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 9.07/5.15        = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_405_power__mult,axiom,
% 9.07/5.15      ! [A: real,M: nat,N: nat] :
% 9.07/5.15        ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 9.07/5.15        = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_406_power__mult,axiom,
% 9.07/5.15      ! [A: int,M: nat,N: nat] :
% 9.07/5.15        ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 9.07/5.15        = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_407_power__mult,axiom,
% 9.07/5.15      ! [A: complex,M: nat,N: nat] :
% 9.07/5.15        ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 9.07/5.15        = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_408_mult__le__mono2,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_409_mult__le__mono1,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_410_mult__le__mono,axiom,
% 9.07/5.15      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.15        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.15       => ( ( ord_less_eq_nat @ K @ L )
% 9.07/5.15         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_411_le__square,axiom,
% 9.07/5.15      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 9.07/5.15  
% 9.07/5.15  thf(fact_412_le__cube,axiom,
% 9.07/5.15      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_413_left__add__mult__distrib,axiom,
% 9.07/5.16      ! [I: nat,U: nat,J: nat,K: nat] :
% 9.07/5.16        ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 9.07/5.16        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_414_add__mult__distrib2,axiom,
% 9.07/5.16      ! [K: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.16        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_415_add__mult__distrib,axiom,
% 9.07/5.16      ! [M: nat,N: nat,K: nat] :
% 9.07/5.16        ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 9.07/5.16        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_416_diff__mult__distrib2,axiom,
% 9.07/5.16      ! [K: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.07/5.16        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_417_diff__mult__distrib,axiom,
% 9.07/5.16      ! [M: nat,N: nat,K: nat] :
% 9.07/5.16        ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 9.07/5.16        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_418_div__mult2__eq,axiom,
% 9.07/5.16      ! [M: nat,N: nat,Q2: nat] :
% 9.07/5.16        ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 9.07/5.16        = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_419_mult__numeral__1,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_420_mult__numeral__1,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_421_mult__numeral__1,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_422_mult__numeral__1,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_423_mult__numeral__1,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_424_mult__numeral__1__right,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_425_mult__numeral__1__right,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_426_mult__numeral__1__right,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_427_mult__numeral__1__right,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_428_mult__numeral__1__right,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_429_mult__diff__mult,axiom,
% 9.07/5.16      ! [X3: complex,Y3: complex,A: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ Y3 ) @ ( times_times_complex @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_complex @ ( times_times_complex @ X3 @ ( minus_minus_complex @ Y3 @ B2 ) ) @ ( times_times_complex @ ( minus_minus_complex @ X3 @ A ) @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_430_mult__diff__mult,axiom,
% 9.07/5.16      ! [X3: real,Y3: real,A: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( times_times_real @ X3 @ Y3 ) @ ( times_times_real @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y3 @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_431_mult__diff__mult,axiom,
% 9.07/5.16      ! [X3: rat,Y3: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y3 ) @ ( times_times_rat @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y3 @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_432_mult__diff__mult,axiom,
% 9.07/5.16      ! [X3: int,Y3: int,A: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( times_times_int @ X3 @ Y3 ) @ ( times_times_int @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y3 @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_433_square__diff__square__factored,axiom,
% 9.07/5.16      ! [X3: complex,Y3: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ ( times_times_complex @ Y3 @ Y3 ) )
% 9.07/5.16        = ( times_times_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_434_square__diff__square__factored,axiom,
% 9.07/5.16      ! [X3: real,Y3: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 9.07/5.16        = ( times_times_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_435_square__diff__square__factored,axiom,
% 9.07/5.16      ! [X3: rat,Y3: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 9.07/5.16        = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_rat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_436_square__diff__square__factored,axiom,
% 9.07/5.16      ! [X3: int,Y3: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 9.07/5.16        = ( times_times_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( minus_minus_int @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_437_eq__add__iff2,axiom,
% 9.07/5.16      ! [A: complex,E: complex,C: complex,B2: complex,D: complex] :
% 9.07/5.16        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( C
% 9.07/5.16          = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_438_eq__add__iff2,axiom,
% 9.07/5.16      ! [A: real,E: real,C: real,B2: real,D: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( C
% 9.07/5.16          = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_439_eq__add__iff2,axiom,
% 9.07/5.16      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( C
% 9.07/5.16          = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_440_eq__add__iff2,axiom,
% 9.07/5.16      ! [A: int,E: int,C: int,B2: int,D: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( C
% 9.07/5.16          = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_441_eq__add__iff1,axiom,
% 9.07/5.16      ! [A: complex,E: complex,C: complex,B2: complex,D: complex] :
% 9.07/5.16        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_complex @ ( times_times_complex @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B2 ) @ E ) @ C )
% 9.07/5.16          = D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_442_eq__add__iff1,axiom,
% 9.07/5.16      ! [A: real,E: real,C: real,B2: real,D: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C )
% 9.07/5.16          = D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_443_eq__add__iff1,axiom,
% 9.07/5.16      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C )
% 9.07/5.16          = D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_444_eq__add__iff1,axiom,
% 9.07/5.16      ! [A: int,E: int,C: int,B2: int,D: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 9.07/5.16          = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C )
% 9.07/5.16          = D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_445_power__add,axiom,
% 9.07/5.16      ! [A: complex,M: nat,N: nat] :
% 9.07/5.16        ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.16        = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_446_power__add,axiom,
% 9.07/5.16      ! [A: real,M: nat,N: nat] :
% 9.07/5.16        ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.16        = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_447_power__add,axiom,
% 9.07/5.16      ! [A: rat,M: nat,N: nat] :
% 9.07/5.16        ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.16        = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_448_power__add,axiom,
% 9.07/5.16      ! [A: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.16        = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_449_power__add,axiom,
% 9.07/5.16      ! [A: int,M: nat,N: nat] :
% 9.07/5.16        ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.16        = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_450_less__mult__imp__div__less,axiom,
% 9.07/5.16      ! [M: nat,I: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 9.07/5.16       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_451_times__div__less__eq__dividend,axiom,
% 9.07/5.16      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_452_div__times__less__eq__dividend,axiom,
% 9.07/5.16      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_453_ordered__ring__class_Ole__add__iff1,axiom,
% 9.07/5.16      ! [A: real,E: real,C: real,B2: real,D: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_454_ordered__ring__class_Ole__add__iff1,axiom,
% 9.07/5.16      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_455_ordered__ring__class_Ole__add__iff1,axiom,
% 9.07/5.16      ! [A: int,E: int,C: int,B2: int,D: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_456_ordered__ring__class_Ole__add__iff2,axiom,
% 9.07/5.16      ! [A: real,E: real,C: real,B2: real,D: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_457_ordered__ring__class_Ole__add__iff2,axiom,
% 9.07/5.16      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_458_ordered__ring__class_Ole__add__iff2,axiom,
% 9.07/5.16      ! [A: int,E: int,C: int,B2: int,D: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_459_less__add__iff1,axiom,
% 9.07/5.16      ! [A: real,E: real,C: real,B2: real,D: real] :
% 9.07/5.16        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_460_less__add__iff1,axiom,
% 9.07/5.16      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_461_less__add__iff1,axiom,
% 9.07/5.16      ! [A: int,E: int,C: int,B2: int,D: int] :
% 9.07/5.16        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B2 ) @ E ) @ C ) @ D ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_462_less__add__iff2,axiom,
% 9.07/5.16      ! [A: real,E: real,C: real,B2: real,D: real] :
% 9.07/5.16        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_463_less__add__iff2,axiom,
% 9.07/5.16      ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_464_less__add__iff2,axiom,
% 9.07/5.16      ! [A: int,E: int,C: int,B2: int,D: int] :
% 9.07/5.16        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
% 9.07/5.16        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A ) @ E ) @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_465_nat__eq__add__iff1,axiom,
% 9.07/5.16      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ J @ I )
% 9.07/5.16       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.07/5.16            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 9.07/5.16            = N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_466_nat__eq__add__iff2,axiom,
% 9.07/5.16      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.07/5.16            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( M
% 9.07/5.16            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_467_nat__le__add__iff1,axiom,
% 9.07/5.16      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ J @ I )
% 9.07/5.16       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_468_nat__le__add__iff2,axiom,
% 9.07/5.16      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_469_nat__diff__add__eq1,axiom,
% 9.07/5.16      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ J @ I )
% 9.07/5.16       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_470_nat__diff__add__eq2,axiom,
% 9.07/5.16      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_471_mult__2,axiom,
% 9.07/5.16      ! [Z: complex] :
% 9.07/5.16        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 9.07/5.16        = ( plus_plus_complex @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_472_mult__2,axiom,
% 9.07/5.16      ! [Z: real] :
% 9.07/5.16        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 9.07/5.16        = ( plus_plus_real @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_473_mult__2,axiom,
% 9.07/5.16      ! [Z: rat] :
% 9.07/5.16        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 9.07/5.16        = ( plus_plus_rat @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_474_mult__2,axiom,
% 9.07/5.16      ! [Z: nat] :
% 9.07/5.16        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 9.07/5.16        = ( plus_plus_nat @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_475_mult__2,axiom,
% 9.07/5.16      ! [Z: int] :
% 9.07/5.16        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 9.07/5.16        = ( plus_plus_int @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_476_mult__2__right,axiom,
% 9.07/5.16      ! [Z: complex] :
% 9.07/5.16        ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( plus_plus_complex @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_477_mult__2__right,axiom,
% 9.07/5.16      ! [Z: real] :
% 9.07/5.16        ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( plus_plus_real @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_478_mult__2__right,axiom,
% 9.07/5.16      ! [Z: rat] :
% 9.07/5.16        ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( plus_plus_rat @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_479_mult__2__right,axiom,
% 9.07/5.16      ! [Z: nat] :
% 9.07/5.16        ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( plus_plus_nat @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_480_mult__2__right,axiom,
% 9.07/5.16      ! [Z: int] :
% 9.07/5.16        ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( plus_plus_int @ Z @ Z ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_481_left__add__twice,axiom,
% 9.07/5.16      ! [A: complex,B2: complex] :
% 9.07/5.16        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_482_left__add__twice,axiom,
% 9.07/5.16      ! [A: real,B2: real] :
% 9.07/5.16        ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_483_left__add__twice,axiom,
% 9.07/5.16      ! [A: rat,B2: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_484_left__add__twice,axiom,
% 9.07/5.16      ! [A: nat,B2: nat] :
% 9.07/5.16        ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_485_left__add__twice,axiom,
% 9.07/5.16      ! [A: int,B2: int] :
% 9.07/5.16        ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 9.07/5.16        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_486_power4__eq__xxxx,axiom,
% 9.07/5.16      ! [X3: complex] :
% 9.07/5.16        ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.07/5.16        = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_487_power4__eq__xxxx,axiom,
% 9.07/5.16      ! [X3: real] :
% 9.07/5.16        ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.07/5.16        = ( times_times_real @ ( times_times_real @ ( times_times_real @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_488_power4__eq__xxxx,axiom,
% 9.07/5.16      ! [X3: rat] :
% 9.07/5.16        ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.07/5.16        = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_489_power4__eq__xxxx,axiom,
% 9.07/5.16      ! [X3: nat] :
% 9.07/5.16        ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.07/5.16        = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_490_power4__eq__xxxx,axiom,
% 9.07/5.16      ! [X3: int] :
% 9.07/5.16        ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.07/5.16        = ( times_times_int @ ( times_times_int @ ( times_times_int @ X3 @ X3 ) @ X3 ) @ X3 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_491_power2__eq__square,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( times_times_complex @ A @ A ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_492_power2__eq__square,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( times_times_real @ A @ A ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_493_power2__eq__square,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( times_times_rat @ A @ A ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_494_power2__eq__square,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( times_times_nat @ A @ A ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_495_power2__eq__square,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( times_times_int @ A @ A ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_496_power__even__eq,axiom,
% 9.07/5.16      ! [A: nat,N: nat] :
% 9.07/5.16        ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.16        = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_497_power__even__eq,axiom,
% 9.07/5.16      ! [A: real,N: nat] :
% 9.07/5.16        ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.16        = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_498_power__even__eq,axiom,
% 9.07/5.16      ! [A: int,N: nat] :
% 9.07/5.16        ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.16        = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_499_power__even__eq,axiom,
% 9.07/5.16      ! [A: complex,N: nat] :
% 9.07/5.16        ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.07/5.16        = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_500_nat__less__add__iff2,axiom,
% 9.07/5.16      ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_501_nat__less__add__iff1,axiom,
% 9.07/5.16      ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ J @ I )
% 9.07/5.16       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.07/5.16          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_502_is__pred__in__set__def,axiom,
% 9.07/5.16      ( vEBT_is_pred_in_set
% 9.07/5.16      = ( ^ [Xs: set_nat,X: nat,Y: nat] :
% 9.07/5.16            ( ( member_nat @ Y @ Xs )
% 9.07/5.16            & ( ord_less_nat @ Y @ X )
% 9.07/5.16            & ! [Z2: nat] :
% 9.07/5.16                ( ( member_nat @ Z2 @ Xs )
% 9.07/5.16               => ( ( ord_less_nat @ Z2 @ X )
% 9.07/5.16                 => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_503_power2__sum,axiom,
% 9.07/5.16      ! [X3: complex,Y3: complex] :
% 9.07/5.16        ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_504_power2__sum,axiom,
% 9.07/5.16      ! [X3: real,Y3: real] :
% 9.07/5.16        ( ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_505_power2__sum,axiom,
% 9.07/5.16      ! [X3: rat,Y3: rat] :
% 9.07/5.16        ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_506_power2__sum,axiom,
% 9.07/5.16      ! [X3: nat,Y3: nat] :
% 9.07/5.16        ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_507_power2__sum,axiom,
% 9.07/5.16      ! [X3: int,Y3: int] :
% 9.07/5.16        ( ( power_power_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_508_power2__diff,axiom,
% 9.07/5.16      ! [X3: complex,Y3: complex] :
% 9.07/5.16        ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_509_power2__diff,axiom,
% 9.07/5.16      ! [X3: real,Y3: real] :
% 9.07/5.16        ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_510_power2__diff,axiom,
% 9.07/5.16      ! [X3: rat,Y3: rat] :
% 9.07/5.16        ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_511_power2__diff,axiom,
% 9.07/5.16      ! [X3: int,Y3: int] :
% 9.07/5.16        ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.16        = ( 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 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_512_mul__def,axiom,
% 9.07/5.16      ( vEBT_VEBT_mul
% 9.07/5.16      = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_513_mul__shift,axiom,
% 9.07/5.16      ! [X3: nat,Y3: nat,Z: nat] :
% 9.07/5.16        ( ( ( times_times_nat @ X3 @ Y3 )
% 9.07/5.16          = Z )
% 9.07/5.16        = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 9.07/5.16          = ( some_nat @ Z ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_514_member__inv,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.07/5.16        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.16       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.07/5.16          & ( ( X3 = Mi )
% 9.07/5.16            | ( X3 = Ma )
% 9.07/5.16            | ( ( ord_less_nat @ X3 @ Ma )
% 9.07/5.16              & ( ord_less_nat @ Mi @ X3 )
% 9.07/5.16              & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.16              & ( 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 ) ) ) ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_515_thisvalid,axiom,
% 9.07/5.16      vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_516_mintlistlength,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.07/5.16        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.07/5.16       => ( ( Mi != Ma )
% 9.07/5.16         => ( ( ord_less_nat @ Mi @ Ma )
% 9.07/5.16            & ? [M4: nat] :
% 9.07/5.16                ( ( ( some_nat @ M4 )
% 9.07/5.16                  = ( vEBT_vebt_mint @ Summary ) )
% 9.07/5.16                & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_517_sum__squares__bound,axiom,
% 9.07/5.16      ! [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 ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_518_sum__squares__bound,axiom,
% 9.07/5.16      ! [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 ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_519__C4_Ohyps_C_I3_J,axiom,
% 9.07/5.16      ! [X3: nat,Px: nat] :
% 9.07/5.16        ( ( ( vEBT_vebt_pred @ summary @ X3 )
% 9.07/5.16          = ( some_nat @ Px ) )
% 9.07/5.16        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X3 @ Px ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_520_mi__ma__2__deg,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.07/5.16        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.07/5.16       => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.07/5.16          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_521_enat__ord__number_I1_J,axiom,
% 9.07/5.16      ! [M: num,N: num] :
% 9.07/5.16        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.07/5.16        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_522_enat__ord__number_I2_J,axiom,
% 9.07/5.16      ! [M: num,N: num] :
% 9.07/5.16        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.07/5.16        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_523_add__diff__cancel__right_H,axiom,
% 9.07/5.16      ! [A: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_524_add__diff__cancel__right_H,axiom,
% 9.07/5.16      ! [A: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_525_add__diff__cancel__right_H,axiom,
% 9.07/5.16      ! [A: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_526_add__diff__cancel__right_H,axiom,
% 9.07/5.16      ! [A: nat,B2: nat] :
% 9.07/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_527_add__diff__cancel__right_H,axiom,
% 9.07/5.16      ! [A: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_528_inthall,axiom,
% 9.07/5.16      ! [Xs2: list_real,P: real > $o,N: nat] :
% 9.07/5.16        ( ! [X5: real] :
% 9.07/5.16            ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
% 9.07/5.16           => ( P @ X5 ) )
% 9.07/5.16       => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 9.07/5.16         => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_529_inthall,axiom,
% 9.07/5.16      ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 9.07/5.16        ( ! [X5: complex] :
% 9.07/5.16            ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
% 9.07/5.16           => ( P @ X5 ) )
% 9.07/5.16       => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 9.07/5.16         => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_530_inthall,axiom,
% 9.07/5.16      ! [Xs2: list_int,P: int > $o,N: nat] :
% 9.07/5.16        ( ! [X5: int] :
% 9.07/5.16            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 9.07/5.16           => ( P @ X5 ) )
% 9.07/5.16       => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.16         => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_531_inthall,axiom,
% 9.07/5.16      ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 9.07/5.16        ( ! [X5: vEBT_VEBT] :
% 9.07/5.16            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.07/5.16           => ( P @ X5 ) )
% 9.07/5.16       => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.16         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_532_inthall,axiom,
% 9.07/5.16      ! [Xs2: list_o,P: $o > $o,N: nat] :
% 9.07/5.16        ( ! [X5: $o] :
% 9.07/5.16            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 9.07/5.16           => ( P @ X5 ) )
% 9.07/5.16       => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.16         => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_533_inthall,axiom,
% 9.07/5.16      ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 9.07/5.16        ( ! [X5: nat] :
% 9.07/5.16            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 9.07/5.16           => ( P @ X5 ) )
% 9.07/5.16       => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.16         => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_534_add__left__cancel,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ A @ B2 )
% 9.07/5.16          = ( plus_plus_real @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_535_add__left__cancel,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ A @ B2 )
% 9.07/5.16          = ( plus_plus_rat @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_536_add__left__cancel,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ( plus_plus_nat @ A @ B2 )
% 9.07/5.16          = ( plus_plus_nat @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_537_add__left__cancel,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ A @ B2 )
% 9.07/5.16          = ( plus_plus_int @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_538_add__right__cancel,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ B2 @ A )
% 9.07/5.16          = ( plus_plus_real @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_539_add__right__cancel,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ B2 @ A )
% 9.07/5.16          = ( plus_plus_rat @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_540_add__right__cancel,axiom,
% 9.07/5.16      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.16        ( ( ( plus_plus_nat @ B2 @ A )
% 9.07/5.16          = ( plus_plus_nat @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_541_add__right__cancel,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ B2 @ A )
% 9.07/5.16          = ( plus_plus_int @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_542__C4_Ohyps_C_I1_J,axiom,
% 9.07/5.16      ! [X6: vEBT_VEBT] :
% 9.07/5.16        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.07/5.16       => ( ( vEBT_invar_vebt @ X6 @ na )
% 9.07/5.16          & ! [Xa: nat,Xb: nat] :
% 9.07/5.16              ( ( ( vEBT_vebt_pred @ X6 @ Xa )
% 9.07/5.16                = ( some_nat @ Xb ) )
% 9.07/5.16              = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ X6 ) @ Xa @ Xb ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_543_mi__eq__ma__no__ch,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.07/5.16        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 9.07/5.16       => ( ( Mi = Ma )
% 9.07/5.16         => ( ! [X6: vEBT_VEBT] :
% 9.07/5.16                ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.16               => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) )
% 9.07/5.16            & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_544_add__le__cancel__left,axiom,
% 9.07/5.16      ! [C: real,A: real,B2: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_545_add__le__cancel__left,axiom,
% 9.07/5.16      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_546_add__le__cancel__left,axiom,
% 9.07/5.16      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_547_add__le__cancel__left,axiom,
% 9.07/5.16      ! [C: int,A: int,B2: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_548_add__le__cancel__right,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_549_add__le__cancel__right,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_550_add__le__cancel__right,axiom,
% 9.07/5.16      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_eq_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_551_add__le__cancel__right,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_552_add__less__cancel__left,axiom,
% 9.07/5.16      ! [C: real,A: real,B2: real] :
% 9.07/5.16        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_553_add__less__cancel__left,axiom,
% 9.07/5.16      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_554_add__less__cancel__left,axiom,
% 9.07/5.16      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_555_add__less__cancel__left,axiom,
% 9.07/5.16      ! [C: int,A: int,B2: int] :
% 9.07/5.16        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_556_add__less__cancel__right,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_557_add__less__cancel__right,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_558_add__less__cancel__right,axiom,
% 9.07/5.16      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_559_add__less__cancel__right,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.16        = ( ord_less_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_560_add__diff__cancel,axiom,
% 9.07/5.16      ! [A: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_561_add__diff__cancel,axiom,
% 9.07/5.16      ! [A: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_562_add__diff__cancel,axiom,
% 9.07/5.16      ! [A: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_563_add__diff__cancel,axiom,
% 9.07/5.16      ! [A: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_564_diff__add__cancel,axiom,
% 9.07/5.16      ! [A: complex,B2: complex] :
% 9.07/5.16        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_565_diff__add__cancel,axiom,
% 9.07/5.16      ! [A: real,B2: real] :
% 9.07/5.16        ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_566_diff__add__cancel,axiom,
% 9.07/5.16      ! [A: rat,B2: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_567_diff__add__cancel,axiom,
% 9.07/5.16      ! [A: int,B2: int] :
% 9.07/5.16        ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_568_add__diff__cancel__left,axiom,
% 9.07/5.16      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( plus_plus_complex @ C @ A ) @ ( plus_plus_complex @ C @ B2 ) )
% 9.07/5.16        = ( minus_minus_complex @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_569_add__diff__cancel__left,axiom,
% 9.07/5.16      ! [C: real,A: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 9.07/5.16        = ( minus_minus_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_570_add__diff__cancel__left,axiom,
% 9.07/5.16      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 9.07/5.16        = ( minus_minus_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_571_add__diff__cancel__left,axiom,
% 9.07/5.16      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 9.07/5.16        = ( minus_minus_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_572_add__diff__cancel__left,axiom,
% 9.07/5.16      ! [C: int,A: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 9.07/5.16        = ( minus_minus_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_573_add__diff__cancel__left_H,axiom,
% 9.07/5.16      ! [A: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ A )
% 9.07/5.16        = B2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_574_add__diff__cancel__left_H,axiom,
% 9.07/5.16      ! [A: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
% 9.07/5.16        = B2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_575_add__diff__cancel__left_H,axiom,
% 9.07/5.16      ! [A: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ A )
% 9.07/5.16        = B2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_576_add__diff__cancel__left_H,axiom,
% 9.07/5.16      ! [A: nat,B2: nat] :
% 9.07/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
% 9.07/5.16        = B2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_577_add__diff__cancel__left_H,axiom,
% 9.07/5.16      ! [A: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ A )
% 9.07/5.16        = B2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_578_add__diff__cancel__right,axiom,
% 9.07/5.16      ! [A: complex,C: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ ( plus_plus_complex @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_complex @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_579_add__diff__cancel__right,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_580_add__diff__cancel__right,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_581_add__diff__cancel__right,axiom,
% 9.07/5.16      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_582_add__diff__cancel__right,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_583_insert__simp__mima,axiom,
% 9.07/5.16      ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.07/5.16        ( ( ( X3 = Mi )
% 9.07/5.16          | ( X3 = Ma ) )
% 9.07/5.16       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.07/5.16         => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.16            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_584_semiring__norm_I13_J,axiom,
% 9.07/5.16      ! [M: num,N: num] :
% 9.07/5.16        ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.07/5.16        = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_585_semiring__norm_I11_J,axiom,
% 9.07/5.16      ! [M: num] :
% 9.07/5.16        ( ( times_times_num @ M @ one )
% 9.07/5.16        = M ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_586_semiring__norm_I12_J,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( times_times_num @ one @ N )
% 9.07/5.16        = N ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_587_pred__max,axiom,
% 9.07/5.16      ! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.07/5.16       => ( ( ord_less_nat @ Ma @ X3 )
% 9.07/5.16         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.16            = ( some_nat @ Ma ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_588_num__double,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( times_times_num @ ( bit0 @ one ) @ N )
% 9.07/5.16        = ( bit0 @ N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_589_power__mult__numeral,axiom,
% 9.07/5.16      ! [A: nat,M: num,N: num] :
% 9.07/5.16        ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_590_power__mult__numeral,axiom,
% 9.07/5.16      ! [A: real,M: num,N: num] :
% 9.07/5.16        ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_591_power__mult__numeral,axiom,
% 9.07/5.16      ! [A: int,M: num,N: num] :
% 9.07/5.16        ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_592_power__mult__numeral,axiom,
% 9.07/5.16      ! [A: complex,M: num,N: num] :
% 9.07/5.16        ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_593_pred__list__to__short,axiom,
% 9.07/5.16      ! [Deg: nat,X3: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.07/5.16       => ( ( ord_less_eq_nat @ X3 @ Ma )
% 9.07/5.16         => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.16           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.16              = none_nat ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_594_enat__less__induct,axiom,
% 9.07/5.16      ! [P: extended_enat > $o,N: extended_enat] :
% 9.07/5.16        ( ! [N3: extended_enat] :
% 9.07/5.16            ( ! [M2: extended_enat] :
% 9.07/5.16                ( ( ord_le72135733267957522d_enat @ M2 @ N3 )
% 9.07/5.16               => ( P @ M2 ) )
% 9.07/5.16           => ( P @ N3 ) )
% 9.07/5.16       => ( P @ N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_595_add__diff__assoc__enat,axiom,
% 9.07/5.16      ! [Z: extended_enat,Y3: extended_enat,X3: extended_enat] :
% 9.07/5.16        ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 9.07/5.16       => ( ( plus_p3455044024723400733d_enat @ X3 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 9.07/5.16          = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_596_L2__set__mult__ineq__lemma,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_597_four__x__squared,axiom,
% 9.07/5.16      ! [X3: real] :
% 9.07/5.16        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.16        = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_598_div__mult2__numeral__eq,axiom,
% 9.07/5.16      ! [A: nat,K: num,L: num] :
% 9.07/5.16        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
% 9.07/5.16        = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_599_div__mult2__numeral__eq,axiom,
% 9.07/5.16      ! [A: int,K: num,L: num] :
% 9.07/5.16        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
% 9.07/5.16        = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_600_vebt__pred_Osimps_I4_J,axiom,
% 9.07/5.16      ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 9.07/5.16        ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 9.07/5.16        = none_nat ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_601_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_602_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_603_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_604_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_605_mult_Oassoc,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_606_mult_Oassoc,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( times_times_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_607_mult_Oassoc,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_608_mult_Oassoc,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_609_mult_Ocommute,axiom,
% 9.07/5.16      ( times_times_real
% 9.07/5.16      = ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_610_mult_Ocommute,axiom,
% 9.07/5.16      ( times_times_rat
% 9.07/5.16      = ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_611_mult_Ocommute,axiom,
% 9.07/5.16      ( times_times_nat
% 9.07/5.16      = ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_612_mult_Ocommute,axiom,
% 9.07/5.16      ( times_times_int
% 9.07/5.16      = ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_613_mult_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
% 9.07/5.16        = ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_614_mult_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( times_times_rat @ B2 @ ( times_times_rat @ A @ C ) )
% 9.07/5.16        = ( times_times_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_615_mult_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.16        ( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
% 9.07/5.16        = ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_616_mult_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
% 9.07/5.16        = ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_617_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_618_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_619_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_620_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_621_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ( plus_plus_real @ I @ K )
% 9.07/5.16          = ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_622_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ( plus_plus_rat @ I @ K )
% 9.07/5.16          = ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_623_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ( plus_plus_nat @ I @ K )
% 9.07/5.16          = ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_624_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ( plus_plus_int @ I @ K )
% 9.07/5.16          = ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_625_group__cancel_Oadd1,axiom,
% 9.07/5.16      ! [A4: real,K: real,A: real,B2: real] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_real @ K @ A ) )
% 9.07/5.16       => ( ( plus_plus_real @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_626_group__cancel_Oadd1,axiom,
% 9.07/5.16      ! [A4: rat,K: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_rat @ K @ A ) )
% 9.07/5.16       => ( ( plus_plus_rat @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_627_group__cancel_Oadd1,axiom,
% 9.07/5.16      ! [A4: nat,K: nat,A: nat,B2: nat] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_nat @ K @ A ) )
% 9.07/5.16       => ( ( plus_plus_nat @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_628_group__cancel_Oadd1,axiom,
% 9.07/5.16      ! [A4: int,K: int,A: int,B2: int] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_int @ K @ A ) )
% 9.07/5.16       => ( ( plus_plus_int @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_629_group__cancel_Oadd2,axiom,
% 9.07/5.16      ! [B5: real,K: real,B2: real,A: real] :
% 9.07/5.16        ( ( B5
% 9.07/5.16          = ( plus_plus_real @ K @ B2 ) )
% 9.07/5.16       => ( ( plus_plus_real @ A @ B5 )
% 9.07/5.16          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_630_group__cancel_Oadd2,axiom,
% 9.07/5.16      ! [B5: rat,K: rat,B2: rat,A: rat] :
% 9.07/5.16        ( ( B5
% 9.07/5.16          = ( plus_plus_rat @ K @ B2 ) )
% 9.07/5.16       => ( ( plus_plus_rat @ A @ B5 )
% 9.07/5.16          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_631_group__cancel_Oadd2,axiom,
% 9.07/5.16      ! [B5: nat,K: nat,B2: nat,A: nat] :
% 9.07/5.16        ( ( B5
% 9.07/5.16          = ( plus_plus_nat @ K @ B2 ) )
% 9.07/5.16       => ( ( plus_plus_nat @ A @ B5 )
% 9.07/5.16          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_632_group__cancel_Oadd2,axiom,
% 9.07/5.16      ! [B5: int,K: int,B2: int,A: int] :
% 9.07/5.16        ( ( B5
% 9.07/5.16          = ( plus_plus_int @ K @ B2 ) )
% 9.07/5.16       => ( ( plus_plus_int @ A @ B5 )
% 9.07/5.16          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_633_add_Oassoc,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_634_add_Oassoc,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_635_add_Oassoc,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_636_add_Oassoc,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_637_add_Oleft__cancel,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ A @ B2 )
% 9.07/5.16          = ( plus_plus_real @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_638_add_Oleft__cancel,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ A @ B2 )
% 9.07/5.16          = ( plus_plus_rat @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_639_add_Oleft__cancel,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ A @ B2 )
% 9.07/5.16          = ( plus_plus_int @ A @ C ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_640_add_Oright__cancel,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ B2 @ A )
% 9.07/5.16          = ( plus_plus_real @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_641_add_Oright__cancel,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ B2 @ A )
% 9.07/5.16          = ( plus_plus_rat @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_642_add_Oright__cancel,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ B2 @ A )
% 9.07/5.16          = ( plus_plus_int @ C @ A ) )
% 9.07/5.16        = ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_643_add_Ocommute,axiom,
% 9.07/5.16      ( plus_plus_real
% 9.07/5.16      = ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_644_add_Ocommute,axiom,
% 9.07/5.16      ( plus_plus_rat
% 9.07/5.16      = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_645_add_Ocommute,axiom,
% 9.07/5.16      ( plus_plus_nat
% 9.07/5.16      = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_646_add_Ocommute,axiom,
% 9.07/5.16      ( plus_plus_int
% 9.07/5.16      = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_647_add_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
% 9.07/5.16        = ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_648_add_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A @ C ) )
% 9.07/5.16        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_649_add_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.16        ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
% 9.07/5.16        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_650_add_Oleft__commute,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
% 9.07/5.16        = ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_651_add__left__imp__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ A @ B2 )
% 9.07/5.16          = ( plus_plus_real @ A @ C ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_652_add__left__imp__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ A @ B2 )
% 9.07/5.16          = ( plus_plus_rat @ A @ C ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_653_add__left__imp__eq,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ( plus_plus_nat @ A @ B2 )
% 9.07/5.16          = ( plus_plus_nat @ A @ C ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_654_add__left__imp__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ A @ B2 )
% 9.07/5.16          = ( plus_plus_int @ A @ C ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_655_add__right__imp__eq,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ B2 @ A )
% 9.07/5.16          = ( plus_plus_real @ C @ A ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_656_add__right__imp__eq,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ B2 @ A )
% 9.07/5.16          = ( plus_plus_rat @ C @ A ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_657_add__right__imp__eq,axiom,
% 9.07/5.16      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.16        ( ( ( plus_plus_nat @ B2 @ A )
% 9.07/5.16          = ( plus_plus_nat @ C @ A ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_658_add__right__imp__eq,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ B2 @ A )
% 9.07/5.16          = ( plus_plus_int @ C @ A ) )
% 9.07/5.16       => ( B2 = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_659_diff__eq__diff__eq,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex,D: complex] :
% 9.07/5.16        ( ( ( minus_minus_complex @ A @ B2 )
% 9.07/5.16          = ( minus_minus_complex @ C @ D ) )
% 9.07/5.16       => ( ( A = B2 )
% 9.07/5.16          = ( C = D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_660_diff__eq__diff__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ( minus_minus_real @ A @ B2 )
% 9.07/5.16          = ( minus_minus_real @ C @ D ) )
% 9.07/5.16       => ( ( A = B2 )
% 9.07/5.16          = ( C = D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_661_diff__eq__diff__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ( minus_minus_rat @ A @ B2 )
% 9.07/5.16          = ( minus_minus_rat @ C @ D ) )
% 9.07/5.16       => ( ( A = B2 )
% 9.07/5.16          = ( C = D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_662_diff__eq__diff__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ( minus_minus_int @ A @ B2 )
% 9.07/5.16          = ( minus_minus_int @ C @ D ) )
% 9.07/5.16       => ( ( A = B2 )
% 9.07/5.16          = ( C = D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_663_diff__right__commute,axiom,
% 9.07/5.16      ! [A: complex,C: complex,B2: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B2 )
% 9.07/5.16        = ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_664_diff__right__commute,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
% 9.07/5.16        = ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_665_diff__right__commute,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 )
% 9.07/5.16        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_666_diff__right__commute,axiom,
% 9.07/5.16      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.16        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
% 9.07/5.16        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_667_diff__right__commute,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 )
% 9.07/5.16        = ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_668_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( ord_less_eq_real @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_669_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( ord_less_eq_rat @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_670_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_671_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( ord_less_eq_int @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_672_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_eq_real @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_673_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_eq_rat @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_674_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_eq_nat @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_675_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_eq_int @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_676_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( ord_less_eq_real @ I @ J )
% 9.07/5.16          & ( ord_less_eq_real @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_677_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( ord_less_eq_rat @ I @ J )
% 9.07/5.16          & ( ord_less_eq_rat @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_678_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16          & ( ord_less_eq_nat @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_679_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( ord_less_eq_int @ I @ J )
% 9.07/5.16          & ( ord_less_eq_int @ K @ L ) )
% 9.07/5.16       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_680_add__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_real @ C @ D )
% 9.07/5.16         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_681_add__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_rat @ C @ D )
% 9.07/5.16         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_682_add__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_nat @ C @ D )
% 9.07/5.16         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_683_add__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_int @ C @ D )
% 9.07/5.16         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_684_add__left__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_685_add__left__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_686_add__left__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_687_add__left__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_688_less__eqE,axiom,
% 9.07/5.16      ! [A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ~ ! [C2: nat] :
% 9.07/5.16              ( B2
% 9.07/5.16             != ( plus_plus_nat @ A @ C2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_689_add__right__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_690_add__right__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_691_add__right__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_692_add__right__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_693_le__iff__add,axiom,
% 9.07/5.16      ( ord_less_eq_nat
% 9.07/5.16      = ( ^ [A5: nat,B4: nat] :
% 9.07/5.16          ? [C3: nat] :
% 9.07/5.16            ( B4
% 9.07/5.16            = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_694_add__le__imp__le__left,axiom,
% 9.07/5.16      ! [C: real,A: real,B2: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_695_add__le__imp__le__left,axiom,
% 9.07/5.16      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_696_add__le__imp__le__left,axiom,
% 9.07/5.16      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_697_add__le__imp__le__left,axiom,
% 9.07/5.16      ! [C: int,A: int,B2: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_698_add__le__imp__le__right,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_699_add__le__imp__le__right,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_700_add__le__imp__le__right,axiom,
% 9.07/5.16      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_701_add__le__imp__le__right,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_702_diff__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,D: real,C: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_real @ D @ C )
% 9.07/5.16         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_703_diff__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,D: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_rat @ D @ C )
% 9.07/5.16         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_704_diff__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,D: int,C: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_int @ D @ C )
% 9.07/5.16         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_705_diff__left__mono,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ B2 @ A )
% 9.07/5.16       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_706_diff__left__mono,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ B2 @ A )
% 9.07/5.16       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_707_diff__left__mono,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.16       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_708_diff__right__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_709_diff__right__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_710_diff__right__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_711_diff__eq__diff__less__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ( minus_minus_real @ A @ B2 )
% 9.07/5.16          = ( minus_minus_real @ C @ D ) )
% 9.07/5.16       => ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_712_diff__eq__diff__less__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ( minus_minus_rat @ A @ B2 )
% 9.07/5.16          = ( minus_minus_rat @ C @ D ) )
% 9.07/5.16       => ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_713_diff__eq__diff__less__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ( minus_minus_int @ A @ B2 )
% 9.07/5.16          = ( minus_minus_int @ C @ D ) )
% 9.07/5.16       => ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_714_add__mono__thms__linordered__field_I5_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( ord_less_real @ I @ J )
% 9.07/5.16          & ( ord_less_real @ K @ L ) )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_715_add__mono__thms__linordered__field_I5_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( ord_less_rat @ I @ J )
% 9.07/5.16          & ( ord_less_rat @ K @ L ) )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_716_add__mono__thms__linordered__field_I5_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( ord_less_nat @ I @ J )
% 9.07/5.16          & ( ord_less_nat @ K @ L ) )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_717_add__mono__thms__linordered__field_I5_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( ord_less_int @ I @ J )
% 9.07/5.16          & ( ord_less_int @ K @ L ) )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_718_add__mono__thms__linordered__field_I2_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_real @ K @ L ) )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_719_add__mono__thms__linordered__field_I2_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_rat @ K @ L ) )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_720_add__mono__thms__linordered__field_I2_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_nat @ K @ L ) )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_721_add__mono__thms__linordered__field_I2_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( I = J )
% 9.07/5.16          & ( ord_less_int @ K @ L ) )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_722_add__mono__thms__linordered__field_I1_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( ord_less_real @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_723_add__mono__thms__linordered__field_I1_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( ord_less_rat @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_724_add__mono__thms__linordered__field_I1_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( ord_less_nat @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_725_add__mono__thms__linordered__field_I1_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( ord_less_int @ I @ J )
% 9.07/5.16          & ( K = L ) )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_726_add__strict__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_real @ C @ D )
% 9.07/5.16         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_727_add__strict__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_rat @ C @ D )
% 9.07/5.16         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_728_add__strict__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.16        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_nat @ C @ D )
% 9.07/5.16         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_729_add__strict__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_int @ C @ D )
% 9.07/5.16         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_730_add__strict__left__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ B2 )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_731_add__strict__left__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_732_add__strict__left__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_733_add__strict__left__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ B2 )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_734_add__strict__right__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ B2 )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_735_add__strict__right__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_736_add__strict__right__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_737_add__strict__right__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ B2 )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_738_add__less__imp__less__left,axiom,
% 9.07/5.16      ! [C: real,A: real,B2: real] :
% 9.07/5.16        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_739_add__less__imp__less__left,axiom,
% 9.07/5.16      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_740_add__less__imp__less__left,axiom,
% 9.07/5.16      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_741_add__less__imp__less__left,axiom,
% 9.07/5.16      ! [C: int,A: int,B2: int] :
% 9.07/5.16        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
% 9.07/5.16       => ( ord_less_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_742_add__less__imp__less__right,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_real @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_743_add__less__imp__less__right,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_rat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_744_add__less__imp__less__right,axiom,
% 9.07/5.16      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_nat @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_745_add__less__imp__less__right,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.16       => ( ord_less_int @ A @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_746_diff__strict__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,D: real,C: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_real @ D @ C )
% 9.07/5.16         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_747_diff__strict__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,D: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_rat @ D @ C )
% 9.07/5.16         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_748_diff__strict__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,D: int,C: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_int @ D @ C )
% 9.07/5.16         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_749_diff__eq__diff__less,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ( minus_minus_real @ A @ B2 )
% 9.07/5.16          = ( minus_minus_real @ C @ D ) )
% 9.07/5.16       => ( ( ord_less_real @ A @ B2 )
% 9.07/5.16          = ( ord_less_real @ C @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_750_diff__eq__diff__less,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ( minus_minus_rat @ A @ B2 )
% 9.07/5.16          = ( minus_minus_rat @ C @ D ) )
% 9.07/5.16       => ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16          = ( ord_less_rat @ C @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_751_diff__eq__diff__less,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ( minus_minus_int @ A @ B2 )
% 9.07/5.16          = ( minus_minus_int @ C @ D ) )
% 9.07/5.16       => ( ( ord_less_int @ A @ B2 )
% 9.07/5.16          = ( ord_less_int @ C @ D ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_752_diff__strict__left__mono,axiom,
% 9.07/5.16      ! [B2: real,A: real,C: real] :
% 9.07/5.16        ( ( ord_less_real @ B2 @ A )
% 9.07/5.16       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_753_diff__strict__left__mono,axiom,
% 9.07/5.16      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_rat @ B2 @ A )
% 9.07/5.16       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_754_diff__strict__left__mono,axiom,
% 9.07/5.16      ! [B2: int,A: int,C: int] :
% 9.07/5.16        ( ( ord_less_int @ B2 @ A )
% 9.07/5.16       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_755_diff__strict__right__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ B2 )
% 9.07/5.16       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_756_diff__strict__right__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_757_diff__strict__right__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ B2 )
% 9.07/5.16       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_758_group__cancel_Osub1,axiom,
% 9.07/5.16      ! [A4: complex,K: complex,A: complex,B2: complex] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_complex @ K @ A ) )
% 9.07/5.16       => ( ( minus_minus_complex @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_complex @ K @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_759_group__cancel_Osub1,axiom,
% 9.07/5.16      ! [A4: real,K: real,A: real,B2: real] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_real @ K @ A ) )
% 9.07/5.16       => ( ( minus_minus_real @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_760_group__cancel_Osub1,axiom,
% 9.07/5.16      ! [A4: rat,K: rat,A: rat,B2: rat] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_rat @ K @ A ) )
% 9.07/5.16       => ( ( minus_minus_rat @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_761_group__cancel_Osub1,axiom,
% 9.07/5.16      ! [A4: int,K: int,A: int,B2: int] :
% 9.07/5.16        ( ( A4
% 9.07/5.16          = ( plus_plus_int @ K @ A ) )
% 9.07/5.16       => ( ( minus_minus_int @ A4 @ B2 )
% 9.07/5.16          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_762_diff__eq__eq,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( ( minus_minus_complex @ A @ B2 )
% 9.07/5.16          = C )
% 9.07/5.16        = ( A
% 9.07/5.16          = ( plus_plus_complex @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_763_diff__eq__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ( minus_minus_real @ A @ B2 )
% 9.07/5.16          = C )
% 9.07/5.16        = ( A
% 9.07/5.16          = ( plus_plus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_764_diff__eq__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ( minus_minus_rat @ A @ B2 )
% 9.07/5.16          = C )
% 9.07/5.16        = ( A
% 9.07/5.16          = ( plus_plus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_765_diff__eq__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ( minus_minus_int @ A @ B2 )
% 9.07/5.16          = C )
% 9.07/5.16        = ( A
% 9.07/5.16          = ( plus_plus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_766_eq__diff__eq,axiom,
% 9.07/5.16      ! [A: complex,C: complex,B2: complex] :
% 9.07/5.16        ( ( A
% 9.07/5.16          = ( minus_minus_complex @ C @ B2 ) )
% 9.07/5.16        = ( ( plus_plus_complex @ A @ B2 )
% 9.07/5.16          = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_767_eq__diff__eq,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( A
% 9.07/5.16          = ( minus_minus_real @ C @ B2 ) )
% 9.07/5.16        = ( ( plus_plus_real @ A @ B2 )
% 9.07/5.16          = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_768_eq__diff__eq,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( A
% 9.07/5.16          = ( minus_minus_rat @ C @ B2 ) )
% 9.07/5.16        = ( ( plus_plus_rat @ A @ B2 )
% 9.07/5.16          = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_769_eq__diff__eq,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( A
% 9.07/5.16          = ( minus_minus_int @ C @ B2 ) )
% 9.07/5.16        = ( ( plus_plus_int @ A @ B2 )
% 9.07/5.16          = C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_770_add__diff__eq,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( plus_plus_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_771_add__diff__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_772_add__diff__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_773_add__diff__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( plus_plus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_774_diff__diff__eq2,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ A @ ( minus_minus_complex @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_775_diff__diff__eq2,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_776_diff__diff__eq2,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_777_diff__diff__eq2,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( minus_minus_int @ A @ ( minus_minus_int @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_778_diff__add__eq,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( plus_plus_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_complex @ ( plus_plus_complex @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_779_diff__add__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_780_diff__add__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_781_diff__add__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( plus_plus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_782_diff__add__eq__diff__diff__swap,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ A @ ( plus_plus_complex @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_complex @ ( minus_minus_complex @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_783_diff__add__eq__diff__diff__swap,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_784_diff__add__eq__diff__diff__swap,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_785_diff__add__eq__diff__diff__swap,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 9.07/5.16        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_786_add__implies__diff,axiom,
% 9.07/5.16      ! [C: complex,B2: complex,A: complex] :
% 9.07/5.16        ( ( ( plus_plus_complex @ C @ B2 )
% 9.07/5.16          = A )
% 9.07/5.16       => ( C
% 9.07/5.16          = ( minus_minus_complex @ A @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_787_add__implies__diff,axiom,
% 9.07/5.16      ! [C: real,B2: real,A: real] :
% 9.07/5.16        ( ( ( plus_plus_real @ C @ B2 )
% 9.07/5.16          = A )
% 9.07/5.16       => ( C
% 9.07/5.16          = ( minus_minus_real @ A @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_788_add__implies__diff,axiom,
% 9.07/5.16      ! [C: rat,B2: rat,A: rat] :
% 9.07/5.16        ( ( ( plus_plus_rat @ C @ B2 )
% 9.07/5.16          = A )
% 9.07/5.16       => ( C
% 9.07/5.16          = ( minus_minus_rat @ A @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_789_add__implies__diff,axiom,
% 9.07/5.16      ! [C: nat,B2: nat,A: nat] :
% 9.07/5.16        ( ( ( plus_plus_nat @ C @ B2 )
% 9.07/5.16          = A )
% 9.07/5.16       => ( C
% 9.07/5.16          = ( minus_minus_nat @ A @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_790_add__implies__diff,axiom,
% 9.07/5.16      ! [C: int,B2: int,A: int] :
% 9.07/5.16        ( ( ( plus_plus_int @ C @ B2 )
% 9.07/5.16          = A )
% 9.07/5.16       => ( C
% 9.07/5.16          = ( minus_minus_int @ A @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_791_diff__diff__eq,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_complex @ A @ ( plus_plus_complex @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_792_diff__diff__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_793_diff__diff__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_794_diff__diff__eq,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_795_diff__diff__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( minus_minus_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( minus_minus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_796_add__mono__thms__linordered__field_I4_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( ord_less_eq_real @ I @ J )
% 9.07/5.16          & ( ord_less_real @ K @ L ) )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_797_add__mono__thms__linordered__field_I4_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( ord_less_eq_rat @ I @ J )
% 9.07/5.16          & ( ord_less_rat @ K @ L ) )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_798_add__mono__thms__linordered__field_I4_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.16          & ( ord_less_nat @ K @ L ) )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_799_add__mono__thms__linordered__field_I4_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( ord_less_eq_int @ I @ J )
% 9.07/5.16          & ( ord_less_int @ K @ L ) )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_800_add__mono__thms__linordered__field_I3_J,axiom,
% 9.07/5.16      ! [I: real,J: real,K: real,L: real] :
% 9.07/5.16        ( ( ( ord_less_real @ I @ J )
% 9.07/5.16          & ( ord_less_eq_real @ K @ L ) )
% 9.07/5.16       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_801_add__mono__thms__linordered__field_I3_J,axiom,
% 9.07/5.16      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.07/5.16        ( ( ( ord_less_rat @ I @ J )
% 9.07/5.16          & ( ord_less_eq_rat @ K @ L ) )
% 9.07/5.16       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_802_add__mono__thms__linordered__field_I3_J,axiom,
% 9.07/5.16      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.07/5.16        ( ( ( ord_less_nat @ I @ J )
% 9.07/5.16          & ( ord_less_eq_nat @ K @ L ) )
% 9.07/5.16       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_803_add__mono__thms__linordered__field_I3_J,axiom,
% 9.07/5.16      ! [I: int,J: int,K: int,L: int] :
% 9.07/5.16        ( ( ( ord_less_int @ I @ J )
% 9.07/5.16          & ( ord_less_eq_int @ K @ L ) )
% 9.07/5.16       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_804_add__le__less__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_real @ C @ D )
% 9.07/5.16         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_805_add__le__less__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_rat @ C @ D )
% 9.07/5.16         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_806_add__le__less__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_nat @ C @ D )
% 9.07/5.16         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_807_add__le__less__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_int @ C @ D )
% 9.07/5.16         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_808_add__less__le__mono,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_real @ C @ D )
% 9.07/5.16         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_809_add__less__le__mono,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_rat @ C @ D )
% 9.07/5.16         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_810_add__less__le__mono,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.16        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_nat @ C @ D )
% 9.07/5.16         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_811_add__less__le__mono,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_int @ C @ D )
% 9.07/5.16         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_812_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16         => ( ( ( minus_minus_nat @ B2 @ A )
% 9.07/5.16              = C )
% 9.07/5.16            = ( B2
% 9.07/5.16              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_813_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 9.07/5.16      ! [A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
% 9.07/5.16          = B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_814_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 9.07/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_815_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
% 9.07/5.16          = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_816_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
% 9.07/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_817_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
% 9.07/5.16          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_818_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 9.07/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_819_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
% 9.07/5.16          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_820_le__add__diff,axiom,
% 9.07/5.16      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_821_diff__add,axiom,
% 9.07/5.16      ! [A: nat,B2: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.16       => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
% 9.07/5.16          = B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_822_le__diff__eq,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_823_le__diff__eq,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_824_le__diff__eq,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_825_diff__le__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_826_diff__le__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_827_diff__le__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_828_less__diff__eq,axiom,
% 9.07/5.16      ! [A: real,C: real,B2: real] :
% 9.07/5.16        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_829_less__diff__eq,axiom,
% 9.07/5.16      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.16        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_830_less__diff__eq,axiom,
% 9.07/5.16      ! [A: int,C: int,B2: int] :
% 9.07/5.16        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 9.07/5.16        = ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_831_diff__less__eq,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( ord_less_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_832_diff__less__eq,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_833_diff__less__eq,axiom,
% 9.07/5.16      ! [A: int,B2: int,C: int] :
% 9.07/5.16        ( ( ord_less_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.07/5.16        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_834__C2_C,axiom,
% 9.07/5.16      ( ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.16          = none_nat )
% 9.07/5.16       => ( ( ( ord_less_nat @ mi @ xa )
% 9.07/5.16           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 9.07/5.16              = ( some_nat @ mi ) ) )
% 9.07/5.16          & ( ~ ( ord_less_nat @ mi @ xa )
% 9.07/5.16           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 9.07/5.16              = none_nat ) ) ) )
% 9.07/5.16      & ( ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.16         != none_nat )
% 9.07/5.16       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 9.07/5.16          = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_835_invar__vebt_Ointros_I4_J,axiom,
% 9.07/5.16      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.07/5.16        ( ! [X5: vEBT_VEBT] :
% 9.07/5.16            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.16           => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.07/5.16       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.07/5.16         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.07/5.16              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.16           => ( ( M = N )
% 9.07/5.16             => ( ( Deg
% 9.07/5.16                  = ( plus_plus_nat @ N @ M ) )
% 9.07/5.16               => ( ! [I3: nat] :
% 9.07/5.16                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.16                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
% 9.07/5.16                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 9.07/5.16                 => ( ( ( Mi = Ma )
% 9.07/5.16                     => ! [X5: vEBT_VEBT] :
% 9.07/5.16                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.16                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 9.07/5.16                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.07/5.16                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.07/5.16                       => ( ( ( Mi != Ma )
% 9.07/5.16                           => ! [I3: nat] :
% 9.07/5.16                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.16                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.07/5.16                                      = I3 )
% 9.07/5.16                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.07/5.16                                  & ! [X5: nat] :
% 9.07/5.16                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 9.07/5.16                                          = I3 )
% 9.07/5.16                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 9.07/5.16                                     => ( ( ord_less_nat @ Mi @ X5 )
% 9.07/5.16                                        & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 9.07/5.16                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_836_both__member__options__from__chilf__to__complete__tree,axiom,
% 9.07/5.16      ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.07/5.16        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.16       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.07/5.16         => ( ( 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 ) ) ) ) )
% 9.07/5.16           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_837_both__member__options__from__complete__tree__to__child,axiom,
% 9.07/5.16      ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.07/5.16        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.07/5.16       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.16         => ( ( 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 ) ) ) ) )
% 9.07/5.16            | ( X3 = Mi )
% 9.07/5.16            | ( X3 = Ma ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_838_invar__vebt_Ointros_I2_J,axiom,
% 9.07/5.16      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.07/5.16        ( ! [X5: vEBT_VEBT] :
% 9.07/5.16            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.16           => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.07/5.16       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.07/5.16         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.07/5.16              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.16           => ( ( M = N )
% 9.07/5.16             => ( ( Deg
% 9.07/5.16                  = ( plus_plus_nat @ N @ M ) )
% 9.07/5.16               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.07/5.16                 => ( ! [X5: vEBT_VEBT] :
% 9.07/5.16                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.16                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 9.07/5.16                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_839_summaxma,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.07/5.16        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 9.07/5.16       => ( ( Mi != Ma )
% 9.07/5.16         => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 9.07/5.16            = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_840_set__n__deg__not__0,axiom,
% 9.07/5.16      ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 9.07/5.16        ( ! [X5: vEBT_VEBT] :
% 9.07/5.16            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.16           => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.07/5.16       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.07/5.16            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.16         => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_841_vebt__maxt_Osimps_I3_J,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.07/5.16        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.07/5.16        = ( some_nat @ Ma ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_842_vebt__mint_Osimps_I3_J,axiom,
% 9.07/5.16      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.07/5.16        ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.07/5.16        = ( some_nat @ Mi ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_843_times__divide__eq__right,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.07/5.16        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_844_times__divide__eq__right,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( times_times_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.07/5.16        = ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_845_times__divide__eq__right,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( times_times_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.16        = ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_846_divide__divide__eq__right,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.07/5.16        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_847_divide__divide__eq__right,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( divide_divide_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.07/5.16        = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_848_divide__divide__eq__right,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.16        = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_849__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
% 9.07/5.16      ord_less_eq_nat @ one_one_nat @ na ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_850_real__divide__square__eq,axiom,
% 9.07/5.16      ! [R: real,A: real] :
% 9.07/5.16        ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 9.07/5.16        = ( divide_divide_real @ A @ R ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_851_VEBT_Oinject_I1_J,axiom,
% 9.07/5.16      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 9.07/5.16        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.07/5.16          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 9.07/5.16        = ( ( X11 = Y11 )
% 9.07/5.16          & ( X12 = Y12 )
% 9.07/5.16          & ( X13 = Y13 )
% 9.07/5.16          & ( X14 = Y14 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_852_mult__1,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( times_times_complex @ one_one_complex @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_853_mult__1,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( times_times_real @ one_one_real @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_854_mult__1,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( times_times_rat @ one_one_rat @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_855_mult__1,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( times_times_nat @ one_one_nat @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_856_mult__1,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( times_times_int @ one_one_int @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_857_mult_Oright__neutral,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( times_times_complex @ A @ one_one_complex )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_858_mult_Oright__neutral,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( times_times_real @ A @ one_one_real )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_859_mult_Oright__neutral,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( times_times_rat @ A @ one_one_rat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_860_mult_Oright__neutral,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( times_times_nat @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_861_mult_Oright__neutral,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( times_times_int @ A @ one_one_int )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_862_times__divide__eq__left,axiom,
% 9.07/5.16      ! [B2: complex,C: complex,A: complex] :
% 9.07/5.16        ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C ) @ A )
% 9.07/5.16        = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_863_times__divide__eq__left,axiom,
% 9.07/5.16      ! [B2: real,C: real,A: real] :
% 9.07/5.16        ( ( times_times_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.07/5.16        = ( divide_divide_real @ ( times_times_real @ B2 @ A ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_864_times__divide__eq__left,axiom,
% 9.07/5.16      ! [B2: rat,C: rat,A: rat] :
% 9.07/5.16        ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.07/5.16        = ( divide_divide_rat @ ( times_times_rat @ B2 @ A ) @ C ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_865_divide__divide__eq__left,axiom,
% 9.07/5.16      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.16        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 9.07/5.16        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_866_divide__divide__eq__left,axiom,
% 9.07/5.16      ! [A: real,B2: real,C: real] :
% 9.07/5.16        ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 9.07/5.16        = ( divide_divide_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_867_divide__divide__eq__left,axiom,
% 9.07/5.16      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.16        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 9.07/5.16        = ( divide_divide_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_868_div__by__1,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_869_div__by__1,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( divide_divide_real @ A @ one_one_real )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_870_div__by__1,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( divide_divide_rat @ A @ one_one_rat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_871_div__by__1,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( divide_divide_nat @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_872_div__by__1,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( divide_divide_int @ A @ one_one_int )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_873_bits__div__by__1,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( divide_divide_nat @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_874_bits__div__by__1,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( divide_divide_int @ A @ one_one_int )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_875_power__one,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( power_power_rat @ one_one_rat @ N )
% 9.07/5.16        = one_one_rat ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_876_power__one,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( power_power_nat @ one_one_nat @ N )
% 9.07/5.16        = one_one_nat ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_877_power__one,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( power_power_real @ one_one_real @ N )
% 9.07/5.16        = one_one_real ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_878_power__one,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( power_power_int @ one_one_int @ N )
% 9.07/5.16        = one_one_int ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_879_power__one,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( power_power_complex @ one_one_complex @ N )
% 9.07/5.16        = one_one_complex ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_880_power__one__right,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( power_power_nat @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_881_power__one__right,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( power_power_real @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_882_power__one__right,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( power_power_int @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_883_power__one__right,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( power_power_complex @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_884_nat__1__eq__mult__iff,axiom,
% 9.07/5.16      ! [M: nat,N: nat] :
% 9.07/5.16        ( ( one_one_nat
% 9.07/5.16          = ( times_times_nat @ M @ N ) )
% 9.07/5.16        = ( ( M = one_one_nat )
% 9.07/5.16          & ( N = one_one_nat ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_885_nat__mult__eq__1__iff,axiom,
% 9.07/5.16      ! [M: nat,N: nat] :
% 9.07/5.16        ( ( ( times_times_nat @ M @ N )
% 9.07/5.16          = one_one_nat )
% 9.07/5.16        = ( ( M = one_one_nat )
% 9.07/5.16          & ( N = one_one_nat ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_886_numeral__eq__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ( numera6690914467698888265omplex @ N )
% 9.07/5.16          = one_one_complex )
% 9.07/5.16        = ( N = one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_887_numeral__eq__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ( numeral_numeral_real @ N )
% 9.07/5.16          = one_one_real )
% 9.07/5.16        = ( N = one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_888_numeral__eq__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ( numeral_numeral_rat @ N )
% 9.07/5.16          = one_one_rat )
% 9.07/5.16        = ( N = one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_889_numeral__eq__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ( numeral_numeral_nat @ N )
% 9.07/5.16          = one_one_nat )
% 9.07/5.16        = ( N = one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_890_numeral__eq__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ( numeral_numeral_int @ N )
% 9.07/5.16          = one_one_int )
% 9.07/5.16        = ( N = one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_891_one__eq__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( one_one_complex
% 9.07/5.16          = ( numera6690914467698888265omplex @ N ) )
% 9.07/5.16        = ( one = N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_892_one__eq__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( one_one_real
% 9.07/5.16          = ( numeral_numeral_real @ N ) )
% 9.07/5.16        = ( one = N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_893_one__eq__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( one_one_rat
% 9.07/5.16          = ( numeral_numeral_rat @ N ) )
% 9.07/5.16        = ( one = N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_894_one__eq__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( one_one_nat
% 9.07/5.16          = ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( one = N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_895_one__eq__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( one_one_int
% 9.07/5.16          = ( numeral_numeral_int @ N ) )
% 9.07/5.16        = ( one = N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_896_power__inject__exp,axiom,
% 9.07/5.16      ! [A: real,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.16       => ( ( ( power_power_real @ A @ M )
% 9.07/5.16            = ( power_power_real @ A @ N ) )
% 9.07/5.16          = ( M = N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_897_power__inject__exp,axiom,
% 9.07/5.16      ! [A: rat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.16       => ( ( ( power_power_rat @ A @ M )
% 9.07/5.16            = ( power_power_rat @ A @ N ) )
% 9.07/5.16          = ( M = N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_898_power__inject__exp,axiom,
% 9.07/5.16      ! [A: nat,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.16       => ( ( ( power_power_nat @ A @ M )
% 9.07/5.16            = ( power_power_nat @ A @ N ) )
% 9.07/5.16          = ( M = N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_899_power__inject__exp,axiom,
% 9.07/5.16      ! [A: int,M: nat,N: nat] :
% 9.07/5.16        ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.16       => ( ( ( power_power_int @ A @ M )
% 9.07/5.16            = ( power_power_int @ A @ N ) )
% 9.07/5.16          = ( M = N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_900_option_Ocollapse,axiom,
% 9.07/5.16      ! [Option: option_nat] :
% 9.07/5.16        ( ( Option != none_nat )
% 9.07/5.16       => ( ( some_nat @ ( the_nat @ Option ) )
% 9.07/5.16          = Option ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_901_option_Ocollapse,axiom,
% 9.07/5.16      ! [Option: option4927543243414619207at_nat] :
% 9.07/5.16        ( ( Option != none_P5556105721700978146at_nat )
% 9.07/5.16       => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 9.07/5.16          = Option ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_902_option_Ocollapse,axiom,
% 9.07/5.16      ! [Option: option_num] :
% 9.07/5.16        ( ( Option != none_num )
% 9.07/5.16       => ( ( some_num @ ( the_num @ Option ) )
% 9.07/5.16          = Option ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_903_power__strict__increasing__iff,axiom,
% 9.07/5.16      ! [B2: real,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_real @ one_one_real @ B2 )
% 9.07/5.16       => ( ( ord_less_real @ ( power_power_real @ B2 @ X3 ) @ ( power_power_real @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_904_power__strict__increasing__iff,axiom,
% 9.07/5.16      ! [B2: rat,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_rat @ one_one_rat @ B2 )
% 9.07/5.16       => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X3 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_905_power__strict__increasing__iff,axiom,
% 9.07/5.16      ! [B2: nat,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_nat @ one_one_nat @ B2 )
% 9.07/5.16       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X3 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_906_power__strict__increasing__iff,axiom,
% 9.07/5.16      ! [B2: int,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_int @ one_one_int @ B2 )
% 9.07/5.16       => ( ( ord_less_int @ ( power_power_int @ B2 @ X3 ) @ ( power_power_int @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_907_one__add__one,axiom,
% 9.07/5.16      ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 9.07/5.16      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_908_one__add__one,axiom,
% 9.07/5.16      ( ( plus_plus_real @ one_one_real @ one_one_real )
% 9.07/5.16      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_909_one__add__one,axiom,
% 9.07/5.16      ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 9.07/5.16      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_910_one__add__one,axiom,
% 9.07/5.16      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 9.07/5.16      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_911_one__add__one,axiom,
% 9.07/5.16      ( ( plus_plus_int @ one_one_int @ one_one_int )
% 9.07/5.16      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_912_power__increasing__iff,axiom,
% 9.07/5.16      ! [B2: real,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_real @ one_one_real @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X3 ) @ ( power_power_real @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_913_power__increasing__iff,axiom,
% 9.07/5.16      ! [B2: rat,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_rat @ one_one_rat @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X3 ) @ ( power_power_rat @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_914_power__increasing__iff,axiom,
% 9.07/5.16      ! [B2: nat,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_nat @ one_one_nat @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X3 ) @ ( power_power_nat @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_915_power__increasing__iff,axiom,
% 9.07/5.16      ! [B2: int,X3: nat,Y3: nat] :
% 9.07/5.16        ( ( ord_less_int @ one_one_int @ B2 )
% 9.07/5.16       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X3 ) @ ( power_power_int @ B2 @ Y3 ) )
% 9.07/5.16          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_916_numeral__plus__one,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 9.07/5.16        = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_917_numeral__plus__one,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 9.07/5.16        = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_918_numeral__plus__one,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 9.07/5.16        = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_919_numeral__plus__one,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 9.07/5.16        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_920_numeral__plus__one,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 9.07/5.16        = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_921_one__plus__numeral,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 9.07/5.16        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_922_one__plus__numeral,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 9.07/5.16        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_923_one__plus__numeral,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 9.07/5.16        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_924_one__plus__numeral,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_925_one__plus__numeral,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 9.07/5.16        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_926_numeral__le__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 9.07/5.16        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_927_numeral__le__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 9.07/5.16        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_928_numeral__le__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 9.07/5.16        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_929_numeral__le__one__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 9.07/5.16        = ( ord_less_eq_num @ N @ one ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_930_one__less__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 9.07/5.16        = ( ord_less_num @ one @ N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_931_one__less__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 9.07/5.16        = ( ord_less_num @ one @ N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_932_one__less__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 9.07/5.16        = ( ord_less_num @ one @ N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_933_one__less__numeral__iff,axiom,
% 9.07/5.16      ! [N: num] :
% 9.07/5.16        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 9.07/5.16        = ( ord_less_num @ one @ N ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_934_one__reorient,axiom,
% 9.07/5.16      ! [X3: complex] :
% 9.07/5.16        ( ( one_one_complex = X3 )
% 9.07/5.16        = ( X3 = one_one_complex ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_935_one__reorient,axiom,
% 9.07/5.16      ! [X3: real] :
% 9.07/5.16        ( ( one_one_real = X3 )
% 9.07/5.16        = ( X3 = one_one_real ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_936_one__reorient,axiom,
% 9.07/5.16      ! [X3: rat] :
% 9.07/5.16        ( ( one_one_rat = X3 )
% 9.07/5.16        = ( X3 = one_one_rat ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_937_one__reorient,axiom,
% 9.07/5.16      ! [X3: nat] :
% 9.07/5.16        ( ( one_one_nat = X3 )
% 9.07/5.16        = ( X3 = one_one_nat ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_938_one__reorient,axiom,
% 9.07/5.16      ! [X3: int] :
% 9.07/5.16        ( ( one_one_int = X3 )
% 9.07/5.16        = ( X3 = one_one_int ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_939_less__eq__real__def,axiom,
% 9.07/5.16      ( ord_less_eq_real
% 9.07/5.16      = ( ^ [X: real,Y: real] :
% 9.07/5.16            ( ( ord_less_real @ X @ Y )
% 9.07/5.16            | ( X = Y ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_940_complete__real,axiom,
% 9.07/5.16      ! [S2: set_real] :
% 9.07/5.16        ( ? [X6: real] : ( member_real @ X6 @ S2 )
% 9.07/5.16       => ( ? [Z3: real] :
% 9.07/5.16            ! [X5: real] :
% 9.07/5.16              ( ( member_real @ X5 @ S2 )
% 9.07/5.16             => ( ord_less_eq_real @ X5 @ Z3 ) )
% 9.07/5.16         => ? [Y4: real] :
% 9.07/5.16              ( ! [X6: real] :
% 9.07/5.16                  ( ( member_real @ X6 @ S2 )
% 9.07/5.16                 => ( ord_less_eq_real @ X6 @ Y4 ) )
% 9.07/5.16              & ! [Z3: real] :
% 9.07/5.16                  ( ! [X5: real] :
% 9.07/5.16                      ( ( member_real @ X5 @ S2 )
% 9.07/5.16                     => ( ord_less_eq_real @ X5 @ Z3 ) )
% 9.07/5.16                 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_941_le__numeral__extra_I4_J,axiom,
% 9.07/5.16      ord_less_eq_real @ one_one_real @ one_one_real ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_942_le__numeral__extra_I4_J,axiom,
% 9.07/5.16      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_943_le__numeral__extra_I4_J,axiom,
% 9.07/5.16      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_944_le__numeral__extra_I4_J,axiom,
% 9.07/5.16      ord_less_eq_int @ one_one_int @ one_one_int ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_945_less__numeral__extra_I4_J,axiom,
% 9.07/5.16      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_946_less__numeral__extra_I4_J,axiom,
% 9.07/5.16      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_947_less__numeral__extra_I4_J,axiom,
% 9.07/5.16      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_948_less__numeral__extra_I4_J,axiom,
% 9.07/5.16      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_949_mult_Ocomm__neutral,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( times_times_complex @ A @ one_one_complex )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_950_mult_Ocomm__neutral,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( times_times_real @ A @ one_one_real )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_951_mult_Ocomm__neutral,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( times_times_rat @ A @ one_one_rat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_952_mult_Ocomm__neutral,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( times_times_nat @ A @ one_one_nat )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_953_mult_Ocomm__neutral,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( times_times_int @ A @ one_one_int )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_954_comm__monoid__mult__class_Omult__1,axiom,
% 9.07/5.16      ! [A: complex] :
% 9.07/5.16        ( ( times_times_complex @ one_one_complex @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_955_comm__monoid__mult__class_Omult__1,axiom,
% 9.07/5.16      ! [A: real] :
% 9.07/5.16        ( ( times_times_real @ one_one_real @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_956_comm__monoid__mult__class_Omult__1,axiom,
% 9.07/5.16      ! [A: rat] :
% 9.07/5.16        ( ( times_times_rat @ one_one_rat @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_957_comm__monoid__mult__class_Omult__1,axiom,
% 9.07/5.16      ! [A: nat] :
% 9.07/5.16        ( ( times_times_nat @ one_one_nat @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_958_comm__monoid__mult__class_Omult__1,axiom,
% 9.07/5.16      ! [A: int] :
% 9.07/5.16        ( ( times_times_int @ one_one_int @ A )
% 9.07/5.16        = A ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_959_nat__mult__1,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( times_times_nat @ one_one_nat @ N )
% 9.07/5.16        = N ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_960_nat__mult__1__right,axiom,
% 9.07/5.16      ! [N: nat] :
% 9.07/5.16        ( ( times_times_nat @ N @ one_one_nat )
% 9.07/5.16        = N ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_961_option_Osel,axiom,
% 9.07/5.16      ! [X2: nat] :
% 9.07/5.16        ( ( the_nat @ ( some_nat @ X2 ) )
% 9.07/5.16        = X2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_962_option_Osel,axiom,
% 9.07/5.16      ! [X2: product_prod_nat_nat] :
% 9.07/5.16        ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
% 9.07/5.16        = X2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_963_option_Osel,axiom,
% 9.07/5.16      ! [X2: num] :
% 9.07/5.16        ( ( the_num @ ( some_num @ X2 ) )
% 9.07/5.16        = X2 ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_964_option_Oexpand,axiom,
% 9.07/5.16      ! [Option: option_nat,Option2: option_nat] :
% 9.07/5.16        ( ( ( Option = none_nat )
% 9.07/5.16          = ( Option2 = none_nat ) )
% 9.07/5.16       => ( ( ( Option != none_nat )
% 9.07/5.16           => ( ( Option2 != none_nat )
% 9.07/5.16             => ( ( the_nat @ Option )
% 9.07/5.16                = ( the_nat @ Option2 ) ) ) )
% 9.07/5.16         => ( Option = Option2 ) ) ) ).
% 9.07/5.16  
% 9.07/5.16  thf(fact_965_option_Oexpand,axiom,
% 9.07/5.16      ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 9.07/5.17        ( ( ( Option = none_P5556105721700978146at_nat )
% 9.07/5.17          = ( Option2 = none_P5556105721700978146at_nat ) )
% 9.07/5.17       => ( ( ( Option != none_P5556105721700978146at_nat )
% 9.07/5.17           => ( ( Option2 != none_P5556105721700978146at_nat )
% 9.07/5.17             => ( ( the_Pr8591224930841456533at_nat @ Option )
% 9.07/5.17                = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 9.07/5.17         => ( Option = Option2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_966_option_Oexpand,axiom,
% 9.07/5.17      ! [Option: option_num,Option2: option_num] :
% 9.07/5.17        ( ( ( Option = none_num )
% 9.07/5.17          = ( Option2 = none_num ) )
% 9.07/5.17       => ( ( ( Option != none_num )
% 9.07/5.17           => ( ( Option2 != none_num )
% 9.07/5.17             => ( ( the_num @ Option )
% 9.07/5.17                = ( the_num @ Option2 ) ) ) )
% 9.07/5.17         => ( Option = Option2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_967_less__half__sum,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( ord_less_real @ A @ B2 )
% 9.07/5.17       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_968_less__half__sum,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.17       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_969_gt__half__sum,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( ord_less_real @ A @ B2 )
% 9.07/5.17       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_970_gt__half__sum,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.17       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_971_one__le__numeral,axiom,
% 9.07/5.17      ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_972_one__le__numeral,axiom,
% 9.07/5.17      ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_973_one__le__numeral,axiom,
% 9.07/5.17      ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_974_one__le__numeral,axiom,
% 9.07/5.17      ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_975_not__numeral__less__one,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_976_not__numeral__less__one,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_977_not__numeral__less__one,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_978_not__numeral__less__one,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_979_less__1__mult,axiom,
% 9.07/5.17      ! [M: real,N: real] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ M )
% 9.07/5.17       => ( ( ord_less_real @ one_one_real @ N )
% 9.07/5.17         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_980_less__1__mult,axiom,
% 9.07/5.17      ! [M: rat,N: rat] :
% 9.07/5.17        ( ( ord_less_rat @ one_one_rat @ M )
% 9.07/5.17       => ( ( ord_less_rat @ one_one_rat @ N )
% 9.07/5.17         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_981_less__1__mult,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ one_one_nat @ M )
% 9.07/5.17       => ( ( ord_less_nat @ one_one_nat @ N )
% 9.07/5.17         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_982_less__1__mult,axiom,
% 9.07/5.17      ! [M: int,N: int] :
% 9.07/5.17        ( ( ord_less_int @ one_one_int @ M )
% 9.07/5.17       => ( ( ord_less_int @ one_one_int @ N )
% 9.07/5.17         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_983_less__add__one,axiom,
% 9.07/5.17      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_984_less__add__one,axiom,
% 9.07/5.17      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_985_less__add__one,axiom,
% 9.07/5.17      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_986_less__add__one,axiom,
% 9.07/5.17      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_987_add__mono1,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( ord_less_real @ A @ B2 )
% 9.07/5.17       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_988_add__mono1,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.17       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_989_add__mono1,axiom,
% 9.07/5.17      ! [A: nat,B2: nat] :
% 9.07/5.17        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.17       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_990_add__mono1,axiom,
% 9.07/5.17      ! [A: int,B2: int] :
% 9.07/5.17        ( ( ord_less_int @ A @ B2 )
% 9.07/5.17       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_991_one__plus__numeral__commute,axiom,
% 9.07/5.17      ! [X3: num] :
% 9.07/5.17        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
% 9.07/5.17        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_992_one__plus__numeral__commute,axiom,
% 9.07/5.17      ! [X3: num] :
% 9.07/5.17        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
% 9.07/5.17        = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_993_one__plus__numeral__commute,axiom,
% 9.07/5.17      ! [X3: num] :
% 9.07/5.17        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
% 9.07/5.17        = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_994_one__plus__numeral__commute,axiom,
% 9.07/5.17      ! [X3: num] :
% 9.07/5.17        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
% 9.07/5.17        = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_995_one__plus__numeral__commute,axiom,
% 9.07/5.17      ! [X3: num] :
% 9.07/5.17        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
% 9.07/5.17        = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_996_numeral__One,axiom,
% 9.07/5.17      ( ( numera6690914467698888265omplex @ one )
% 9.07/5.17      = one_one_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_997_numeral__One,axiom,
% 9.07/5.17      ( ( numeral_numeral_real @ one )
% 9.07/5.17      = one_one_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_998_numeral__One,axiom,
% 9.07/5.17      ( ( numeral_numeral_rat @ one )
% 9.07/5.17      = one_one_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_999_numeral__One,axiom,
% 9.07/5.17      ( ( numeral_numeral_nat @ one )
% 9.07/5.17      = one_one_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1000_numeral__One,axiom,
% 9.07/5.17      ( ( numeral_numeral_int @ one )
% 9.07/5.17      = one_one_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1001_one__le__power,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.07/5.17       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1002_one__le__power,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.07/5.17       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1003_one__le__power,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.07/5.17       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1004_one__le__power,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.07/5.17       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1005_left__right__inverse__power,axiom,
% 9.07/5.17      ! [X3: complex,Y3: complex,N: nat] :
% 9.07/5.17        ( ( ( times_times_complex @ X3 @ Y3 )
% 9.07/5.17          = one_one_complex )
% 9.07/5.17       => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 9.07/5.17          = one_one_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1006_left__right__inverse__power,axiom,
% 9.07/5.17      ! [X3: real,Y3: real,N: nat] :
% 9.07/5.17        ( ( ( times_times_real @ X3 @ Y3 )
% 9.07/5.17          = one_one_real )
% 9.07/5.17       => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 9.07/5.17          = one_one_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1007_left__right__inverse__power,axiom,
% 9.07/5.17      ! [X3: rat,Y3: rat,N: nat] :
% 9.07/5.17        ( ( ( times_times_rat @ X3 @ Y3 )
% 9.07/5.17          = one_one_rat )
% 9.07/5.17       => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 9.07/5.17          = one_one_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1008_left__right__inverse__power,axiom,
% 9.07/5.17      ! [X3: nat,Y3: nat,N: nat] :
% 9.07/5.17        ( ( ( times_times_nat @ X3 @ Y3 )
% 9.07/5.17          = one_one_nat )
% 9.07/5.17       => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 9.07/5.17          = one_one_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1009_left__right__inverse__power,axiom,
% 9.07/5.17      ! [X3: int,Y3: int,N: nat] :
% 9.07/5.17        ( ( ( times_times_int @ X3 @ Y3 )
% 9.07/5.17          = one_one_int )
% 9.07/5.17       => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 9.07/5.17          = one_one_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1010_power__one__over,axiom,
% 9.07/5.17      ! [A: complex,N: nat] :
% 9.07/5.17        ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 9.07/5.17        = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1011_power__one__over,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 9.07/5.17        = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1012_power__one__over,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 9.07/5.17        = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1013_numerals_I1_J,axiom,
% 9.07/5.17      ( ( numeral_numeral_nat @ one )
% 9.07/5.17      = one_one_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1014_option_Oexhaust__sel,axiom,
% 9.07/5.17      ! [Option: option_nat] :
% 9.07/5.17        ( ( Option != none_nat )
% 9.07/5.17       => ( Option
% 9.07/5.17          = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1015_option_Oexhaust__sel,axiom,
% 9.07/5.17      ! [Option: option4927543243414619207at_nat] :
% 9.07/5.17        ( ( Option != none_P5556105721700978146at_nat )
% 9.07/5.17       => ( Option
% 9.07/5.17          = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1016_option_Oexhaust__sel,axiom,
% 9.07/5.17      ! [Option: option_num] :
% 9.07/5.17        ( ( Option != none_num )
% 9.07/5.17       => ( Option
% 9.07/5.17          = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1017_linordered__field__no__ub,axiom,
% 9.07/5.17      ! [X6: real] :
% 9.07/5.17      ? [X_1: real] : ( ord_less_real @ X6 @ X_1 ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1018_linordered__field__no__ub,axiom,
% 9.07/5.17      ! [X6: rat] :
% 9.07/5.17      ? [X_1: rat] : ( ord_less_rat @ X6 @ X_1 ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1019_linordered__field__no__lb,axiom,
% 9.07/5.17      ! [X6: real] :
% 9.07/5.17      ? [Y4: real] : ( ord_less_real @ Y4 @ X6 ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1020_linordered__field__no__lb,axiom,
% 9.07/5.17      ! [X6: rat] :
% 9.07/5.17      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X6 ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1021_power__less__power__Suc,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.17       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1022_power__less__power__Suc,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.17       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1023_power__less__power__Suc,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.17       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1024_power__less__power__Suc,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.17       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1025_power__gt1__lemma,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.17       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1026_power__gt1__lemma,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.17       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1027_power__gt1__lemma,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.17       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1028_power__gt1__lemma,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.17       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1029_square__diff__one__factored,axiom,
% 9.07/5.17      ! [X3: complex] :
% 9.07/5.17        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
% 9.07/5.17        = ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1030_square__diff__one__factored,axiom,
% 9.07/5.17      ! [X3: real] :
% 9.07/5.17        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
% 9.07/5.17        = ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1031_square__diff__one__factored,axiom,
% 9.07/5.17      ! [X3: rat] :
% 9.07/5.17        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
% 9.07/5.17        = ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1032_square__diff__one__factored,axiom,
% 9.07/5.17      ! [X3: int] :
% 9.07/5.17        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
% 9.07/5.17        = ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1033_power__strict__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: real] :
% 9.07/5.17        ( ( ord_less_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.17         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1034_power__strict__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: rat] :
% 9.07/5.17        ( ( ord_less_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.17         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1035_power__strict__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: nat] :
% 9.07/5.17        ( ( ord_less_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.17         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1036_power__strict__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: int] :
% 9.07/5.17        ( ( ord_less_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.17         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1037_power__less__imp__less__exp,axiom,
% 9.07/5.17      ! [A: real,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.17       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 9.07/5.17         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1038_power__less__imp__less__exp,axiom,
% 9.07/5.17      ! [A: rat,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.17       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 9.07/5.17         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1039_power__less__imp__less__exp,axiom,
% 9.07/5.17      ! [A: nat,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.17       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.07/5.17         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1040_power__less__imp__less__exp,axiom,
% 9.07/5.17      ! [A: int,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.17       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.07/5.17         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1041_power__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: real] :
% 9.07/5.17        ( ( ord_less_eq_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_eq_real @ one_one_real @ A )
% 9.07/5.17         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1042_power__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: rat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.07/5.17         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1043_power__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.07/5.17         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1044_power__increasing,axiom,
% 9.07/5.17      ! [N: nat,N4: nat,A: int] :
% 9.07/5.17        ( ( ord_less_eq_nat @ N @ N4 )
% 9.07/5.17       => ( ( ord_less_eq_int @ one_one_int @ A )
% 9.07/5.17         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1045_power__le__imp__le__exp,axiom,
% 9.07/5.17      ! [A: real,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.17       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 9.07/5.17         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1046_power__le__imp__le__exp,axiom,
% 9.07/5.17      ! [A: rat,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.17       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 9.07/5.17         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1047_power__le__imp__le__exp,axiom,
% 9.07/5.17      ! [A: nat,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.17       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.07/5.17         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1048_power__le__imp__le__exp,axiom,
% 9.07/5.17      ! [A: int,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.17       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.07/5.17         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1049_one__power2,axiom,
% 9.07/5.17      ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17      = one_one_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1050_one__power2,axiom,
% 9.07/5.17      ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17      = one_one_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1051_one__power2,axiom,
% 9.07/5.17      ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17      = one_one_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1052_one__power2,axiom,
% 9.07/5.17      ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17      = one_one_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1053_one__power2,axiom,
% 9.07/5.17      ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17      = one_one_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1054_nat__1__add__1,axiom,
% 9.07/5.17      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 9.07/5.17      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1055_times__divide__times__eq,axiom,
% 9.07/5.17      ! [X3: complex,Y3: complex,Z: complex,W: complex] :
% 9.07/5.17        ( ( times_times_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 9.07/5.17        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1056_times__divide__times__eq,axiom,
% 9.07/5.17      ! [X3: real,Y3: real,Z: real,W: real] :
% 9.07/5.17        ( ( times_times_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 9.07/5.17        = ( divide_divide_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1057_times__divide__times__eq,axiom,
% 9.07/5.17      ! [X3: rat,Y3: rat,Z: rat,W: rat] :
% 9.07/5.17        ( ( times_times_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 9.07/5.17        = ( divide_divide_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1058_divide__divide__times__eq,axiom,
% 9.07/5.17      ! [X3: complex,Y3: complex,Z: complex,W: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 9.07/5.17        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1059_divide__divide__times__eq,axiom,
% 9.07/5.17      ! [X3: real,Y3: real,Z: real,W: real] :
% 9.07/5.17        ( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 9.07/5.17        = ( divide_divide_real @ ( times_times_real @ X3 @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1060_divide__divide__times__eq,axiom,
% 9.07/5.17      ! [X3: rat,Y3: rat,Z: rat,W: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 9.07/5.17        = ( divide_divide_rat @ ( times_times_rat @ X3 @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1061_divide__divide__eq__left_H,axiom,
% 9.07/5.17      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B2 ) @ C )
% 9.07/5.17        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1062_divide__divide__eq__left_H,axiom,
% 9.07/5.17      ! [A: real,B2: real,C: real] :
% 9.07/5.17        ( ( divide_divide_real @ ( divide_divide_real @ A @ B2 ) @ C )
% 9.07/5.17        = ( divide_divide_real @ A @ ( times_times_real @ C @ B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1063_divide__divide__eq__left_H,axiom,
% 9.07/5.17      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B2 ) @ C )
% 9.07/5.17        = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1064_add__divide__distrib,axiom,
% 9.07/5.17      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B2 ) @ C )
% 9.07/5.17        = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1065_add__divide__distrib,axiom,
% 9.07/5.17      ! [A: real,B2: real,C: real] :
% 9.07/5.17        ( ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ C )
% 9.07/5.17        = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1066_add__divide__distrib,axiom,
% 9.07/5.17      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B2 ) @ C )
% 9.07/5.17        = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1067_diff__divide__distrib,axiom,
% 9.07/5.17      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B2 ) @ C )
% 9.07/5.17        = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B2 @ C ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1068_diff__divide__distrib,axiom,
% 9.07/5.17      ! [A: real,B2: real,C: real] :
% 9.07/5.17        ( ( divide_divide_real @ ( minus_minus_real @ A @ B2 ) @ C )
% 9.07/5.17        = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1069_diff__divide__distrib,axiom,
% 9.07/5.17      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B2 ) @ C )
% 9.07/5.17        = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1070_ex__power__ivl1,axiom,
% 9.07/5.17      ! [B2: nat,K: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.07/5.17       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 9.07/5.17         => ? [N3: nat] :
% 9.07/5.17              ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
% 9.07/5.17              & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1071_ex__power__ivl2,axiom,
% 9.07/5.17      ! [B2: nat,K: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.07/5.17       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.07/5.17         => ? [N3: nat] :
% 9.07/5.17              ( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
% 9.07/5.17              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1072_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.07/5.17      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B2: product_prod_nat_nat] :
% 9.07/5.17        ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B2 ) )
% 9.07/5.17        = ( some_P7363390416028606310at_nat @ ( F @ A @ B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1073_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.07/5.17      ! [F: num > num > num,A: num,B2: num] :
% 9.07/5.17        ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B2 ) )
% 9.07/5.17        = ( some_num @ ( F @ A @ B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1074_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.07/5.17      ! [F: nat > nat > nat,A: nat,B2: nat] :
% 9.07/5.17        ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B2 ) )
% 9.07/5.17        = ( some_nat @ ( F @ A @ B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1075_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.07/5.17      ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 9.07/5.17        ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 9.07/5.17        = none_P5556105721700978146at_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1076_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.07/5.17      ! [Uu: num > num > num,Uv: option_num] :
% 9.07/5.17        ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 9.07/5.17        = none_num ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1077_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.07/5.17      ! [Uu: nat > nat > nat,Uv: option_nat] :
% 9.07/5.17        ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 9.07/5.17        = none_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1078_vebt__mint_Osimps_I2_J,axiom,
% 9.07/5.17      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.07/5.17        ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.07/5.17        = none_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1079_vebt__maxt_Osimps_I2_J,axiom,
% 9.07/5.17      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.07/5.17        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.07/5.17        = none_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1080_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.07/5.17      ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 9.07/5.17        ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 9.07/5.17        = none_P5556105721700978146at_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1081_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.07/5.17      ! [Uw: num > num > num,V: num] :
% 9.07/5.17        ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 9.07/5.17        = none_num ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1082_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.07/5.17      ! [Uw: nat > nat > nat,V: nat] :
% 9.07/5.17        ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 9.07/5.17        = none_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1083_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.07/5.17      ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 9.07/5.17        ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb2 )
% 9.07/5.17          = Y3 )
% 9.07/5.17       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 9.07/5.17           => ( Y3 != none_P5556105721700978146at_nat ) )
% 9.07/5.17         => ( ( ? [V2: product_prod_nat_nat] :
% 9.07/5.17                  ( Xa2
% 9.07/5.17                  = ( some_P7363390416028606310at_nat @ V2 ) )
% 9.07/5.17             => ( ( Xb2 = none_P5556105721700978146at_nat )
% 9.07/5.17               => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 9.07/5.17           => ~ ! [A3: product_prod_nat_nat] :
% 9.07/5.17                  ( ( Xa2
% 9.07/5.17                    = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.07/5.17                 => ! [B3: product_prod_nat_nat] :
% 9.07/5.17                      ( ( Xb2
% 9.07/5.17                        = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.07/5.17                     => ( Y3
% 9.07/5.17                       != ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1084_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.07/5.17      ! [X3: num > num > num,Xa2: option_num,Xb2: option_num,Y3: option_num] :
% 9.07/5.17        ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb2 )
% 9.07/5.17          = Y3 )
% 9.07/5.17       => ( ( ( Xa2 = none_num )
% 9.07/5.17           => ( Y3 != none_num ) )
% 9.07/5.17         => ( ( ? [V2: num] :
% 9.07/5.17                  ( Xa2
% 9.07/5.17                  = ( some_num @ V2 ) )
% 9.07/5.17             => ( ( Xb2 = none_num )
% 9.07/5.17               => ( Y3 != none_num ) ) )
% 9.07/5.17           => ~ ! [A3: num] :
% 9.07/5.17                  ( ( Xa2
% 9.07/5.17                    = ( some_num @ A3 ) )
% 9.07/5.17                 => ! [B3: num] :
% 9.07/5.17                      ( ( Xb2
% 9.07/5.17                        = ( some_num @ B3 ) )
% 9.07/5.17                     => ( Y3
% 9.07/5.17                       != ( some_num @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1085_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.07/5.17      ! [X3: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y3: option_nat] :
% 9.07/5.17        ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb2 )
% 9.07/5.17          = Y3 )
% 9.07/5.17       => ( ( ( Xa2 = none_nat )
% 9.07/5.17           => ( Y3 != none_nat ) )
% 9.07/5.17         => ( ( ? [V2: nat] :
% 9.07/5.17                  ( Xa2
% 9.07/5.17                  = ( some_nat @ V2 ) )
% 9.07/5.17             => ( ( Xb2 = none_nat )
% 9.07/5.17               => ( Y3 != none_nat ) ) )
% 9.07/5.17           => ~ ! [A3: nat] :
% 9.07/5.17                  ( ( Xa2
% 9.07/5.17                    = ( some_nat @ A3 ) )
% 9.07/5.17                 => ! [B3: nat] :
% 9.07/5.17                      ( ( Xb2
% 9.07/5.17                        = ( some_nat @ B3 ) )
% 9.07/5.17                     => ( Y3
% 9.07/5.17                       != ( some_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1086_nested__mint,axiom,
% 9.07/5.17      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.07/5.17       => ( ( N
% 9.07/5.17            = ( suc @ ( suc @ Va ) ) )
% 9.07/5.17         => ( ~ ( ord_less_nat @ Ma @ Mi )
% 9.07/5.17           => ( ( Ma != Mi )
% 9.07/5.17             => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1087_real__average__minus__second,axiom,
% 9.07/5.17      ! [B2: real,A: real] :
% 9.07/5.17        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 9.07/5.17        = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1088_real__average__minus__first,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 9.07/5.17        = ( divide_divide_real @ ( minus_minus_real @ B2 @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1089__C1_C,axiom,
% 9.07/5.17      ( ( ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17           != none_nat )
% 9.07/5.17          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 9.07/5.17          = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17      & ( ~ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17             != none_nat )
% 9.07/5.17            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 9.07/5.17          = ( if_option_nat
% 9.07/5.17            @ ( ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.17              = none_nat )
% 9.07/5.17            @ ( if_option_nat @ ( ord_less_nat @ mi @ xa ) @ ( some_nat @ mi ) @ none_nat )
% 9.07/5.17            @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_pred @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1090_invar__vebt_Ointros_I5_J,axiom,
% 9.07/5.17      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.07/5.17        ( ! [X5: vEBT_VEBT] :
% 9.07/5.17            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.17           => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.07/5.17       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.07/5.17         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.07/5.17              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.17           => ( ( M
% 9.07/5.17                = ( suc @ N ) )
% 9.07/5.17             => ( ( Deg
% 9.07/5.17                  = ( plus_plus_nat @ N @ M ) )
% 9.07/5.17               => ( ! [I3: nat] :
% 9.07/5.17                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.17                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
% 9.07/5.17                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 9.07/5.17                 => ( ( ( Mi = Ma )
% 9.07/5.17                     => ! [X5: vEBT_VEBT] :
% 9.07/5.17                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.17                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 9.07/5.17                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.07/5.17                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.07/5.17                       => ( ( ( Mi != Ma )
% 9.07/5.17                           => ! [I3: nat] :
% 9.07/5.17                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.17                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.07/5.17                                      = I3 )
% 9.07/5.17                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.07/5.17                                  & ! [X5: nat] :
% 9.07/5.17                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 9.07/5.17                                          = I3 )
% 9.07/5.17                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 9.07/5.17                                     => ( ( ord_less_nat @ Mi @ X5 )
% 9.07/5.17                                        & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 9.07/5.17                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1091_pred__less__length__list,axiom,
% 9.07/5.17      ! [Deg: nat,X3: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.07/5.17       => ( ( ord_less_eq_nat @ X3 @ Ma )
% 9.07/5.17         => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17           => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.17              = ( if_option_nat
% 9.07/5.17                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17                   != none_nat )
% 9.07/5.17                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( 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 ) ) ) ) ) )
% 9.07/5.17                @ ( if_option_nat
% 9.07/5.17                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.17                    = none_nat )
% 9.07/5.17                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
% 9.07/5.17                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1092_pred__lesseq__max,axiom,
% 9.07/5.17      ! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.07/5.17       => ( ( ord_less_eq_nat @ X3 @ Ma )
% 9.07/5.17         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.17            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17              @ ( if_option_nat
% 9.07/5.17                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17                   != none_nat )
% 9.07/5.17                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( 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 ) ) ) ) ) )
% 9.07/5.17                @ ( if_option_nat
% 9.07/5.17                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.17                    = none_nat )
% 9.07/5.17                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
% 9.07/5.17                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.07/5.17              @ none_nat ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1093_divmod__step__eq,axiom,
% 9.07/5.17      ! [L: num,R: nat,Q2: nat] :
% 9.07/5.17        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
% 9.07/5.17         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 9.07/5.17            = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L ) ) ) ) )
% 9.07/5.17        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R )
% 9.07/5.17         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 9.07/5.17            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1094_divmod__step__eq,axiom,
% 9.07/5.17      ! [L: num,R: int,Q2: int] :
% 9.07/5.17        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
% 9.07/5.17         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.07/5.17            = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L ) ) ) ) )
% 9.07/5.17        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R )
% 9.07/5.17         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.07/5.17            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1095_divmod__step__eq,axiom,
% 9.07/5.17      ! [L: num,R: code_integer,Q2: code_integer] :
% 9.07/5.17        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
% 9.07/5.17         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 9.07/5.17            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 9.07/5.17        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R )
% 9.07/5.17         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 9.07/5.17            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1096_maxt__corr__help__empty,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT,N: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.17       => ( ( ( vEBT_vebt_maxt @ T )
% 9.07/5.17            = none_nat )
% 9.07/5.17         => ( ( vEBT_VEBT_set_vebt @ T )
% 9.07/5.17            = bot_bot_set_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1097_mint__corr__help__empty,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT,N: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.17       => ( ( ( vEBT_vebt_mint @ T )
% 9.07/5.17            = none_nat )
% 9.07/5.17         => ( ( vEBT_VEBT_set_vebt @ T )
% 9.07/5.17            = bot_bot_set_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1098_all__set__conv__all__nth,axiom,
% 9.07/5.17      ! [Xs2: list_int,P: int > $o] :
% 9.07/5.17        ( ( ! [X: int] :
% 9.07/5.17              ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X ) ) )
% 9.07/5.17        = ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.17             => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1099_all__set__conv__all__nth,axiom,
% 9.07/5.17      ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.07/5.17        ( ( ! [X: vEBT_VEBT] :
% 9.07/5.17              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X ) ) )
% 9.07/5.17        = ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.17             => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1100_all__set__conv__all__nth,axiom,
% 9.07/5.17      ! [Xs2: list_o,P: $o > $o] :
% 9.07/5.17        ( ( ! [X: $o] :
% 9.07/5.17              ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X ) ) )
% 9.07/5.17        = ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.17             => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1101_all__set__conv__all__nth,axiom,
% 9.07/5.17      ! [Xs2: list_nat,P: nat > $o] :
% 9.07/5.17        ( ( ! [X: nat] :
% 9.07/5.17              ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X ) ) )
% 9.07/5.17        = ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.17             => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1102_even__odd__cases,axiom,
% 9.07/5.17      ! [X3: nat] :
% 9.07/5.17        ( ! [N3: nat] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( plus_plus_nat @ N3 @ N3 ) )
% 9.07/5.17       => ~ ! [N3: nat] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1103_set__vebt_H__def,axiom,
% 9.07/5.17      ( vEBT_VEBT_set_vebt
% 9.07/5.17      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1104_deg__SUcn__Node,axiom,
% 9.07/5.17      ! [Tree: vEBT_VEBT,N: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 9.07/5.17       => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.07/5.17            ( Tree
% 9.07/5.17            = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1105_zdiv__numeral__Bit0,axiom,
% 9.07/5.17      ! [V: num,W: num] :
% 9.07/5.17        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.07/5.17        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1106_old_Onat_Oinject,axiom,
% 9.07/5.17      ! [Nat: nat,Nat2: nat] :
% 9.07/5.17        ( ( ( suc @ Nat )
% 9.07/5.17          = ( suc @ Nat2 ) )
% 9.07/5.17        = ( Nat = Nat2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1107_nat_Oinject,axiom,
% 9.07/5.17      ! [X2: nat,Y2: nat] :
% 9.07/5.17        ( ( ( suc @ X2 )
% 9.07/5.17          = ( suc @ Y2 ) )
% 9.07/5.17        = ( X2 = Y2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1108_lessI,axiom,
% 9.07/5.17      ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1109_Suc__mono,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ N )
% 9.07/5.17       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1110_Suc__less__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.07/5.17        = ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1111_Suc__le__mono,axiom,
% 9.07/5.17      ! [N: nat,M: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 9.07/5.17        = ( ord_less_eq_nat @ N @ M ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1112_add__Suc__right,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 9.07/5.17        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1113_diff__Suc__Suc,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.07/5.17        = ( minus_minus_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1114_Suc__diff__diff,axiom,
% 9.07/5.17      ! [M: nat,N: nat,K: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 9.07/5.17        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1115_mult__Suc__right,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( times_times_nat @ M @ ( suc @ N ) )
% 9.07/5.17        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1116_diff__Suc__1,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 9.07/5.17        = N ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1117_diff__Suc__diff__eq2,axiom,
% 9.07/5.17      ! [K: nat,J: nat,I: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.17       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 9.07/5.17          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1118_diff__Suc__diff__eq1,axiom,
% 9.07/5.17      ! [K: nat,J: nat,I: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ K @ J )
% 9.07/5.17       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 9.07/5.17          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1119_Suc__numeral,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ( ( suc @ ( numeral_numeral_nat @ N ) )
% 9.07/5.17        = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1120_add__2__eq__Suc_H,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17        = ( suc @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1121_add__2__eq__Suc,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.07/5.17        = ( suc @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1122_div2__Suc__Suc,axiom,
% 9.07/5.17      ! [M: nat] :
% 9.07/5.17        ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.17        = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1123_Suc__1,axiom,
% 9.07/5.17      ( ( suc @ one_one_nat )
% 9.07/5.17      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1124_mult__commute__abs,axiom,
% 9.07/5.17      ! [C: real] :
% 9.07/5.17        ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
% 9.07/5.17        = ( times_times_real @ C ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1125_mult__commute__abs,axiom,
% 9.07/5.17      ! [C: rat] :
% 9.07/5.17        ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
% 9.07/5.17        = ( times_times_rat @ C ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1126_mult__commute__abs,axiom,
% 9.07/5.17      ! [C: nat] :
% 9.07/5.17        ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
% 9.07/5.17        = ( times_times_nat @ C ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1127_mult__commute__abs,axiom,
% 9.07/5.17      ! [C: int] :
% 9.07/5.17        ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
% 9.07/5.17        = ( times_times_int @ C ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1128_n__not__Suc__n,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( N
% 9.07/5.17       != ( suc @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1129_Suc__inject,axiom,
% 9.07/5.17      ! [X3: nat,Y3: nat] :
% 9.07/5.17        ( ( ( suc @ X3 )
% 9.07/5.17          = ( suc @ Y3 ) )
% 9.07/5.17       => ( X3 = Y3 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1130_lambda__one,axiom,
% 9.07/5.17      ( ( ^ [X: complex] : X )
% 9.07/5.17      = ( times_times_complex @ one_one_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1131_lambda__one,axiom,
% 9.07/5.17      ( ( ^ [X: real] : X )
% 9.07/5.17      = ( times_times_real @ one_one_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1132_lambda__one,axiom,
% 9.07/5.17      ( ( ^ [X: rat] : X )
% 9.07/5.17      = ( times_times_rat @ one_one_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1133_lambda__one,axiom,
% 9.07/5.17      ( ( ^ [X: nat] : X )
% 9.07/5.17      = ( times_times_nat @ one_one_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1134_lambda__one,axiom,
% 9.07/5.17      ( ( ^ [X: int] : X )
% 9.07/5.17      = ( times_times_int @ one_one_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1135_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.07/5.17      ! [X3: produc8306885398267862888on_nat] :
% 9.07/5.17        ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.07/5.17       => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.07/5.17         => ~ ! [F2: nat > nat > nat,A3: nat,B3: nat] :
% 9.07/5.17                ( X3
% 9.07/5.17               != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1136_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.07/5.17      ! [X3: produc5542196010084753463at_nat] :
% 9.07/5.17        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.07/5.17       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.07/5.17         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 9.07/5.17                ( X3
% 9.07/5.17               != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1137_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.07/5.17      ! [X3: produc1193250871479095198on_num] :
% 9.07/5.17        ( ! [Uu2: num > num > num,Uv2: option_num] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.07/5.17       => ( ! [Uw2: num > num > num,V2: num] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.07/5.17         => ~ ! [F2: num > num > num,A3: num,B3: num] :
% 9.07/5.17                ( X3
% 9.07/5.17               != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1138_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.07/5.17      ! [X3: produc2233624965454879586on_nat] :
% 9.07/5.17        ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.07/5.17       => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.07/5.17         => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
% 9.07/5.17                ( X3
% 9.07/5.17               != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1139_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.07/5.17      ! [X3: produc5491161045314408544at_nat] :
% 9.07/5.17        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.07/5.17       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.07/5.17         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 9.07/5.17                ( X3
% 9.07/5.17               != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1140_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.07/5.17      ! [X3: produc7036089656553540234on_num] :
% 9.07/5.17        ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 9.07/5.17            ( X3
% 9.07/5.17           != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.07/5.17       => ( ! [Uw2: num > num > $o,V2: num] :
% 9.07/5.17              ( X3
% 9.07/5.17             != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.07/5.17         => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
% 9.07/5.17                ( X3
% 9.07/5.17               != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1141_Nat_OlessE,axiom,
% 9.07/5.17      ! [I: nat,K: nat] :
% 9.07/5.17        ( ( ord_less_nat @ I @ K )
% 9.07/5.17       => ( ( K
% 9.07/5.17           != ( suc @ I ) )
% 9.07/5.17         => ~ ! [J2: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I @ J2 )
% 9.07/5.17               => ( K
% 9.07/5.17                 != ( suc @ J2 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1142_Suc__lessD,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ ( suc @ M ) @ N )
% 9.07/5.17       => ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1143_Suc__lessE,axiom,
% 9.07/5.17      ! [I: nat,K: nat] :
% 9.07/5.17        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 9.07/5.17       => ~ ! [J2: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I @ J2 )
% 9.07/5.17             => ( K
% 9.07/5.17               != ( suc @ J2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1144_Suc__lessI,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ N )
% 9.07/5.17       => ( ( ( suc @ M )
% 9.07/5.17           != N )
% 9.07/5.17         => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1145_less__SucE,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.07/5.17       => ( ~ ( ord_less_nat @ M @ N )
% 9.07/5.17         => ( M = N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1146_less__SucI,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ N )
% 9.07/5.17       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1147_Ex__less__Suc,axiom,
% 9.07/5.17      ! [N: nat,P: nat > $o] :
% 9.07/5.17        ( ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.07/5.17              & ( P @ I4 ) ) )
% 9.07/5.17        = ( ( P @ N )
% 9.07/5.17          | ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ N )
% 9.07/5.17              & ( P @ I4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1148_less__Suc__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.07/5.17        = ( ( ord_less_nat @ M @ N )
% 9.07/5.17          | ( M = N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1149_not__less__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ~ ( ord_less_nat @ M @ N ) )
% 9.07/5.17        = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1150_All__less__Suc,axiom,
% 9.07/5.17      ! [N: nat,P: nat > $o] :
% 9.07/5.17        ( ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.07/5.17             => ( P @ I4 ) ) )
% 9.07/5.17        = ( ( P @ N )
% 9.07/5.17          & ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ N )
% 9.07/5.17             => ( P @ I4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1151_Suc__less__eq2,axiom,
% 9.07/5.17      ! [N: nat,M: nat] :
% 9.07/5.17        ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.07/5.17        = ( ? [M5: nat] :
% 9.07/5.17              ( ( M
% 9.07/5.17                = ( suc @ M5 ) )
% 9.07/5.17              & ( ord_less_nat @ N @ M5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1152_less__antisym,axiom,
% 9.07/5.17      ! [N: nat,M: nat] :
% 9.07/5.17        ( ~ ( ord_less_nat @ N @ M )
% 9.07/5.17       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.07/5.17         => ( M = N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1153_Suc__less__SucD,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.07/5.17       => ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1154_less__trans__Suc,axiom,
% 9.07/5.17      ! [I: nat,J: nat,K: nat] :
% 9.07/5.17        ( ( ord_less_nat @ I @ J )
% 9.07/5.17       => ( ( ord_less_nat @ J @ K )
% 9.07/5.17         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1155_less__Suc__induct,axiom,
% 9.07/5.17      ! [I: nat,J: nat,P: nat > nat > $o] :
% 9.07/5.17        ( ( ord_less_nat @ I @ J )
% 9.07/5.17       => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 9.07/5.17         => ( ! [I3: nat,J2: nat,K3: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I3 @ J2 )
% 9.07/5.17               => ( ( ord_less_nat @ J2 @ K3 )
% 9.07/5.17                 => ( ( P @ I3 @ J2 )
% 9.07/5.17                   => ( ( P @ J2 @ K3 )
% 9.07/5.17                     => ( P @ I3 @ K3 ) ) ) ) )
% 9.07/5.17           => ( P @ I @ J ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1156_strict__inc__induct,axiom,
% 9.07/5.17      ! [I: nat,J: nat,P: nat > $o] :
% 9.07/5.17        ( ( ord_less_nat @ I @ J )
% 9.07/5.17       => ( ! [I3: nat] :
% 9.07/5.17              ( ( J
% 9.07/5.17                = ( suc @ I3 ) )
% 9.07/5.17             => ( P @ I3 ) )
% 9.07/5.17         => ( ! [I3: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I3 @ J )
% 9.07/5.17               => ( ( P @ ( suc @ I3 ) )
% 9.07/5.17                 => ( P @ I3 ) ) )
% 9.07/5.17           => ( P @ I ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1157_not__less__less__Suc__eq,axiom,
% 9.07/5.17      ! [N: nat,M: nat] :
% 9.07/5.17        ( ~ ( ord_less_nat @ N @ M )
% 9.07/5.17       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.07/5.17          = ( N = M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1158_Suc__leD,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.07/5.17       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1159_le__SucE,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.07/5.17       => ( ~ ( ord_less_eq_nat @ M @ N )
% 9.07/5.17         => ( M
% 9.07/5.17            = ( suc @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1160_le__SucI,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.17       => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1161_Suc__le__D,axiom,
% 9.07/5.17      ! [N: nat,M6: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
% 9.07/5.17       => ? [M4: nat] :
% 9.07/5.17            ( M6
% 9.07/5.17            = ( suc @ M4 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1162_le__Suc__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.07/5.17        = ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.17          | ( M
% 9.07/5.17            = ( suc @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1163_Suc__n__not__le__n,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1164_not__less__eq__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 9.07/5.17        = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1165_full__nat__induct,axiom,
% 9.07/5.17      ! [P: nat > $o,N: nat] :
% 9.07/5.17        ( ! [N3: nat] :
% 9.07/5.17            ( ! [M2: nat] :
% 9.07/5.17                ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
% 9.07/5.17               => ( P @ M2 ) )
% 9.07/5.17           => ( P @ N3 ) )
% 9.07/5.17       => ( P @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1166_nat__induct__at__least,axiom,
% 9.07/5.17      ! [M: nat,N: nat,P: nat > $o] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.17       => ( ( P @ M )
% 9.07/5.17         => ( ! [N3: nat] :
% 9.07/5.17                ( ( ord_less_eq_nat @ M @ N3 )
% 9.07/5.17               => ( ( P @ N3 )
% 9.07/5.17                 => ( P @ ( suc @ N3 ) ) ) )
% 9.07/5.17           => ( P @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1167_transitive__stepwise__le,axiom,
% 9.07/5.17      ! [M: nat,N: nat,R2: nat > nat > $o] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.17       => ( ! [X5: nat] : ( R2 @ X5 @ X5 )
% 9.07/5.17         => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 9.07/5.17                ( ( R2 @ X5 @ Y4 )
% 9.07/5.17               => ( ( R2 @ Y4 @ Z4 )
% 9.07/5.17                 => ( R2 @ X5 @ Z4 ) ) )
% 9.07/5.17           => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
% 9.07/5.17             => ( R2 @ M @ N ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1168_nat__arith_Osuc1,axiom,
% 9.07/5.17      ! [A4: nat,K: nat,A: nat] :
% 9.07/5.17        ( ( A4
% 9.07/5.17          = ( plus_plus_nat @ K @ A ) )
% 9.07/5.17       => ( ( suc @ A4 )
% 9.07/5.17          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1169_add__Suc,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.07/5.17        = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1170_add__Suc__shift,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.07/5.17        = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1171_zero__induct__lemma,axiom,
% 9.07/5.17      ! [P: nat > $o,K: nat,I: nat] :
% 9.07/5.17        ( ( P @ K )
% 9.07/5.17       => ( ! [N3: nat] :
% 9.07/5.17              ( ( P @ ( suc @ N3 ) )
% 9.07/5.17             => ( P @ N3 ) )
% 9.07/5.17         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1172_Suc__mult__cancel1,axiom,
% 9.07/5.17      ! [K: nat,M: nat,N: nat] :
% 9.07/5.17        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 9.07/5.17          = ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.07/5.17        = ( M = N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1173_numeral__code_I2_J,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 9.07/5.17        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1174_numeral__code_I2_J,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 9.07/5.17        = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1175_numeral__code_I2_J,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 9.07/5.17        = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1176_numeral__code_I2_J,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 9.07/5.17        = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1177_numeral__code_I2_J,axiom,
% 9.07/5.17      ! [N: num] :
% 9.07/5.17        ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 9.07/5.17        = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1178_set__vebt__def,axiom,
% 9.07/5.17      ( vEBT_set_vebt
% 9.07/5.17      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1179_real__arch__pow,axiom,
% 9.07/5.17      ! [X3: real,Y3: real] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ X3 )
% 9.07/5.17       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X3 @ N3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1180_power__numeral__even,axiom,
% 9.07/5.17      ! [Z: complex,W: num] :
% 9.07/5.17        ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 9.07/5.17        = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1181_power__numeral__even,axiom,
% 9.07/5.17      ! [Z: real,W: num] :
% 9.07/5.17        ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 9.07/5.17        = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1182_power__numeral__even,axiom,
% 9.07/5.17      ! [Z: rat,W: num] :
% 9.07/5.17        ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 9.07/5.17        = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1183_power__numeral__even,axiom,
% 9.07/5.17      ! [Z: nat,W: num] :
% 9.07/5.17        ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 9.07/5.17        = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1184_power__numeral__even,axiom,
% 9.07/5.17      ! [Z: int,W: num] :
% 9.07/5.17        ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 9.07/5.17        = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1185_set__conv__nth,axiom,
% 9.07/5.17      ( set_complex2
% 9.07/5.17      = ( ^ [Xs: list_complex] :
% 9.07/5.17            ( collect_complex
% 9.07/5.17            @ ^ [Uu3: complex] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_complex @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1186_set__conv__nth,axiom,
% 9.07/5.17      ( set_real2
% 9.07/5.17      = ( ^ [Xs: list_real] :
% 9.07/5.17            ( collect_real
% 9.07/5.17            @ ^ [Uu3: real] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_real @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1187_set__conv__nth,axiom,
% 9.07/5.17      ( set_list_nat2
% 9.07/5.17      = ( ^ [Xs: list_list_nat] :
% 9.07/5.17            ( collect_list_nat
% 9.07/5.17            @ ^ [Uu3: list_nat] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_list_nat @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1188_set__conv__nth,axiom,
% 9.07/5.17      ( set_int2
% 9.07/5.17      = ( ^ [Xs: list_int] :
% 9.07/5.17            ( collect_int
% 9.07/5.17            @ ^ [Uu3: int] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_int @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1189_set__conv__nth,axiom,
% 9.07/5.17      ( set_VEBT_VEBT2
% 9.07/5.17      = ( ^ [Xs: list_VEBT_VEBT] :
% 9.07/5.17            ( collect_VEBT_VEBT
% 9.07/5.17            @ ^ [Uu3: vEBT_VEBT] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1190_set__conv__nth,axiom,
% 9.07/5.17      ( set_o2
% 9.07/5.17      = ( ^ [Xs: list_o] :
% 9.07/5.17            ( collect_o
% 9.07/5.17            @ ^ [Uu3: $o] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_o @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1191_set__conv__nth,axiom,
% 9.07/5.17      ( set_nat2
% 9.07/5.17      = ( ^ [Xs: list_nat] :
% 9.07/5.17            ( collect_nat
% 9.07/5.17            @ ^ [Uu3: nat] :
% 9.07/5.17              ? [I4: nat] :
% 9.07/5.17                ( ( Uu3
% 9.07/5.17                  = ( nth_nat @ Xs @ I4 ) )
% 9.07/5.17                & ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1192_lift__Suc__mono__less,axiom,
% 9.07/5.17      ! [F: nat > real,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1193_lift__Suc__mono__less,axiom,
% 9.07/5.17      ! [F: nat > rat,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1194_lift__Suc__mono__less,axiom,
% 9.07/5.17      ! [F: nat > num,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1195_lift__Suc__mono__less,axiom,
% 9.07/5.17      ! [F: nat > nat,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1196_lift__Suc__mono__less,axiom,
% 9.07/5.17      ! [F: nat > int,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1197_lift__Suc__mono__less__iff,axiom,
% 9.07/5.17      ! [F: nat > real,N: nat,M: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 9.07/5.17          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1198_lift__Suc__mono__less__iff,axiom,
% 9.07/5.17      ! [F: nat > rat,N: nat,M: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 9.07/5.17          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1199_lift__Suc__mono__less__iff,axiom,
% 9.07/5.17      ! [F: nat > num,N: nat,M: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 9.07/5.17          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1200_lift__Suc__mono__less__iff,axiom,
% 9.07/5.17      ! [F: nat > nat,N: nat,M: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 9.07/5.17          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1201_lift__Suc__mono__less__iff,axiom,
% 9.07/5.17      ! [F: nat > int,N: nat,M: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 9.07/5.17          = ( ord_less_nat @ N @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1202_power__Suc2,axiom,
% 9.07/5.17      ! [A: complex,N: nat] :
% 9.07/5.17        ( ( power_power_complex @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1203_power__Suc2,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( power_power_real @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1204_power__Suc2,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1205_power__Suc2,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1206_power__Suc2,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( power_power_int @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1207_power__Suc,axiom,
% 9.07/5.17      ! [A: complex,N: nat] :
% 9.07/5.17        ( ( power_power_complex @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1208_power__Suc,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( power_power_real @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1209_power__Suc,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1210_power__Suc,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1211_power__Suc,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( power_power_int @ A @ ( suc @ N ) )
% 9.07/5.17        = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1212_lift__Suc__mono__le,axiom,
% 9.07/5.17      ! [F: nat > set_int,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1213_lift__Suc__mono__le,axiom,
% 9.07/5.17      ! [F: nat > rat,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1214_lift__Suc__mono__le,axiom,
% 9.07/5.17      ! [F: nat > num,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1215_lift__Suc__mono__le,axiom,
% 9.07/5.17      ! [F: nat > nat,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1216_lift__Suc__mono__le,axiom,
% 9.07/5.17      ! [F: nat > int,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1217_lift__Suc__antimono__le,axiom,
% 9.07/5.17      ! [F: nat > set_int,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1218_lift__Suc__antimono__le,axiom,
% 9.07/5.17      ! [F: nat > rat,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1219_lift__Suc__antimono__le,axiom,
% 9.07/5.17      ! [F: nat > num,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1220_lift__Suc__antimono__le,axiom,
% 9.07/5.17      ! [F: nat > nat,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1221_lift__Suc__antimono__le,axiom,
% 9.07/5.17      ! [F: nat > int,N: nat,N5: nat] :
% 9.07/5.17        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.07/5.17       => ( ( ord_less_eq_nat @ N @ N5 )
% 9.07/5.17         => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1222_Suc__leI,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ N )
% 9.07/5.17       => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1223_Suc__le__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.07/5.17        = ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1224_dec__induct,axiom,
% 9.07/5.17      ! [I: nat,J: nat,P: nat > $o] :
% 9.07/5.17        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.17       => ( ( P @ I )
% 9.07/5.17         => ( ! [N3: nat] :
% 9.07/5.17                ( ( ord_less_eq_nat @ I @ N3 )
% 9.07/5.17               => ( ( ord_less_nat @ N3 @ J )
% 9.07/5.17                 => ( ( P @ N3 )
% 9.07/5.17                   => ( P @ ( suc @ N3 ) ) ) ) )
% 9.07/5.17           => ( P @ J ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1225_inc__induct,axiom,
% 9.07/5.17      ! [I: nat,J: nat,P: nat > $o] :
% 9.07/5.17        ( ( ord_less_eq_nat @ I @ J )
% 9.07/5.17       => ( ( P @ J )
% 9.07/5.17         => ( ! [N3: nat] :
% 9.07/5.17                ( ( ord_less_eq_nat @ I @ N3 )
% 9.07/5.17               => ( ( ord_less_nat @ N3 @ J )
% 9.07/5.17                 => ( ( P @ ( suc @ N3 ) )
% 9.07/5.17                   => ( P @ N3 ) ) ) )
% 9.07/5.17           => ( P @ I ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1226_Suc__le__lessD,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.07/5.17       => ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1227_le__less__Suc__eq,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.17       => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.07/5.17          = ( N = M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1228_less__Suc__eq__le,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.07/5.17        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1229_less__eq__Suc__le,axiom,
% 9.07/5.17      ( ord_less_nat
% 9.07/5.17      = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1230_le__imp__less__Suc,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.17       => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1231_less__imp__Suc__add,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ N )
% 9.07/5.17       => ? [K3: nat] :
% 9.07/5.17            ( N
% 9.07/5.17            = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1232_less__iff__Suc__add,axiom,
% 9.07/5.17      ( ord_less_nat
% 9.07/5.17      = ( ^ [M3: nat,N2: nat] :
% 9.07/5.17          ? [K2: nat] :
% 9.07/5.17            ( N2
% 9.07/5.17            = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1233_less__add__Suc2,axiom,
% 9.07/5.17      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1234_less__add__Suc1,axiom,
% 9.07/5.17      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1235_less__natE,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ M @ N )
% 9.07/5.17       => ~ ! [Q3: nat] :
% 9.07/5.17              ( N
% 9.07/5.17             != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1236_diff__less__Suc,axiom,
% 9.07/5.17      ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1237_Suc__diff__Suc,axiom,
% 9.07/5.17      ! [N: nat,M: nat] :
% 9.07/5.17        ( ( ord_less_nat @ N @ M )
% 9.07/5.17       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 9.07/5.17          = ( minus_minus_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1238_Suc__mult__less__cancel1,axiom,
% 9.07/5.17      ! [K: nat,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.07/5.17        = ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1239_Suc__diff__le,axiom,
% 9.07/5.17      ! [N: nat,M: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ N @ M )
% 9.07/5.17       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.07/5.17          = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1240_Suc__mult__le__cancel1,axiom,
% 9.07/5.17      ! [K: nat,M: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.07/5.17        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1241_mult__Suc,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( times_times_nat @ ( suc @ M ) @ N )
% 9.07/5.17        = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1242_Suc__div__le__mono,axiom,
% 9.07/5.17      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1243_Suc__eq__plus1__left,axiom,
% 9.07/5.17      ( suc
% 9.07/5.17      = ( plus_plus_nat @ one_one_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1244_plus__1__eq__Suc,axiom,
% 9.07/5.17      ( ( plus_plus_nat @ one_one_nat )
% 9.07/5.17      = suc ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1245_Suc__eq__plus1,axiom,
% 9.07/5.17      ( suc
% 9.07/5.17      = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1246_diff__Suc__eq__diff__pred,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.07/5.17        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1247_power__gt1,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( ord_less_real @ one_one_real @ A )
% 9.07/5.17       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1248_power__gt1,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( ord_less_rat @ one_one_rat @ A )
% 9.07/5.17       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1249_power__gt1,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ one_one_nat @ A )
% 9.07/5.17       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1250_power__gt1,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( ord_less_int @ one_one_int @ A )
% 9.07/5.17       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1251_subset__code_I1_J,axiom,
% 9.07/5.17      ! [Xs2: list_real,B5: set_real] :
% 9.07/5.17        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
% 9.07/5.17        = ( ! [X: real] :
% 9.07/5.17              ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 9.07/5.17             => ( member_real @ X @ B5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1252_subset__code_I1_J,axiom,
% 9.07/5.17      ! [Xs2: list_complex,B5: set_complex] :
% 9.07/5.17        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B5 )
% 9.07/5.17        = ( ! [X: complex] :
% 9.07/5.17              ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 9.07/5.17             => ( member_complex @ X @ B5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1253_subset__code_I1_J,axiom,
% 9.07/5.17      ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
% 9.07/5.17        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
% 9.07/5.17        = ( ! [X: vEBT_VEBT] :
% 9.07/5.17              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.07/5.17             => ( member_VEBT_VEBT @ X @ B5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1254_subset__code_I1_J,axiom,
% 9.07/5.17      ! [Xs2: list_nat,B5: set_nat] :
% 9.07/5.17        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
% 9.07/5.17        = ( ! [X: nat] :
% 9.07/5.17              ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 9.07/5.17             => ( member_nat @ X @ B5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1255_subset__code_I1_J,axiom,
% 9.07/5.17      ! [Xs2: list_int,B5: set_int] :
% 9.07/5.17        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
% 9.07/5.17        = ( ! [X: int] :
% 9.07/5.17              ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 9.07/5.17             => ( member_int @ X @ B5 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1256_Ex__list__of__length,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17      ? [Xs3: list_VEBT_VEBT] :
% 9.07/5.17        ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 9.07/5.17        = N ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1257_Ex__list__of__length,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17      ? [Xs3: list_o] :
% 9.07/5.17        ( ( size_size_list_o @ Xs3 )
% 9.07/5.17        = N ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1258_Ex__list__of__length,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17      ? [Xs3: list_nat] :
% 9.07/5.17        ( ( size_size_list_nat @ Xs3 )
% 9.07/5.17        = N ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1259_neq__if__length__neq,axiom,
% 9.07/5.17      ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.07/5.17        ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.07/5.17         != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.07/5.17       => ( Xs2 != Ys ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1260_neq__if__length__neq,axiom,
% 9.07/5.17      ! [Xs2: list_o,Ys: list_o] :
% 9.07/5.17        ( ( ( size_size_list_o @ Xs2 )
% 9.07/5.17         != ( size_size_list_o @ Ys ) )
% 9.07/5.17       => ( Xs2 != Ys ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1261_neq__if__length__neq,axiom,
% 9.07/5.17      ! [Xs2: list_nat,Ys: list_nat] :
% 9.07/5.17        ( ( ( size_size_list_nat @ Xs2 )
% 9.07/5.17         != ( size_size_list_nat @ Ys ) )
% 9.07/5.17       => ( Xs2 != Ys ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1262_div__nat__eqI,axiom,
% 9.07/5.17      ! [N: nat,Q2: nat,M: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 9.07/5.17       => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 9.07/5.17         => ( ( divide_divide_nat @ M @ N )
% 9.07/5.17            = Q2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1263_Suc__nat__number__of__add,axiom,
% 9.07/5.17      ! [V: num,N: nat] :
% 9.07/5.17        ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 9.07/5.17        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1264_two__realpow__ge__one,axiom,
% 9.07/5.17      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1265_power__odd__eq,axiom,
% 9.07/5.17      ! [A: complex,N: nat] :
% 9.07/5.17        ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.07/5.17        = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1266_power__odd__eq,axiom,
% 9.07/5.17      ! [A: real,N: nat] :
% 9.07/5.17        ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.07/5.17        = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1267_power__odd__eq,axiom,
% 9.07/5.17      ! [A: rat,N: nat] :
% 9.07/5.17        ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.07/5.17        = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1268_power__odd__eq,axiom,
% 9.07/5.17      ! [A: nat,N: nat] :
% 9.07/5.17        ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.07/5.17        = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1269_power__odd__eq,axiom,
% 9.07/5.17      ! [A: int,N: nat] :
% 9.07/5.17        ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.07/5.17        = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1270_length__induct,axiom,
% 9.07/5.17      ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 9.07/5.17        ( ! [Xs3: list_VEBT_VEBT] :
% 9.07/5.17            ( ! [Ys2: list_VEBT_VEBT] :
% 9.07/5.17                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 9.07/5.17               => ( P @ Ys2 ) )
% 9.07/5.17           => ( P @ Xs3 ) )
% 9.07/5.17       => ( P @ Xs2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1271_length__induct,axiom,
% 9.07/5.17      ! [P: list_o > $o,Xs2: list_o] :
% 9.07/5.17        ( ! [Xs3: list_o] :
% 9.07/5.17            ( ! [Ys2: list_o] :
% 9.07/5.17                ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 9.07/5.17               => ( P @ Ys2 ) )
% 9.07/5.17           => ( P @ Xs3 ) )
% 9.07/5.17       => ( P @ Xs2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1272_length__induct,axiom,
% 9.07/5.17      ! [P: list_nat > $o,Xs2: list_nat] :
% 9.07/5.17        ( ! [Xs3: list_nat] :
% 9.07/5.17            ( ! [Ys2: list_nat] :
% 9.07/5.17                ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 9.07/5.17               => ( P @ Ys2 ) )
% 9.07/5.17           => ( P @ Xs3 ) )
% 9.07/5.17       => ( P @ Xs2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1273_vebt__pred_Osimps_I7_J,axiom,
% 9.07/5.17      ! [Ma: nat,X3: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.07/5.17        ( ( ( ord_less_nat @ Ma @ X3 )
% 9.07/5.17         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.17            = ( some_nat @ Ma ) ) )
% 9.07/5.17        & ( ~ ( ord_less_nat @ Ma @ X3 )
% 9.07/5.17         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.17            = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17              @ ( if_option_nat
% 9.07/5.17                @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17                   != none_nat )
% 9.07/5.17                  & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( 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 ) ) ) ) ) )
% 9.07/5.17                @ ( if_option_nat
% 9.07/5.17                  @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.17                    = none_nat )
% 9.07/5.17                  @ ( if_option_nat @ ( ord_less_nat @ Mi @ X3 ) @ ( some_nat @ Mi ) @ none_nat )
% 9.07/5.17                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.07/5.17              @ none_nat ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1274_nth__equalityI,axiom,
% 9.07/5.17      ! [Xs2: list_int,Ys: list_int] :
% 9.07/5.17        ( ( ( size_size_list_int @ Xs2 )
% 9.07/5.17          = ( size_size_list_int @ Ys ) )
% 9.07/5.17       => ( ! [I3: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.17             => ( ( nth_int @ Xs2 @ I3 )
% 9.07/5.17                = ( nth_int @ Ys @ I3 ) ) )
% 9.07/5.17         => ( Xs2 = Ys ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1275_nth__equalityI,axiom,
% 9.07/5.17      ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.07/5.17        ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.07/5.17          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.07/5.17       => ( ! [I3: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.17             => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 9.07/5.17                = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
% 9.07/5.17         => ( Xs2 = Ys ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1276_nth__equalityI,axiom,
% 9.07/5.17      ! [Xs2: list_o,Ys: list_o] :
% 9.07/5.17        ( ( ( size_size_list_o @ Xs2 )
% 9.07/5.17          = ( size_size_list_o @ Ys ) )
% 9.07/5.17       => ( ! [I3: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.17             => ( ( nth_o @ Xs2 @ I3 )
% 9.07/5.17                = ( nth_o @ Ys @ I3 ) ) )
% 9.07/5.17         => ( Xs2 = Ys ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1277_nth__equalityI,axiom,
% 9.07/5.17      ! [Xs2: list_nat,Ys: list_nat] :
% 9.07/5.17        ( ( ( size_size_list_nat @ Xs2 )
% 9.07/5.17          = ( size_size_list_nat @ Ys ) )
% 9.07/5.17       => ( ! [I3: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.17             => ( ( nth_nat @ Xs2 @ I3 )
% 9.07/5.17                = ( nth_nat @ Ys @ I3 ) ) )
% 9.07/5.17         => ( Xs2 = Ys ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1278_Skolem__list__nth,axiom,
% 9.07/5.17      ! [K: nat,P: nat > int > $o] :
% 9.07/5.17        ( ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17             => ? [X7: int] : ( P @ I4 @ X7 ) ) )
% 9.07/5.17        = ( ? [Xs: list_int] :
% 9.07/5.17              ( ( ( size_size_list_int @ Xs )
% 9.07/5.17                = K )
% 9.07/5.17              & ! [I4: nat] :
% 9.07/5.17                  ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17                 => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1279_Skolem__list__nth,axiom,
% 9.07/5.17      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 9.07/5.17        ( ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17             => ? [X7: vEBT_VEBT] : ( P @ I4 @ X7 ) ) )
% 9.07/5.17        = ( ? [Xs: list_VEBT_VEBT] :
% 9.07/5.17              ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.07/5.17                = K )
% 9.07/5.17              & ! [I4: nat] :
% 9.07/5.17                  ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17                 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1280_Skolem__list__nth,axiom,
% 9.07/5.17      ! [K: nat,P: nat > $o > $o] :
% 9.07/5.17        ( ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17             => ? [X7: $o] : ( P @ I4 @ X7 ) ) )
% 9.07/5.17        = ( ? [Xs: list_o] :
% 9.07/5.17              ( ( ( size_size_list_o @ Xs )
% 9.07/5.17                = K )
% 9.07/5.17              & ! [I4: nat] :
% 9.07/5.17                  ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17                 => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1281_Skolem__list__nth,axiom,
% 9.07/5.17      ! [K: nat,P: nat > nat > $o] :
% 9.07/5.17        ( ( ! [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17             => ? [X7: nat] : ( P @ I4 @ X7 ) ) )
% 9.07/5.17        = ( ? [Xs: list_nat] :
% 9.07/5.17              ( ( ( size_size_list_nat @ Xs )
% 9.07/5.17                = K )
% 9.07/5.17              & ! [I4: nat] :
% 9.07/5.17                  ( ( ord_less_nat @ I4 @ K )
% 9.07/5.17                 => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1282_list__eq__iff__nth__eq,axiom,
% 9.07/5.17      ( ( ^ [Y6: list_int,Z5: list_int] : ( Y6 = Z5 ) )
% 9.07/5.17      = ( ^ [Xs: list_int,Ys3: list_int] :
% 9.07/5.17            ( ( ( size_size_list_int @ Xs )
% 9.07/5.17              = ( size_size_list_int @ Ys3 ) )
% 9.07/5.17            & ! [I4: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.07/5.17               => ( ( nth_int @ Xs @ I4 )
% 9.07/5.17                  = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1283_list__eq__iff__nth__eq,axiom,
% 9.07/5.17      ( ( ^ [Y6: list_VEBT_VEBT,Z5: list_VEBT_VEBT] : ( Y6 = Z5 ) )
% 9.07/5.17      = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 9.07/5.17            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.07/5.17              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 9.07/5.17            & ! [I4: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.07/5.17               => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 9.07/5.17                  = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1284_list__eq__iff__nth__eq,axiom,
% 9.07/5.17      ( ( ^ [Y6: list_o,Z5: list_o] : ( Y6 = Z5 ) )
% 9.07/5.17      = ( ^ [Xs: list_o,Ys3: list_o] :
% 9.07/5.17            ( ( ( size_size_list_o @ Xs )
% 9.07/5.17              = ( size_size_list_o @ Ys3 ) )
% 9.07/5.17            & ! [I4: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.07/5.17               => ( ( nth_o @ Xs @ I4 )
% 9.07/5.17                  = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1285_list__eq__iff__nth__eq,axiom,
% 9.07/5.17      ( ( ^ [Y6: list_nat,Z5: list_nat] : ( Y6 = Z5 ) )
% 9.07/5.17      = ( ^ [Xs: list_nat,Ys3: list_nat] :
% 9.07/5.17            ( ( ( size_size_list_nat @ Xs )
% 9.07/5.17              = ( size_size_list_nat @ Ys3 ) )
% 9.07/5.17            & ! [I4: nat] :
% 9.07/5.17                ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.07/5.17               => ( ( nth_nat @ Xs @ I4 )
% 9.07/5.17                  = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1286_invar__vebt_Ointros_I3_J,axiom,
% 9.07/5.17      ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.07/5.17        ( ! [X5: vEBT_VEBT] :
% 9.07/5.17            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.17           => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.07/5.17       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.07/5.17         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.07/5.17              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.17           => ( ( M
% 9.07/5.17                = ( suc @ N ) )
% 9.07/5.17             => ( ( Deg
% 9.07/5.17                  = ( plus_plus_nat @ N @ M ) )
% 9.07/5.17               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.07/5.17                 => ( ! [X5: vEBT_VEBT] :
% 9.07/5.17                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.07/5.17                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 9.07/5.17                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1287_discrete,axiom,
% 9.07/5.17      ( ord_less_nat
% 9.07/5.17      = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1288_discrete,axiom,
% 9.07/5.17      ( ord_less_int
% 9.07/5.17      = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1289_nth__mem,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_real] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 9.07/5.17       => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1290_nth__mem,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_complex] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 9.07/5.17       => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1291_nth__mem,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_int] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.17       => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1292_nth__mem,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_VEBT_VEBT] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.17       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1293_nth__mem,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_o] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.17       => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1294_nth__mem,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_nat] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.17       => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1295_list__ball__nth,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_int,P: int > $o] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.17       => ( ! [X5: int] :
% 9.07/5.17              ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X5 ) )
% 9.07/5.17         => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1296_list__ball__nth,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.17       => ( ! [X5: vEBT_VEBT] :
% 9.07/5.17              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X5 ) )
% 9.07/5.17         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1297_list__ball__nth,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_o,P: $o > $o] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.17       => ( ! [X5: $o] :
% 9.07/5.17              ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X5 ) )
% 9.07/5.17         => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1298_list__ball__nth,axiom,
% 9.07/5.17      ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 9.07/5.17        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.17       => ( ! [X5: nat] :
% 9.07/5.17              ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 9.07/5.17             => ( P @ X5 ) )
% 9.07/5.17         => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1299_in__set__conv__nth,axiom,
% 9.07/5.17      ! [X3: real,Xs2: list_real] :
% 9.07/5.17        ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 9.07/5.17        = ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
% 9.07/5.17              & ( ( nth_real @ Xs2 @ I4 )
% 9.07/5.17                = X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1300_in__set__conv__nth,axiom,
% 9.07/5.17      ! [X3: complex,Xs2: list_complex] :
% 9.07/5.17        ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 9.07/5.17        = ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 9.07/5.17              & ( ( nth_complex @ Xs2 @ I4 )
% 9.07/5.17                = X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1301_in__set__conv__nth,axiom,
% 9.07/5.17      ! [X3: int,Xs2: list_int] :
% 9.07/5.17        ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 9.07/5.17        = ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.17              & ( ( nth_int @ Xs2 @ I4 )
% 9.07/5.17                = X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1302_in__set__conv__nth,axiom,
% 9.07/5.17      ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 9.07/5.17        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.07/5.17        = ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.17              & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 9.07/5.17                = X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1303_in__set__conv__nth,axiom,
% 9.07/5.17      ! [X3: $o,Xs2: list_o] :
% 9.07/5.17        ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 9.07/5.17        = ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.17              & ( ( nth_o @ Xs2 @ I4 )
% 9.07/5.17                = X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1304_in__set__conv__nth,axiom,
% 9.07/5.17      ! [X3: nat,Xs2: list_nat] :
% 9.07/5.17        ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 9.07/5.17        = ( ? [I4: nat] :
% 9.07/5.17              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.17              & ( ( nth_nat @ Xs2 @ I4 )
% 9.07/5.17                = X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1305_all__nth__imp__all__set,axiom,
% 9.07/5.17      ! [Xs2: list_real,P: real > $o,X3: real] :
% 9.07/5.17        ( ! [I3: nat] :
% 9.07/5.17            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
% 9.07/5.17           => ( P @ ( nth_real @ Xs2 @ I3 ) ) )
% 9.07/5.17       => ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 9.07/5.17         => ( P @ X3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1306_all__nth__imp__all__set,axiom,
% 9.07/5.17      ! [Xs2: list_complex,P: complex > $o,X3: complex] :
% 9.07/5.17        ( ! [I3: nat] :
% 9.07/5.17            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 9.07/5.17           => ( P @ ( nth_complex @ Xs2 @ I3 ) ) )
% 9.07/5.17       => ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 9.07/5.17         => ( P @ X3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1307_all__nth__imp__all__set,axiom,
% 9.07/5.17      ! [Xs2: list_int,P: int > $o,X3: int] :
% 9.07/5.17        ( ! [I3: nat] :
% 9.07/5.17            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 9.07/5.17           => ( P @ ( nth_int @ Xs2 @ I3 ) ) )
% 9.07/5.17       => ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 9.07/5.17         => ( P @ X3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1308_all__nth__imp__all__set,axiom,
% 9.07/5.17      ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 9.07/5.17        ( ! [I3: nat] :
% 9.07/5.17            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.07/5.17           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
% 9.07/5.17       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.07/5.17         => ( P @ X3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1309_all__nth__imp__all__set,axiom,
% 9.07/5.17      ! [Xs2: list_o,P: $o > $o,X3: $o] :
% 9.07/5.17        ( ! [I3: nat] :
% 9.07/5.17            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 9.07/5.17           => ( P @ ( nth_o @ Xs2 @ I3 ) ) )
% 9.07/5.17       => ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 9.07/5.17         => ( P @ X3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1310_all__nth__imp__all__set,axiom,
% 9.07/5.17      ! [Xs2: list_nat,P: nat > $o,X3: nat] :
% 9.07/5.17        ( ! [I3: nat] :
% 9.07/5.17            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
% 9.07/5.17           => ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
% 9.07/5.17       => ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 9.07/5.17         => ( P @ X3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1311_vebt__member_Osimps_I5_J,axiom,
% 9.07/5.17      ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.07/5.17        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.17        = ( ( X3 != Mi )
% 9.07/5.17         => ( ( X3 != Ma )
% 9.07/5.17           => ( ~ ( ord_less_nat @ X3 @ Mi )
% 9.07/5.17              & ( ~ ( ord_less_nat @ X3 @ Mi )
% 9.07/5.17               => ( ~ ( ord_less_nat @ Ma @ X3 )
% 9.07/5.17                  & ( ~ ( ord_less_nat @ Ma @ X3 )
% 9.07/5.17                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17                       => ( 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 ) ) ) ) ) )
% 9.07/5.17                      & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1312_vebt__insert_Osimps_I4_J,axiom,
% 9.07/5.17      ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.07/5.17        ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X3 )
% 9.07/5.17        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1313_buildup__gives__empty,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 9.07/5.17        = bot_bot_set_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1314_double__not__eq__Suc__double,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.07/5.17       != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1315_Suc__double__not__eq__double,axiom,
% 9.07/5.17      ! [M: nat,N: nat] :
% 9.07/5.17        ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.07/5.17       != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1316_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 9.07/5.17      ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 9.07/5.17        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X3 )
% 9.07/5.17        = ( ( X3 = Mi )
% 9.07/5.17          | ( X3 = Ma )
% 9.07/5.17          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17             => ( 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 ) ) ) ) ) )
% 9.07/5.17            & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1317_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 9.07/5.17      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 9.07/5.17        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X3 )
% 9.07/5.17        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17           => ( 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 ) ) ) ) ) )
% 9.07/5.17          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1318_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 9.07/5.17      ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
% 9.07/5.17        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X3 )
% 9.07/5.17        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.07/5.17           => ( 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 ) ) ) ) ) )
% 9.07/5.17          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1319_vebt__pred_Oelims,axiom,
% 9.07/5.17      ! [X3: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 9.07/5.17        ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
% 9.07/5.17          = Y3 )
% 9.07/5.17       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.07/5.17                ( X3
% 9.07/5.17                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.07/5.17           => ( ( Xa2 = zero_zero_nat )
% 9.07/5.17             => ( Y3 != none_nat ) ) )
% 9.07/5.17         => ( ! [A3: $o] :
% 9.07/5.17                ( ? [Uw2: $o] :
% 9.07/5.17                    ( X3
% 9.07/5.17                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 9.07/5.17               => ( ( Xa2
% 9.07/5.17                    = ( suc @ zero_zero_nat ) )
% 9.07/5.17                 => ~ ( ( A3
% 9.07/5.17                       => ( Y3
% 9.07/5.17                          = ( some_nat @ zero_zero_nat ) ) )
% 9.07/5.17                      & ( ~ A3
% 9.07/5.17                       => ( Y3 = none_nat ) ) ) ) )
% 9.07/5.17           => ( ! [A3: $o,B3: $o] :
% 9.07/5.17                  ( ( X3
% 9.07/5.17                    = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.07/5.17                 => ( ? [Va2: nat] :
% 9.07/5.17                        ( Xa2
% 9.07/5.17                        = ( suc @ ( suc @ Va2 ) ) )
% 9.07/5.17                   => ~ ( ( B3
% 9.07/5.17                         => ( Y3
% 9.07/5.17                            = ( some_nat @ one_one_nat ) ) )
% 9.07/5.17                        & ( ~ B3
% 9.07/5.17                         => ( ( A3
% 9.07/5.17                             => ( Y3
% 9.07/5.17                                = ( some_nat @ zero_zero_nat ) ) )
% 9.07/5.17                            & ( ~ A3
% 9.07/5.17                             => ( Y3 = none_nat ) ) ) ) ) ) )
% 9.07/5.17             => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 9.07/5.17                      ( X3
% 9.07/5.17                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 9.07/5.17                 => ( Y3 != none_nat ) )
% 9.07/5.17               => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 9.07/5.17                        ( X3
% 9.07/5.17                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 9.07/5.17                   => ( Y3 != none_nat ) )
% 9.07/5.17                 => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 9.07/5.17                          ( X3
% 9.07/5.17                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 9.07/5.17                     => ( Y3 != none_nat ) )
% 9.07/5.17                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.07/5.17                          ( ( X3
% 9.07/5.17                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.07/5.17                         => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 9.07/5.17                               => ( Y3
% 9.07/5.17                                  = ( some_nat @ Ma2 ) ) )
% 9.07/5.17                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.07/5.17                               => ( Y3
% 9.07/5.17                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.07/5.17                                    @ ( if_option_nat
% 9.07/5.17                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17                                         != none_nat )
% 9.07/5.17                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.07/5.17                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.07/5.17                                      @ ( if_option_nat
% 9.07/5.17                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.07/5.17                                          = none_nat )
% 9.07/5.17                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 9.07/5.17                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.07/5.17                                    @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1320_low__def,axiom,
% 9.07/5.17      ( vEBT_VEBT_low
% 9.07/5.17      = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1321_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 9.07/5.17      ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1322_valid__tree__deg__neq__0,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT] :
% 9.07/5.17        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1323_valid__0__not,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT] :
% 9.07/5.17        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1324_buildup__nothing__in__min__max,axiom,
% 9.07/5.17      ! [N: nat,X3: nat] :
% 9.07/5.17        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1325_buildup__nothing__in__leaf,axiom,
% 9.07/5.17      ! [N: nat,X3: nat] :
% 9.07/5.17        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1326_deg__not__0,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT,N: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.17       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1327_Leaf__0__not,axiom,
% 9.07/5.17      ! [A: $o,B2: $o] :
% 9.07/5.17        ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1328_deg__1__Leafy,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT,N: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ T @ N )
% 9.07/5.17       => ( ( N = one_one_nat )
% 9.07/5.17         => ? [A3: $o,B3: $o] :
% 9.07/5.17              ( T
% 9.07/5.17              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1329_deg__1__Leaf,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.07/5.17       => ? [A3: $o,B3: $o] :
% 9.07/5.17            ( T
% 9.07/5.17            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1330_deg1Leaf,axiom,
% 9.07/5.17      ! [T: vEBT_VEBT] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.07/5.17        = ( ? [A5: $o,B4: $o] :
% 9.07/5.17              ( T
% 9.07/5.17              = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1331_both__member__options__def,axiom,
% 9.07/5.17      ( vEBT_V8194947554948674370ptions
% 9.07/5.17      = ( ^ [T2: vEBT_VEBT,X: nat] :
% 9.07/5.17            ( ( vEBT_V5719532721284313246member @ T2 @ X )
% 9.07/5.17            | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1332_buildup__gives__valid,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.17       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1333_mod__mod__trivial,axiom,
% 9.07/5.17      ! [A: nat,B2: nat] :
% 9.07/5.17        ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 9.07/5.17        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1334_mod__mod__trivial,axiom,
% 9.07/5.17      ! [A: int,B2: int] :
% 9.07/5.17        ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 9.07/5.17        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1335_mod__mod__trivial,axiom,
% 9.07/5.17      ! [A: code_integer,B2: code_integer] :
% 9.07/5.17        ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 9.07/5.17        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1336_member__valid__both__member__options,axiom,
% 9.07/5.17      ! [Tree: vEBT_VEBT,N: nat,X3: nat] :
% 9.07/5.17        ( ( vEBT_invar_vebt @ Tree @ N )
% 9.07/5.17       => ( ( vEBT_vebt_member @ Tree @ X3 )
% 9.07/5.17         => ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
% 9.07/5.17            | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1337_le__zero__eq,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 9.07/5.17        = ( N = zero_zero_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1338_not__gr__zero,axiom,
% 9.07/5.17      ! [N: nat] :
% 9.07/5.17        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 9.07/5.17        = ( N = zero_zero_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1339_mult__zero__left,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( times_times_complex @ zero_zero_complex @ A )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1340_mult__zero__left,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( times_times_real @ zero_zero_real @ A )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1341_mult__zero__left,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( times_times_rat @ zero_zero_rat @ A )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1342_mult__zero__left,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( times_times_nat @ zero_zero_nat @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1343_mult__zero__left,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( times_times_int @ zero_zero_int @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1344_mult__zero__right,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( times_times_complex @ A @ zero_zero_complex )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1345_mult__zero__right,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( times_times_real @ A @ zero_zero_real )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1346_mult__zero__right,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( times_times_rat @ A @ zero_zero_rat )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1347_mult__zero__right,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( times_times_nat @ A @ zero_zero_nat )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1348_mult__zero__right,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( times_times_int @ A @ zero_zero_int )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1349_mult__eq__0__iff,axiom,
% 9.07/5.17      ! [A: complex,B2: complex] :
% 9.07/5.17        ( ( ( times_times_complex @ A @ B2 )
% 9.07/5.17          = zero_zero_complex )
% 9.07/5.17        = ( ( A = zero_zero_complex )
% 9.07/5.17          | ( B2 = zero_zero_complex ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1350_mult__eq__0__iff,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( ( times_times_real @ A @ B2 )
% 9.07/5.17          = zero_zero_real )
% 9.07/5.17        = ( ( A = zero_zero_real )
% 9.07/5.17          | ( B2 = zero_zero_real ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1351_mult__eq__0__iff,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( ( times_times_rat @ A @ B2 )
% 9.07/5.17          = zero_zero_rat )
% 9.07/5.17        = ( ( A = zero_zero_rat )
% 9.07/5.17          | ( B2 = zero_zero_rat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1352_mult__eq__0__iff,axiom,
% 9.07/5.17      ! [A: nat,B2: nat] :
% 9.07/5.17        ( ( ( times_times_nat @ A @ B2 )
% 9.07/5.17          = zero_zero_nat )
% 9.07/5.17        = ( ( A = zero_zero_nat )
% 9.07/5.17          | ( B2 = zero_zero_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1353_mult__eq__0__iff,axiom,
% 9.07/5.17      ! [A: int,B2: int] :
% 9.07/5.17        ( ( ( times_times_int @ A @ B2 )
% 9.07/5.17          = zero_zero_int )
% 9.07/5.17        = ( ( A = zero_zero_int )
% 9.07/5.17          | ( B2 = zero_zero_int ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1354_mult__cancel__left,axiom,
% 9.07/5.17      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.17        ( ( ( times_times_complex @ C @ A )
% 9.07/5.17          = ( times_times_complex @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_complex )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1355_mult__cancel__left,axiom,
% 9.07/5.17      ! [C: real,A: real,B2: real] :
% 9.07/5.17        ( ( ( times_times_real @ C @ A )
% 9.07/5.17          = ( times_times_real @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_real )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1356_mult__cancel__left,axiom,
% 9.07/5.17      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.17        ( ( ( times_times_rat @ C @ A )
% 9.07/5.17          = ( times_times_rat @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_rat )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1357_mult__cancel__left,axiom,
% 9.07/5.17      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.17        ( ( ( times_times_nat @ C @ A )
% 9.07/5.17          = ( times_times_nat @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_nat )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1358_mult__cancel__left,axiom,
% 9.07/5.17      ! [C: int,A: int,B2: int] :
% 9.07/5.17        ( ( ( times_times_int @ C @ A )
% 9.07/5.17          = ( times_times_int @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_int )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1359_mult__cancel__right,axiom,
% 9.07/5.17      ! [A: complex,C: complex,B2: complex] :
% 9.07/5.17        ( ( ( times_times_complex @ A @ C )
% 9.07/5.17          = ( times_times_complex @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_complex )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1360_mult__cancel__right,axiom,
% 9.07/5.17      ! [A: real,C: real,B2: real] :
% 9.07/5.17        ( ( ( times_times_real @ A @ C )
% 9.07/5.17          = ( times_times_real @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_real )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1361_mult__cancel__right,axiom,
% 9.07/5.17      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.17        ( ( ( times_times_rat @ A @ C )
% 9.07/5.17          = ( times_times_rat @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_rat )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1362_mult__cancel__right,axiom,
% 9.07/5.17      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.17        ( ( ( times_times_nat @ A @ C )
% 9.07/5.17          = ( times_times_nat @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_nat )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1363_mult__cancel__right,axiom,
% 9.07/5.17      ! [A: int,C: int,B2: int] :
% 9.07/5.17        ( ( ( times_times_int @ A @ C )
% 9.07/5.17          = ( times_times_int @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_int )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1364_add__0,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1365_add__0,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1366_add__0,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1367_add__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1368_add__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1369_zero__eq__add__iff__both__eq__0,axiom,
% 9.07/5.17      ! [X3: nat,Y3: nat] :
% 9.07/5.17        ( ( zero_zero_nat
% 9.07/5.17          = ( plus_plus_nat @ X3 @ Y3 ) )
% 9.07/5.17        = ( ( X3 = zero_zero_nat )
% 9.07/5.17          & ( Y3 = zero_zero_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1370_add__eq__0__iff__both__eq__0,axiom,
% 9.07/5.17      ! [X3: nat,Y3: nat] :
% 9.07/5.17        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.07/5.17          = zero_zero_nat )
% 9.07/5.17        = ( ( X3 = zero_zero_nat )
% 9.07/5.17          & ( Y3 = zero_zero_nat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1371_add__cancel__right__right,axiom,
% 9.07/5.17      ! [A: complex,B2: complex] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_complex @ A @ B2 ) )
% 9.07/5.17        = ( B2 = zero_zero_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1372_add__cancel__right__right,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_real @ A @ B2 ) )
% 9.07/5.17        = ( B2 = zero_zero_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1373_add__cancel__right__right,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_rat @ A @ B2 ) )
% 9.07/5.17        = ( B2 = zero_zero_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1374_add__cancel__right__right,axiom,
% 9.07/5.17      ! [A: nat,B2: nat] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_nat @ A @ B2 ) )
% 9.07/5.17        = ( B2 = zero_zero_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1375_add__cancel__right__right,axiom,
% 9.07/5.17      ! [A: int,B2: int] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_int @ A @ B2 ) )
% 9.07/5.17        = ( B2 = zero_zero_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1376_add__cancel__right__left,axiom,
% 9.07/5.17      ! [A: complex,B2: complex] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_complex @ B2 @ A ) )
% 9.07/5.17        = ( B2 = zero_zero_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1377_add__cancel__right__left,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_real @ B2 @ A ) )
% 9.07/5.17        = ( B2 = zero_zero_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1378_add__cancel__right__left,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_rat @ B2 @ A ) )
% 9.07/5.17        = ( B2 = zero_zero_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1379_add__cancel__right__left,axiom,
% 9.07/5.17      ! [A: nat,B2: nat] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_nat @ B2 @ A ) )
% 9.07/5.17        = ( B2 = zero_zero_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1380_add__cancel__right__left,axiom,
% 9.07/5.17      ! [A: int,B2: int] :
% 9.07/5.17        ( ( A
% 9.07/5.17          = ( plus_plus_int @ B2 @ A ) )
% 9.07/5.17        = ( B2 = zero_zero_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1381_add__cancel__left__right,axiom,
% 9.07/5.17      ! [A: complex,B2: complex] :
% 9.07/5.17        ( ( ( plus_plus_complex @ A @ B2 )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1382_add__cancel__left__right,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( ( plus_plus_real @ A @ B2 )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1383_add__cancel__left__right,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( ( plus_plus_rat @ A @ B2 )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1384_add__cancel__left__right,axiom,
% 9.07/5.17      ! [A: nat,B2: nat] :
% 9.07/5.17        ( ( ( plus_plus_nat @ A @ B2 )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1385_add__cancel__left__right,axiom,
% 9.07/5.17      ! [A: int,B2: int] :
% 9.07/5.17        ( ( ( plus_plus_int @ A @ B2 )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1386_add__cancel__left__left,axiom,
% 9.07/5.17      ! [B2: complex,A: complex] :
% 9.07/5.17        ( ( ( plus_plus_complex @ B2 @ A )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_complex ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1387_add__cancel__left__left,axiom,
% 9.07/5.17      ! [B2: real,A: real] :
% 9.07/5.17        ( ( ( plus_plus_real @ B2 @ A )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1388_add__cancel__left__left,axiom,
% 9.07/5.17      ! [B2: rat,A: rat] :
% 9.07/5.17        ( ( ( plus_plus_rat @ B2 @ A )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1389_add__cancel__left__left,axiom,
% 9.07/5.17      ! [B2: nat,A: nat] :
% 9.07/5.17        ( ( ( plus_plus_nat @ B2 @ A )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_nat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1390_add__cancel__left__left,axiom,
% 9.07/5.17      ! [B2: int,A: int] :
% 9.07/5.17        ( ( ( plus_plus_int @ B2 @ A )
% 9.07/5.17          = A )
% 9.07/5.17        = ( B2 = zero_zero_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1391_double__zero__sym,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( zero_zero_real
% 9.07/5.17          = ( plus_plus_real @ A @ A ) )
% 9.07/5.17        = ( A = zero_zero_real ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1392_double__zero__sym,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( zero_zero_rat
% 9.07/5.17          = ( plus_plus_rat @ A @ A ) )
% 9.07/5.17        = ( A = zero_zero_rat ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1393_double__zero__sym,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( zero_zero_int
% 9.07/5.17          = ( plus_plus_int @ A @ A ) )
% 9.07/5.17        = ( A = zero_zero_int ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1394_add_Oright__neutral,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1395_add_Oright__neutral,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1396_add_Oright__neutral,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1397_add_Oright__neutral,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1398_add_Oright__neutral,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1399_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( minus_minus_complex @ A @ A )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1400_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( minus_minus_real @ A @ A )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1401_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( minus_minus_rat @ A @ A )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1402_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ A @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1403_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( minus_minus_int @ A @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1404_diff__zero,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1405_diff__zero,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1406_diff__zero,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1407_diff__zero,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1408_diff__zero,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1409_zero__diff,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1410_diff__0__right,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1411_diff__0__right,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1412_diff__0__right,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1413_diff__0__right,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1414_diff__self,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( minus_minus_complex @ A @ A )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1415_diff__self,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( minus_minus_real @ A @ A )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1416_diff__self,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( minus_minus_rat @ A @ A )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1417_diff__self,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( minus_minus_int @ A @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1418_divide__eq__0__iff,axiom,
% 9.07/5.17      ! [A: complex,B2: complex] :
% 9.07/5.17        ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 9.07/5.17          = zero_zero_complex )
% 9.07/5.17        = ( ( A = zero_zero_complex )
% 9.07/5.17          | ( B2 = zero_zero_complex ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1419_divide__eq__0__iff,axiom,
% 9.07/5.17      ! [A: real,B2: real] :
% 9.07/5.17        ( ( ( divide_divide_real @ A @ B2 )
% 9.07/5.17          = zero_zero_real )
% 9.07/5.17        = ( ( A = zero_zero_real )
% 9.07/5.17          | ( B2 = zero_zero_real ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1420_divide__eq__0__iff,axiom,
% 9.07/5.17      ! [A: rat,B2: rat] :
% 9.07/5.17        ( ( ( divide_divide_rat @ A @ B2 )
% 9.07/5.17          = zero_zero_rat )
% 9.07/5.17        = ( ( A = zero_zero_rat )
% 9.07/5.17          | ( B2 = zero_zero_rat ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1421_divide__cancel__left,axiom,
% 9.07/5.17      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.17        ( ( ( divide1717551699836669952omplex @ C @ A )
% 9.07/5.17          = ( divide1717551699836669952omplex @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_complex )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1422_divide__cancel__left,axiom,
% 9.07/5.17      ! [C: real,A: real,B2: real] :
% 9.07/5.17        ( ( ( divide_divide_real @ C @ A )
% 9.07/5.17          = ( divide_divide_real @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_real )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1423_divide__cancel__left,axiom,
% 9.07/5.17      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.17        ( ( ( divide_divide_rat @ C @ A )
% 9.07/5.17          = ( divide_divide_rat @ C @ B2 ) )
% 9.07/5.17        = ( ( C = zero_zero_rat )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1424_divide__cancel__right,axiom,
% 9.07/5.17      ! [A: complex,C: complex,B2: complex] :
% 9.07/5.17        ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.07/5.17          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_complex )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1425_divide__cancel__right,axiom,
% 9.07/5.17      ! [A: real,C: real,B2: real] :
% 9.07/5.17        ( ( ( divide_divide_real @ A @ C )
% 9.07/5.17          = ( divide_divide_real @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_real )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1426_divide__cancel__right,axiom,
% 9.07/5.17      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.17        ( ( ( divide_divide_rat @ A @ C )
% 9.07/5.17          = ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.17        = ( ( C = zero_zero_rat )
% 9.07/5.17          | ( A = B2 ) ) ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1427_division__ring__divide__zero,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1428_division__ring__divide__zero,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1429_division__ring__divide__zero,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1430_div__0,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1431_div__0,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( divide_divide_real @ zero_zero_real @ A )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1432_div__0,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1433_div__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1434_div__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1435_div__by__0,axiom,
% 9.07/5.17      ! [A: complex] :
% 9.07/5.17        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.07/5.17        = zero_zero_complex ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1436_div__by__0,axiom,
% 9.07/5.17      ! [A: real] :
% 9.07/5.17        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.07/5.17        = zero_zero_real ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1437_div__by__0,axiom,
% 9.07/5.17      ! [A: rat] :
% 9.07/5.17        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.07/5.17        = zero_zero_rat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1438_div__by__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1439_div__by__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1440_bits__div__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1441_bits__div__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1442_bits__div__by__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1443_bits__div__by__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1444_mod__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1445_mod__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1446_mod__0,axiom,
% 9.07/5.17      ! [A: code_integer] :
% 9.07/5.17        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.07/5.17        = zero_z3403309356797280102nteger ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1447_mod__by__0,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1448_mod__by__0,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1449_mod__by__0,axiom,
% 9.07/5.17      ! [A: code_integer] :
% 9.07/5.17        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 9.07/5.17        = A ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1450_mod__self,axiom,
% 9.07/5.17      ! [A: nat] :
% 9.07/5.17        ( ( modulo_modulo_nat @ A @ A )
% 9.07/5.17        = zero_zero_nat ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1451_mod__self,axiom,
% 9.07/5.17      ! [A: int] :
% 9.07/5.17        ( ( modulo_modulo_int @ A @ A )
% 9.07/5.17        = zero_zero_int ) ).
% 9.07/5.17  
% 9.07/5.17  thf(fact_1452_mod__self,axiom,
% 9.07/5.17      ! [A: code_integer] :
% 9.07/5.17        ( ( modulo364778990260209775nteger @ A @ A )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1453_bits__mod__0,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1454_bits__mod__0,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1455_bits__mod__0,axiom,
% 9.07/5.18      ! [A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1456_less__nat__zero__code,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1457_neq0__conv,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( N != zero_zero_nat )
% 9.07/5.18        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1458_bot__nat__0_Onot__eq__extremum,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( A != zero_zero_nat )
% 9.07/5.18        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1459_le0,axiom,
% 9.07/5.18      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1460_bot__nat__0_Oextremum,axiom,
% 9.07/5.18      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1461_mod__add__self1,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1462_mod__add__self1,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1463_mod__add__self1,axiom,
% 9.07/5.18      ! [B2: code_integer,A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1464_mod__add__self2,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1465_mod__add__self2,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1466_mod__add__self2,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1467_minus__mod__self2,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1468_minus__mod__self2,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1469_Nat_Oadd__0__right,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 9.07/5.18        = M ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1470_add__is__0,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ( plus_plus_nat @ M @ N )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18        = ( ( M = zero_zero_nat )
% 9.07/5.18          & ( N = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1471_diff__self__eq__0,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( minus_minus_nat @ M @ M )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1472_diff__0__eq__0,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( minus_minus_nat @ zero_zero_nat @ N )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1473_mult__is__0,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ( times_times_nat @ M @ N )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18        = ( ( M = zero_zero_nat )
% 9.07/5.18          | ( N = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1474_mult__0__right,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( times_times_nat @ M @ zero_zero_nat )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1475_mult__cancel1,axiom,
% 9.07/5.18      ! [K: nat,M: nat,N: nat] :
% 9.07/5.18        ( ( ( times_times_nat @ K @ M )
% 9.07/5.18          = ( times_times_nat @ K @ N ) )
% 9.07/5.18        = ( ( M = N )
% 9.07/5.18          | ( K = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1476_mult__cancel2,axiom,
% 9.07/5.18      ! [M: nat,K: nat,N: nat] :
% 9.07/5.18        ( ( ( times_times_nat @ M @ K )
% 9.07/5.18          = ( times_times_nat @ N @ K ) )
% 9.07/5.18        = ( ( M = N )
% 9.07/5.18          | ( K = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1477_mod__less,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ M @ N )
% 9.07/5.18       => ( ( modulo_modulo_nat @ M @ N )
% 9.07/5.18          = M ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1478_add__le__same__cancel1,axiom,
% 9.07/5.18      ! [B2: real,A: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1479_add__le__same__cancel1,axiom,
% 9.07/5.18      ! [B2: rat,A: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1480_add__le__same__cancel1,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1481_add__le__same__cancel1,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1482_add__le__same__cancel2,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1483_add__le__same__cancel2,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1484_add__le__same__cancel2,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1485_add__le__same__cancel2,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1486_le__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1487_le__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1488_le__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1489_le__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1490_le__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1491_le__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1492_le__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1493_le__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1494_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.07/5.18        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1495_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.07/5.18        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1496_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.07/5.18        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1497_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.07/5.18        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1498_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.07/5.18        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1499_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.07/5.18        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1500_diff__ge__0__iff__ge,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_real @ B2 @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1501_diff__ge__0__iff__ge,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_rat @ B2 @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1502_diff__ge__0__iff__ge,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_eq_int @ B2 @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1503_add__less__same__cancel1,axiom,
% 9.07/5.18      ! [B2: real,A: real] :
% 9.07/5.18        ( ( ord_less_real @ ( plus_plus_real @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1504_add__less__same__cancel1,axiom,
% 9.07/5.18      ! [B2: rat,A: rat] :
% 9.07/5.18        ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1505_add__less__same__cancel1,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1506_add__less__same__cancel1,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 9.07/5.18        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1507_add__less__same__cancel2,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1508_add__less__same__cancel2,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1509_add__less__same__cancel2,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1510_add__less__same__cancel2,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1511_less__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1512_less__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1513_less__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1514_less__add__same__cancel1,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1515_less__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_real @ zero_zero_real @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1516_less__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1517_less__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1518_less__add__same__cancel2,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 9.07/5.18        = ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1519_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.07/5.18        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1520_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.07/5.18        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1521_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.07/5.18        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1522_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.07/5.18        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1523_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.07/5.18        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1524_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.07/5.18        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1525_diff__gt__0__iff__gt,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_real @ B2 @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1526_diff__gt__0__iff__gt,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_rat @ B2 @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1527_diff__gt__0__iff__gt,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B2 ) )
% 9.07/5.18        = ( ord_less_int @ B2 @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1528_mult__cancel__left1,axiom,
% 9.07/5.18      ! [C: complex,B2: complex] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_complex @ C @ B2 ) )
% 9.07/5.18        = ( ( C = zero_zero_complex )
% 9.07/5.18          | ( B2 = one_one_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1529_mult__cancel__left1,axiom,
% 9.07/5.18      ! [C: real,B2: real] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_real @ C @ B2 ) )
% 9.07/5.18        = ( ( C = zero_zero_real )
% 9.07/5.18          | ( B2 = one_one_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1530_mult__cancel__left1,axiom,
% 9.07/5.18      ! [C: rat,B2: rat] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_rat @ C @ B2 ) )
% 9.07/5.18        = ( ( C = zero_zero_rat )
% 9.07/5.18          | ( B2 = one_one_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1531_mult__cancel__left1,axiom,
% 9.07/5.18      ! [C: int,B2: int] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_int @ C @ B2 ) )
% 9.07/5.18        = ( ( C = zero_zero_int )
% 9.07/5.18          | ( B2 = one_one_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1532_mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: complex,A: complex] :
% 9.07/5.18        ( ( ( times_times_complex @ C @ A )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_complex )
% 9.07/5.18          | ( A = one_one_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1533_mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: real,A: real] :
% 9.07/5.18        ( ( ( times_times_real @ C @ A )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_real )
% 9.07/5.18          | ( A = one_one_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1534_mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: rat,A: rat] :
% 9.07/5.18        ( ( ( times_times_rat @ C @ A )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_rat )
% 9.07/5.18          | ( A = one_one_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1535_mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: int,A: int] :
% 9.07/5.18        ( ( ( times_times_int @ C @ A )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_int )
% 9.07/5.18          | ( A = one_one_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1536_mult__cancel__right1,axiom,
% 9.07/5.18      ! [C: complex,B2: complex] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_complex @ B2 @ C ) )
% 9.07/5.18        = ( ( C = zero_zero_complex )
% 9.07/5.18          | ( B2 = one_one_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1537_mult__cancel__right1,axiom,
% 9.07/5.18      ! [C: real,B2: real] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_real @ B2 @ C ) )
% 9.07/5.18        = ( ( C = zero_zero_real )
% 9.07/5.18          | ( B2 = one_one_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1538_mult__cancel__right1,axiom,
% 9.07/5.18      ! [C: rat,B2: rat] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_rat @ B2 @ C ) )
% 9.07/5.18        = ( ( C = zero_zero_rat )
% 9.07/5.18          | ( B2 = one_one_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1539_mult__cancel__right1,axiom,
% 9.07/5.18      ! [C: int,B2: int] :
% 9.07/5.18        ( ( C
% 9.07/5.18          = ( times_times_int @ B2 @ C ) )
% 9.07/5.18        = ( ( C = zero_zero_int )
% 9.07/5.18          | ( B2 = one_one_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1540_mult__cancel__right2,axiom,
% 9.07/5.18      ! [A: complex,C: complex] :
% 9.07/5.18        ( ( ( times_times_complex @ A @ C )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_complex )
% 9.07/5.18          | ( A = one_one_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1541_mult__cancel__right2,axiom,
% 9.07/5.18      ! [A: real,C: real] :
% 9.07/5.18        ( ( ( times_times_real @ A @ C )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_real )
% 9.07/5.18          | ( A = one_one_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1542_mult__cancel__right2,axiom,
% 9.07/5.18      ! [A: rat,C: rat] :
% 9.07/5.18        ( ( ( times_times_rat @ A @ C )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_rat )
% 9.07/5.18          | ( A = one_one_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1543_mult__cancel__right2,axiom,
% 9.07/5.18      ! [A: int,C: int] :
% 9.07/5.18        ( ( ( times_times_int @ A @ C )
% 9.07/5.18          = C )
% 9.07/5.18        = ( ( C = zero_zero_int )
% 9.07/5.18          | ( A = one_one_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1544_sum__squares__eq__zero__iff,axiom,
% 9.07/5.18      ! [X3: real,Y3: real] :
% 9.07/5.18        ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 9.07/5.18          = zero_zero_real )
% 9.07/5.18        = ( ( X3 = zero_zero_real )
% 9.07/5.18          & ( Y3 = zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1545_sum__squares__eq__zero__iff,axiom,
% 9.07/5.18      ! [X3: rat,Y3: rat] :
% 9.07/5.18        ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 9.07/5.18          = zero_zero_rat )
% 9.07/5.18        = ( ( X3 = zero_zero_rat )
% 9.07/5.18          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1546_sum__squares__eq__zero__iff,axiom,
% 9.07/5.18      ! [X3: int,Y3: int] :
% 9.07/5.18        ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 9.07/5.18          = zero_zero_int )
% 9.07/5.18        = ( ( X3 = zero_zero_int )
% 9.07/5.18          & ( Y3 = zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1547_diff__numeral__special_I9_J,axiom,
% 9.07/5.18      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 9.07/5.18      = zero_zero_complex ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1548_diff__numeral__special_I9_J,axiom,
% 9.07/5.18      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 9.07/5.18      = zero_zero_real ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1549_diff__numeral__special_I9_J,axiom,
% 9.07/5.18      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 9.07/5.18      = zero_zero_rat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1550_diff__numeral__special_I9_J,axiom,
% 9.07/5.18      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 9.07/5.18      = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1551_mult__divide__mult__cancel__left__if,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( ( C = zero_zero_complex )
% 9.07/5.18         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 9.07/5.18            = zero_zero_complex ) )
% 9.07/5.18        & ( ( C != zero_zero_complex )
% 9.07/5.18         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 9.07/5.18            = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1552_mult__divide__mult__cancel__left__if,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( ( C = zero_zero_real )
% 9.07/5.18         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.18            = zero_zero_real ) )
% 9.07/5.18        & ( ( C != zero_zero_real )
% 9.07/5.18         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.18            = ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1553_mult__divide__mult__cancel__left__if,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( ( C = zero_zero_rat )
% 9.07/5.18         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.18            = zero_zero_rat ) )
% 9.07/5.18        & ( ( C != zero_zero_rat )
% 9.07/5.18         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.18            = ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1554_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( C != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 9.07/5.18          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1555_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( C != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.18          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1556_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( C != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.18          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1557_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( C != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B2 @ C ) )
% 9.07/5.18          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1558_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( C != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.18          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1559_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( C != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.18          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1560_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( C != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
% 9.07/5.18          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1561_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( C != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.18          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1562_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( C != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.18          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1563_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( C != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B2 ) )
% 9.07/5.18          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1564_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( C != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.18          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1565_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( C != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.18          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1566_div__mult__mult1__if,axiom,
% 9.07/5.18      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.18        ( ( ( C = zero_zero_nat )
% 9.07/5.18         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 9.07/5.18            = zero_zero_nat ) )
% 9.07/5.18        & ( ( C != zero_zero_nat )
% 9.07/5.18         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 9.07/5.18            = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1567_div__mult__mult1__if,axiom,
% 9.07/5.18      ! [C: int,A: int,B2: int] :
% 9.07/5.18        ( ( ( C = zero_zero_int )
% 9.07/5.18         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.18            = zero_zero_int ) )
% 9.07/5.18        & ( ( C != zero_zero_int )
% 9.07/5.18         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.18            = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1568_div__mult__mult2,axiom,
% 9.07/5.18      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.18        ( ( C != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 9.07/5.18          = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1569_div__mult__mult2,axiom,
% 9.07/5.18      ! [C: int,A: int,B2: int] :
% 9.07/5.18        ( ( C != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.18          = ( divide_divide_int @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1570_div__mult__mult1,axiom,
% 9.07/5.18      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.18        ( ( C != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 9.07/5.18          = ( divide_divide_nat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1571_div__mult__mult1,axiom,
% 9.07/5.18      ! [C: int,A: int,B2: int] :
% 9.07/5.18        ( ( C != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.18          = ( divide_divide_int @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1572_nonzero__mult__div__cancel__left,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( A != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ A )
% 9.07/5.18          = B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1573_nonzero__mult__div__cancel__left,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( A != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ A )
% 9.07/5.18          = B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1574_nonzero__mult__div__cancel__left,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( A != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ A )
% 9.07/5.18          = B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1575_nonzero__mult__div__cancel__left,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( A != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ A )
% 9.07/5.18          = B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1576_nonzero__mult__div__cancel__left,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( A != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ A )
% 9.07/5.18          = B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1577_nonzero__mult__div__cancel__right,axiom,
% 9.07/5.18      ! [B2: complex,A: complex] :
% 9.07/5.18        ( ( B2 != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B2 ) @ B2 )
% 9.07/5.18          = A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1578_nonzero__mult__div__cancel__right,axiom,
% 9.07/5.18      ! [B2: real,A: real] :
% 9.07/5.18        ( ( B2 != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ ( times_times_real @ A @ B2 ) @ B2 )
% 9.07/5.18          = A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1579_nonzero__mult__div__cancel__right,axiom,
% 9.07/5.18      ! [B2: rat,A: rat] :
% 9.07/5.18        ( ( B2 != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B2 ) @ B2 )
% 9.07/5.18          = A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1580_nonzero__mult__div__cancel__right,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( B2 != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 9.07/5.18          = A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1581_nonzero__mult__div__cancel__right,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( B2 != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 9.07/5.18          = A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1582_diff__add__zero,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1583_divide__eq__1__iff,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 9.07/5.18          = one_one_complex )
% 9.07/5.18        = ( ( B2 != zero_zero_complex )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1584_divide__eq__1__iff,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ( divide_divide_real @ A @ B2 )
% 9.07/5.18          = one_one_real )
% 9.07/5.18        = ( ( B2 != zero_zero_real )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1585_divide__eq__1__iff,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ( divide_divide_rat @ A @ B2 )
% 9.07/5.18          = one_one_rat )
% 9.07/5.18        = ( ( B2 != zero_zero_rat )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1586_one__eq__divide__iff,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( one_one_complex
% 9.07/5.18          = ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.07/5.18        = ( ( B2 != zero_zero_complex )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1587_one__eq__divide__iff,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( one_one_real
% 9.07/5.18          = ( divide_divide_real @ A @ B2 ) )
% 9.07/5.18        = ( ( B2 != zero_zero_real )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1588_one__eq__divide__iff,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( one_one_rat
% 9.07/5.18          = ( divide_divide_rat @ A @ B2 ) )
% 9.07/5.18        = ( ( B2 != zero_zero_rat )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1589_divide__self,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( A != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.07/5.18          = one_one_complex ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1590_divide__self,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( A != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ A @ A )
% 9.07/5.18          = one_one_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1591_divide__self,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( A != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ A @ A )
% 9.07/5.18          = one_one_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1592_divide__self__if,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( ( A = zero_zero_complex )
% 9.07/5.18         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.07/5.18            = zero_zero_complex ) )
% 9.07/5.18        & ( ( A != zero_zero_complex )
% 9.07/5.18         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.07/5.18            = one_one_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1593_divide__self__if,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ( A = zero_zero_real )
% 9.07/5.18         => ( ( divide_divide_real @ A @ A )
% 9.07/5.18            = zero_zero_real ) )
% 9.07/5.18        & ( ( A != zero_zero_real )
% 9.07/5.18         => ( ( divide_divide_real @ A @ A )
% 9.07/5.18            = one_one_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1594_divide__self__if,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ( A = zero_zero_rat )
% 9.07/5.18         => ( ( divide_divide_rat @ A @ A )
% 9.07/5.18            = zero_zero_rat ) )
% 9.07/5.18        & ( ( A != zero_zero_rat )
% 9.07/5.18         => ( ( divide_divide_rat @ A @ A )
% 9.07/5.18            = one_one_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1595_divide__eq__eq__1,axiom,
% 9.07/5.18      ! [B2: real,A: real] :
% 9.07/5.18        ( ( ( divide_divide_real @ B2 @ A )
% 9.07/5.18          = one_one_real )
% 9.07/5.18        = ( ( A != zero_zero_real )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1596_divide__eq__eq__1,axiom,
% 9.07/5.18      ! [B2: rat,A: rat] :
% 9.07/5.18        ( ( ( divide_divide_rat @ B2 @ A )
% 9.07/5.18          = one_one_rat )
% 9.07/5.18        = ( ( A != zero_zero_rat )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1597_eq__divide__eq__1,axiom,
% 9.07/5.18      ! [B2: real,A: real] :
% 9.07/5.18        ( ( one_one_real
% 9.07/5.18          = ( divide_divide_real @ B2 @ A ) )
% 9.07/5.18        = ( ( A != zero_zero_real )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1598_eq__divide__eq__1,axiom,
% 9.07/5.18      ! [B2: rat,A: rat] :
% 9.07/5.18        ( ( one_one_rat
% 9.07/5.18          = ( divide_divide_rat @ B2 @ A ) )
% 9.07/5.18        = ( ( A != zero_zero_rat )
% 9.07/5.18          & ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1599_one__divide__eq__0__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ( divide_divide_real @ one_one_real @ A )
% 9.07/5.18          = zero_zero_real )
% 9.07/5.18        = ( A = zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1600_one__divide__eq__0__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 9.07/5.18          = zero_zero_rat )
% 9.07/5.18        = ( A = zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1601_zero__eq__1__divide__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( zero_zero_real
% 9.07/5.18          = ( divide_divide_real @ one_one_real @ A ) )
% 9.07/5.18        = ( A = zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1602_zero__eq__1__divide__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( zero_zero_rat
% 9.07/5.18          = ( divide_divide_rat @ one_one_rat @ A ) )
% 9.07/5.18        = ( A = zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1603_div__self,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( A != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.07/5.18          = one_one_complex ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1604_div__self,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( A != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ A @ A )
% 9.07/5.18          = one_one_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1605_div__self,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( A != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ A @ A )
% 9.07/5.18          = one_one_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1606_div__self,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( A != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ A @ A )
% 9.07/5.18          = one_one_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1607_div__self,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( A != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ A @ A )
% 9.07/5.18          = one_one_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1608_power__0__Suc,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 9.07/5.18        = zero_zero_rat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1609_power__0__Suc,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1610_power__0__Suc,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 9.07/5.18        = zero_zero_real ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1611_power__0__Suc,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1612_power__0__Suc,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 9.07/5.18        = zero_zero_complex ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1613_power__zero__numeral,axiom,
% 9.07/5.18      ! [K: num] :
% 9.07/5.18        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 9.07/5.18        = zero_zero_rat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1614_power__zero__numeral,axiom,
% 9.07/5.18      ! [K: num] :
% 9.07/5.18        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1615_power__zero__numeral,axiom,
% 9.07/5.18      ! [K: num] :
% 9.07/5.18        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 9.07/5.18        = zero_zero_real ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1616_power__zero__numeral,axiom,
% 9.07/5.18      ! [K: num] :
% 9.07/5.18        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1617_power__zero__numeral,axiom,
% 9.07/5.18      ! [K: num] :
% 9.07/5.18        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 9.07/5.18        = zero_zero_complex ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1618_mod__mult__self1__is__0,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A ) @ B2 )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1619_mod__mult__self1__is__0,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A ) @ B2 )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1620_mod__mult__self1__is__0,axiom,
% 9.07/5.18      ! [B2: code_integer,A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ B2 )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1621_mod__mult__self2__is__0,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1622_mod__mult__self2__is__0,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1623_mod__mult__self2__is__0,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1624_mod__by__1,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1625_mod__by__1,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1626_mod__by__1,axiom,
% 9.07/5.18      ! [A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1627_bits__mod__by__1,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1628_bits__mod__by__1,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1629_bits__mod__by__1,axiom,
% 9.07/5.18      ! [A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1630_mod__div__trivial,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1631_mod__div__trivial,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1632_mod__div__trivial,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1633_bits__mod__div__trivial,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1634_bits__mod__div__trivial,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1635_bits__mod__div__trivial,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1636_power__Suc0__right,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1637_power__Suc0__right,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1638_power__Suc0__right,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1639_power__Suc0__right,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1640_mod__mult__self1,axiom,
% 9.07/5.18      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1641_mod__mult__self1,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1642_mod__mult__self1,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1643_mod__mult__self2,axiom,
% 9.07/5.18      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1644_mod__mult__self2,axiom,
% 9.07/5.18      ! [A: int,B2: int,C: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1645_mod__mult__self2,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1646_mod__mult__self3,axiom,
% 9.07/5.18      ! [C: nat,B2: nat,A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1647_mod__mult__self3,axiom,
% 9.07/5.18      ! [C: int,B2: int,A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1648_mod__mult__self3,axiom,
% 9.07/5.18      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B2 ) @ A ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1649_mod__mult__self4,axiom,
% 9.07/5.18      ! [B2: nat,C: nat,A: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1650_mod__mult__self4,axiom,
% 9.07/5.18      ! [B2: int,C: int,A: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1651_mod__mult__self4,axiom,
% 9.07/5.18      ! [B2: code_integer,C: code_integer,A: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C ) @ A ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1652_zero__less__Suc,axiom,
% 9.07/5.18      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1653_less__Suc0,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = ( N = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1654_add__gr__0,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.07/5.18          | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1655_mult__eq__1__iff,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ( times_times_nat @ M @ N )
% 9.07/5.18          = ( suc @ zero_zero_nat ) )
% 9.07/5.18        = ( ( M
% 9.07/5.18            = ( suc @ zero_zero_nat ) )
% 9.07/5.18          & ( N
% 9.07/5.18            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1656_one__eq__mult__iff,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ( suc @ zero_zero_nat )
% 9.07/5.18          = ( times_times_nat @ M @ N ) )
% 9.07/5.18        = ( ( M
% 9.07/5.18            = ( suc @ zero_zero_nat ) )
% 9.07/5.18          & ( N
% 9.07/5.18            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1657_zero__less__diff,axiom,
% 9.07/5.18      ! [N: nat,M: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 9.07/5.18        = ( ord_less_nat @ M @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1658_div__by__Suc__0,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = M ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1659_nat__mult__less__cancel__disj,axiom,
% 9.07/5.18      ! [K: nat,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.18          & ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1660_nat__0__less__mult__iff,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.07/5.18          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1661_mult__less__cancel2,axiom,
% 9.07/5.18      ! [M: nat,K: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.18          & ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1662_diff__is__0__eq,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ( minus_minus_nat @ M @ N )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1663_diff__is__0__eq_H,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.18       => ( ( minus_minus_nat @ M @ N )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1664_div__less,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ M @ N )
% 9.07/5.18       => ( ( divide_divide_nat @ M @ N )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1665_less__one,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ N @ one_one_nat )
% 9.07/5.18        = ( N = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1666_power__Suc__0,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.07/5.18        = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1667_nat__power__eq__Suc__0__iff,axiom,
% 9.07/5.18      ! [X3: nat,M: nat] :
% 9.07/5.18        ( ( ( power_power_nat @ X3 @ M )
% 9.07/5.18          = ( suc @ zero_zero_nat ) )
% 9.07/5.18        = ( ( M = zero_zero_nat )
% 9.07/5.18          | ( X3
% 9.07/5.18            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1668_nat__zero__less__power__iff,axiom,
% 9.07/5.18      ! [X3: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.07/5.18          | ( N = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1669_nat__mult__div__cancel__disj,axiom,
% 9.07/5.18      ! [K: nat,M: nat,N: nat] :
% 9.07/5.18        ( ( ( K = zero_zero_nat )
% 9.07/5.18         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.07/5.18            = zero_zero_nat ) )
% 9.07/5.18        & ( ( K != zero_zero_nat )
% 9.07/5.18         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.07/5.18            = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1670_mod__by__Suc__0,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1671_zero__le__divide__1__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.07/5.18        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1672_zero__le__divide__1__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.07/5.18        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1673_divide__le__0__1__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.07/5.18        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1674_divide__le__0__1__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.07/5.18        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1675_zero__less__divide__1__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.07/5.18        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1676_zero__less__divide__1__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.07/5.18        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1677_less__divide__eq__1__pos,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.18       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1678_less__divide__eq__1__pos,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.18       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1679_less__divide__eq__1__neg,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.18       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_real @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1680_less__divide__eq__1__neg,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.18       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_rat @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1681_divide__less__eq__1__pos,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.18       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 9.07/5.18          = ( ord_less_real @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1682_divide__less__eq__1__pos,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.18       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 9.07/5.18          = ( ord_less_rat @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1683_divide__less__eq__1__neg,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.18       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 9.07/5.18          = ( ord_less_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1684_divide__less__eq__1__neg,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.18       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 9.07/5.18          = ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1685_divide__less__0__1__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.07/5.18        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1686_divide__less__0__1__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.07/5.18        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1687_divide__eq__eq__numeral1_I1_J,axiom,
% 9.07/5.18      ! [B2: complex,W: num,A: complex] :
% 9.07/5.18        ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
% 9.07/5.18          = A )
% 9.07/5.18        = ( ( ( ( numera6690914467698888265omplex @ W )
% 9.07/5.18             != zero_zero_complex )
% 9.07/5.18           => ( B2
% 9.07/5.18              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 9.07/5.18          & ( ( ( numera6690914467698888265omplex @ W )
% 9.07/5.18              = zero_zero_complex )
% 9.07/5.18           => ( A = zero_zero_complex ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1688_divide__eq__eq__numeral1_I1_J,axiom,
% 9.07/5.18      ! [B2: real,W: num,A: real] :
% 9.07/5.18        ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
% 9.07/5.18          = A )
% 9.07/5.18        = ( ( ( ( numeral_numeral_real @ W )
% 9.07/5.18             != zero_zero_real )
% 9.07/5.18           => ( B2
% 9.07/5.18              = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 9.07/5.18          & ( ( ( numeral_numeral_real @ W )
% 9.07/5.18              = zero_zero_real )
% 9.07/5.18           => ( A = zero_zero_real ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1689_divide__eq__eq__numeral1_I1_J,axiom,
% 9.07/5.18      ! [B2: rat,W: num,A: rat] :
% 9.07/5.18        ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
% 9.07/5.18          = A )
% 9.07/5.18        = ( ( ( ( numeral_numeral_rat @ W )
% 9.07/5.18             != zero_zero_rat )
% 9.07/5.18           => ( B2
% 9.07/5.18              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 9.07/5.18          & ( ( ( numeral_numeral_rat @ W )
% 9.07/5.18              = zero_zero_rat )
% 9.07/5.18           => ( A = zero_zero_rat ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1690_eq__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.18      ! [A: complex,B2: complex,W: num] :
% 9.07/5.18        ( ( A
% 9.07/5.18          = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
% 9.07/5.18        = ( ( ( ( numera6690914467698888265omplex @ W )
% 9.07/5.18             != zero_zero_complex )
% 9.07/5.18           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 9.07/5.18              = B2 ) )
% 9.07/5.18          & ( ( ( numera6690914467698888265omplex @ W )
% 9.07/5.18              = zero_zero_complex )
% 9.07/5.18           => ( A = zero_zero_complex ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1691_eq__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.18      ! [A: real,B2: real,W: num] :
% 9.07/5.18        ( ( A
% 9.07/5.18          = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
% 9.07/5.18        = ( ( ( ( numeral_numeral_real @ W )
% 9.07/5.18             != zero_zero_real )
% 9.07/5.18           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 9.07/5.18              = B2 ) )
% 9.07/5.18          & ( ( ( numeral_numeral_real @ W )
% 9.07/5.18              = zero_zero_real )
% 9.07/5.18           => ( A = zero_zero_real ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1692_eq__divide__eq__numeral1_I1_J,axiom,
% 9.07/5.18      ! [A: rat,B2: rat,W: num] :
% 9.07/5.18        ( ( A
% 9.07/5.18          = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
% 9.07/5.18        = ( ( ( ( numeral_numeral_rat @ W )
% 9.07/5.18             != zero_zero_rat )
% 9.07/5.18           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 9.07/5.18              = B2 ) )
% 9.07/5.18          & ( ( ( numeral_numeral_rat @ W )
% 9.07/5.18              = zero_zero_rat )
% 9.07/5.18           => ( A = zero_zero_rat ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1693_nonzero__divide__mult__cancel__left,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( A != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B2 ) )
% 9.07/5.18          = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1694_nonzero__divide__mult__cancel__left,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( A != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B2 ) )
% 9.07/5.18          = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1695_nonzero__divide__mult__cancel__left,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( A != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B2 ) )
% 9.07/5.18          = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1696_nonzero__divide__mult__cancel__right,axiom,
% 9.07/5.18      ! [B2: complex,A: complex] :
% 9.07/5.18        ( ( B2 != zero_zero_complex )
% 9.07/5.18       => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A @ B2 ) )
% 9.07/5.18          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1697_nonzero__divide__mult__cancel__right,axiom,
% 9.07/5.18      ! [B2: real,A: real] :
% 9.07/5.18        ( ( B2 != zero_zero_real )
% 9.07/5.18       => ( ( divide_divide_real @ B2 @ ( times_times_real @ A @ B2 ) )
% 9.07/5.18          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1698_nonzero__divide__mult__cancel__right,axiom,
% 9.07/5.18      ! [B2: rat,A: rat] :
% 9.07/5.18        ( ( B2 != zero_zero_rat )
% 9.07/5.18       => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A @ B2 ) )
% 9.07/5.18          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1699_div__mult__self1,axiom,
% 9.07/5.18      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.18        ( ( B2 != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B2 ) ) @ B2 )
% 9.07/5.18          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1700_div__mult__self1,axiom,
% 9.07/5.18      ! [B2: int,A: int,C: int] :
% 9.07/5.18        ( ( B2 != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B2 ) ) @ B2 )
% 9.07/5.18          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1701_div__mult__self2,axiom,
% 9.07/5.18      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.18        ( ( B2 != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B2 @ C ) ) @ B2 )
% 9.07/5.18          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1702_div__mult__self2,axiom,
% 9.07/5.18      ! [B2: int,A: int,C: int] :
% 9.07/5.18        ( ( B2 != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B2 @ C ) ) @ B2 )
% 9.07/5.18          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1703_div__mult__self3,axiom,
% 9.07/5.18      ! [B2: nat,C: nat,A: nat] :
% 9.07/5.18        ( ( B2 != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B2 ) @ A ) @ B2 )
% 9.07/5.18          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1704_div__mult__self3,axiom,
% 9.07/5.18      ! [B2: int,C: int,A: int] :
% 9.07/5.18        ( ( B2 != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B2 ) @ A ) @ B2 )
% 9.07/5.18          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1705_div__mult__self4,axiom,
% 9.07/5.18      ! [B2: nat,C: nat,A: nat] :
% 9.07/5.18        ( ( B2 != zero_zero_nat )
% 9.07/5.18       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C ) @ A ) @ B2 )
% 9.07/5.18          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1706_div__mult__self4,axiom,
% 9.07/5.18      ! [B2: int,C: int,A: int] :
% 9.07/5.18        ( ( B2 != zero_zero_int )
% 9.07/5.18       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C ) @ A ) @ B2 )
% 9.07/5.18          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1707_power__eq__0__iff,axiom,
% 9.07/5.18      ! [A: rat,N: nat] :
% 9.07/5.18        ( ( ( power_power_rat @ A @ N )
% 9.07/5.18          = zero_zero_rat )
% 9.07/5.18        = ( ( A = zero_zero_rat )
% 9.07/5.18          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1708_power__eq__0__iff,axiom,
% 9.07/5.18      ! [A: nat,N: nat] :
% 9.07/5.18        ( ( ( power_power_nat @ A @ N )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18        = ( ( A = zero_zero_nat )
% 9.07/5.18          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1709_power__eq__0__iff,axiom,
% 9.07/5.18      ! [A: real,N: nat] :
% 9.07/5.18        ( ( ( power_power_real @ A @ N )
% 9.07/5.18          = zero_zero_real )
% 9.07/5.18        = ( ( A = zero_zero_real )
% 9.07/5.18          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1710_power__eq__0__iff,axiom,
% 9.07/5.18      ! [A: int,N: nat] :
% 9.07/5.18        ( ( ( power_power_int @ A @ N )
% 9.07/5.18          = zero_zero_int )
% 9.07/5.18        = ( ( A = zero_zero_int )
% 9.07/5.18          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1711_power__eq__0__iff,axiom,
% 9.07/5.18      ! [A: complex,N: nat] :
% 9.07/5.18        ( ( ( power_power_complex @ A @ N )
% 9.07/5.18          = zero_zero_complex )
% 9.07/5.18        = ( ( A = zero_zero_complex )
% 9.07/5.18          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1712_Suc__pred,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.07/5.18          = N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1713_one__le__mult__iff,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 9.07/5.18        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.07/5.18          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1714_nat__mult__le__cancel__disj,axiom,
% 9.07/5.18      ! [K: nat,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.18         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1715_mult__le__cancel2,axiom,
% 9.07/5.18      ! [M: nat,K: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.07/5.18        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.18         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1716_div__mult__self1__is__m,axiom,
% 9.07/5.18      ! [N: nat,M: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 9.07/5.18          = M ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1717_div__mult__self__is__m,axiom,
% 9.07/5.18      ! [N: nat,M: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 9.07/5.18          = M ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1718_Suc__mod__mult__self1,axiom,
% 9.07/5.18      ! [M: nat,K: nat,N: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 9.07/5.18        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1719_Suc__mod__mult__self2,axiom,
% 9.07/5.18      ! [M: nat,N: nat,K: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 9.07/5.18        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1720_Suc__mod__mult__self3,axiom,
% 9.07/5.18      ! [K: nat,N: nat,M: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 9.07/5.18        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1721_Suc__mod__mult__self4,axiom,
% 9.07/5.18      ! [N: nat,K: nat,M: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 9.07/5.18        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1722_le__divide__eq__1__pos,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.18       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1723_le__divide__eq__1__pos,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.18       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1724_le__divide__eq__1__neg,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.18       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1725_le__divide__eq__1__neg,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.18       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 9.07/5.18          = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1726_divide__le__eq__1__pos,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.18       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 9.07/5.18          = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1727_divide__le__eq__1__pos,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.18       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 9.07/5.18          = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1728_divide__le__eq__1__neg,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.18       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 9.07/5.18          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1729_divide__le__eq__1__neg,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.18       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 9.07/5.18          = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1730_power__strict__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: real,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.18       => ( ( ord_less_real @ B2 @ one_one_real )
% 9.07/5.18         => ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1731_power__strict__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: rat,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.18       => ( ( ord_less_rat @ B2 @ one_one_rat )
% 9.07/5.18         => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1732_power__strict__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: nat,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.18       => ( ( ord_less_nat @ B2 @ one_one_nat )
% 9.07/5.18         => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1733_power__strict__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: int,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.18       => ( ( ord_less_int @ B2 @ one_one_int )
% 9.07/5.18         => ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1734_power__mono__iff,axiom,
% 9.07/5.18      ! [A: real,B2: real,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.18       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.18         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 9.07/5.18              = ( ord_less_eq_real @ A @ B2 ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1735_power__mono__iff,axiom,
% 9.07/5.18      ! [A: rat,B2: rat,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.18       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.18         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 9.07/5.18              = ( ord_less_eq_rat @ A @ B2 ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1736_power__mono__iff,axiom,
% 9.07/5.18      ! [A: nat,B2: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.18       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.18         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 9.07/5.18              = ( ord_less_eq_nat @ A @ B2 ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1737_power__mono__iff,axiom,
% 9.07/5.18      ! [A: int,B2: int,N: nat] :
% 9.07/5.18        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.18       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.18         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 9.07/5.18              = ( ord_less_eq_int @ A @ B2 ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1738_zero__eq__power2,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_rat )
% 9.07/5.18        = ( A = zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1739_zero__eq__power2,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18        = ( A = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1740_zero__eq__power2,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_real )
% 9.07/5.18        = ( A = zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1741_zero__eq__power2,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_int )
% 9.07/5.18        = ( A = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1742_zero__eq__power2,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_complex )
% 9.07/5.18        = ( A = zero_zero_complex ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1743_one__mod__two__eq__one,axiom,
% 9.07/5.18      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18      = one_one_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1744_one__mod__two__eq__one,axiom,
% 9.07/5.18      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18      = one_one_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1745_one__mod__two__eq__one,axiom,
% 9.07/5.18      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.18      = one_one_Code_integer ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1746_bits__one__mod__two__eq__one,axiom,
% 9.07/5.18      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18      = one_one_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1747_bits__one__mod__two__eq__one,axiom,
% 9.07/5.18      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18      = one_one_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1748_bits__one__mod__two__eq__one,axiom,
% 9.07/5.18      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.18      = one_one_Code_integer ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1749_mod2__Suc__Suc,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18        = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1750_Suc__diff__1,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.07/5.18          = N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1751_Suc__times__numeral__mod__eq,axiom,
% 9.07/5.18      ! [K: num,N: nat] :
% 9.07/5.18        ( ( ( numeral_numeral_nat @ K )
% 9.07/5.18         != one_one_nat )
% 9.07/5.18       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 9.07/5.18          = one_one_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1752_one__div__two__eq__zero,axiom,
% 9.07/5.18      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18      = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1753_one__div__two__eq__zero,axiom,
% 9.07/5.18      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18      = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1754_bits__1__div__2,axiom,
% 9.07/5.18      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18      = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1755_bits__1__div__2,axiom,
% 9.07/5.18      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18      = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1756_power__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: real,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.18       => ( ( ord_less_real @ B2 @ one_one_real )
% 9.07/5.18         => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1757_power__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: rat,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.18       => ( ( ord_less_rat @ B2 @ one_one_rat )
% 9.07/5.18         => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1758_power__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: nat,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.18       => ( ( ord_less_nat @ B2 @ one_one_nat )
% 9.07/5.18         => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1759_power__decreasing__iff,axiom,
% 9.07/5.18      ! [B2: int,M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.18       => ( ( ord_less_int @ B2 @ one_one_int )
% 9.07/5.18         => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N ) )
% 9.07/5.18            = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1760_power2__less__eq__zero__iff,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 9.07/5.18        = ( A = zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1761_power2__less__eq__zero__iff,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 9.07/5.18        = ( A = zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1762_power2__less__eq__zero__iff,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.07/5.18        = ( A = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1763_power2__eq__iff__nonneg,axiom,
% 9.07/5.18      ! [X3: real,Y3: real] :
% 9.07/5.18        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.18       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.07/5.18         => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18              = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18            = ( X3 = Y3 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1764_power2__eq__iff__nonneg,axiom,
% 9.07/5.18      ! [X3: rat,Y3: rat] :
% 9.07/5.18        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.18       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.07/5.18         => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18              = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18            = ( X3 = Y3 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1765_power2__eq__iff__nonneg,axiom,
% 9.07/5.18      ! [X3: nat,Y3: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.07/5.18       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.07/5.18         => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18              = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18            = ( X3 = Y3 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1766_power2__eq__iff__nonneg,axiom,
% 9.07/5.18      ! [X3: int,Y3: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.07/5.18       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.07/5.18         => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18              = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18            = ( X3 = Y3 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1767_zero__less__power2,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18        = ( A != zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1768_zero__less__power2,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18        = ( A != zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1769_zero__less__power2,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18        = ( A != zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1770_sum__power2__eq__zero__iff,axiom,
% 9.07/5.18      ! [X3: rat,Y3: rat] :
% 9.07/5.18        ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18          = zero_zero_rat )
% 9.07/5.18        = ( ( X3 = zero_zero_rat )
% 9.07/5.18          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1771_sum__power2__eq__zero__iff,axiom,
% 9.07/5.18      ! [X3: real,Y3: real] :
% 9.07/5.18        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18          = zero_zero_real )
% 9.07/5.18        = ( ( X3 = zero_zero_real )
% 9.07/5.18          & ( Y3 = zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1772_sum__power2__eq__zero__iff,axiom,
% 9.07/5.18      ! [X3: int,Y3: int] :
% 9.07/5.18        ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18          = zero_zero_int )
% 9.07/5.18        = ( ( X3 = zero_zero_int )
% 9.07/5.18          & ( Y3 = zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1773_not__mod__2__eq__0__eq__1,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18         != zero_zero_nat )
% 9.07/5.18        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = one_one_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1774_not__mod__2__eq__0__eq__1,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18         != zero_zero_int )
% 9.07/5.18        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18          = one_one_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1775_not__mod__2__eq__0__eq__1,axiom,
% 9.07/5.18      ! [A: code_integer] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.18         != zero_z3403309356797280102nteger )
% 9.07/5.18        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.18          = one_one_Code_integer ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1776_not__mod__2__eq__1__eq__0,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18         != one_one_nat )
% 9.07/5.18        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1777_not__mod__2__eq__1__eq__0,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18         != one_one_int )
% 9.07/5.18        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1778_not__mod__2__eq__1__eq__0,axiom,
% 9.07/5.18      ! [A: code_integer] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.18         != one_one_Code_integer )
% 9.07/5.18        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_z3403309356797280102nteger ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1779_not__mod2__eq__Suc__0__eq__0,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18         != ( suc @ zero_zero_nat ) )
% 9.07/5.18        = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1780_add__self__mod__2,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1781_mod2__gr__0,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.18        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.18          = one_one_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1782_zero__reorient,axiom,
% 9.07/5.18      ! [X3: complex] :
% 9.07/5.18        ( ( zero_zero_complex = X3 )
% 9.07/5.18        = ( X3 = zero_zero_complex ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1783_zero__reorient,axiom,
% 9.07/5.18      ! [X3: real] :
% 9.07/5.18        ( ( zero_zero_real = X3 )
% 9.07/5.18        = ( X3 = zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1784_zero__reorient,axiom,
% 9.07/5.18      ! [X3: rat] :
% 9.07/5.18        ( ( zero_zero_rat = X3 )
% 9.07/5.18        = ( X3 = zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1785_zero__reorient,axiom,
% 9.07/5.18      ! [X3: nat] :
% 9.07/5.18        ( ( zero_zero_nat = X3 )
% 9.07/5.18        = ( X3 = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1786_zero__reorient,axiom,
% 9.07/5.18      ! [X3: int] :
% 9.07/5.18        ( ( zero_zero_int = X3 )
% 9.07/5.18        = ( X3 = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1787_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.07/5.18       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1788_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.18       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B2 ) @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1789_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.18       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ A ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1790_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.18       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B2 ) @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1791_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.18       => ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1792_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.07/5.18      ! [B2: code_integer,A: code_integer] :
% 9.07/5.18        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.07/5.18       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1793_mod__eq__self__iff__div__eq__0,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ A @ B2 )
% 9.07/5.18          = A )
% 9.07/5.18        = ( ( divide_divide_nat @ A @ B2 )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1794_mod__eq__self__iff__div__eq__0,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ A @ B2 )
% 9.07/5.18          = A )
% 9.07/5.18        = ( ( divide_divide_int @ A @ B2 )
% 9.07/5.18          = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1795_mod__eq__self__iff__div__eq__0,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 9.07/5.18          = A )
% 9.07/5.18        = ( ( divide6298287555418463151nteger @ A @ B2 )
% 9.07/5.18          = zero_z3403309356797280102nteger ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1796_mod__Suc,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.07/5.18            = N )
% 9.07/5.18         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.07/5.18            = zero_zero_nat ) )
% 9.07/5.18        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.07/5.18           != N )
% 9.07/5.18         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.07/5.18            = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1797_mod__less__divisor,axiom,
% 9.07/5.18      ! [N: nat,M: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1798_mod__eq__0D,axiom,
% 9.07/5.18      ! [M: nat,D: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ M @ D )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18       => ? [Q3: nat] :
% 9.07/5.18            ( M
% 9.07/5.18            = ( times_times_nat @ D @ Q3 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1799_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 9.07/5.18      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 9.07/5.18        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1800_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 9.07/5.18      vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1801_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 9.07/5.18      ! [Uv: $o] :
% 9.07/5.18        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1802_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 9.07/5.18      ! [Uu: $o] :
% 9.07/5.18        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1803_vebt__member_Osimps_I1_J,axiom,
% 9.07/5.18      ! [A: $o,B2: $o,X3: nat] :
% 9.07/5.18        ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 9.07/5.18        = ( ( ( X3 = zero_zero_nat )
% 9.07/5.18           => A )
% 9.07/5.18          & ( ( X3 != zero_zero_nat )
% 9.07/5.18           => ( ( ( X3 = one_one_nat )
% 9.07/5.18               => B2 )
% 9.07/5.18              & ( X3 = one_one_nat ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1804_vebt__insert_Osimps_I1_J,axiom,
% 9.07/5.18      ! [X3: nat,A: $o,B2: $o] :
% 9.07/5.18        ( ( ( X3 = zero_zero_nat )
% 9.07/5.18         => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 9.07/5.18            = ( vEBT_Leaf @ $true @ B2 ) ) )
% 9.07/5.18        & ( ( X3 != zero_zero_nat )
% 9.07/5.18         => ( ( ( X3 = one_one_nat )
% 9.07/5.18             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 9.07/5.18                = ( vEBT_Leaf @ A @ $true ) ) )
% 9.07/5.18            & ( ( X3 != one_one_nat )
% 9.07/5.18             => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B2 ) @ X3 )
% 9.07/5.18                = ( vEBT_Leaf @ A @ B2 ) ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1805_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.07/5.18      ! [B2: code_integer,A: code_integer] :
% 9.07/5.18        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.07/5.18       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1806_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.07/5.18      ! [B2: nat,A: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.18       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1807_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.07/5.18      ! [B2: int,A: int] :
% 9.07/5.18        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.18       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1808_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.07/5.18       => ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 9.07/5.18         => ( ( modulo364778990260209775nteger @ A @ B2 )
% 9.07/5.18            = A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1809_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.18       => ( ( ord_less_nat @ A @ B2 )
% 9.07/5.18         => ( ( modulo_modulo_nat @ A @ B2 )
% 9.07/5.18            = A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1810_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.18       => ( ( ord_less_int @ A @ B2 )
% 9.07/5.18         => ( ( modulo_modulo_int @ A @ B2 )
% 9.07/5.18            = A ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1811_cong__exp__iff__simps_I2_J,axiom,
% 9.07/5.18      ! [N: num,Q2: num] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1812_cong__exp__iff__simps_I2_J,axiom,
% 9.07/5.18      ! [N: num,Q2: num] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.07/5.18          = zero_zero_int )
% 9.07/5.18        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 9.07/5.18          = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1813_cong__exp__iff__simps_I2_J,axiom,
% 9.07/5.18      ! [N: num,Q2: num] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.07/5.18          = zero_z3403309356797280102nteger )
% 9.07/5.18        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 9.07/5.18          = zero_z3403309356797280102nteger ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1814_cong__exp__iff__simps_I1_J,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 9.07/5.18        = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1815_cong__exp__iff__simps_I1_J,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 9.07/5.18        = zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1816_cong__exp__iff__simps_I1_J,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 9.07/5.18        = zero_z3403309356797280102nteger ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1817_VEBT__internal_Onaive__member_Ocases,axiom,
% 9.07/5.18      ! [X3: produc9072475918466114483BT_nat] :
% 9.07/5.18        ( ! [A3: $o,B3: $o,X5: nat] :
% 9.07/5.18            ( X3
% 9.07/5.18           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
% 9.07/5.18       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 9.07/5.18              ( X3
% 9.07/5.18             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 9.07/5.18         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 9.07/5.18                ( X3
% 9.07/5.18               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X5 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1818_mod__le__divisor,axiom,
% 9.07/5.18      ! [N: nat,M: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1819_div__less__mono,axiom,
% 9.07/5.18      ! [A4: nat,B5: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ A4 @ B5 )
% 9.07/5.18       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18         => ( ( ( modulo_modulo_nat @ A4 @ N )
% 9.07/5.18              = zero_zero_nat )
% 9.07/5.18           => ( ( ( modulo_modulo_nat @ B5 @ N )
% 9.07/5.18                = zero_zero_nat )
% 9.07/5.18             => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1820_invar__vebt_Ointros_I1_J,axiom,
% 9.07/5.18      ! [A: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1821_power__0__left,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ( N = zero_zero_nat )
% 9.07/5.18         => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.07/5.18            = one_one_rat ) )
% 9.07/5.18        & ( ( N != zero_zero_nat )
% 9.07/5.18         => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.07/5.18            = zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1822_power__0__left,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ( N = zero_zero_nat )
% 9.07/5.18         => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.07/5.18            = one_one_nat ) )
% 9.07/5.18        & ( ( N != zero_zero_nat )
% 9.07/5.18         => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.07/5.18            = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1823_power__0__left,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ( N = zero_zero_nat )
% 9.07/5.18         => ( ( power_power_real @ zero_zero_real @ N )
% 9.07/5.18            = one_one_real ) )
% 9.07/5.18        & ( ( N != zero_zero_nat )
% 9.07/5.18         => ( ( power_power_real @ zero_zero_real @ N )
% 9.07/5.18            = zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1824_power__0__left,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ( N = zero_zero_nat )
% 9.07/5.18         => ( ( power_power_int @ zero_zero_int @ N )
% 9.07/5.18            = one_one_int ) )
% 9.07/5.18        & ( ( N != zero_zero_nat )
% 9.07/5.18         => ( ( power_power_int @ zero_zero_int @ N )
% 9.07/5.18            = zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1825_power__0__left,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ( N = zero_zero_nat )
% 9.07/5.18         => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.07/5.18            = one_one_complex ) )
% 9.07/5.18        & ( ( N != zero_zero_nat )
% 9.07/5.18         => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.07/5.18            = zero_zero_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1826_zero__power,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.07/5.18          = zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1827_zero__power,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.07/5.18          = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1828_zero__power,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( power_power_real @ zero_zero_real @ N )
% 9.07/5.18          = zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1829_zero__power,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( power_power_int @ zero_zero_int @ N )
% 9.07/5.18          = zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1830_zero__power,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18       => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.07/5.18          = zero_zero_complex ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1831_mod__mult__eq,axiom,
% 9.07/5.18      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1832_mod__mult__eq,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1833_mod__mult__eq,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1834_mod__mult__cong,axiom,
% 9.07/5.18      ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ A @ C )
% 9.07/5.18          = ( modulo_modulo_nat @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo_modulo_nat @ B2 @ C )
% 9.07/5.18            = ( modulo_modulo_nat @ B @ C ) )
% 9.07/5.18         => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1835_mod__mult__cong,axiom,
% 9.07/5.18      ! [A: int,C: int,A2: int,B2: int,B: int] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ A @ C )
% 9.07/5.18          = ( modulo_modulo_int @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo_modulo_int @ B2 @ C )
% 9.07/5.18            = ( modulo_modulo_int @ B @ C ) )
% 9.07/5.18         => ( ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo_modulo_int @ ( times_times_int @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1836_mod__mult__cong,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.07/5.18          = ( modulo364778990260209775nteger @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 9.07/5.18            = ( modulo364778990260209775nteger @ B @ C ) )
% 9.07/5.18         => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1837_mod__mult__mult2,axiom,
% 9.07/5.18      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 9.07/5.18        = ( times_times_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1838_mod__mult__mult2,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.18        = ( times_times_int @ ( modulo_modulo_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1839_mod__mult__mult2,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.07/5.18        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1840_mult__mod__right,axiom,
% 9.07/5.18      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.18        ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.07/5.18        = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1841_mult__mod__right,axiom,
% 9.07/5.18      ! [C: int,A: int,B2: int] :
% 9.07/5.18        ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 9.07/5.18        = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1842_mult__mod__right,axiom,
% 9.07/5.18      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1843_mod__mult__left__eq,axiom,
% 9.07/5.18      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1844_mod__mult__left__eq,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1845_mod__mult__left__eq,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1846_mod__mult__right__eq,axiom,
% 9.07/5.18      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1847_mod__mult__right__eq,axiom,
% 9.07/5.18      ! [A: int,B2: int,C: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( times_times_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1848_mod__mult__right__eq,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1849_mod__add__eq,axiom,
% 9.07/5.18      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1850_mod__add__eq,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1851_mod__add__eq,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1852_mod__add__cong,axiom,
% 9.07/5.18      ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
% 9.07/5.18        ( ( ( modulo_modulo_nat @ A @ C )
% 9.07/5.18          = ( modulo_modulo_nat @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo_modulo_nat @ B2 @ C )
% 9.07/5.18            = ( modulo_modulo_nat @ B @ C ) )
% 9.07/5.18         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1853_mod__add__cong,axiom,
% 9.07/5.18      ! [A: int,C: int,A2: int,B2: int,B: int] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ A @ C )
% 9.07/5.18          = ( modulo_modulo_int @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo_modulo_int @ B2 @ C )
% 9.07/5.18            = ( modulo_modulo_int @ B @ C ) )
% 9.07/5.18         => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1854_mod__add__cong,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.07/5.18          = ( modulo364778990260209775nteger @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 9.07/5.18            = ( modulo364778990260209775nteger @ B @ C ) )
% 9.07/5.18         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1855_mod__add__left__eq,axiom,
% 9.07/5.18      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1856_mod__add__left__eq,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1857_mod__add__left__eq,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1858_mod__add__right__eq,axiom,
% 9.07/5.18      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1859_mod__add__right__eq,axiom,
% 9.07/5.18      ! [A: int,B2: int,C: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1860_mod__add__right__eq,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1861_mod__diff__eq,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1862_mod__diff__eq,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1863_mod__diff__cong,axiom,
% 9.07/5.18      ! [A: int,C: int,A2: int,B2: int,B: int] :
% 9.07/5.18        ( ( ( modulo_modulo_int @ A @ C )
% 9.07/5.18          = ( modulo_modulo_int @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo_modulo_int @ B2 @ C )
% 9.07/5.18            = ( modulo_modulo_int @ B @ C ) )
% 9.07/5.18         => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1864_mod__diff__cong,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
% 9.07/5.18        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.07/5.18          = ( modulo364778990260209775nteger @ A2 @ C ) )
% 9.07/5.18       => ( ( ( modulo364778990260209775nteger @ B2 @ C )
% 9.07/5.18            = ( modulo364778990260209775nteger @ B @ C ) )
% 9.07/5.18         => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 9.07/5.18            = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B ) @ C ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1865_mod__diff__left__eq,axiom,
% 9.07/5.18      ! [A: int,C: int,B2: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1866_mod__diff__left__eq,axiom,
% 9.07/5.18      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B2 ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1867_mod__diff__right__eq,axiom,
% 9.07/5.18      ! [A: int,B2: int,C: int] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo_modulo_int @ ( minus_minus_int @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1868_mod__diff__right__eq,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1869_power__mod,axiom,
% 9.07/5.18      ! [A: nat,B2: nat,N: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B2 ) @ N ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1870_power__mod,axiom,
% 9.07/5.18      ! [A: int,B2: int,N: nat] :
% 9.07/5.18        ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B2 ) @ N ) @ B2 )
% 9.07/5.18        = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1871_power__mod,axiom,
% 9.07/5.18      ! [A: code_integer,B2: code_integer,N: nat] :
% 9.07/5.18        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ N ) @ B2 )
% 9.07/5.18        = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1872_VEBT_Oexhaust,axiom,
% 9.07/5.18      ! [Y3: vEBT_VEBT] :
% 9.07/5.18        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 9.07/5.18            ( Y3
% 9.07/5.18           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 9.07/5.18       => ~ ! [X21: $o,X222: $o] :
% 9.07/5.18              ( Y3
% 9.07/5.18             != ( vEBT_Leaf @ X21 @ X222 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1873_VEBT_Odistinct_I1_J,axiom,
% 9.07/5.18      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X212: $o,X223: $o] :
% 9.07/5.18        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.07/5.18       != ( vEBT_Leaf @ X212 @ X223 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1874_VEBT__internal_Ovalid_H_Ocases,axiom,
% 9.07/5.18      ! [X3: produc9072475918466114483BT_nat] :
% 9.07/5.18        ( ! [Uu2: $o,Uv2: $o,D3: nat] :
% 9.07/5.18            ( X3
% 9.07/5.18           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
% 9.07/5.18       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 9.07/5.18              ( X3
% 9.07/5.18             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1875_mod__Suc__eq,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 9.07/5.18        = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1876_mod__Suc__Suc__eq,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 9.07/5.18        = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1877_mod__less__eq__dividend,axiom,
% 9.07/5.18      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1878_vebt__pred_Osimps_I1_J,axiom,
% 9.07/5.18      ! [Uu: $o,Uv: $o] :
% 9.07/5.18        ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 9.07/5.18        = none_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1879_le__numeral__extra_I3_J,axiom,
% 9.07/5.18      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1880_le__numeral__extra_I3_J,axiom,
% 9.07/5.18      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1881_le__numeral__extra_I3_J,axiom,
% 9.07/5.18      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1882_le__numeral__extra_I3_J,axiom,
% 9.07/5.18      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1883_zero__le,axiom,
% 9.07/5.18      ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1884_less__numeral__extra_I3_J,axiom,
% 9.07/5.18      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1885_less__numeral__extra_I3_J,axiom,
% 9.07/5.18      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1886_less__numeral__extra_I3_J,axiom,
% 9.07/5.18      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1887_less__numeral__extra_I3_J,axiom,
% 9.07/5.18      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1888_field__lbound__gt__zero,axiom,
% 9.07/5.18      ! [D1: real,D22: real] :
% 9.07/5.18        ( ( ord_less_real @ zero_zero_real @ D1 )
% 9.07/5.18       => ( ( ord_less_real @ zero_zero_real @ D22 )
% 9.07/5.18         => ? [E2: real] :
% 9.07/5.18              ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.07/5.18              & ( ord_less_real @ E2 @ D1 )
% 9.07/5.18              & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1889_field__lbound__gt__zero,axiom,
% 9.07/5.18      ! [D1: rat,D22: rat] :
% 9.07/5.18        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 9.07/5.18       => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 9.07/5.18         => ? [E2: rat] :
% 9.07/5.18              ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.07/5.18              & ( ord_less_rat @ E2 @ D1 )
% 9.07/5.18              & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1890_zero__less__iff__neq__zero,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.18        = ( N != zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1891_gr__implies__not__zero,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ M @ N )
% 9.07/5.18       => ( N != zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1892_not__less__zero,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1893_gr__zeroI,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( N != zero_zero_nat )
% 9.07/5.18       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1894_zero__neq__numeral,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( zero_zero_complex
% 9.07/5.18       != ( numera6690914467698888265omplex @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1895_zero__neq__numeral,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( zero_zero_real
% 9.07/5.18       != ( numeral_numeral_real @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1896_zero__neq__numeral,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( zero_zero_rat
% 9.07/5.18       != ( numeral_numeral_rat @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1897_zero__neq__numeral,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( zero_zero_nat
% 9.07/5.18       != ( numeral_numeral_nat @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1898_zero__neq__numeral,axiom,
% 9.07/5.18      ! [N: num] :
% 9.07/5.18        ( zero_zero_int
% 9.07/5.18       != ( numeral_numeral_int @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1899_mult__not__zero,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( ( times_times_complex @ A @ B2 )
% 9.07/5.18         != zero_zero_complex )
% 9.07/5.18       => ( ( A != zero_zero_complex )
% 9.07/5.18          & ( B2 != zero_zero_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1900_mult__not__zero,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ( times_times_real @ A @ B2 )
% 9.07/5.18         != zero_zero_real )
% 9.07/5.18       => ( ( A != zero_zero_real )
% 9.07/5.18          & ( B2 != zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1901_mult__not__zero,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ( times_times_rat @ A @ B2 )
% 9.07/5.18         != zero_zero_rat )
% 9.07/5.18       => ( ( A != zero_zero_rat )
% 9.07/5.18          & ( B2 != zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1902_mult__not__zero,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ( times_times_nat @ A @ B2 )
% 9.07/5.18         != zero_zero_nat )
% 9.07/5.18       => ( ( A != zero_zero_nat )
% 9.07/5.18          & ( B2 != zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1903_mult__not__zero,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ( times_times_int @ A @ B2 )
% 9.07/5.18         != zero_zero_int )
% 9.07/5.18       => ( ( A != zero_zero_int )
% 9.07/5.18          & ( B2 != zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1904_divisors__zero,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( ( times_times_complex @ A @ B2 )
% 9.07/5.18          = zero_zero_complex )
% 9.07/5.18       => ( ( A = zero_zero_complex )
% 9.07/5.18          | ( B2 = zero_zero_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1905_divisors__zero,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( ( times_times_real @ A @ B2 )
% 9.07/5.18          = zero_zero_real )
% 9.07/5.18       => ( ( A = zero_zero_real )
% 9.07/5.18          | ( B2 = zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1906_divisors__zero,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( ( times_times_rat @ A @ B2 )
% 9.07/5.18          = zero_zero_rat )
% 9.07/5.18       => ( ( A = zero_zero_rat )
% 9.07/5.18          | ( B2 = zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1907_divisors__zero,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( ( times_times_nat @ A @ B2 )
% 9.07/5.18          = zero_zero_nat )
% 9.07/5.18       => ( ( A = zero_zero_nat )
% 9.07/5.18          | ( B2 = zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1908_divisors__zero,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( ( times_times_int @ A @ B2 )
% 9.07/5.18          = zero_zero_int )
% 9.07/5.18       => ( ( A = zero_zero_int )
% 9.07/5.18          | ( B2 = zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1909_no__zero__divisors,axiom,
% 9.07/5.18      ! [A: complex,B2: complex] :
% 9.07/5.18        ( ( A != zero_zero_complex )
% 9.07/5.18       => ( ( B2 != zero_zero_complex )
% 9.07/5.18         => ( ( times_times_complex @ A @ B2 )
% 9.07/5.18           != zero_zero_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1910_no__zero__divisors,axiom,
% 9.07/5.18      ! [A: real,B2: real] :
% 9.07/5.18        ( ( A != zero_zero_real )
% 9.07/5.18       => ( ( B2 != zero_zero_real )
% 9.07/5.18         => ( ( times_times_real @ A @ B2 )
% 9.07/5.18           != zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1911_no__zero__divisors,axiom,
% 9.07/5.18      ! [A: rat,B2: rat] :
% 9.07/5.18        ( ( A != zero_zero_rat )
% 9.07/5.18       => ( ( B2 != zero_zero_rat )
% 9.07/5.18         => ( ( times_times_rat @ A @ B2 )
% 9.07/5.18           != zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1912_no__zero__divisors,axiom,
% 9.07/5.18      ! [A: nat,B2: nat] :
% 9.07/5.18        ( ( A != zero_zero_nat )
% 9.07/5.18       => ( ( B2 != zero_zero_nat )
% 9.07/5.18         => ( ( times_times_nat @ A @ B2 )
% 9.07/5.18           != zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1913_no__zero__divisors,axiom,
% 9.07/5.18      ! [A: int,B2: int] :
% 9.07/5.18        ( ( A != zero_zero_int )
% 9.07/5.18       => ( ( B2 != zero_zero_int )
% 9.07/5.18         => ( ( times_times_int @ A @ B2 )
% 9.07/5.18           != zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1914_mult__left__cancel,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( C != zero_zero_complex )
% 9.07/5.18       => ( ( ( times_times_complex @ C @ A )
% 9.07/5.18            = ( times_times_complex @ C @ B2 ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1915_mult__left__cancel,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( C != zero_zero_real )
% 9.07/5.18       => ( ( ( times_times_real @ C @ A )
% 9.07/5.18            = ( times_times_real @ C @ B2 ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1916_mult__left__cancel,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( C != zero_zero_rat )
% 9.07/5.18       => ( ( ( times_times_rat @ C @ A )
% 9.07/5.18            = ( times_times_rat @ C @ B2 ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1917_mult__left__cancel,axiom,
% 9.07/5.18      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.18        ( ( C != zero_zero_nat )
% 9.07/5.18       => ( ( ( times_times_nat @ C @ A )
% 9.07/5.18            = ( times_times_nat @ C @ B2 ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1918_mult__left__cancel,axiom,
% 9.07/5.18      ! [C: int,A: int,B2: int] :
% 9.07/5.18        ( ( C != zero_zero_int )
% 9.07/5.18       => ( ( ( times_times_int @ C @ A )
% 9.07/5.18            = ( times_times_int @ C @ B2 ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1919_mult__right__cancel,axiom,
% 9.07/5.18      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.18        ( ( C != zero_zero_complex )
% 9.07/5.18       => ( ( ( times_times_complex @ A @ C )
% 9.07/5.18            = ( times_times_complex @ B2 @ C ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1920_mult__right__cancel,axiom,
% 9.07/5.18      ! [C: real,A: real,B2: real] :
% 9.07/5.18        ( ( C != zero_zero_real )
% 9.07/5.18       => ( ( ( times_times_real @ A @ C )
% 9.07/5.18            = ( times_times_real @ B2 @ C ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1921_mult__right__cancel,axiom,
% 9.07/5.18      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.18        ( ( C != zero_zero_rat )
% 9.07/5.18       => ( ( ( times_times_rat @ A @ C )
% 9.07/5.18            = ( times_times_rat @ B2 @ C ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1922_mult__right__cancel,axiom,
% 9.07/5.18      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.18        ( ( C != zero_zero_nat )
% 9.07/5.18       => ( ( ( times_times_nat @ A @ C )
% 9.07/5.18            = ( times_times_nat @ B2 @ C ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1923_mult__right__cancel,axiom,
% 9.07/5.18      ! [C: int,A: int,B2: int] :
% 9.07/5.18        ( ( C != zero_zero_int )
% 9.07/5.18       => ( ( ( times_times_int @ A @ C )
% 9.07/5.18            = ( times_times_int @ B2 @ C ) )
% 9.07/5.18          = ( A = B2 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1924_zero__neq__one,axiom,
% 9.07/5.18      zero_zero_complex != one_one_complex ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1925_zero__neq__one,axiom,
% 9.07/5.18      zero_zero_real != one_one_real ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1926_zero__neq__one,axiom,
% 9.07/5.18      zero_zero_rat != one_one_rat ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1927_zero__neq__one,axiom,
% 9.07/5.18      zero_zero_nat != one_one_nat ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1928_zero__neq__one,axiom,
% 9.07/5.18      zero_zero_int != one_one_int ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1929_verit__sum__simplify,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1930_verit__sum__simplify,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1931_verit__sum__simplify,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1932_verit__sum__simplify,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1933_verit__sum__simplify,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1934_add_Ogroup__left__neutral,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1935_add_Ogroup__left__neutral,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1936_add_Ogroup__left__neutral,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1937_add_Ogroup__left__neutral,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1938_add_Ocomm__neutral,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1939_add_Ocomm__neutral,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1940_add_Ocomm__neutral,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1941_add_Ocomm__neutral,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1942_add_Ocomm__neutral,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1943_comm__monoid__add__class_Oadd__0,axiom,
% 9.07/5.18      ! [A: complex] :
% 9.07/5.18        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1944_comm__monoid__add__class_Oadd__0,axiom,
% 9.07/5.18      ! [A: real] :
% 9.07/5.18        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1945_comm__monoid__add__class_Oadd__0,axiom,
% 9.07/5.18      ! [A: rat] :
% 9.07/5.18        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1946_comm__monoid__add__class_Oadd__0,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1947_comm__monoid__add__class_Oadd__0,axiom,
% 9.07/5.18      ! [A: int] :
% 9.07/5.18        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.07/5.18        = A ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1948_eq__iff__diff__eq__0,axiom,
% 9.07/5.18      ( ( ^ [Y6: complex,Z5: complex] : ( Y6 = Z5 ) )
% 9.07/5.18      = ( ^ [A5: complex,B4: complex] :
% 9.07/5.18            ( ( minus_minus_complex @ A5 @ B4 )
% 9.07/5.18            = zero_zero_complex ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1949_eq__iff__diff__eq__0,axiom,
% 9.07/5.18      ( ( ^ [Y6: real,Z5: real] : ( Y6 = Z5 ) )
% 9.07/5.18      = ( ^ [A5: real,B4: real] :
% 9.07/5.18            ( ( minus_minus_real @ A5 @ B4 )
% 9.07/5.18            = zero_zero_real ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1950_eq__iff__diff__eq__0,axiom,
% 9.07/5.18      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 9.07/5.18      = ( ^ [A5: rat,B4: rat] :
% 9.07/5.18            ( ( minus_minus_rat @ A5 @ B4 )
% 9.07/5.18            = zero_zero_rat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1951_eq__iff__diff__eq__0,axiom,
% 9.07/5.18      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.07/5.18      = ( ^ [A5: int,B4: int] :
% 9.07/5.18            ( ( minus_minus_int @ A5 @ B4 )
% 9.07/5.18            = zero_zero_int ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1952_power__not__zero,axiom,
% 9.07/5.18      ! [A: rat,N: nat] :
% 9.07/5.18        ( ( A != zero_zero_rat )
% 9.07/5.18       => ( ( power_power_rat @ A @ N )
% 9.07/5.18         != zero_zero_rat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1953_power__not__zero,axiom,
% 9.07/5.18      ! [A: nat,N: nat] :
% 9.07/5.18        ( ( A != zero_zero_nat )
% 9.07/5.18       => ( ( power_power_nat @ A @ N )
% 9.07/5.18         != zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1954_power__not__zero,axiom,
% 9.07/5.18      ! [A: real,N: nat] :
% 9.07/5.18        ( ( A != zero_zero_real )
% 9.07/5.18       => ( ( power_power_real @ A @ N )
% 9.07/5.18         != zero_zero_real ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1955_power__not__zero,axiom,
% 9.07/5.18      ! [A: int,N: nat] :
% 9.07/5.18        ( ( A != zero_zero_int )
% 9.07/5.18       => ( ( power_power_int @ A @ N )
% 9.07/5.18         != zero_zero_int ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1956_power__not__zero,axiom,
% 9.07/5.18      ! [A: complex,N: nat] :
% 9.07/5.18        ( ( A != zero_zero_complex )
% 9.07/5.18       => ( ( power_power_complex @ A @ N )
% 9.07/5.18         != zero_zero_complex ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1957_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 9.07/5.18      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 9.07/5.18        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1958_num_Osize_I4_J,axiom,
% 9.07/5.18      ( ( size_size_num @ one )
% 9.07/5.18      = zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1959_nat_Odistinct_I1_J,axiom,
% 9.07/5.18      ! [X2: nat] :
% 9.07/5.18        ( zero_zero_nat
% 9.07/5.18       != ( suc @ X2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1960_old_Onat_Odistinct_I2_J,axiom,
% 9.07/5.18      ! [Nat2: nat] :
% 9.07/5.18        ( ( suc @ Nat2 )
% 9.07/5.18       != zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1961_old_Onat_Odistinct_I1_J,axiom,
% 9.07/5.18      ! [Nat2: nat] :
% 9.07/5.18        ( zero_zero_nat
% 9.07/5.18       != ( suc @ Nat2 ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1962_nat_OdiscI,axiom,
% 9.07/5.18      ! [Nat: nat,X2: nat] :
% 9.07/5.18        ( ( Nat
% 9.07/5.18          = ( suc @ X2 ) )
% 9.07/5.18       => ( Nat != zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1963_old_Onat_Oexhaust,axiom,
% 9.07/5.18      ! [Y3: nat] :
% 9.07/5.18        ( ( Y3 != zero_zero_nat )
% 9.07/5.18       => ~ ! [Nat3: nat] :
% 9.07/5.18              ( Y3
% 9.07/5.18             != ( suc @ Nat3 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1964_nat__induct,axiom,
% 9.07/5.18      ! [P: nat > $o,N: nat] :
% 9.07/5.18        ( ( P @ zero_zero_nat )
% 9.07/5.18       => ( ! [N3: nat] :
% 9.07/5.18              ( ( P @ N3 )
% 9.07/5.18             => ( P @ ( suc @ N3 ) ) )
% 9.07/5.18         => ( P @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1965_diff__induct,axiom,
% 9.07/5.18      ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.07/5.18        ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 9.07/5.18       => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 9.07/5.18         => ( ! [X5: nat,Y4: nat] :
% 9.07/5.18                ( ( P @ X5 @ Y4 )
% 9.07/5.18               => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
% 9.07/5.18           => ( P @ M @ N ) ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1966_zero__induct,axiom,
% 9.07/5.18      ! [P: nat > $o,K: nat] :
% 9.07/5.18        ( ( P @ K )
% 9.07/5.18       => ( ! [N3: nat] :
% 9.07/5.18              ( ( P @ ( suc @ N3 ) )
% 9.07/5.18             => ( P @ N3 ) )
% 9.07/5.18         => ( P @ zero_zero_nat ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1967_Suc__neq__Zero,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( ( suc @ M )
% 9.07/5.18       != zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1968_Zero__neq__Suc,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( zero_zero_nat
% 9.07/5.18       != ( suc @ M ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1969_Zero__not__Suc,axiom,
% 9.07/5.18      ! [M: nat] :
% 9.07/5.18        ( zero_zero_nat
% 9.07/5.18       != ( suc @ M ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1970_not0__implies__Suc,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( N != zero_zero_nat )
% 9.07/5.18       => ? [M4: nat] :
% 9.07/5.18            ( N
% 9.07/5.18            = ( suc @ M4 ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1971_infinite__descent0,axiom,
% 9.07/5.18      ! [P: nat > $o,N: nat] :
% 9.07/5.18        ( ( P @ zero_zero_nat )
% 9.07/5.18       => ( ! [N3: nat] :
% 9.07/5.18              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.07/5.18             => ( ~ ( P @ N3 )
% 9.07/5.18               => ? [M2: nat] :
% 9.07/5.18                    ( ( ord_less_nat @ M2 @ N3 )
% 9.07/5.18                    & ~ ( P @ M2 ) ) ) )
% 9.07/5.18         => ( P @ N ) ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1972_gr__implies__not0,axiom,
% 9.07/5.18      ! [M: nat,N: nat] :
% 9.07/5.18        ( ( ord_less_nat @ M @ N )
% 9.07/5.18       => ( N != zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1973_less__zeroE,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1974_not__less0,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1975_not__gr0,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 9.07/5.18        = ( N = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1976_gr0I,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( N != zero_zero_nat )
% 9.07/5.18       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1977_bot__nat__0_Oextremum__strict,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1978_le__0__eq,axiom,
% 9.07/5.18      ! [N: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 9.07/5.18        = ( N = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1979_bot__nat__0_Oextremum__uniqueI,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.18       => ( A = zero_zero_nat ) ) ).
% 9.07/5.18  
% 9.07/5.18  thf(fact_1980_bot__nat__0_Oextremum__unique,axiom,
% 9.07/5.18      ! [A: nat] :
% 9.07/5.18        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.18        = ( A = zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1981_less__eq__nat_Osimps_I1_J,axiom,
% 9.07/5.19      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1982_add__eq__self__zero,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ( plus_plus_nat @ M @ N )
% 9.07/5.19          = M )
% 9.07/5.19       => ( N = zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1983_plus__nat_Oadd__0,axiom,
% 9.07/5.19      ! [N: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ zero_zero_nat @ N )
% 9.07/5.19        = N ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1984_diffs0__imp__equal,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ( minus_minus_nat @ M @ N )
% 9.07/5.19          = zero_zero_nat )
% 9.07/5.19       => ( ( ( minus_minus_nat @ N @ M )
% 9.07/5.19            = zero_zero_nat )
% 9.07/5.19         => ( M = N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1985_minus__nat_Odiff__0,axiom,
% 9.07/5.19      ! [M: nat] :
% 9.07/5.19        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 9.07/5.19        = M ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1986_mult__0,axiom,
% 9.07/5.19      ! [N: nat] :
% 9.07/5.19        ( ( times_times_nat @ zero_zero_nat @ N )
% 9.07/5.19        = zero_zero_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1987_nat__mult__eq__cancel__disj,axiom,
% 9.07/5.19      ! [K: nat,M: nat,N: nat] :
% 9.07/5.19        ( ( ( times_times_nat @ K @ M )
% 9.07/5.19          = ( times_times_nat @ K @ N ) )
% 9.07/5.19        = ( ( K = zero_zero_nat )
% 9.07/5.19          | ( M = N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1988_split__mod,axiom,
% 9.07/5.19      ! [P: nat > $o,M: nat,N: nat] :
% 9.07/5.19        ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 9.07/5.19        = ( ( ( N = zero_zero_nat )
% 9.07/5.19           => ( P @ M ) )
% 9.07/5.19          & ( ( N != zero_zero_nat )
% 9.07/5.19           => ! [I4: nat,J3: nat] :
% 9.07/5.19                ( ( ord_less_nat @ J3 @ N )
% 9.07/5.19               => ( ( M
% 9.07/5.19                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 9.07/5.19                 => ( P @ J3 ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1989_power__eq__imp__eq__base,axiom,
% 9.07/5.19      ! [A: real,N: nat,B2: real] :
% 9.07/5.19        ( ( ( power_power_real @ A @ N )
% 9.07/5.19          = ( power_power_real @ B2 @ N ) )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.19           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19             => ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1990_power__eq__imp__eq__base,axiom,
% 9.07/5.19      ! [A: rat,N: nat,B2: rat] :
% 9.07/5.19        ( ( ( power_power_rat @ A @ N )
% 9.07/5.19          = ( power_power_rat @ B2 @ N ) )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.19           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19             => ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1991_power__eq__imp__eq__base,axiom,
% 9.07/5.19      ! [A: nat,N: nat,B2: nat] :
% 9.07/5.19        ( ( ( power_power_nat @ A @ N )
% 9.07/5.19          = ( power_power_nat @ B2 @ N ) )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.19           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19             => ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1992_power__eq__imp__eq__base,axiom,
% 9.07/5.19      ! [A: int,N: nat,B2: int] :
% 9.07/5.19        ( ( ( power_power_int @ A @ N )
% 9.07/5.19          = ( power_power_int @ B2 @ N ) )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.19           => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19             => ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1993_power__eq__iff__eq__base,axiom,
% 9.07/5.19      ! [N: nat,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.19           => ( ( ( power_power_real @ A @ N )
% 9.07/5.19                = ( power_power_real @ B2 @ N ) )
% 9.07/5.19              = ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1994_power__eq__iff__eq__base,axiom,
% 9.07/5.19      ! [N: nat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.19           => ( ( ( power_power_rat @ A @ N )
% 9.07/5.19                = ( power_power_rat @ B2 @ N ) )
% 9.07/5.19              = ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1995_power__eq__iff__eq__base,axiom,
% 9.07/5.19      ! [N: nat,A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.19           => ( ( ( power_power_nat @ A @ N )
% 9.07/5.19                = ( power_power_nat @ B2 @ N ) )
% 9.07/5.19              = ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1996_power__eq__iff__eq__base,axiom,
% 9.07/5.19      ! [N: nat,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.19           => ( ( ( power_power_int @ A @ N )
% 9.07/5.19                = ( power_power_int @ B2 @ N ) )
% 9.07/5.19              = ( A = B2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1997_lambda__zero,axiom,
% 9.07/5.19      ( ( ^ [H: complex] : zero_zero_complex )
% 9.07/5.19      = ( times_times_complex @ zero_zero_complex ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1998_lambda__zero,axiom,
% 9.07/5.19      ( ( ^ [H: real] : zero_zero_real )
% 9.07/5.19      = ( times_times_real @ zero_zero_real ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_1999_lambda__zero,axiom,
% 9.07/5.19      ( ( ^ [H: rat] : zero_zero_rat )
% 9.07/5.19      = ( times_times_rat @ zero_zero_rat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2000_lambda__zero,axiom,
% 9.07/5.19      ( ( ^ [H: nat] : zero_zero_nat )
% 9.07/5.19      = ( times_times_nat @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2001_lambda__zero,axiom,
% 9.07/5.19      ( ( ^ [H: int] : zero_zero_int )
% 9.07/5.19      = ( times_times_int @ zero_zero_int ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2002_vebt__insert_Osimps_I2_J,axiom,
% 9.07/5.19      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 9.07/5.19        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X3 )
% 9.07/5.19        = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2003_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 9.07/5.19      ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
% 9.07/5.19        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
% 9.07/5.19        = ( ( X3 = Mi )
% 9.07/5.19          | ( X3 = Ma ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2004_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.07/5.19      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 9.07/5.19       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.07/5.19          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2005_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.07/5.19      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.07/5.19          = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2006_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.07/5.19          = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2007_cong__exp__iff__simps_I9_J,axiom,
% 9.07/5.19      ! [M: num,Q2: num,N: num] :
% 9.07/5.19        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.07/5.19          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 9.07/5.19        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 9.07/5.19          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2008_cong__exp__iff__simps_I9_J,axiom,
% 9.07/5.19      ! [M: num,Q2: num,N: num] :
% 9.07/5.19        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.07/5.19          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 9.07/5.19        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 9.07/5.19          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2009_cong__exp__iff__simps_I9_J,axiom,
% 9.07/5.19      ! [M: num,Q2: num,N: num] :
% 9.07/5.19        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.07/5.19          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 9.07/5.19        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 9.07/5.19          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2010_cong__exp__iff__simps_I4_J,axiom,
% 9.07/5.19      ! [M: num,N: num] :
% 9.07/5.19        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 9.07/5.19        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2011_cong__exp__iff__simps_I4_J,axiom,
% 9.07/5.19      ! [M: num,N: num] :
% 9.07/5.19        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 9.07/5.19        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2012_cong__exp__iff__simps_I4_J,axiom,
% 9.07/5.19      ! [M: num,N: num] :
% 9.07/5.19        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 9.07/5.19        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2013_mod__eqE,axiom,
% 9.07/5.19      ! [A: int,C: int,B2: int] :
% 9.07/5.19        ( ( ( modulo_modulo_int @ A @ C )
% 9.07/5.19          = ( modulo_modulo_int @ B2 @ C ) )
% 9.07/5.19       => ~ ! [D3: int] :
% 9.07/5.19              ( B2
% 9.07/5.19             != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2014_mod__eqE,axiom,
% 9.07/5.19      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.07/5.19          = ( modulo364778990260209775nteger @ B2 @ C ) )
% 9.07/5.19       => ~ ! [D3: code_integer] :
% 9.07/5.19              ( B2
% 9.07/5.19             != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2015_div__add1__eq,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.07/5.19        = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2016_div__add1__eq,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.07/5.19        = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2017_div__add1__eq,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.19        ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 9.07/5.19        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2018_Suc__times__mod__eq,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.07/5.19       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 9.07/5.19          = one_one_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2019_mod__induct,axiom,
% 9.07/5.19      ! [P: nat > $o,N: nat,P4: nat,M: nat] :
% 9.07/5.19        ( ( P @ N )
% 9.07/5.19       => ( ( ord_less_nat @ N @ P4 )
% 9.07/5.19         => ( ( ord_less_nat @ M @ P4 )
% 9.07/5.19           => ( ! [N3: nat] :
% 9.07/5.19                  ( ( ord_less_nat @ N3 @ P4 )
% 9.07/5.19                 => ( ( P @ N3 )
% 9.07/5.19                   => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
% 9.07/5.19             => ( P @ M ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2020_mod__Suc__le__divisor,axiom,
% 9.07/5.19      ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2021_power__strict__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,N: nat] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19           => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2022_power__strict__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,N: nat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19           => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2023_power__strict__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19           => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2024_power__strict__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,N: nat] :
% 9.07/5.19        ( ( ord_less_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19           => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2025_mod__if,axiom,
% 9.07/5.19      ( modulo_modulo_nat
% 9.07/5.19      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2026_mod__geq,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ~ ( ord_less_nat @ M @ N )
% 9.07/5.19       => ( ( modulo_modulo_nat @ M @ N )
% 9.07/5.19          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2027_vebt__insert_Ocases,axiom,
% 9.07/5.19      ! [X3: produc9072475918466114483BT_nat] :
% 9.07/5.19        ( ! [A3: $o,B3: $o,X5: nat] :
% 9.07/5.19            ( X3
% 9.07/5.19           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
% 9.07/5.19       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 9.07/5.19              ( X3
% 9.07/5.19             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X5 ) )
% 9.07/5.19         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X5 ) )
% 9.07/5.19           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                  ( X3
% 9.07/5.19                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) )
% 9.07/5.19             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                    ( X3
% 9.07/5.19                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2028_vebt__member_Ocases,axiom,
% 9.07/5.19      ! [X3: produc9072475918466114483BT_nat] :
% 9.07/5.19        ( ! [A3: $o,B3: $o,X5: nat] :
% 9.07/5.19            ( X3
% 9.07/5.19           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
% 9.07/5.19       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
% 9.07/5.19              ( X3
% 9.07/5.19             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
% 9.07/5.19         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
% 9.07/5.19           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                  ( X3
% 9.07/5.19                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
% 9.07/5.19             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                    ( X3
% 9.07/5.19                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2029_le__mod__geq,axiom,
% 9.07/5.19      ! [N: nat,M: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ N @ M )
% 9.07/5.19       => ( ( modulo_modulo_nat @ M @ N )
% 9.07/5.19          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2030_nat__mod__eq__iff,axiom,
% 9.07/5.19      ! [X3: nat,N: nat,Y3: nat] :
% 9.07/5.19        ( ( ( modulo_modulo_nat @ X3 @ N )
% 9.07/5.19          = ( modulo_modulo_nat @ Y3 @ N ) )
% 9.07/5.19        = ( ? [Q1: nat,Q22: nat] :
% 9.07/5.19              ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N @ Q1 ) )
% 9.07/5.19              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2031_VEBT__internal_OminNull_Ocases,axiom,
% 9.07/5.19      ! [X3: vEBT_VEBT] :
% 9.07/5.19        ( ( X3
% 9.07/5.19         != ( vEBT_Leaf @ $false @ $false ) )
% 9.07/5.19       => ( ! [Uv2: $o] :
% 9.07/5.19              ( X3
% 9.07/5.19             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.07/5.19         => ( ! [Uu2: $o] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.07/5.19           => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.07/5.19                  ( X3
% 9.07/5.19                 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.07/5.19             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.07/5.19                    ( X3
% 9.07/5.19                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2032_vebt__pred_Osimps_I2_J,axiom,
% 9.07/5.19      ! [A: $o,Uw: $o] :
% 9.07/5.19        ( ( A
% 9.07/5.19         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 9.07/5.19            = ( some_nat @ zero_zero_nat ) ) )
% 9.07/5.19        & ( ~ A
% 9.07/5.19         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 9.07/5.19            = none_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2033_vebt__mint_Osimps_I1_J,axiom,
% 9.07/5.19      ! [A: $o,B2: $o] :
% 9.07/5.19        ( ( A
% 9.07/5.19         => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 9.07/5.19            = ( some_nat @ zero_zero_nat ) ) )
% 9.07/5.19        & ( ~ A
% 9.07/5.19         => ( ( B2
% 9.07/5.19             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 9.07/5.19                = ( some_nat @ one_one_nat ) ) )
% 9.07/5.19            & ( ~ B2
% 9.07/5.19             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B2 ) )
% 9.07/5.19                = none_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2034_not__numeral__le__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2035_not__numeral__le__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2036_not__numeral__le__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2037_not__numeral__le__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2038_zero__le__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2039_zero__le__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2040_zero__le__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2041_zero__le__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2042_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2043_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2044_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2045_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2046_zero__le__mult__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2047_zero__le__mult__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2048_zero__le__mult__iff,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19            & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19            & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2049_mult__nonneg__nonpos2,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2050_mult__nonneg__nonpos2,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2051_mult__nonneg__nonpos2,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 9.07/5.19         => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2052_mult__nonneg__nonpos2,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2053_mult__nonpos__nonneg,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2054_mult__nonpos__nonneg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2055_mult__nonpos__nonneg,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.19         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2056_mult__nonpos__nonneg,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2057_mult__nonneg__nonpos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2058_mult__nonneg__nonpos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2059_mult__nonneg__nonpos,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 9.07/5.19         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2060_mult__nonneg__nonpos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2061_mult__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.19         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2062_mult__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.19         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2063_mult__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.19         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2064_mult__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.19         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2065_split__mult__neg__le,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 9.07/5.19          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
% 9.07/5.19       => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2066_split__mult__neg__le,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 9.07/5.19          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
% 9.07/5.19       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2067_split__mult__neg__le,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19            & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
% 9.07/5.19          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.19            & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
% 9.07/5.19       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2068_split__mult__neg__le,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19            & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 9.07/5.19          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19            & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
% 9.07/5.19       => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2069_mult__le__0__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 9.07/5.19        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 9.07/5.19          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2070_mult__le__0__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 9.07/5.19        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 9.07/5.19          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2071_mult__le__0__iff,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 9.07/5.19        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19            & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
% 9.07/5.19          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19            & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2072_mult__right__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2073_mult__right__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2074_mult__right__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2075_mult__right__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2076_mult__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: real,A: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2077_mult__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2078_mult__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: int,A: int,C: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2079_mult__left__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2080_mult__left__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2081_mult__left__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2082_mult__left__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2083_mult__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2084_mult__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2085_mult__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2086_mult__left__mono__neg,axiom,
% 9.07/5.19      ! [B2: real,A: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2087_mult__left__mono__neg,axiom,
% 9.07/5.19      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2088_mult__left__mono__neg,axiom,
% 9.07/5.19      ! [B2: int,A: int,C: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.07/5.19         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2089_split__mult__pos__le,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 9.07/5.19       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2090_split__mult__pos__le,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 9.07/5.19       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2091_split__mult__pos__le,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19            & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19            & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 9.07/5.19       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2092_zero__le__square,axiom,
% 9.07/5.19      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2093_zero__le__square,axiom,
% 9.07/5.19      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2094_zero__le__square,axiom,
% 9.07/5.19      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2095_mult__mono_H,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2096_mult__mono_H,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2097_mult__mono_H,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2098_mult__mono_H,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2099_mult__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2100_mult__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2101_mult__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2102_mult__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2103_not__numeral__less__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2104_not__numeral__less__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2105_not__numeral__less__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2106_not__numeral__less__zero,axiom,
% 9.07/5.19      ! [N: num] :
% 9.07/5.19        ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2107_zero__less__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2108_zero__less__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2109_zero__less__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2110_zero__less__numeral,axiom,
% 9.07/5.19      ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2111_not__one__le__zero,axiom,
% 9.07/5.19      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2112_not__one__le__zero,axiom,
% 9.07/5.19      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2113_not__one__le__zero,axiom,
% 9.07/5.19      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2114_not__one__le__zero,axiom,
% 9.07/5.19      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2115_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2116_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2117_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2118_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2119_zero__less__one__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2120_zero__less__one__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2121_zero__less__one__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2122_zero__less__one__class_Ozero__le__one,axiom,
% 9.07/5.19      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2123_add__decreasing,axiom,
% 9.07/5.19      ! [A: real,C: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ B2 )
% 9.07/5.19         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2124_add__decreasing,axiom,
% 9.07/5.19      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ B2 )
% 9.07/5.19         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2125_add__decreasing,axiom,
% 9.07/5.19      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_eq_nat @ C @ B2 )
% 9.07/5.19         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2126_add__decreasing,axiom,
% 9.07/5.19      ! [A: int,C: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ C @ B2 )
% 9.07/5.19         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2127_add__increasing,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ B2 @ C )
% 9.07/5.19         => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2128_add__increasing,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2129_add__increasing,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.07/5.19         => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2130_add__increasing,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ B2 @ C )
% 9.07/5.19         => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2131_add__decreasing2,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2132_add__decreasing2,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2133_add__decreasing2,axiom,
% 9.07/5.19      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2134_add__decreasing2,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2135_add__increasing2,axiom,
% 9.07/5.19      ! [C: real,B2: real,A: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19       => ( ( ord_less_eq_real @ B2 @ A )
% 9.07/5.19         => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2136_add__increasing2,axiom,
% 9.07/5.19      ! [C: rat,B2: rat,A: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19       => ( ( ord_less_eq_rat @ B2 @ A )
% 9.07/5.19         => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2137_add__increasing2,axiom,
% 9.07/5.19      ! [C: nat,B2: nat,A: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19       => ( ( ord_less_eq_nat @ B2 @ A )
% 9.07/5.19         => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2138_add__increasing2,axiom,
% 9.07/5.19      ! [C: int,B2: int,A: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19       => ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.19         => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2139_add__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.19         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2140_add__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.19         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2141_add__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.19         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2142_add__nonneg__nonneg,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.19         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2143_add__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2144_add__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2145_add__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 9.07/5.19         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2146_add__nonpos__nonpos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2147_add__nonneg__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.07/5.19         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_real )
% 9.07/5.19            = ( ( X3 = zero_zero_real )
% 9.07/5.19              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2148_add__nonneg__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.07/5.19         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_rat )
% 9.07/5.19            = ( ( X3 = zero_zero_rat )
% 9.07/5.19              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2149_add__nonneg__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: nat,Y3: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.07/5.19         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_nat )
% 9.07/5.19            = ( ( X3 = zero_zero_nat )
% 9.07/5.19              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2150_add__nonneg__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: int,Y3: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.07/5.19         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_int )
% 9.07/5.19            = ( ( X3 = zero_zero_int )
% 9.07/5.19              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2151_add__nonpos__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.07/5.19         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_real )
% 9.07/5.19            = ( ( X3 = zero_zero_real )
% 9.07/5.19              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2152_add__nonpos__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.07/5.19         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_rat )
% 9.07/5.19            = ( ( X3 = zero_zero_rat )
% 9.07/5.19              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2153_add__nonpos__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: nat,Y3: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 9.07/5.19         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_nat )
% 9.07/5.19            = ( ( X3 = zero_zero_nat )
% 9.07/5.19              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2154_add__nonpos__eq__0__iff,axiom,
% 9.07/5.19      ! [X3: int,Y3: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.07/5.19         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 9.07/5.19              = zero_zero_int )
% 9.07/5.19            = ( ( X3 = zero_zero_int )
% 9.07/5.19              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2155_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2156_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2157_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2158_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2159_mult__less__cancel__right__disj,axiom,
% 9.07/5.19      ! [A: real,C: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19            & ( ord_less_real @ A @ B2 ) )
% 9.07/5.19          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19            & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2160_mult__less__cancel__right__disj,axiom,
% 9.07/5.19      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19            & ( ord_less_rat @ A @ B2 ) )
% 9.07/5.19          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2161_mult__less__cancel__right__disj,axiom,
% 9.07/5.19      ! [A: int,C: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19            & ( ord_less_int @ A @ B2 ) )
% 9.07/5.19          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.19            & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2162_mult__strict__right__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2163_mult__strict__right__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2164_mult__strict__right__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2165_mult__strict__right__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2166_mult__strict__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: real,A: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2167_mult__strict__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2168_mult__strict__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: int,A: int,C: int] :
% 9.07/5.19        ( ( ord_less_int @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2169_mult__less__cancel__left__disj,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19            & ( ord_less_real @ A @ B2 ) )
% 9.07/5.19          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19            & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2170_mult__less__cancel__left__disj,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19            & ( ord_less_rat @ A @ B2 ) )
% 9.07/5.19          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2171_mult__less__cancel__left__disj,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19            & ( ord_less_int @ A @ B2 ) )
% 9.07/5.19          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.19            & ( ord_less_int @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2172_mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2173_mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2174_mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2175_mult__strict__left__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2176_mult__strict__left__mono__neg,axiom,
% 9.07/5.19      ! [B2: real,A: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2177_mult__strict__left__mono__neg,axiom,
% 9.07/5.19      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2178_mult__strict__left__mono__neg,axiom,
% 9.07/5.19      ! [B2: int,A: int,C: int] :
% 9.07/5.19        ( ( ord_less_int @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2179_mult__less__cancel__left__pos,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_real @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2180_mult__less__cancel__left__pos,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2181_mult__less__cancel__left__pos,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2182_mult__less__cancel__left__neg,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_real @ B2 @ A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2183_mult__less__cancel__left__neg,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_rat @ B2 @ A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2184_mult__less__cancel__left__neg,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_int @ B2 @ A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2185_zero__less__mult__pos2,axiom,
% 9.07/5.19      ! [B2: real,A: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2186_zero__less__mult__pos2,axiom,
% 9.07/5.19      ! [B2: rat,A: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A ) )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2187_zero__less__mult__pos2,axiom,
% 9.07/5.19      ! [B2: nat,A: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19         => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2188_zero__less__mult__pos2,axiom,
% 9.07/5.19      ! [B2: int,A: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19         => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2189_zero__less__mult__pos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2190_zero__less__mult__pos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2191_zero__less__mult__pos,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19         => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2192_zero__less__mult__pos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19         => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2193_zero__less__mult__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_real @ zero_zero_real @ B2 ) )
% 9.07/5.19          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2194_zero__less__mult__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 9.07/5.19          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2195_zero__less__mult__iff,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19            & ( ord_less_int @ zero_zero_int @ B2 ) )
% 9.07/5.19          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.07/5.19            & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2196_mult__pos__neg2,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2197_mult__pos__neg2,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ B2 @ A ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2198_mult__pos__neg2,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2199_mult__pos__neg2,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2200_mult__pos__pos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2201_mult__pos__pos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2202_mult__pos__pos,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.19         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2203_mult__pos__pos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.19         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2204_mult__pos__neg,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2205_mult__pos__neg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2206_mult__pos__neg,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2207_mult__pos__neg,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2208_mult__neg__pos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.19         => ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2209_mult__neg__pos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.19         => ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2210_mult__neg__pos,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2211_mult__neg__pos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.19         => ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2212_mult__less__0__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_real @ B2 @ zero_zero_real ) )
% 9.07/5.19          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2213_mult__less__0__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( times_times_rat @ A @ B2 ) @ zero_zero_rat )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 9.07/5.19          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2214_mult__less__0__iff,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
% 9.07/5.19        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19            & ( ord_less_int @ B2 @ zero_zero_int ) )
% 9.07/5.19          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.07/5.19            & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2215_not__square__less__zero,axiom,
% 9.07/5.19      ! [A: real] :
% 9.07/5.19        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2216_not__square__less__zero,axiom,
% 9.07/5.19      ! [A: rat] :
% 9.07/5.19        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2217_not__square__less__zero,axiom,
% 9.07/5.19      ! [A: int] :
% 9.07/5.19        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2218_mult__neg__neg,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2219_mult__neg__neg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2220_mult__neg__neg,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2221_vebt__maxt_Osimps_I1_J,axiom,
% 9.07/5.19      ! [B2: $o,A: $o] :
% 9.07/5.19        ( ( B2
% 9.07/5.19         => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 9.07/5.19            = ( some_nat @ one_one_nat ) ) )
% 9.07/5.19        & ( ~ B2
% 9.07/5.19         => ( ( A
% 9.07/5.19             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 9.07/5.19                = ( some_nat @ zero_zero_nat ) ) )
% 9.07/5.19            & ( ~ A
% 9.07/5.19             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B2 ) )
% 9.07/5.19                = none_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2222_le__iff__diff__le__0,axiom,
% 9.07/5.19      ( ord_less_eq_real
% 9.07/5.19      = ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2223_le__iff__diff__le__0,axiom,
% 9.07/5.19      ( ord_less_eq_rat
% 9.07/5.19      = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2224_le__iff__diff__le__0,axiom,
% 9.07/5.19      ( ord_less_eq_int
% 9.07/5.19      = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2225_less__numeral__extra_I1_J,axiom,
% 9.07/5.19      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2226_less__numeral__extra_I1_J,axiom,
% 9.07/5.19      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2227_less__numeral__extra_I1_J,axiom,
% 9.07/5.19      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2228_less__numeral__extra_I1_J,axiom,
% 9.07/5.19      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2229_not__one__less__zero,axiom,
% 9.07/5.19      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2230_not__one__less__zero,axiom,
% 9.07/5.19      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2231_not__one__less__zero,axiom,
% 9.07/5.19      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2232_not__one__less__zero,axiom,
% 9.07/5.19      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2233_zero__less__one,axiom,
% 9.07/5.19      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2234_zero__less__one,axiom,
% 9.07/5.19      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2235_zero__less__one,axiom,
% 9.07/5.19      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2236_zero__less__one,axiom,
% 9.07/5.19      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2237_add__less__zeroD,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.07/5.19          | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2238_add__less__zeroD,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 9.07/5.19          | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2239_add__less__zeroD,axiom,
% 9.07/5.19      ! [X3: int,Y3: int] :
% 9.07/5.19        ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y3 ) @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_int @ X3 @ zero_zero_int )
% 9.07/5.19          | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2240_add__neg__neg,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2241_add__neg__neg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2242_add__neg__neg,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.07/5.19       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 9.07/5.19         => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2243_add__neg__neg,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.07/5.19         => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2244_add__pos__pos,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2245_add__pos__pos,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2246_add__pos__pos,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.19         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2247_add__pos__pos,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.19         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2248_canonically__ordered__monoid__add__class_OlessE,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ~ ! [C2: nat] :
% 9.07/5.19              ( ( B2
% 9.07/5.19                = ( plus_plus_nat @ A @ C2 ) )
% 9.07/5.19             => ( C2 = zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2249_pos__add__strict,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ( ord_less_real @ B2 @ C )
% 9.07/5.19         => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2250_pos__add__strict,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ( ord_less_rat @ B2 @ C )
% 9.07/5.19         => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2251_pos__add__strict,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ( ord_less_nat @ B2 @ C )
% 9.07/5.19         => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2252_pos__add__strict,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ( ord_less_int @ B2 @ C )
% 9.07/5.19         => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2253_divide__right__mono__neg,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2254_divide__right__mono__neg,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2255_divide__nonpos__nonpos,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2256_divide__nonpos__nonpos,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2257_divide__nonpos__nonneg,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.07/5.19         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2258_divide__nonpos__nonneg,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.07/5.19         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2259_divide__nonneg__nonpos,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2260_divide__nonneg__nonpos,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2261_divide__nonneg__nonneg,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.07/5.19         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2262_divide__nonneg__nonneg,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.07/5.19         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2263_zero__le__divide__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2264_zero__le__divide__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
% 9.07/5.19          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2265_divide__right__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2266_divide__right__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2267_divide__le__0__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 9.07/5.19        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
% 9.07/5.19          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2268_divide__le__0__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 9.07/5.19        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
% 9.07/5.19          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2269_less__iff__diff__less__0,axiom,
% 9.07/5.19      ( ord_less_real
% 9.07/5.19      = ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2270_less__iff__diff__less__0,axiom,
% 9.07/5.19      ( ord_less_rat
% 9.07/5.19      = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2271_less__iff__diff__less__0,axiom,
% 9.07/5.19      ( ord_less_int
% 9.07/5.19      = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2272_divide__strict__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: real,A: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2273_divide__strict__right__mono__neg,axiom,
% 9.07/5.19      ! [B2: rat,A: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ B2 @ A )
% 9.07/5.19       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2274_divide__strict__right__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2275_divide__strict__right__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2276_zero__less__divide__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_real @ zero_zero_real @ B2 ) )
% 9.07/5.19          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2277_zero__less__divide__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_rat @ zero_zero_rat @ B2 ) )
% 9.07/5.19          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2278_divide__less__cancel,axiom,
% 9.07/5.19      ! [A: real,C: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19           => ( ord_less_real @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19           => ( ord_less_real @ B2 @ A ) )
% 9.07/5.19          & ( C != zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2279_divide__less__cancel,axiom,
% 9.07/5.19      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19           => ( ord_less_rat @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19           => ( ord_less_rat @ B2 @ A ) )
% 9.07/5.19          & ( C != zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2280_divide__less__0__iff,axiom,
% 9.07/5.19      ! [A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( divide_divide_real @ A @ B2 ) @ zero_zero_real )
% 9.07/5.19        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19            & ( ord_less_real @ B2 @ zero_zero_real ) )
% 9.07/5.19          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.19            & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2281_divide__less__0__iff,axiom,
% 9.07/5.19      ! [A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B2 ) @ zero_zero_rat )
% 9.07/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19            & ( ord_less_rat @ B2 @ zero_zero_rat ) )
% 9.07/5.19          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.19            & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2282_divide__pos__pos,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2283_divide__pos__pos,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2284_divide__pos__neg,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.07/5.19       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2285_divide__pos__neg,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.07/5.19       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2286_divide__neg__pos,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.07/5.19         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2287_divide__neg__pos,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.07/5.19         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2288_divide__neg__neg,axiom,
% 9.07/5.19      ! [X3: real,Y3: real] :
% 9.07/5.19        ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.07/5.19         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2289_divide__neg__neg,axiom,
% 9.07/5.19      ! [X3: rat,Y3: rat] :
% 9.07/5.19        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.07/5.19         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2290_zero__le__power,axiom,
% 9.07/5.19      ! [A: real,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2291_zero__le__power,axiom,
% 9.07/5.19      ! [A: rat,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2292_zero__le__power,axiom,
% 9.07/5.19      ! [A: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2293_zero__le__power,axiom,
% 9.07/5.19      ! [A: int,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2294_power__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2295_power__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2296_power__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2297_power__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,N: nat] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2298_zero__less__power,axiom,
% 9.07/5.19      ! [A: real,N: nat] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2299_zero__less__power,axiom,
% 9.07/5.19      ! [A: rat,N: nat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2300_zero__less__power,axiom,
% 9.07/5.19      ! [A: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2301_zero__less__power,axiom,
% 9.07/5.19      ! [A: int,N: nat] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2302_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 9.07/5.19      ! [X3: vEBT_VEBT] :
% 9.07/5.19        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 9.07/5.19       => ( ! [Uv2: $o] :
% 9.07/5.19              ( X3
% 9.07/5.19             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.07/5.19         => ( ! [Uu2: $o] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.07/5.19           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.07/5.19                  ( X3
% 9.07/5.19                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2303_frac__eq__eq,axiom,
% 9.07/5.19      ! [Y3: complex,Z: complex,X3: complex,W: complex] :
% 9.07/5.19        ( ( Y3 != zero_zero_complex )
% 9.07/5.19       => ( ( Z != zero_zero_complex )
% 9.07/5.19         => ( ( ( divide1717551699836669952omplex @ X3 @ Y3 )
% 9.07/5.19              = ( divide1717551699836669952omplex @ W @ Z ) )
% 9.07/5.19            = ( ( times_times_complex @ X3 @ Z )
% 9.07/5.19              = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2304_frac__eq__eq,axiom,
% 9.07/5.19      ! [Y3: real,Z: real,X3: real,W: real] :
% 9.07/5.19        ( ( Y3 != zero_zero_real )
% 9.07/5.19       => ( ( Z != zero_zero_real )
% 9.07/5.19         => ( ( ( divide_divide_real @ X3 @ Y3 )
% 9.07/5.19              = ( divide_divide_real @ W @ Z ) )
% 9.07/5.19            = ( ( times_times_real @ X3 @ Z )
% 9.07/5.19              = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2305_frac__eq__eq,axiom,
% 9.07/5.19      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 9.07/5.19        ( ( Y3 != zero_zero_rat )
% 9.07/5.19       => ( ( Z != zero_zero_rat )
% 9.07/5.19         => ( ( ( divide_divide_rat @ X3 @ Y3 )
% 9.07/5.19              = ( divide_divide_rat @ W @ Z ) )
% 9.07/5.19            = ( ( times_times_rat @ X3 @ Z )
% 9.07/5.19              = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2306_divide__eq__eq,axiom,
% 9.07/5.19      ! [B2: complex,C: complex,A: complex] :
% 9.07/5.19        ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 9.07/5.19          = A )
% 9.07/5.19        = ( ( ( C != zero_zero_complex )
% 9.07/5.19           => ( B2
% 9.07/5.19              = ( times_times_complex @ A @ C ) ) )
% 9.07/5.19          & ( ( C = zero_zero_complex )
% 9.07/5.19           => ( A = zero_zero_complex ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2307_divide__eq__eq,axiom,
% 9.07/5.19      ! [B2: real,C: real,A: real] :
% 9.07/5.19        ( ( ( divide_divide_real @ B2 @ C )
% 9.07/5.19          = A )
% 9.07/5.19        = ( ( ( C != zero_zero_real )
% 9.07/5.19           => ( B2
% 9.07/5.19              = ( times_times_real @ A @ C ) ) )
% 9.07/5.19          & ( ( C = zero_zero_real )
% 9.07/5.19           => ( A = zero_zero_real ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2308_divide__eq__eq,axiom,
% 9.07/5.19      ! [B2: rat,C: rat,A: rat] :
% 9.07/5.19        ( ( ( divide_divide_rat @ B2 @ C )
% 9.07/5.19          = A )
% 9.07/5.19        = ( ( ( C != zero_zero_rat )
% 9.07/5.19           => ( B2
% 9.07/5.19              = ( times_times_rat @ A @ C ) ) )
% 9.07/5.19          & ( ( C = zero_zero_rat )
% 9.07/5.19           => ( A = zero_zero_rat ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2309_eq__divide__eq,axiom,
% 9.07/5.19      ! [A: complex,B2: complex,C: complex] :
% 9.07/5.19        ( ( A
% 9.07/5.19          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.07/5.19        = ( ( ( C != zero_zero_complex )
% 9.07/5.19           => ( ( times_times_complex @ A @ C )
% 9.07/5.19              = B2 ) )
% 9.07/5.19          & ( ( C = zero_zero_complex )
% 9.07/5.19           => ( A = zero_zero_complex ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2310_eq__divide__eq,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real] :
% 9.07/5.19        ( ( A
% 9.07/5.19          = ( divide_divide_real @ B2 @ C ) )
% 9.07/5.19        = ( ( ( C != zero_zero_real )
% 9.07/5.19           => ( ( times_times_real @ A @ C )
% 9.07/5.19              = B2 ) )
% 9.07/5.19          & ( ( C = zero_zero_real )
% 9.07/5.19           => ( A = zero_zero_real ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2311_eq__divide__eq,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.19        ( ( A
% 9.07/5.19          = ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.19        = ( ( ( C != zero_zero_rat )
% 9.07/5.19           => ( ( times_times_rat @ A @ C )
% 9.07/5.19              = B2 ) )
% 9.07/5.19          & ( ( C = zero_zero_rat )
% 9.07/5.19           => ( A = zero_zero_rat ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2312_divide__eq__imp,axiom,
% 9.07/5.19      ! [C: complex,B2: complex,A: complex] :
% 9.07/5.19        ( ( C != zero_zero_complex )
% 9.07/5.19       => ( ( B2
% 9.07/5.19            = ( times_times_complex @ A @ C ) )
% 9.07/5.19         => ( ( divide1717551699836669952omplex @ B2 @ C )
% 9.07/5.19            = A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2313_divide__eq__imp,axiom,
% 9.07/5.19      ! [C: real,B2: real,A: real] :
% 9.07/5.19        ( ( C != zero_zero_real )
% 9.07/5.19       => ( ( B2
% 9.07/5.19            = ( times_times_real @ A @ C ) )
% 9.07/5.19         => ( ( divide_divide_real @ B2 @ C )
% 9.07/5.19            = A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2314_divide__eq__imp,axiom,
% 9.07/5.19      ! [C: rat,B2: rat,A: rat] :
% 9.07/5.19        ( ( C != zero_zero_rat )
% 9.07/5.19       => ( ( B2
% 9.07/5.19            = ( times_times_rat @ A @ C ) )
% 9.07/5.19         => ( ( divide_divide_rat @ B2 @ C )
% 9.07/5.19            = A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2315_eq__divide__imp,axiom,
% 9.07/5.19      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.19        ( ( C != zero_zero_complex )
% 9.07/5.19       => ( ( ( times_times_complex @ A @ C )
% 9.07/5.19            = B2 )
% 9.07/5.19         => ( A
% 9.07/5.19            = ( divide1717551699836669952omplex @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2316_eq__divide__imp,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( C != zero_zero_real )
% 9.07/5.19       => ( ( ( times_times_real @ A @ C )
% 9.07/5.19            = B2 )
% 9.07/5.19         => ( A
% 9.07/5.19            = ( divide_divide_real @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2317_eq__divide__imp,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( C != zero_zero_rat )
% 9.07/5.19       => ( ( ( times_times_rat @ A @ C )
% 9.07/5.19            = B2 )
% 9.07/5.19         => ( A
% 9.07/5.19            = ( divide_divide_rat @ B2 @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2318_nonzero__divide__eq__eq,axiom,
% 9.07/5.19      ! [C: complex,B2: complex,A: complex] :
% 9.07/5.19        ( ( C != zero_zero_complex )
% 9.07/5.19       => ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 9.07/5.19            = A )
% 9.07/5.19          = ( B2
% 9.07/5.19            = ( times_times_complex @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2319_nonzero__divide__eq__eq,axiom,
% 9.07/5.19      ! [C: real,B2: real,A: real] :
% 9.07/5.19        ( ( C != zero_zero_real )
% 9.07/5.19       => ( ( ( divide_divide_real @ B2 @ C )
% 9.07/5.19            = A )
% 9.07/5.19          = ( B2
% 9.07/5.19            = ( times_times_real @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2320_nonzero__divide__eq__eq,axiom,
% 9.07/5.19      ! [C: rat,B2: rat,A: rat] :
% 9.07/5.19        ( ( C != zero_zero_rat )
% 9.07/5.19       => ( ( ( divide_divide_rat @ B2 @ C )
% 9.07/5.19            = A )
% 9.07/5.19          = ( B2
% 9.07/5.19            = ( times_times_rat @ A @ C ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2321_nonzero__eq__divide__eq,axiom,
% 9.07/5.19      ! [C: complex,A: complex,B2: complex] :
% 9.07/5.19        ( ( C != zero_zero_complex )
% 9.07/5.19       => ( ( A
% 9.07/5.19            = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.07/5.19          = ( ( times_times_complex @ A @ C )
% 9.07/5.19            = B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2322_nonzero__eq__divide__eq,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( C != zero_zero_real )
% 9.07/5.19       => ( ( A
% 9.07/5.19            = ( divide_divide_real @ B2 @ C ) )
% 9.07/5.19          = ( ( times_times_real @ A @ C )
% 9.07/5.19            = B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2323_nonzero__eq__divide__eq,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( C != zero_zero_rat )
% 9.07/5.19       => ( ( A
% 9.07/5.19            = ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.19          = ( ( times_times_rat @ A @ C )
% 9.07/5.19            = B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2324_right__inverse__eq,axiom,
% 9.07/5.19      ! [B2: complex,A: complex] :
% 9.07/5.19        ( ( B2 != zero_zero_complex )
% 9.07/5.19       => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 9.07/5.19            = one_one_complex )
% 9.07/5.19          = ( A = B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2325_right__inverse__eq,axiom,
% 9.07/5.19      ! [B2: real,A: real] :
% 9.07/5.19        ( ( B2 != zero_zero_real )
% 9.07/5.19       => ( ( ( divide_divide_real @ A @ B2 )
% 9.07/5.19            = one_one_real )
% 9.07/5.19          = ( A = B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2326_right__inverse__eq,axiom,
% 9.07/5.19      ! [B2: rat,A: rat] :
% 9.07/5.19        ( ( B2 != zero_zero_rat )
% 9.07/5.19       => ( ( ( divide_divide_rat @ A @ B2 )
% 9.07/5.19            = one_one_rat )
% 9.07/5.19          = ( A = B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2327_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 9.07/5.19      ! [X3: vEBT_VEBT] :
% 9.07/5.19        ( ( vEBT_VEBT_minNull @ X3 )
% 9.07/5.19       => ( ( X3
% 9.07/5.19           != ( vEBT_Leaf @ $false @ $false ) )
% 9.07/5.19         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2328_power__0,axiom,
% 9.07/5.19      ! [A: rat] :
% 9.07/5.19        ( ( power_power_rat @ A @ zero_zero_nat )
% 9.07/5.19        = one_one_rat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2329_power__0,axiom,
% 9.07/5.19      ! [A: nat] :
% 9.07/5.19        ( ( power_power_nat @ A @ zero_zero_nat )
% 9.07/5.19        = one_one_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2330_power__0,axiom,
% 9.07/5.19      ! [A: real] :
% 9.07/5.19        ( ( power_power_real @ A @ zero_zero_nat )
% 9.07/5.19        = one_one_real ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2331_power__0,axiom,
% 9.07/5.19      ! [A: int] :
% 9.07/5.19        ( ( power_power_int @ A @ zero_zero_nat )
% 9.07/5.19        = one_one_int ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2332_power__0,axiom,
% 9.07/5.19      ! [A: complex] :
% 9.07/5.19        ( ( power_power_complex @ A @ zero_zero_nat )
% 9.07/5.19        = one_one_complex ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2333_less__Suc__eq__0__disj,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.07/5.19        = ( ( M = zero_zero_nat )
% 9.07/5.19          | ? [J3: nat] :
% 9.07/5.19              ( ( M
% 9.07/5.19                = ( suc @ J3 ) )
% 9.07/5.19              & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2334_gr0__implies__Suc,axiom,
% 9.07/5.19      ! [N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19       => ? [M4: nat] :
% 9.07/5.19            ( N
% 9.07/5.19            = ( suc @ M4 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2335_All__less__Suc2,axiom,
% 9.07/5.19      ! [N: nat,P: nat > $o] :
% 9.07/5.19        ( ( ! [I4: nat] :
% 9.07/5.19              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.07/5.19             => ( P @ I4 ) ) )
% 9.07/5.19        = ( ( P @ zero_zero_nat )
% 9.07/5.19          & ! [I4: nat] :
% 9.07/5.19              ( ( ord_less_nat @ I4 @ N )
% 9.07/5.19             => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2336_gr0__conv__Suc,axiom,
% 9.07/5.19      ! [N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19        = ( ? [M3: nat] :
% 9.07/5.19              ( N
% 9.07/5.19              = ( suc @ M3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2337_Ex__less__Suc2,axiom,
% 9.07/5.19      ! [N: nat,P: nat > $o] :
% 9.07/5.19        ( ( ? [I4: nat] :
% 9.07/5.19              ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.07/5.19              & ( P @ I4 ) ) )
% 9.07/5.19        = ( ( P @ zero_zero_nat )
% 9.07/5.19          | ? [I4: nat] :
% 9.07/5.19              ( ( ord_less_nat @ I4 @ N )
% 9.07/5.19              & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2338_add__is__1,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ( plus_plus_nat @ M @ N )
% 9.07/5.19          = ( suc @ zero_zero_nat ) )
% 9.07/5.19        = ( ( ( M
% 9.07/5.19              = ( suc @ zero_zero_nat ) )
% 9.07/5.19            & ( N = zero_zero_nat ) )
% 9.07/5.19          | ( ( M = zero_zero_nat )
% 9.07/5.19            & ( N
% 9.07/5.19              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2339_one__is__add,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ( suc @ zero_zero_nat )
% 9.07/5.19          = ( plus_plus_nat @ M @ N ) )
% 9.07/5.19        = ( ( ( M
% 9.07/5.19              = ( suc @ zero_zero_nat ) )
% 9.07/5.19            & ( N = zero_zero_nat ) )
% 9.07/5.19          | ( ( M = zero_zero_nat )
% 9.07/5.19            & ( N
% 9.07/5.19              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2340_ex__least__nat__le,axiom,
% 9.07/5.19      ! [P: nat > $o,N: nat] :
% 9.07/5.19        ( ( P @ N )
% 9.07/5.19       => ( ~ ( P @ zero_zero_nat )
% 9.07/5.19         => ? [K3: nat] :
% 9.07/5.19              ( ( ord_less_eq_nat @ K3 @ N )
% 9.07/5.19              & ! [I2: nat] :
% 9.07/5.19                  ( ( ord_less_nat @ I2 @ K3 )
% 9.07/5.19                 => ~ ( P @ I2 ) )
% 9.07/5.19              & ( P @ K3 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2341_option_Osize_I4_J,axiom,
% 9.07/5.19      ! [X2: nat] :
% 9.07/5.19        ( ( size_size_option_nat @ ( some_nat @ X2 ) )
% 9.07/5.19        = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2342_option_Osize_I4_J,axiom,
% 9.07/5.19      ! [X2: product_prod_nat_nat] :
% 9.07/5.19        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
% 9.07/5.19        = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2343_option_Osize_I4_J,axiom,
% 9.07/5.19      ! [X2: num] :
% 9.07/5.19        ( ( size_size_option_num @ ( some_num @ X2 ) )
% 9.07/5.19        = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2344_less__imp__add__positive,axiom,
% 9.07/5.19      ! [I: nat,J: nat] :
% 9.07/5.19        ( ( ord_less_nat @ I @ J )
% 9.07/5.19       => ? [K3: nat] :
% 9.07/5.19            ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 9.07/5.19            & ( ( plus_plus_nat @ I @ K3 )
% 9.07/5.19              = J ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2345_option_Osize_I3_J,axiom,
% 9.07/5.19      ( ( size_size_option_nat @ none_nat )
% 9.07/5.19      = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2346_option_Osize_I3_J,axiom,
% 9.07/5.19      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 9.07/5.19      = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2347_option_Osize_I3_J,axiom,
% 9.07/5.19      ( ( size_size_option_num @ none_num )
% 9.07/5.19      = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2348_diff__less,axiom,
% 9.07/5.19      ! [N: nat,M: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.07/5.19         => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2349_nat__mult__less__cancel1,axiom,
% 9.07/5.19      ! [K: nat,M: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.19       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.07/5.19          = ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2350_nat__mult__eq__cancel1,axiom,
% 9.07/5.19      ! [K: nat,M: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.19       => ( ( ( times_times_nat @ K @ M )
% 9.07/5.19            = ( times_times_nat @ K @ N ) )
% 9.07/5.19          = ( M = N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2351_mult__less__mono2,axiom,
% 9.07/5.19      ! [I: nat,J: nat,K: nat] :
% 9.07/5.19        ( ( ord_less_nat @ I @ J )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2352_mult__less__mono1,axiom,
% 9.07/5.19      ! [I: nat,J: nat,K: nat] :
% 9.07/5.19        ( ( ord_less_nat @ I @ J )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.07/5.19         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2353_One__nat__def,axiom,
% 9.07/5.19      ( one_one_nat
% 9.07/5.19      = ( suc @ zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2354_divmod__digit__0_I2_J,axiom,
% 9.07/5.19      ! [B2: nat,A: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.07/5.19         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
% 9.07/5.19            = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2355_divmod__digit__0_I2_J,axiom,
% 9.07/5.19      ! [B2: int,A: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.07/5.19         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
% 9.07/5.19            = ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2356_divmod__digit__0_I2_J,axiom,
% 9.07/5.19      ! [B2: code_integer,A: code_integer] :
% 9.07/5.19        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.07/5.19       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.07/5.19         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
% 9.07/5.19            = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2357_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( ( divide_divide_nat @ M @ N )
% 9.07/5.19          = zero_zero_nat )
% 9.07/5.19        = ( ( ord_less_nat @ M @ N )
% 9.07/5.19          | ( N = zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2358_diff__add__0,axiom,
% 9.07/5.19      ! [N: nat,M: nat] :
% 9.07/5.19        ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 9.07/5.19        = zero_zero_nat ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2359_bits__stable__imp__add__self,axiom,
% 9.07/5.19      ! [A: nat] :
% 9.07/5.19        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.07/5.19          = A )
% 9.07/5.19       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.07/5.19          = zero_zero_nat ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2360_bits__stable__imp__add__self,axiom,
% 9.07/5.19      ! [A: int] :
% 9.07/5.19        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.07/5.19          = A )
% 9.07/5.19       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.07/5.19          = zero_zero_int ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2361_bits__stable__imp__add__self,axiom,
% 9.07/5.19      ! [A: code_integer] :
% 9.07/5.19        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.07/5.19          = A )
% 9.07/5.19       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.07/5.19          = zero_z3403309356797280102nteger ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2362_nat__power__less__imp__less,axiom,
% 9.07/5.19      ! [I: nat,M: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ I )
% 9.07/5.19       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.07/5.19         => ( ord_less_nat @ M @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2363_mult__eq__self__implies__10,axiom,
% 9.07/5.19      ! [M: nat,N: nat] :
% 9.07/5.19        ( ( M
% 9.07/5.19          = ( times_times_nat @ M @ N ) )
% 9.07/5.19       => ( ( N = one_one_nat )
% 9.07/5.19          | ( M = zero_zero_nat ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2364_vebt__insert_Osimps_I3_J,axiom,
% 9.07/5.19      ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 9.07/5.19        ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X3 )
% 9.07/5.19        = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2365_vebt__member_Osimps_I3_J,axiom,
% 9.07/5.19      ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
% 9.07/5.19        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2366_verit__le__mono__div,axiom,
% 9.07/5.19      ! [A4: nat,B5: nat,N: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A4 @ B5 )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.07/5.19         => ( ord_less_eq_nat
% 9.07/5.19            @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
% 9.07/5.19              @ ( if_nat
% 9.07/5.19                @ ( ( modulo_modulo_nat @ B5 @ N )
% 9.07/5.19                  = zero_zero_nat )
% 9.07/5.19                @ one_one_nat
% 9.07/5.19                @ zero_zero_nat ) )
% 9.07/5.19            @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2367_VEBT__internal_Omembermima_Ocases,axiom,
% 9.07/5.19      ! [X3: produc9072475918466114483BT_nat] :
% 9.07/5.19        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 9.07/5.19            ( X3
% 9.07/5.19           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 9.07/5.19       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 9.07/5.19              ( X3
% 9.07/5.19             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 9.07/5.19         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
% 9.07/5.19           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                  ( X3
% 9.07/5.19                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X5 ) )
% 9.07/5.19             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                    ( X3
% 9.07/5.19                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2368_vebt__pred_Ocases,axiom,
% 9.07/5.19      ! [X3: produc9072475918466114483BT_nat] :
% 9.07/5.19        ( ! [Uu2: $o,Uv2: $o] :
% 9.07/5.19            ( X3
% 9.07/5.19           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 9.07/5.19       => ( ! [A3: $o,Uw2: $o] :
% 9.07/5.19              ( X3
% 9.07/5.19             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 9.07/5.19         => ( ! [A3: $o,B3: $o,Va2: nat] :
% 9.07/5.19                ( X3
% 9.07/5.19               != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 9.07/5.19           => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 9.07/5.19                  ( X3
% 9.07/5.19                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 9.07/5.19             => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 9.07/5.19                    ( X3
% 9.07/5.19                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
% 9.07/5.19               => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 9.07/5.19                      ( X3
% 9.07/5.19                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
% 9.07/5.19                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.07/5.19                        ( X3
% 9.07/5.19                       != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2369_vebt__pred_Osimps_I3_J,axiom,
% 9.07/5.19      ! [B2: $o,A: $o,Va: nat] :
% 9.07/5.19        ( ( B2
% 9.07/5.19         => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
% 9.07/5.19            = ( some_nat @ one_one_nat ) ) )
% 9.07/5.19        & ( ~ B2
% 9.07/5.19         => ( ( A
% 9.07/5.19             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
% 9.07/5.19                = ( some_nat @ zero_zero_nat ) ) )
% 9.07/5.19            & ( ~ A
% 9.07/5.19             => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B2 ) @ ( suc @ ( suc @ Va ) ) )
% 9.07/5.19                = none_nat ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2370_divmod__digit__0_I1_J,axiom,
% 9.07/5.19      ! [B2: nat,A: nat] :
% 9.07/5.19        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.07/5.19         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.07/5.19            = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2371_divmod__digit__0_I1_J,axiom,
% 9.07/5.19      ! [B2: int,A: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.07/5.19         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.07/5.19            = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2372_divmod__digit__0_I1_J,axiom,
% 9.07/5.19      ! [B2: code_integer,A: code_integer] :
% 9.07/5.19        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.07/5.19       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.07/5.19         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.07/5.19            = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2373_cong__exp__iff__simps_I8_J,axiom,
% 9.07/5.19      ! [M: num,Q2: num] :
% 9.07/5.19        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.07/5.19       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2374_cong__exp__iff__simps_I8_J,axiom,
% 9.07/5.19      ! [M: num,Q2: num] :
% 9.07/5.19        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.07/5.19       != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2375_cong__exp__iff__simps_I8_J,axiom,
% 9.07/5.19      ! [M: num,Q2: num] :
% 9.07/5.19        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.07/5.19       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2376_cong__exp__iff__simps_I6_J,axiom,
% 9.07/5.19      ! [Q2: num,N: num] :
% 9.07/5.19        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.07/5.19       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2377_cong__exp__iff__simps_I6_J,axiom,
% 9.07/5.19      ! [Q2: num,N: num] :
% 9.07/5.19        ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.07/5.19       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2378_cong__exp__iff__simps_I6_J,axiom,
% 9.07/5.19      ! [Q2: num,N: num] :
% 9.07/5.19        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.07/5.19       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2379_div__mult1__eq,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.07/5.19        = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B2 @ C ) ) @ C ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2380_div__mult1__eq,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.07/5.19        = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B2 @ C ) ) @ C ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2381_div__mult1__eq,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.19        ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 9.07/5.19        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B2 @ C ) ) @ C ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2382_cancel__div__mod__rules_I2_J,axiom,
% 9.07/5.19      ! [B2: nat,A: nat,C: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 9.07/5.19        = ( plus_plus_nat @ A @ C ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2383_cancel__div__mod__rules_I2_J,axiom,
% 9.07/5.19      ! [B2: int,A: int,C: int] :
% 9.07/5.19        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 9.07/5.19        = ( plus_plus_int @ A @ C ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2384_cancel__div__mod__rules_I2_J,axiom,
% 9.07/5.19      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.07/5.19        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 9.07/5.19        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2385_cancel__div__mod__rules_I1_J,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) @ C )
% 9.07/5.19        = ( plus_plus_nat @ A @ C ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2386_cancel__div__mod__rules_I1_J,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int] :
% 9.07/5.19        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) @ C )
% 9.07/5.19        = ( plus_plus_int @ A @ C ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2387_cancel__div__mod__rules_I1_J,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.07/5.19        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ C )
% 9.07/5.19        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2388_mod__div__decomp,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( A
% 9.07/5.19        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2389_mod__div__decomp,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( A
% 9.07/5.19        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2390_mod__div__decomp,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( A
% 9.07/5.19        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2391_div__mult__mod__eq,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2392_div__mult__mod__eq,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2393_div__mult__mod__eq,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2394_mod__div__mult__eq,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2395_mod__div__mult__eq,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2396_mod__div__mult__eq,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2397_mod__mult__div__eq,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2398_mod__mult__div__eq,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2399_mod__mult__div__eq,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2400_mult__div__mod__eq,axiom,
% 9.07/5.19      ! [B2: nat,A: nat] :
% 9.07/5.19        ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2401_mult__div__mod__eq,axiom,
% 9.07/5.19      ! [B2: int,A: int] :
% 9.07/5.19        ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) @ ( modulo_modulo_int @ A @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2402_mult__div__mod__eq,axiom,
% 9.07/5.19      ! [B2: code_integer,A: code_integer] :
% 9.07/5.19        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.07/5.19        = A ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2403_minus__div__mult__eq__mod,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) )
% 9.07/5.19        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2404_minus__div__mult__eq__mod,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) )
% 9.07/5.19        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2405_minus__div__mult__eq__mod,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) )
% 9.07/5.19        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2406_minus__mod__eq__div__mult,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.07/5.19        = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2407_minus__mod__eq__div__mult,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 9.07/5.19        = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2408_minus__mod__eq__div__mult,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.07/5.19        = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2409_minus__mod__eq__mult__div,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.07/5.19        = ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2410_minus__mod__eq__mult__div,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) )
% 9.07/5.19        = ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2411_minus__mod__eq__mult__div,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.07/5.19        = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2412_minus__mult__div__eq__mod,axiom,
% 9.07/5.19      ! [A: nat,B2: nat] :
% 9.07/5.19        ( ( minus_minus_nat @ A @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A @ B2 ) ) )
% 9.07/5.19        = ( modulo_modulo_nat @ A @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2413_minus__mult__div__eq__mod,axiom,
% 9.07/5.19      ! [A: int,B2: int] :
% 9.07/5.19        ( ( minus_minus_int @ A @ ( times_times_int @ B2 @ ( divide_divide_int @ A @ B2 ) ) )
% 9.07/5.19        = ( modulo_modulo_int @ A @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2414_minus__mult__div__eq__mod,axiom,
% 9.07/5.19      ! [A: code_integer,B2: code_integer] :
% 9.07/5.19        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A @ B2 ) ) )
% 9.07/5.19        = ( modulo364778990260209775nteger @ A @ B2 ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2415_mod__eq__nat1E,axiom,
% 9.07/5.19      ! [M: nat,Q2: nat,N: nat] :
% 9.07/5.19        ( ( ( modulo_modulo_nat @ M @ Q2 )
% 9.07/5.19          = ( modulo_modulo_nat @ N @ Q2 ) )
% 9.07/5.19       => ( ( ord_less_eq_nat @ N @ M )
% 9.07/5.19         => ~ ! [S3: nat] :
% 9.07/5.19                ( M
% 9.07/5.19               != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2416_mod__eq__nat2E,axiom,
% 9.07/5.19      ! [M: nat,Q2: nat,N: nat] :
% 9.07/5.19        ( ( ( modulo_modulo_nat @ M @ Q2 )
% 9.07/5.19          = ( modulo_modulo_nat @ N @ Q2 ) )
% 9.07/5.19       => ( ( ord_less_eq_nat @ M @ N )
% 9.07/5.19         => ~ ! [S3: nat] :
% 9.07/5.19                ( N
% 9.07/5.19               != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2417_nat__mod__eq__lemma,axiom,
% 9.07/5.19      ! [X3: nat,N: nat,Y3: nat] :
% 9.07/5.19        ( ( ( modulo_modulo_nat @ X3 @ N )
% 9.07/5.19          = ( modulo_modulo_nat @ Y3 @ N ) )
% 9.07/5.19       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.07/5.19         => ? [Q3: nat] :
% 9.07/5.19              ( X3
% 9.07/5.19              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2418_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 9.07/5.19      ! [X3: vEBT_VEBT,Y3: $o] :
% 9.07/5.19        ( ( ( vEBT_VEBT_minNull @ X3 )
% 9.07/5.19          = Y3 )
% 9.07/5.19       => ( ( ( X3
% 9.07/5.19              = ( vEBT_Leaf @ $false @ $false ) )
% 9.07/5.19           => ~ Y3 )
% 9.07/5.19         => ( ( ? [Uv2: $o] :
% 9.07/5.19                  ( X3
% 9.07/5.19                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.07/5.19             => Y3 )
% 9.07/5.19           => ( ( ? [Uu2: $o] :
% 9.07/5.19                    ( X3
% 9.07/5.19                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.07/5.19               => Y3 )
% 9.07/5.19             => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.07/5.19                      ( X3
% 9.07/5.19                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.07/5.19                 => ~ Y3 )
% 9.07/5.19               => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.07/5.19                        ( X3
% 9.07/5.19                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.07/5.19                   => Y3 ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2419_div__mod__decomp,axiom,
% 9.07/5.19      ! [A4: nat,N: nat] :
% 9.07/5.19        ( A4
% 9.07/5.19        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N ) @ N ) @ ( modulo_modulo_nat @ A4 @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2420_mod__mult2__eq,axiom,
% 9.07/5.19      ! [M: nat,N: nat,Q2: nat] :
% 9.07/5.19        ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 9.07/5.19        = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2421_modulo__nat__def,axiom,
% 9.07/5.19      ( modulo_modulo_nat
% 9.07/5.19      = ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2422_mult__less__le__imp__less,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_real @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19           => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2423_mult__less__le__imp__less,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_rat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2424_mult__less__le__imp__less,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_nat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.07/5.19             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2425_mult__less__le__imp__less,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_eq_int @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19           => ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2426_mult__le__less__imp__less,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ C @ D )
% 9.07/5.19         => ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.19           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2427_mult__le__less__imp__less,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ C @ D )
% 9.07/5.19         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.19           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2428_mult__le__less__imp__less,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ C @ D )
% 9.07/5.19         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.19           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2429_mult__le__less__imp__less,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ C @ D )
% 9.07/5.19         => ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.19           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2430_mult__right__le__imp__le,axiom,
% 9.07/5.19      ! [A: real,C: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2431_mult__right__le__imp__le,axiom,
% 9.07/5.19      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2432_mult__right__le__imp__le,axiom,
% 9.07/5.19      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2433_mult__right__le__imp__le,axiom,
% 9.07/5.19      ! [A: int,C: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2434_mult__left__le__imp__le,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2435_mult__left__le__imp__le,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2436_mult__left__le__imp__le,axiom,
% 9.07/5.19      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 9.07/5.19       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2437_mult__left__le__imp__le,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2438_mult__le__cancel__left__pos,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.19       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2439_mult__le__cancel__left__pos,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.19       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2440_mult__le__cancel__left__pos,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.19       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2441_mult__le__cancel__left__neg,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.19       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_eq_real @ B2 @ A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2442_mult__le__cancel__left__neg,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.19       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_eq_rat @ B2 @ A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2443_mult__le__cancel__left__neg,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.19       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19          = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2444_mult__less__cancel__right,axiom,
% 9.07/5.19      ! [A: real,C: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19           => ( ord_less_real @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.07/5.19           => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2445_mult__less__cancel__right,axiom,
% 9.07/5.19      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19           => ( ord_less_rat @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.07/5.19           => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2446_mult__less__cancel__right,axiom,
% 9.07/5.19      ! [A: int,C: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.19        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19           => ( ord_less_int @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.07/5.19           => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2447_mult__strict__mono_H,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.19           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2448_mult__strict__mono_H,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.19           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2449_mult__strict__mono_H,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.19           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2450_mult__strict__mono_H,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.19        ( ( ord_less_int @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_int @ C @ D )
% 9.07/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.19           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2451_mult__right__less__imp__less,axiom,
% 9.07/5.19      ! [A: real,C: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19         => ( ord_less_real @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2452_mult__right__less__imp__less,axiom,
% 9.07/5.19      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2453_mult__right__less__imp__less,axiom,
% 9.07/5.19      ! [A: nat,C: nat,B2: nat] :
% 9.07/5.19        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2454_mult__right__less__imp__less,axiom,
% 9.07/5.19      ! [A: int,C: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.19       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19         => ( ord_less_int @ A @ B2 ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2455_mult__less__cancel__left,axiom,
% 9.07/5.19      ! [C: real,A: real,B2: real] :
% 9.07/5.19        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19           => ( ord_less_real @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.07/5.19           => ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2456_mult__less__cancel__left,axiom,
% 9.07/5.19      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.19        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19           => ( ord_less_rat @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.07/5.19           => ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2457_mult__less__cancel__left,axiom,
% 9.07/5.19      ! [C: int,A: int,B2: int] :
% 9.07/5.19        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.19        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.19           => ( ord_less_int @ A @ B2 ) )
% 9.07/5.19          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.07/5.19           => ( ord_less_int @ B2 @ A ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2458_mult__strict__mono,axiom,
% 9.07/5.19      ! [A: real,B2: real,C: real,D: real] :
% 9.07/5.19        ( ( ord_less_real @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_real @ C @ D )
% 9.07/5.19         => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.19             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2459_mult__strict__mono,axiom,
% 9.07/5.19      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.07/5.19        ( ( ord_less_rat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_rat @ C @ D )
% 9.07/5.19         => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.19             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2460_mult__strict__mono,axiom,
% 9.07/5.19      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.07/5.19        ( ( ord_less_nat @ A @ B2 )
% 9.07/5.19       => ( ( ord_less_nat @ C @ D )
% 9.07/5.19         => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.19           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.19             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.19  
% 9.07/5.19  thf(fact_2461_mult__strict__mono,axiom,
% 9.07/5.19      ! [A: int,B2: int,C: int,D: int] :
% 9.07/5.20        ( ( ord_less_int @ A @ B2 )
% 9.07/5.20       => ( ( ord_less_int @ C @ D )
% 9.07/5.20         => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.20           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.20             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2462_mult__left__less__imp__less,axiom,
% 9.07/5.20      ! [C: real,A: real,B2: real] :
% 9.07/5.20        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.07/5.20         => ( ord_less_real @ A @ B2 ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2463_mult__left__less__imp__less,axiom,
% 9.07/5.20      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.07/5.20         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2464_mult__left__less__imp__less,axiom,
% 9.07/5.20      ! [C: nat,A: nat,B2: nat] :
% 9.07/5.20        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
% 9.07/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.07/5.20         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2465_mult__left__less__imp__less,axiom,
% 9.07/5.20      ! [C: int,A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.07/5.20         => ( ord_less_int @ A @ B2 ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2466_mult__le__cancel__right,axiom,
% 9.07/5.20      ! [A: real,C: real,B2: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.07/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.20           => ( ord_less_eq_real @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.20           => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2467_mult__le__cancel__right,axiom,
% 9.07/5.20      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.07/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.20           => ( ord_less_eq_rat @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.20           => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2468_mult__le__cancel__right,axiom,
% 9.07/5.20      ! [A: int,C: int,B2: int] :
% 9.07/5.20        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.07/5.20        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.20           => ( ord_less_eq_int @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.20           => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2469_mult__le__cancel__left,axiom,
% 9.07/5.20      ! [C: real,A: real,B2: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.07/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.20           => ( ord_less_eq_real @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.20           => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2470_mult__le__cancel__left,axiom,
% 9.07/5.20      ! [C: rat,A: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.07/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.20           => ( ord_less_eq_rat @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.20           => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2471_mult__le__cancel__left,axiom,
% 9.07/5.20      ! [C: int,A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.07/5.20        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.07/5.20           => ( ord_less_eq_int @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.07/5.20           => ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2472_field__le__epsilon,axiom,
% 9.07/5.20      ! [X3: real,Y3: real] :
% 9.07/5.20        ( ! [E2: real] :
% 9.07/5.20            ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.07/5.20           => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 9.07/5.20       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2473_field__le__epsilon,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat] :
% 9.07/5.20        ( ! [E2: rat] :
% 9.07/5.20            ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.07/5.20           => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 9.07/5.20       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2474_add__strict__increasing2,axiom,
% 9.07/5.20      ! [A: real,B2: real,C: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.20       => ( ( ord_less_real @ B2 @ C )
% 9.07/5.20         => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2475_add__strict__increasing2,axiom,
% 9.07/5.20      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.20       => ( ( ord_less_rat @ B2 @ C )
% 9.07/5.20         => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2476_add__strict__increasing2,axiom,
% 9.07/5.20      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.20       => ( ( ord_less_nat @ B2 @ C )
% 9.07/5.20         => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2477_add__strict__increasing2,axiom,
% 9.07/5.20      ! [A: int,B2: int,C: int] :
% 9.07/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.20       => ( ( ord_less_int @ B2 @ C )
% 9.07/5.20         => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2478_add__strict__increasing,axiom,
% 9.07/5.20      ! [A: real,B2: real,C: real] :
% 9.07/5.20        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.20       => ( ( ord_less_eq_real @ B2 @ C )
% 9.07/5.20         => ( ord_less_real @ B2 @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2479_add__strict__increasing,axiom,
% 9.07/5.20      ! [A: rat,B2: rat,C: rat] :
% 9.07/5.20        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.20       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.07/5.20         => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2480_add__strict__increasing,axiom,
% 9.07/5.20      ! [A: nat,B2: nat,C: nat] :
% 9.07/5.20        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.20       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.07/5.20         => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2481_add__strict__increasing,axiom,
% 9.07/5.20      ! [A: int,B2: int,C: int] :
% 9.07/5.20        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.20       => ( ( ord_less_eq_int @ B2 @ C )
% 9.07/5.20         => ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2482_add__pos__nonneg,axiom,
% 9.07/5.20      ! [A: real,B2: real] :
% 9.07/5.20        ( ( ord_less_real @ zero_zero_real @ A )
% 9.07/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.07/5.20         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2483_add__pos__nonneg,axiom,
% 9.07/5.20      ! [A: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.07/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.07/5.20         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2484_add__pos__nonneg,axiom,
% 9.07/5.20      ! [A: nat,B2: nat] :
% 9.07/5.20        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.07/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.07/5.20         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2485_add__pos__nonneg,axiom,
% 9.07/5.20      ! [A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_int @ zero_zero_int @ A )
% 9.07/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.07/5.20         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2486_add__nonpos__neg,axiom,
% 9.07/5.20      ! [A: real,B2: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.07/5.20       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 9.07/5.20         => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2487_add__nonpos__neg,axiom,
% 9.07/5.20      ! [A: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.07/5.20       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 9.07/5.20         => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2488_add__nonpos__neg,axiom,
% 9.07/5.20      ! [A: nat,B2: nat] :
% 9.07/5.20        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.07/5.20       => ( ( ord_less_nat @ B2 @ zero_zero_nat )
% 9.07/5.20         => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2489_add__nonpos__neg,axiom,
% 9.07/5.20      ! [A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.07/5.20       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.07/5.20         => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2490_add__nonneg__pos,axiom,
% 9.07/5.20      ! [A: real,B2: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.07/5.20       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.07/5.20         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2491_add__nonneg__pos,axiom,
% 9.07/5.20      ! [A: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.07/5.20       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.07/5.20         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2492_add__nonneg__pos,axiom,
% 9.07/5.20      ! [A: nat,B2: nat] :
% 9.07/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.20       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.20         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2493_add__nonneg__pos,axiom,
% 9.07/5.20      ! [A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.20       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.20         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2494_add__neg__nonpos,axiom,
% 9.07/5.20      ! [A: real,B2: real] :
% 9.07/5.20        ( ( ord_less_real @ A @ zero_zero_real )
% 9.07/5.20       => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
% 9.07/5.20         => ( ord_less_real @ ( plus_plus_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2495_add__neg__nonpos,axiom,
% 9.07/5.20      ! [A: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.07/5.20       => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
% 9.07/5.20         => ( ord_less_rat @ ( plus_plus_rat @ A @ B2 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2496_add__neg__nonpos,axiom,
% 9.07/5.20      ! [A: nat,B2: nat] :
% 9.07/5.20        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.07/5.20       => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
% 9.07/5.20         => ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2497_add__neg__nonpos,axiom,
% 9.07/5.20      ! [A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_int @ A @ zero_zero_int )
% 9.07/5.20       => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 9.07/5.20         => ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2498_divide__nonpos__pos,axiom,
% 9.07/5.20      ! [X3: real,Y3: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.07/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.07/5.20         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2499_divide__nonpos__pos,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.07/5.20       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.07/5.20         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2500_divide__nonpos__neg,axiom,
% 9.07/5.20      ! [X3: real,Y3: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.07/5.20       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.07/5.20         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2501_divide__nonpos__neg,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.07/5.20       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.07/5.20         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2502_divide__nonneg__pos,axiom,
% 9.07/5.20      ! [X3: real,Y3: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.07/5.20         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2503_divide__nonneg__pos,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.20       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.07/5.20         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2504_divide__nonneg__neg,axiom,
% 9.07/5.20      ! [X3: real,Y3: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.20       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.07/5.20         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2505_divide__nonneg__neg,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.20       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.07/5.20         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2506_divide__le__cancel,axiom,
% 9.07/5.20      ! [A: real,C: real,B2: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B2 @ C ) )
% 9.07/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.07/5.20           => ( ord_less_eq_real @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.07/5.20           => ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2507_divide__le__cancel,axiom,
% 9.07/5.20      ! [A: rat,C: rat,B2: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B2 @ C ) )
% 9.07/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.07/5.20           => ( ord_less_eq_rat @ A @ B2 ) )
% 9.07/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.07/5.20           => ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2508_frac__less2,axiom,
% 9.07/5.20      ! [X3: real,Y3: real,W: real,Z: real] :
% 9.07/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.07/5.20       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.07/5.20         => ( ( ord_less_real @ zero_zero_real @ W )
% 9.07/5.20           => ( ( ord_less_real @ W @ Z )
% 9.07/5.20             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2509_frac__less2,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat,W: rat,Z: rat] :
% 9.07/5.20        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.07/5.20       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.07/5.20         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 9.07/5.20           => ( ( ord_less_rat @ W @ Z )
% 9.07/5.20             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2510_frac__less,axiom,
% 9.07/5.20      ! [X3: real,Y3: real,W: real,Z: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.20       => ( ( ord_less_real @ X3 @ Y3 )
% 9.07/5.20         => ( ( ord_less_real @ zero_zero_real @ W )
% 9.07/5.20           => ( ( ord_less_eq_real @ W @ Z )
% 9.07/5.20             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2511_frac__less,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat,W: rat,Z: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.20       => ( ( ord_less_rat @ X3 @ Y3 )
% 9.07/5.20         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 9.07/5.20           => ( ( ord_less_eq_rat @ W @ Z )
% 9.07/5.20             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2512_frac__le,axiom,
% 9.07/5.20      ! [Y3: real,X3: real,W: real,Z: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.07/5.20       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.07/5.20         => ( ( ord_less_real @ zero_zero_real @ W )
% 9.07/5.20           => ( ( ord_less_eq_real @ W @ Z )
% 9.07/5.20             => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2513_frac__le,axiom,
% 9.07/5.20      ! [Y3: rat,X3: rat,W: rat,Z: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.07/5.20       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.07/5.20         => ( ( ord_less_rat @ zero_zero_rat @ W )
% 9.07/5.20           => ( ( ord_less_eq_rat @ W @ Z )
% 9.07/5.20             => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2514_div__positive,axiom,
% 9.07/5.20      ! [B2: nat,A: nat] :
% 9.07/5.20        ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.07/5.20       => ( ( ord_less_eq_nat @ B2 @ A )
% 9.07/5.20         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2515_div__positive,axiom,
% 9.07/5.20      ! [B2: int,A: int] :
% 9.07/5.20        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.07/5.20       => ( ( ord_less_eq_int @ B2 @ A )
% 9.07/5.20         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2516_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.07/5.20      ! [A: nat,B2: nat] :
% 9.07/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.07/5.20       => ( ( ord_less_nat @ A @ B2 )
% 9.07/5.20         => ( ( divide_divide_nat @ A @ B2 )
% 9.07/5.20            = zero_zero_nat ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2517_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.07/5.20      ! [A: int,B2: int] :
% 9.07/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.07/5.20       => ( ( ord_less_int @ A @ B2 )
% 9.07/5.20         => ( ( divide_divide_int @ A @ B2 )
% 9.07/5.20            = zero_zero_int ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2518_mult__left__le__one__le,axiom,
% 9.07/5.20      ! [X3: real,Y3: real] :
% 9.07/5.20        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.07/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.07/5.20         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.07/5.20           => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2519_mult__left__le__one__le,axiom,
% 9.07/5.20      ! [X3: rat,Y3: rat] :
% 9.07/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.07/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.07/5.20         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 9.07/5.20           => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 9.07/5.20  
% 9.07/5.20  thf(fact_2520_mult__left__le__one__le,axiom,
% 9.07/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.20         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 9.10/5.20           => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2521_mult__right__le__one__le,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.20         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.20           => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2522_mult__right__le__one__le,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2523_mult__right__le__one__le,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.20         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 9.10/5.20           => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2524_mult__le__one,axiom,
% 9.10/5.20      ! [A: real,B2: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ A @ one_one_real )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_real @ B2 @ one_one_real )
% 9.10/5.20           => ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ one_one_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2525_mult__le__one,axiom,
% 9.10/5.20      ! [A: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B2 ) @ one_one_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2526_mult__le__one,axiom,
% 9.10/5.20      ! [A: nat,B2: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
% 9.10/5.20           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2527_mult__le__one,axiom,
% 9.10/5.20      ! [A: int,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ A @ one_one_int )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_int @ B2 @ one_one_int )
% 9.10/5.20           => ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2528_mult__left__le,axiom,
% 9.10/5.20      ! [C: real,A: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ C @ one_one_real )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.20         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2529_mult__left__le,axiom,
% 9.10/5.20      ! [C: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.20         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2530_mult__left__le,axiom,
% 9.10/5.20      ! [C: nat,A: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2531_mult__left__le,axiom,
% 9.10/5.20      ! [C: int,A: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ C @ one_one_int )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2532_sum__squares__le__zero__iff,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 9.10/5.20        = ( ( X3 = zero_zero_real )
% 9.10/5.20          & ( Y3 = zero_zero_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2533_sum__squares__le__zero__iff,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 9.10/5.20        = ( ( X3 = zero_zero_rat )
% 9.10/5.20          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2534_sum__squares__le__zero__iff,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 9.10/5.20        = ( ( X3 = zero_zero_int )
% 9.10/5.20          & ( Y3 = zero_zero_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2535_sum__squares__ge__zero,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2536_sum__squares__ge__zero,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2537_sum__squares__ge__zero,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2538_power__less__imp__less__base,axiom,
% 9.10/5.20      ! [A: real,N: nat,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.10/5.20         => ( ord_less_real @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2539_power__less__imp__less__base,axiom,
% 9.10/5.20      ! [A: rat,N: nat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.10/5.20         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2540_power__less__imp__less__base,axiom,
% 9.10/5.20      ! [A: nat,N: nat,B2: nat] :
% 9.10/5.20        ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.10/5.20         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2541_power__less__imp__less__base,axiom,
% 9.10/5.20      ! [A: int,N: nat,B2: int] :
% 9.10/5.20        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.20         => ( ord_less_int @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2542_sum__squares__gt__zero__iff,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 9.10/5.20        = ( ( X3 != zero_zero_real )
% 9.10/5.20          | ( Y3 != zero_zero_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2543_sum__squares__gt__zero__iff,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 9.10/5.20        = ( ( X3 != zero_zero_rat )
% 9.10/5.20          | ( Y3 != zero_zero_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2544_sum__squares__gt__zero__iff,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 9.10/5.20        = ( ( X3 != zero_zero_int )
% 9.10/5.20          | ( Y3 != zero_zero_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2545_not__sum__squares__lt__zero,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2546_not__sum__squares__lt__zero,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2547_not__sum__squares__lt__zero,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2548_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.10/5.20      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.10/5.20       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.10/5.20          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2549_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.10/5.20      ! [C: int,A: int,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.20       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.20          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2550_zero__less__two,axiom,
% 9.10/5.20      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2551_zero__less__two,axiom,
% 9.10/5.20      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2552_zero__less__two,axiom,
% 9.10/5.20      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2553_zero__less__two,axiom,
% 9.10/5.20      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2554_divide__strict__left__mono__neg,axiom,
% 9.10/5.20      ! [A: real,B2: real,C: real] :
% 9.10/5.20        ( ( ord_less_real @ A @ B2 )
% 9.10/5.20       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2555_divide__strict__left__mono__neg,axiom,
% 9.10/5.20      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.20       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2556_divide__strict__left__mono,axiom,
% 9.10/5.20      ! [B2: real,A: real,C: real] :
% 9.10/5.20        ( ( ord_less_real @ B2 @ A )
% 9.10/5.20       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2557_divide__strict__left__mono,axiom,
% 9.10/5.20      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.20       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2558_mult__imp__less__div__pos,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.20       => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X3 )
% 9.10/5.20         => ( ord_less_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2559_mult__imp__less__div__pos,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20       => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X3 )
% 9.10/5.20         => ( ord_less_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2560_mult__imp__div__pos__less,axiom,
% 9.10/5.20      ! [Y3: real,X3: real,Z: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.20       => ( ( ord_less_real @ X3 @ ( times_times_real @ Z @ Y3 ) )
% 9.10/5.20         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2561_mult__imp__div__pos__less,axiom,
% 9.10/5.20      ! [Y3: rat,X3: rat,Z: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20       => ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) )
% 9.10/5.20         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2562_pos__less__divide__eq,axiom,
% 9.10/5.20      ! [C: real,A: real,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20       => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2563_pos__less__divide__eq,axiom,
% 9.10/5.20      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2564_pos__divide__less__eq,axiom,
% 9.10/5.20      ! [C: real,B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2565_pos__divide__less__eq,axiom,
% 9.10/5.20      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2566_neg__less__divide__eq,axiom,
% 9.10/5.20      ! [C: real,A: real,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20       => ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2567_neg__less__divide__eq,axiom,
% 9.10/5.20      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2568_neg__divide__less__eq,axiom,
% 9.10/5.20      ! [C: real,B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20       => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2569_neg__divide__less__eq,axiom,
% 9.10/5.20      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20       => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2570_less__divide__eq,axiom,
% 9.10/5.20      ! [A: real,B2: real,C: real] :
% 9.10/5.20        ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2571_less__divide__eq,axiom,
% 9.10/5.20      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2572_divide__less__eq,axiom,
% 9.10/5.20      ! [B2: real,C: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2573_divide__less__eq,axiom,
% 9.10/5.20      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2574_less__divide__eq__1,axiom,
% 9.10/5.20      ! [B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20            & ( ord_less_real @ A @ B2 ) )
% 9.10/5.20          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.20            & ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2575_less__divide__eq__1,axiom,
% 9.10/5.20      ! [B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20            & ( ord_less_rat @ A @ B2 ) )
% 9.10/5.20          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.20            & ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2576_divide__less__eq__1,axiom,
% 9.10/5.20      ! [B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20            & ( ord_less_real @ B2 @ A ) )
% 9.10/5.20          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.20            & ( ord_less_real @ A @ B2 ) )
% 9.10/5.20          | ( A = zero_zero_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2577_divide__less__eq__1,axiom,
% 9.10/5.20      ! [B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20            & ( ord_less_rat @ B2 @ A ) )
% 9.10/5.20          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.20            & ( ord_less_rat @ A @ B2 ) )
% 9.10/5.20          | ( A = zero_zero_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2578_power__le__one,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.20       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.10/5.20         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2579_power__le__one,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.20       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.10/5.20         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2580_power__le__one,axiom,
% 9.10/5.20      ! [A: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.10/5.20         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2581_power__le__one,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.10/5.20         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2582_divide__eq__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: complex,C: complex,W: num] :
% 9.10/5.20        ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 9.10/5.20          = ( numera6690914467698888265omplex @ W ) )
% 9.10/5.20        = ( ( ( C != zero_zero_complex )
% 9.10/5.20           => ( B2
% 9.10/5.20              = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 9.10/5.20          & ( ( C = zero_zero_complex )
% 9.10/5.20           => ( ( numera6690914467698888265omplex @ W )
% 9.10/5.20              = zero_zero_complex ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2583_divide__eq__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: real,C: real,W: num] :
% 9.10/5.20        ( ( ( divide_divide_real @ B2 @ C )
% 9.10/5.20          = ( numeral_numeral_real @ W ) )
% 9.10/5.20        = ( ( ( C != zero_zero_real )
% 9.10/5.20           => ( B2
% 9.10/5.20              = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.10/5.20          & ( ( C = zero_zero_real )
% 9.10/5.20           => ( ( numeral_numeral_real @ W )
% 9.10/5.20              = zero_zero_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2584_divide__eq__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: rat,C: rat,W: num] :
% 9.10/5.20        ( ( ( divide_divide_rat @ B2 @ C )
% 9.10/5.20          = ( numeral_numeral_rat @ W ) )
% 9.10/5.20        = ( ( ( C != zero_zero_rat )
% 9.10/5.20           => ( B2
% 9.10/5.20              = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.10/5.20          & ( ( C = zero_zero_rat )
% 9.10/5.20           => ( ( numeral_numeral_rat @ W )
% 9.10/5.20              = zero_zero_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2585_eq__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: complex,C: complex] :
% 9.10/5.20        ( ( ( numera6690914467698888265omplex @ W )
% 9.10/5.20          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.10/5.20        = ( ( ( C != zero_zero_complex )
% 9.10/5.20           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 9.10/5.20              = B2 ) )
% 9.10/5.20          & ( ( C = zero_zero_complex )
% 9.10/5.20           => ( ( numera6690914467698888265omplex @ W )
% 9.10/5.20              = zero_zero_complex ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2586_eq__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: real,C: real] :
% 9.10/5.20        ( ( ( numeral_numeral_real @ W )
% 9.10/5.20          = ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( C != zero_zero_real )
% 9.10/5.20           => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 9.10/5.20              = B2 ) )
% 9.10/5.20          & ( ( C = zero_zero_real )
% 9.10/5.20           => ( ( numeral_numeral_real @ W )
% 9.10/5.20              = zero_zero_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2587_eq__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: rat,C: rat] :
% 9.10/5.20        ( ( ( numeral_numeral_rat @ W )
% 9.10/5.20          = ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( C != zero_zero_rat )
% 9.10/5.20           => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 9.10/5.20              = B2 ) )
% 9.10/5.20          & ( ( C = zero_zero_rat )
% 9.10/5.20           => ( ( numeral_numeral_rat @ W )
% 9.10/5.20              = zero_zero_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2588_add__divide__eq__if__simps_I2_J,axiom,
% 9.10/5.20      ! [Z: complex,A: complex,B2: complex] :
% 9.10/5.20        ( ( ( Z = zero_zero_complex )
% 9.10/5.20         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 9.10/5.20            = B2 ) )
% 9.10/5.20        & ( ( Z != zero_zero_complex )
% 9.10/5.20         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 9.10/5.20            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2589_add__divide__eq__if__simps_I2_J,axiom,
% 9.10/5.20      ! [Z: real,A: real,B2: real] :
% 9.10/5.20        ( ( ( Z = zero_zero_real )
% 9.10/5.20         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 9.10/5.20            = B2 ) )
% 9.10/5.20        & ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 9.10/5.20            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2590_add__divide__eq__if__simps_I2_J,axiom,
% 9.10/5.20      ! [Z: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ( Z = zero_zero_rat )
% 9.10/5.20         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 9.10/5.20            = B2 ) )
% 9.10/5.20        & ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 9.10/5.20            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2591_add__divide__eq__if__simps_I1_J,axiom,
% 9.10/5.20      ! [Z: complex,A: complex,B2: complex] :
% 9.10/5.20        ( ( ( Z = zero_zero_complex )
% 9.10/5.20         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 9.10/5.20            = A ) )
% 9.10/5.20        & ( ( Z != zero_zero_complex )
% 9.10/5.20         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 9.10/5.20            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2592_add__divide__eq__if__simps_I1_J,axiom,
% 9.10/5.20      ! [Z: real,A: real,B2: real] :
% 9.10/5.20        ( ( ( Z = zero_zero_real )
% 9.10/5.20         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 9.10/5.20            = A ) )
% 9.10/5.20        & ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 9.10/5.20            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2593_add__divide__eq__if__simps_I1_J,axiom,
% 9.10/5.20      ! [Z: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ( Z = zero_zero_rat )
% 9.10/5.20         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 9.10/5.20            = A ) )
% 9.10/5.20        & ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 9.10/5.20            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2594_add__frac__eq,axiom,
% 9.10/5.20      ! [Y3: complex,Z: complex,X3: complex,W: complex] :
% 9.10/5.20        ( ( Y3 != zero_zero_complex )
% 9.10/5.20       => ( ( Z != zero_zero_complex )
% 9.10/5.20         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 9.10/5.20            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2595_add__frac__eq,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real,W: real] :
% 9.10/5.20        ( ( Y3 != zero_zero_real )
% 9.10/5.20       => ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 9.10/5.20            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2596_add__frac__eq,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 9.10/5.20        ( ( Y3 != zero_zero_rat )
% 9.10/5.20       => ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 9.10/5.20            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2597_add__frac__num,axiom,
% 9.10/5.20      ! [Y3: complex,X3: complex,Z: complex] :
% 9.10/5.20        ( ( Y3 != zero_zero_complex )
% 9.10/5.20       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ Z )
% 9.10/5.20          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2598_add__frac__num,axiom,
% 9.10/5.20      ! [Y3: real,X3: real,Z: real] :
% 9.10/5.20        ( ( Y3 != zero_zero_real )
% 9.10/5.20       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z )
% 9.10/5.20          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2599_add__frac__num,axiom,
% 9.10/5.20      ! [Y3: rat,X3: rat,Z: rat] :
% 9.10/5.20        ( ( Y3 != zero_zero_rat )
% 9.10/5.20       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z )
% 9.10/5.20          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2600_add__num__frac,axiom,
% 9.10/5.20      ! [Y3: complex,Z: complex,X3: complex] :
% 9.10/5.20        ( ( Y3 != zero_zero_complex )
% 9.10/5.20       => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 9.10/5.20          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2601_add__num__frac,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real] :
% 9.10/5.20        ( ( Y3 != zero_zero_real )
% 9.10/5.20       => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.10/5.20          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2602_add__num__frac,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat] :
% 9.10/5.20        ( ( Y3 != zero_zero_rat )
% 9.10/5.20       => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) )
% 9.10/5.20          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2603_add__divide__eq__iff,axiom,
% 9.10/5.20      ! [Z: complex,X3: complex,Y3: complex] :
% 9.10/5.20        ( ( Z != zero_zero_complex )
% 9.10/5.20       => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 9.10/5.20          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2604_add__divide__eq__iff,axiom,
% 9.10/5.20      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.20        ( ( Z != zero_zero_real )
% 9.10/5.20       => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y3 @ Z ) )
% 9.10/5.20          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2605_add__divide__eq__iff,axiom,
% 9.10/5.20      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.20        ( ( Z != zero_zero_rat )
% 9.10/5.20       => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z ) )
% 9.10/5.20          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2606_divide__add__eq__iff,axiom,
% 9.10/5.20      ! [Z: complex,X3: complex,Y3: complex] :
% 9.10/5.20        ( ( Z != zero_zero_complex )
% 9.10/5.20       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y3 )
% 9.10/5.20          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2607_divide__add__eq__iff,axiom,
% 9.10/5.20      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.20        ( ( Z != zero_zero_real )
% 9.10/5.20       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z ) @ Y3 )
% 9.10/5.20          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2608_divide__add__eq__iff,axiom,
% 9.10/5.20      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.20        ( ( Z != zero_zero_rat )
% 9.10/5.20       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z ) @ Y3 )
% 9.10/5.20          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2609_power__le__imp__le__base,axiom,
% 9.10/5.20      ! [A: real,N: nat,B2: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.10/5.20         => ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2610_power__le__imp__le__base,axiom,
% 9.10/5.20      ! [A: rat,N: nat,B2: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.10/5.20         => ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2611_power__le__imp__le__base,axiom,
% 9.10/5.20      ! [A: nat,N: nat,B2: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.10/5.20         => ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2612_power__le__imp__le__base,axiom,
% 9.10/5.20      ! [A: int,N: nat,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.20         => ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2613_power__inject__base,axiom,
% 9.10/5.20      ! [A: real,N: nat,B2: real] :
% 9.10/5.20        ( ( ( power_power_real @ A @ ( suc @ N ) )
% 9.10/5.20          = ( power_power_real @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.20         => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.10/5.20           => ( A = B2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2614_power__inject__base,axiom,
% 9.10/5.20      ! [A: rat,N: nat,B2: rat] :
% 9.10/5.20        ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 9.10/5.20          = ( power_power_rat @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.20         => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.10/5.20           => ( A = B2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2615_power__inject__base,axiom,
% 9.10/5.20      ! [A: nat,N: nat,B2: nat] :
% 9.10/5.20        ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 9.10/5.20          = ( power_power_nat @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20         => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
% 9.10/5.20           => ( A = B2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2616_power__inject__base,axiom,
% 9.10/5.20      ! [A: int,N: nat,B2: int] :
% 9.10/5.20        ( ( ( power_power_int @ A @ ( suc @ N ) )
% 9.10/5.20          = ( power_power_int @ B2 @ ( suc @ N ) ) )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20         => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.20           => ( A = B2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2617_div__add__self1,axiom,
% 9.10/5.20      ! [B2: nat,A: nat] :
% 9.10/5.20        ( ( B2 != zero_zero_nat )
% 9.10/5.20       => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
% 9.10/5.20          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2618_div__add__self1,axiom,
% 9.10/5.20      ! [B2: int,A: int] :
% 9.10/5.20        ( ( B2 != zero_zero_int )
% 9.10/5.20       => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
% 9.10/5.20          = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2619_div__add__self2,axiom,
% 9.10/5.20      ! [B2: nat,A: nat] :
% 9.10/5.20        ( ( B2 != zero_zero_nat )
% 9.10/5.20       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
% 9.10/5.20          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2620_div__add__self2,axiom,
% 9.10/5.20      ! [B2: int,A: int] :
% 9.10/5.20        ( ( B2 != zero_zero_int )
% 9.10/5.20       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
% 9.10/5.20          = ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2621_add__divide__eq__if__simps_I4_J,axiom,
% 9.10/5.20      ! [Z: complex,A: complex,B2: complex] :
% 9.10/5.20        ( ( ( Z = zero_zero_complex )
% 9.10/5.20         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 9.10/5.20            = A ) )
% 9.10/5.20        & ( ( Z != zero_zero_complex )
% 9.10/5.20         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B2 @ Z ) )
% 9.10/5.20            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2622_add__divide__eq__if__simps_I4_J,axiom,
% 9.10/5.20      ! [Z: real,A: real,B2: real] :
% 9.10/5.20        ( ( ( Z = zero_zero_real )
% 9.10/5.20         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 9.10/5.20            = A ) )
% 9.10/5.20        & ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B2 @ Z ) )
% 9.10/5.20            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2623_add__divide__eq__if__simps_I4_J,axiom,
% 9.10/5.20      ! [Z: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ( Z = zero_zero_rat )
% 9.10/5.20         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 9.10/5.20            = A ) )
% 9.10/5.20        & ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B2 @ Z ) )
% 9.10/5.20            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B2 ) @ Z ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2624_diff__frac__eq,axiom,
% 9.10/5.20      ! [Y3: complex,Z: complex,X3: complex,W: complex] :
% 9.10/5.20        ( ( Y3 != zero_zero_complex )
% 9.10/5.20       => ( ( Z != zero_zero_complex )
% 9.10/5.20         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 9.10/5.20            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2625_diff__frac__eq,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real,W: real] :
% 9.10/5.20        ( ( Y3 != zero_zero_real )
% 9.10/5.20       => ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 9.10/5.20            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2626_diff__frac__eq,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 9.10/5.20        ( ( Y3 != zero_zero_rat )
% 9.10/5.20       => ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 9.10/5.20            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2627_diff__divide__eq__iff,axiom,
% 9.10/5.20      ! [Z: complex,X3: complex,Y3: complex] :
% 9.10/5.20        ( ( Z != zero_zero_complex )
% 9.10/5.20       => ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 9.10/5.20          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2628_diff__divide__eq__iff,axiom,
% 9.10/5.20      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.20        ( ( Z != zero_zero_real )
% 9.10/5.20       => ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y3 @ Z ) )
% 9.10/5.20          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2629_diff__divide__eq__iff,axiom,
% 9.10/5.20      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.20        ( ( Z != zero_zero_rat )
% 9.10/5.20       => ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z ) )
% 9.10/5.20          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2630_divide__diff__eq__iff,axiom,
% 9.10/5.20      ! [Z: complex,X3: complex,Y3: complex] :
% 9.10/5.20        ( ( Z != zero_zero_complex )
% 9.10/5.20       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z ) @ Y3 )
% 9.10/5.20          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2631_divide__diff__eq__iff,axiom,
% 9.10/5.20      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.20        ( ( Z != zero_zero_real )
% 9.10/5.20       => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z ) @ Y3 )
% 9.10/5.20          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2632_divide__diff__eq__iff,axiom,
% 9.10/5.20      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.20        ( ( Z != zero_zero_rat )
% 9.10/5.20       => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z ) @ Y3 )
% 9.10/5.20          = ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2633_mod__double__modulus,axiom,
% 9.10/5.20      ! [M: code_integer,X3: code_integer] :
% 9.10/5.20        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 9.10/5.20       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 9.10/5.20         => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20              = ( modulo364778990260209775nteger @ X3 @ M ) )
% 9.10/5.20            | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M ) @ M ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2634_mod__double__modulus,axiom,
% 9.10/5.20      ! [M: nat,X3: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.10/5.20         => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20              = ( modulo_modulo_nat @ X3 @ M ) )
% 9.10/5.20            | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20              = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M ) @ M ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2635_mod__double__modulus,axiom,
% 9.10/5.20      ! [M: int,X3: int] :
% 9.10/5.20        ( ( ord_less_int @ zero_zero_int @ M )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.20         => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20              = ( modulo_modulo_int @ X3 @ M ) )
% 9.10/5.20            | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20              = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M ) @ M ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2636_divmod__digit__1_I2_J,axiom,
% 9.10/5.20      ! [A: code_integer,B2: code_integer] :
% 9.10/5.20        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.20       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.10/5.20         => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.10/5.20           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.10/5.20              = ( modulo364778990260209775nteger @ A @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2637_divmod__digit__1_I2_J,axiom,
% 9.10/5.20      ! [A: nat,B2: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.10/5.20           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.10/5.20              = ( modulo_modulo_nat @ A @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2638_divmod__digit__1_I2_J,axiom,
% 9.10/5.20      ! [A: int,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.10/5.20           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
% 9.10/5.20              = ( modulo_modulo_int @ A @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2639_vebt__member_Osimps_I4_J,axiom,
% 9.10/5.20      ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 9.10/5.20        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2640_numeral__1__eq__Suc__0,axiom,
% 9.10/5.20      ( ( numeral_numeral_nat @ one )
% 9.10/5.20      = ( suc @ zero_zero_nat ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2641_num_Osize_I5_J,axiom,
% 9.10/5.20      ! [X2: num] :
% 9.10/5.20        ( ( size_size_num @ ( bit0 @ X2 ) )
% 9.10/5.20        = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2642_ex__least__nat__less,axiom,
% 9.10/5.20      ! [P: nat > $o,N: nat] :
% 9.10/5.20        ( ( P @ N )
% 9.10/5.20       => ( ~ ( P @ zero_zero_nat )
% 9.10/5.20         => ? [K3: nat] :
% 9.10/5.20              ( ( ord_less_nat @ K3 @ N )
% 9.10/5.20              & ! [I2: nat] :
% 9.10/5.20                  ( ( ord_less_eq_nat @ I2 @ K3 )
% 9.10/5.20                 => ~ ( P @ I2 ) )
% 9.10/5.20              & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2643_diff__Suc__less,axiom,
% 9.10/5.20      ! [N: nat,I: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2644_one__less__mult,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.20       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.10/5.20         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2645_n__less__m__mult__n,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.10/5.20         => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2646_n__less__n__mult__m,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.10/5.20         => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2647_length__pos__if__in__set,axiom,
% 9.10/5.20      ! [X3: real,Xs2: list_real] :
% 9.10/5.20        ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2648_length__pos__if__in__set,axiom,
% 9.10/5.20      ! [X3: complex,Xs2: list_complex] :
% 9.10/5.20        ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2649_length__pos__if__in__set,axiom,
% 9.10/5.20      ! [X3: int,Xs2: list_int] :
% 9.10/5.20        ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2650_length__pos__if__in__set,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 9.10/5.20        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2651_length__pos__if__in__set,axiom,
% 9.10/5.20      ! [X3: $o,Xs2: list_o] :
% 9.10/5.20        ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2652_length__pos__if__in__set,axiom,
% 9.10/5.20      ! [X3: nat,Xs2: list_nat] :
% 9.10/5.20        ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2653_nat__induct__non__zero,axiom,
% 9.10/5.20      ! [N: nat,P: nat > $o] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( P @ one_one_nat )
% 9.10/5.20         => ( ! [N3: nat] :
% 9.10/5.20                ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.10/5.20               => ( ( P @ N3 )
% 9.10/5.20                 => ( P @ ( suc @ N3 ) ) ) )
% 9.10/5.20           => ( P @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2654_nat__mult__le__cancel1,axiom,
% 9.10/5.20      ! [K: nat,M: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.20       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.10/5.20          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2655_nat__diff__split,axiom,
% 9.10/5.20      ! [P: nat > $o,A: nat,B2: nat] :
% 9.10/5.20        ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_nat @ A @ B2 )
% 9.10/5.20           => ( P @ zero_zero_nat ) )
% 9.10/5.20          & ! [D2: nat] :
% 9.10/5.20              ( ( A
% 9.10/5.20                = ( plus_plus_nat @ B2 @ D2 ) )
% 9.10/5.20             => ( P @ D2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2656_nat__diff__split__asm,axiom,
% 9.10/5.20      ! [P: nat > $o,A: nat,B2: nat] :
% 9.10/5.20        ( ( P @ ( minus_minus_nat @ A @ B2 ) )
% 9.10/5.20        = ( ~ ( ( ( ord_less_nat @ A @ B2 )
% 9.10/5.20                & ~ ( P @ zero_zero_nat ) )
% 9.10/5.20              | ? [D2: nat] :
% 9.10/5.20                  ( ( A
% 9.10/5.20                    = ( plus_plus_nat @ B2 @ D2 ) )
% 9.10/5.20                  & ~ ( P @ D2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2657_power__gt__expt,axiom,
% 9.10/5.20      ! [N: nat,K: nat] :
% 9.10/5.20        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.20       => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2658_div__greater__zero__iff,axiom,
% 9.10/5.20      ! [M: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.20        = ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2659_div__le__mono2,axiom,
% 9.10/5.20      ! [M: nat,N: nat,K: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.20       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.20         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2660_nat__one__le__power,axiom,
% 9.10/5.20      ! [I: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 9.10/5.20       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2661_div__less__iff__less__mult,axiom,
% 9.10/5.20      ! [Q2: nat,M: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 9.10/5.20       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 9.10/5.20          = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2662_nat__mult__div__cancel1,axiom,
% 9.10/5.20      ! [K: nat,M: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.20       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.10/5.20          = ( divide_divide_nat @ M @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2663_div__eq__dividend__iff,axiom,
% 9.10/5.20      ! [M: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.20       => ( ( ( divide_divide_nat @ M @ N )
% 9.10/5.20            = M )
% 9.10/5.20          = ( N = one_one_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2664_div__less__dividend,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ one_one_nat @ N )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.20         => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2665_nat__induct2,axiom,
% 9.10/5.20      ! [P: nat > $o,N: nat] :
% 9.10/5.20        ( ( P @ zero_zero_nat )
% 9.10/5.20       => ( ( P @ one_one_nat )
% 9.10/5.20         => ( ! [N3: nat] :
% 9.10/5.20                ( ( P @ N3 )
% 9.10/5.20               => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.20           => ( P @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2666_vebt__mint_Ocases,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT] :
% 9.10/5.20        ( ! [A3: $o,B3: $o] :
% 9.10/5.20            ( X3
% 9.10/5.20           != ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20              ( X3
% 9.10/5.20             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.20         => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.20                ( X3
% 9.10/5.20               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2667_field__le__mult__one__interval,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ! [Z4: real] :
% 9.10/5.20            ( ( ord_less_real @ zero_zero_real @ Z4 )
% 9.10/5.20           => ( ( ord_less_real @ Z4 @ one_one_real )
% 9.10/5.20             => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X3 ) @ Y3 ) ) )
% 9.10/5.20       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2668_field__le__mult__one__interval,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ! [Z4: rat] :
% 9.10/5.20            ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 9.10/5.20           => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 9.10/5.20             => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X3 ) @ Y3 ) ) )
% 9.10/5.20       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2669_mult__less__cancel__right2,axiom,
% 9.10/5.20      ! [A: real,C: real] :
% 9.10/5.20        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 9.10/5.20        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ A @ one_one_real ) )
% 9.10/5.20          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2670_mult__less__cancel__right2,axiom,
% 9.10/5.20      ! [A: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.10/5.20        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.10/5.20          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2671_mult__less__cancel__right2,axiom,
% 9.10/5.20      ! [A: int,C: int] :
% 9.10/5.20        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 9.10/5.20        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_int @ A @ one_one_int ) )
% 9.10/5.20          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2672_mult__less__cancel__right1,axiom,
% 9.10/5.20      ! [C: real,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ C @ ( times_times_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ one_one_real @ B2 ) )
% 9.10/5.20          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2673_mult__less__cancel__right1,axiom,
% 9.10/5.20      ! [C: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ one_one_rat @ B2 ) )
% 9.10/5.20          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2674_mult__less__cancel__right1,axiom,
% 9.10/5.20      ! [C: int,B2: int] :
% 9.10/5.20        ( ( ord_less_int @ C @ ( times_times_int @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_int @ one_one_int @ B2 ) )
% 9.10/5.20          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2675_mult__less__cancel__left2,axiom,
% 9.10/5.20      ! [C: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 9.10/5.20        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ A @ one_one_real ) )
% 9.10/5.20          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2676_mult__less__cancel__left2,axiom,
% 9.10/5.20      ! [C: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.10/5.20        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.10/5.20          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2677_mult__less__cancel__left2,axiom,
% 9.10/5.20      ! [C: int,A: int] :
% 9.10/5.20        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 9.10/5.20        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_int @ A @ one_one_int ) )
% 9.10/5.20          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2678_mult__less__cancel__left1,axiom,
% 9.10/5.20      ! [C: real,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ C @ ( times_times_real @ C @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ one_one_real @ B2 ) )
% 9.10/5.20          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2679_mult__less__cancel__left1,axiom,
% 9.10/5.20      ! [C: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ one_one_rat @ B2 ) )
% 9.10/5.20          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2680_mult__less__cancel__left1,axiom,
% 9.10/5.20      ! [C: int,B2: int] :
% 9.10/5.20        ( ( ord_less_int @ C @ ( times_times_int @ C @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_int @ one_one_int @ B2 ) )
% 9.10/5.20          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2681_mult__le__cancel__right2,axiom,
% 9.10/5.20      ! [A: real,C: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.10/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2682_mult__le__cancel__right2,axiom,
% 9.10/5.20      ! [A: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.10/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2683_mult__le__cancel__right2,axiom,
% 9.10/5.20      ! [A: int,C: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 9.10/5.20        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.10/5.20          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2684_mult__le__cancel__right1,axiom,
% 9.10/5.20      ! [C: real,B2: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ C @ ( times_times_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 9.10/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2685_mult__le__cancel__right1,axiom,
% 9.10/5.20      ! [C: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 9.10/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2686_mult__le__cancel__right1,axiom,
% 9.10/5.20      ! [C: int,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ C @ ( times_times_int @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 9.10/5.20          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2687_mult__le__cancel__left2,axiom,
% 9.10/5.20      ! [C: real,A: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.10/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2688_mult__le__cancel__left2,axiom,
% 9.10/5.20      ! [C: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.10/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2689_mult__le__cancel__left2,axiom,
% 9.10/5.20      ! [C: int,A: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 9.10/5.20        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.10/5.20          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2690_mult__le__cancel__left1,axiom,
% 9.10/5.20      ! [C: real,B2: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ one_one_real @ B2 ) )
% 9.10/5.20          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20           => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2691_mult__le__cancel__left1,axiom,
% 9.10/5.20      ! [C: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
% 9.10/5.20          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2692_mult__le__cancel__left1,axiom,
% 9.10/5.20      ! [C: int,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B2 ) )
% 9.10/5.20        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.10/5.20           => ( ord_less_eq_int @ one_one_int @ B2 ) )
% 9.10/5.20          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.10/5.20           => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2693_divide__left__mono__neg,axiom,
% 9.10/5.20      ! [A: real,B2: real,C: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.20       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.10/5.20         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2694_divide__left__mono__neg,axiom,
% 9.10/5.20      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.20       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.10/5.20         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2695_mult__imp__le__div__pos,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.20       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X3 )
% 9.10/5.20         => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2696_mult__imp__le__div__pos,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X3 )
% 9.10/5.20         => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2697_mult__imp__div__pos__le,axiom,
% 9.10/5.20      ! [Y3: real,X3: real,Z: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.20       => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z @ Y3 ) )
% 9.10/5.20         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2698_mult__imp__div__pos__le,axiom,
% 9.10/5.20      ! [Y3: rat,X3: rat,Z: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20       => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z @ Y3 ) )
% 9.10/5.20         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2699_pos__le__divide__eq,axiom,
% 9.10/5.20      ! [C: real,A: real,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2700_pos__le__divide__eq,axiom,
% 9.10/5.20      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2701_pos__divide__le__eq,axiom,
% 9.10/5.20      ! [C: real,B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2702_pos__divide__le__eq,axiom,
% 9.10/5.20      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2703_neg__le__divide__eq,axiom,
% 9.10/5.20      ! [C: real,A: real,B2: real] :
% 9.10/5.20        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2704_neg__le__divide__eq,axiom,
% 9.10/5.20      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.20        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20          = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2705_neg__divide__le__eq,axiom,
% 9.10/5.20      ! [C: real,B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20       => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2706_neg__divide__le__eq,axiom,
% 9.10/5.20      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.10/5.20          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2707_divide__left__mono,axiom,
% 9.10/5.20      ! [B2: real,A: real,C: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ B2 @ A )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.10/5.20         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2708_divide__left__mono,axiom,
% 9.10/5.20      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.10/5.20         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.20           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2709_le__divide__eq,axiom,
% 9.10/5.20      ! [A: real,B2: real,C: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2710_le__divide__eq,axiom,
% 9.10/5.20      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2711_divide__le__eq,axiom,
% 9.10/5.20      ! [B2: real,C: real,A: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ A )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2712_divide__le__eq,axiom,
% 9.10/5.20      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ A )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2713_le__divide__eq__1,axiom,
% 9.10/5.20      ! [B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20            & ( ord_less_eq_real @ A @ B2 ) )
% 9.10/5.20          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.20            & ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2714_le__divide__eq__1,axiom,
% 9.10/5.20      ! [B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20            & ( ord_less_eq_rat @ A @ B2 ) )
% 9.10/5.20          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.20            & ( ord_less_eq_rat @ B2 @ A ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2715_divide__le__eq__1,axiom,
% 9.10/5.20      ! [B2: real,A: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A ) @ one_one_real )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20            & ( ord_less_eq_real @ B2 @ A ) )
% 9.10/5.20          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.20            & ( ord_less_eq_real @ A @ B2 ) )
% 9.10/5.20          | ( A = zero_zero_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2716_divide__le__eq__1,axiom,
% 9.10/5.20      ! [B2: rat,A: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A ) @ one_one_rat )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20            & ( ord_less_eq_rat @ B2 @ A ) )
% 9.10/5.20          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.20            & ( ord_less_eq_rat @ A @ B2 ) )
% 9.10/5.20          | ( A = zero_zero_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2717_convex__bound__le,axiom,
% 9.10/5.20      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ X3 @ A )
% 9.10/5.20       => ( ( ord_less_eq_real @ Y3 @ A )
% 9.10/5.20         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.10/5.20           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.10/5.20             => ( ( ( plus_plus_real @ U @ V )
% 9.10/5.20                  = one_one_real )
% 9.10/5.20               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2718_convex__bound__le,axiom,
% 9.10/5.20      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ X3 @ A )
% 9.10/5.20       => ( ( ord_less_eq_rat @ Y3 @ A )
% 9.10/5.20         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.10/5.20           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.10/5.20             => ( ( ( plus_plus_rat @ U @ V )
% 9.10/5.20                  = one_one_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2719_convex__bound__le,axiom,
% 9.10/5.20      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ X3 @ A )
% 9.10/5.20       => ( ( ord_less_eq_int @ Y3 @ A )
% 9.10/5.20         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.10/5.20           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.10/5.20             => ( ( ( plus_plus_int @ U @ V )
% 9.10/5.20                  = one_one_int )
% 9.10/5.20               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2720_divide__less__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: real,C: real,W: num] :
% 9.10/5.20        ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2721_divide__less__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: rat,C: rat,W: num] :
% 9.10/5.20        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2722_less__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: real,C: real] :
% 9.10/5.20        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2723_less__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2724_frac__le__eq,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real,W: real] :
% 9.10/5.20        ( ( Y3 != zero_zero_real )
% 9.10/5.20       => ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 9.10/5.20            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2725_frac__le__eq,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 9.10/5.20        ( ( Y3 != zero_zero_rat )
% 9.10/5.20       => ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 9.10/5.20            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2726_divmod__digit__1_I1_J,axiom,
% 9.10/5.20      ! [A: code_integer,B2: code_integer] :
% 9.10/5.20        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.20       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.10/5.20         => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.10/5.20           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
% 9.10/5.20              = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2727_divmod__digit__1_I1_J,axiom,
% 9.10/5.20      ! [A: nat,B2: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.10/5.20           => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
% 9.10/5.20              = ( divide_divide_nat @ A @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2728_divmod__digit__1_I1_J,axiom,
% 9.10/5.20      ! [A: int,B2: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.20         => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
% 9.10/5.20           => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
% 9.10/5.20              = ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2729_frac__less__eq,axiom,
% 9.10/5.20      ! [Y3: real,Z: real,X3: real,W: real] :
% 9.10/5.20        ( ( Y3 != zero_zero_real )
% 9.10/5.20       => ( ( Z != zero_zero_real )
% 9.10/5.20         => ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 9.10/5.20            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2730_frac__less__eq,axiom,
% 9.10/5.20      ! [Y3: rat,Z: rat,X3: rat,W: rat] :
% 9.10/5.20        ( ( Y3 != zero_zero_rat )
% 9.10/5.20       => ( ( Z != zero_zero_rat )
% 9.10/5.20         => ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 9.10/5.20            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2731_power__Suc__less,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20       => ( ( ord_less_real @ A @ one_one_real )
% 9.10/5.20         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2732_power__Suc__less,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.10/5.20         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2733_power__Suc__less,axiom,
% 9.10/5.20      ! [A: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.10/5.20         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2734_power__Suc__less,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_int @ zero_zero_int @ A )
% 9.10/5.20       => ( ( ord_less_int @ A @ one_one_int )
% 9.10/5.20         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2735_power__Suc__le__self,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.20       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.10/5.20         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2736_power__Suc__le__self,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.20       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.10/5.20         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2737_power__Suc__le__self,axiom,
% 9.10/5.20      ! [A: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.10/5.20         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2738_power__Suc__le__self,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.10/5.20         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2739_power__Suc__less__one,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20       => ( ( ord_less_real @ A @ one_one_real )
% 9.10/5.20         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2740_power__Suc__less__one,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.10/5.20         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2741_power__Suc__less__one,axiom,
% 9.10/5.20      ! [A: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.10/5.20         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2742_power__Suc__less__one,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_int @ zero_zero_int @ A )
% 9.10/5.20       => ( ( ord_less_int @ A @ one_one_int )
% 9.10/5.20         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2743_power__strict__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: real] :
% 9.10/5.20        ( ( ord_less_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20         => ( ( ord_less_real @ A @ one_one_real )
% 9.10/5.20           => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2744_power__strict__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: rat] :
% 9.10/5.20        ( ( ord_less_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20         => ( ( ord_less_rat @ A @ one_one_rat )
% 9.10/5.20           => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2745_power__strict__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: nat] :
% 9.10/5.20        ( ( ord_less_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.10/5.20         => ( ( ord_less_nat @ A @ one_one_nat )
% 9.10/5.20           => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2746_power__strict__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: int] :
% 9.10/5.20        ( ( ord_less_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.10/5.20         => ( ( ord_less_int @ A @ one_one_int )
% 9.10/5.20           => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2747_power__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: real] :
% 9.10/5.20        ( ( ord_less_eq_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.20         => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.10/5.20           => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2748_power__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: rat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.20         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.10/5.20           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2749_power__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.10/5.20         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.10/5.20           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2750_power__decreasing,axiom,
% 9.10/5.20      ! [N: nat,N4: nat,A: int] :
% 9.10/5.20        ( ( ord_less_eq_nat @ N @ N4 )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.20         => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.10/5.20           => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2751_vebt__mint_Oelims,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.10/5.20        ( ( ( vEBT_vebt_mint @ X3 )
% 9.10/5.20          = Y3 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ~ ( ( A3
% 9.10/5.20                   => ( Y3
% 9.10/5.20                      = ( some_nat @ zero_zero_nat ) ) )
% 9.10/5.20                  & ( ~ A3
% 9.10/5.20                   => ( ( B3
% 9.10/5.20                       => ( Y3
% 9.10/5.20                          = ( some_nat @ one_one_nat ) ) )
% 9.10/5.20                      & ( ~ B3
% 9.10/5.20                       => ( Y3 = none_nat ) ) ) ) ) )
% 9.10/5.20         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20                  ( X3
% 9.10/5.20                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.20             => ( Y3 != none_nat ) )
% 9.10/5.20           => ~ ! [Mi2: nat] :
% 9.10/5.20                  ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.10/5.20                 => ( Y3
% 9.10/5.20                   != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2752_zero__power2,axiom,
% 9.10/5.20      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20      = zero_zero_rat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2753_zero__power2,axiom,
% 9.10/5.20      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20      = zero_zero_nat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2754_zero__power2,axiom,
% 9.10/5.20      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20      = zero_zero_real ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2755_zero__power2,axiom,
% 9.10/5.20      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20      = zero_zero_int ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2756_zero__power2,axiom,
% 9.10/5.20      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20      = zero_zero_complex ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2757_vebt__maxt_Oelims,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.10/5.20        ( ( ( vEBT_vebt_maxt @ X3 )
% 9.10/5.20          = Y3 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ~ ( ( B3
% 9.10/5.20                   => ( Y3
% 9.10/5.20                      = ( some_nat @ one_one_nat ) ) )
% 9.10/5.20                  & ( ~ B3
% 9.10/5.20                   => ( ( A3
% 9.10/5.20                       => ( Y3
% 9.10/5.20                          = ( some_nat @ zero_zero_nat ) ) )
% 9.10/5.20                      & ( ~ A3
% 9.10/5.20                       => ( Y3 = none_nat ) ) ) ) ) )
% 9.10/5.20         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20                  ( X3
% 9.10/5.20                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.20             => ( Y3 != none_nat ) )
% 9.10/5.20           => ~ ! [Mi2: nat,Ma2: nat] :
% 9.10/5.20                  ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.10/5.20                 => ( Y3
% 9.10/5.20                   != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2758_self__le__power,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2759_self__le__power,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2760_self__le__power,axiom,
% 9.10/5.20      ! [A: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2761_self__le__power,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2762_one__less__power,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2763_one__less__power,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_rat @ one_one_rat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2764_one__less__power,axiom,
% 9.10/5.20      ! [A: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ one_one_nat @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2765_one__less__power,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_int @ one_one_int @ A )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2766_numeral__2__eq__2,axiom,
% 9.10/5.20      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.10/5.20      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2767_pos2,axiom,
% 9.10/5.20      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2768_power__diff,axiom,
% 9.10/5.20      ! [A: complex,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_complex )
% 9.10/5.20       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.20            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2769_power__diff,axiom,
% 9.10/5.20      ! [A: real,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_real )
% 9.10/5.20       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.20            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2770_power__diff,axiom,
% 9.10/5.20      ! [A: rat,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_rat )
% 9.10/5.20       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.20            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2771_power__diff,axiom,
% 9.10/5.20      ! [A: nat,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_nat )
% 9.10/5.20       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.20            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2772_power__diff,axiom,
% 9.10/5.20      ! [A: int,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_int )
% 9.10/5.20       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.20            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2773_div__if,axiom,
% 9.10/5.20      ( divide_divide_nat
% 9.10/5.20      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.20            ( if_nat
% 9.10/5.20            @ ( ( ord_less_nat @ M3 @ N2 )
% 9.10/5.20              | ( N2 = zero_zero_nat ) )
% 9.10/5.20            @ zero_zero_nat
% 9.10/5.20            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2774_div__geq,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ~ ( ord_less_nat @ M @ N )
% 9.10/5.20         => ( ( divide_divide_nat @ M @ N )
% 9.10/5.20            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2775_Suc__pred_H,axiom,
% 9.10/5.20      ! [N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( N
% 9.10/5.20          = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2776_Suc__diff__eq__diff__pred,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.10/5.20          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2777_add__eq__if,axiom,
% 9.10/5.20      ( plus_plus_nat
% 9.10/5.20      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2778_less__eq__div__iff__mult__less__eq,axiom,
% 9.10/5.20      ! [Q2: nat,M: nat,N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 9.10/5.20       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 9.10/5.20          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2779_dividend__less__times__div,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2780_dividend__less__div__times,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2781_split__div,axiom,
% 9.10/5.20      ! [P: nat > $o,M: nat,N: nat] :
% 9.10/5.20        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.20        = ( ( ( N = zero_zero_nat )
% 9.10/5.20           => ( P @ zero_zero_nat ) )
% 9.10/5.20          & ( ( N != zero_zero_nat )
% 9.10/5.20           => ! [I4: nat,J3: nat] :
% 9.10/5.20                ( ( ord_less_nat @ J3 @ N )
% 9.10/5.20               => ( ( M
% 9.10/5.20                    = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 9.10/5.20                 => ( P @ I4 ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2782_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.20        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.10/5.20          = Y3 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ( Y3
% 9.10/5.20                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.20                       => A3 )
% 9.10/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.10/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.10/5.20                           => B3 )
% 9.10/5.20                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.10/5.20         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20                  ( X3
% 9.10/5.20                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.10/5.20             => Y3 )
% 9.10/5.20           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                  ( ? [S3: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 9.10/5.20                 => ( Y3
% 9.10/5.20                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                           => ( 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 ) ) ) ) ) )
% 9.10/5.20                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2783_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.20        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.20                   => A3 )
% 9.10/5.20                  & ( ( Xa2 != zero_zero_nat )
% 9.10/5.20                   => ( ( ( Xa2 = one_one_nat )
% 9.10/5.20                       => B3 )
% 9.10/5.20                      & ( Xa2 = one_one_nat ) ) ) ) )
% 9.10/5.20         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                ( ? [S3: vEBT_VEBT] :
% 9.10/5.20                    ( X3
% 9.10/5.20                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 9.10/5.20               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                     => ( 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 ) ) ) ) ) )
% 9.10/5.20                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2784_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.20        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.20                 => A3 )
% 9.10/5.20                & ( ( Xa2 != zero_zero_nat )
% 9.10/5.20                 => ( ( ( Xa2 = one_one_nat )
% 9.10/5.20                     => B3 )
% 9.10/5.20                    & ( Xa2 = one_one_nat ) ) ) ) )
% 9.10/5.20         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20                ( X3
% 9.10/5.20               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.10/5.20           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                  ( ? [S3: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 9.10/5.20                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                     => ( 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 ) ) ) ) ) )
% 9.10/5.20                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2785_mult__eq__if,axiom,
% 9.10/5.20      ( times_times_nat
% 9.10/5.20      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2786_vebt__pred_Osimps_I5_J,axiom,
% 9.10/5.20      ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 9.10/5.20        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
% 9.10/5.20        = none_nat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2787_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.20        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.10/5.20          = Y3 )
% 9.10/5.20       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.10/5.20                ( X3
% 9.10/5.20                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.20           => Y3 )
% 9.10/5.20         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.10/5.20                  ( X3
% 9.10/5.20                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.10/5.20             => Y3 )
% 9.10/5.20           => ( ! [Mi2: nat,Ma2: nat] :
% 9.10/5.20                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.10/5.20                 => ( Y3
% 9.10/5.20                    = ( ~ ( ( Xa2 = Mi2 )
% 9.10/5.20                          | ( Xa2 = Ma2 ) ) ) ) )
% 9.10/5.20             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                    ( ? [Vc2: vEBT_VEBT] :
% 9.10/5.20                        ( X3
% 9.10/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.10/5.20                   => ( Y3
% 9.10/5.20                      = ( ~ ( ( Xa2 = Mi2 )
% 9.10/5.20                            | ( Xa2 = Ma2 )
% 9.10/5.20                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                               => ( 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 ) ) ) ) ) )
% 9.10/5.20                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 9.10/5.20               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                      ( ? [Vd2: vEBT_VEBT] :
% 9.10/5.20                          ( X3
% 9.10/5.20                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.10/5.20                     => ( Y3
% 9.10/5.20                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                               => ( 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 ) ) ) ) ) )
% 9.10/5.20                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2788_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.20        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.10/5.20       => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.20              ( X3
% 9.10/5.20             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.20         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.10/5.20                ( X3
% 9.10/5.20               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.10/5.20           => ( ! [Mi2: nat,Ma2: nat] :
% 9.10/5.20                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.10/5.20                 => ( ( Xa2 = Mi2 )
% 9.10/5.20                    | ( Xa2 = Ma2 ) ) )
% 9.10/5.20             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                    ( ? [Vc2: vEBT_VEBT] :
% 9.10/5.20                        ( X3
% 9.10/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.10/5.20                   => ( ( Xa2 = Mi2 )
% 9.10/5.20                      | ( Xa2 = Ma2 )
% 9.10/5.20                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                         => ( 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 ) ) ) ) ) )
% 9.10/5.20                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.10/5.20               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                      ( ? [Vd2: vEBT_VEBT] :
% 9.10/5.20                          ( X3
% 9.10/5.20                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.10/5.20                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                         => ( 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 ) ) ) ) ) )
% 9.10/5.20                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2789_convex__bound__lt,axiom,
% 9.10/5.20      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 9.10/5.20        ( ( ord_less_real @ X3 @ A )
% 9.10/5.20       => ( ( ord_less_real @ Y3 @ A )
% 9.10/5.20         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.10/5.20           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.10/5.20             => ( ( ( plus_plus_real @ U @ V )
% 9.10/5.20                  = one_one_real )
% 9.10/5.20               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2790_convex__bound__lt,axiom,
% 9.10/5.20      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 9.10/5.20        ( ( ord_less_rat @ X3 @ A )
% 9.10/5.20       => ( ( ord_less_rat @ Y3 @ A )
% 9.10/5.20         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.10/5.20           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.10/5.20             => ( ( ( plus_plus_rat @ U @ V )
% 9.10/5.20                  = one_one_rat )
% 9.10/5.20               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2791_convex__bound__lt,axiom,
% 9.10/5.20      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 9.10/5.20        ( ( ord_less_int @ X3 @ A )
% 9.10/5.20       => ( ( ord_less_int @ Y3 @ A )
% 9.10/5.20         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.10/5.20           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.10/5.20             => ( ( ( plus_plus_int @ U @ V )
% 9.10/5.20                  = one_one_int )
% 9.10/5.20               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2792_le__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: real,C: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2793_le__divide__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [W: num,B2: rat,C: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2794_divide__le__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: real,C: real,W: num] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( numeral_numeral_real @ W ) )
% 9.10/5.20        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.20           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.20               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2795_divide__le__eq__numeral_I1_J,axiom,
% 9.10/5.20      ! [B2: rat,C: rat,W: num] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( numeral_numeral_rat @ W ) )
% 9.10/5.20        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 9.10/5.20          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.20           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B2 ) )
% 9.10/5.20              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.20               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2796_half__gt__zero,axiom,
% 9.10/5.20      ! [A: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.20       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2797_half__gt__zero,axiom,
% 9.10/5.20      ! [A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.20       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2798_half__gt__zero__iff,axiom,
% 9.10/5.20      ! [A: real] :
% 9.10/5.20        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.20        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2799_half__gt__zero__iff,axiom,
% 9.10/5.20      ! [A: rat] :
% 9.10/5.20        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.10/5.20        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2800_scaling__mono,axiom,
% 9.10/5.20      ! [U: real,V: real,R: real,S: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ U @ V )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 9.10/5.20         => ( ( ord_less_eq_real @ R @ S )
% 9.10/5.20           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2801_scaling__mono,axiom,
% 9.10/5.20      ! [U: rat,V: rat,R: rat,S: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ U @ V )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 9.10/5.20         => ( ( ord_less_eq_rat @ R @ S )
% 9.10/5.20           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2802_zero__le__power2,axiom,
% 9.10/5.20      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2803_zero__le__power2,axiom,
% 9.10/5.20      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2804_zero__le__power2,axiom,
% 9.10/5.20      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2805_power2__eq__imp__eq,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20          = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.20         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.20           => ( X3 = Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2806_power2__eq__imp__eq,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20          = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.10/5.20         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20           => ( X3 = Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2807_power2__eq__imp__eq,axiom,
% 9.10/5.20      ! [X3: nat,Y3: nat] :
% 9.10/5.20        ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20          = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.10/5.20         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.10/5.20           => ( X3 = Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2808_power2__eq__imp__eq,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20          = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.20         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.20           => ( X3 = Y3 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2809_power2__le__imp__le,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.20         => ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2810_power2__le__imp__le,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20         => ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2811_power2__le__imp__le,axiom,
% 9.10/5.20      ! [X3: nat,Y3: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.10/5.20         => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2812_power2__le__imp__le,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.20         => ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2813_power2__less__0,axiom,
% 9.10/5.20      ! [A: real] :
% 9.10/5.20        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2814_power2__less__0,axiom,
% 9.10/5.20      ! [A: rat] :
% 9.10/5.20        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2815_power2__less__0,axiom,
% 9.10/5.20      ! [A: int] :
% 9.10/5.20        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2816_exp__add__not__zero__imp__left,axiom,
% 9.10/5.20      ! [M: nat,N: nat] :
% 9.10/5.20        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.20         != zero_zero_nat )
% 9.10/5.20       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.10/5.20         != zero_zero_nat ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2817_exp__add__not__zero__imp__left,axiom,
% 9.10/5.20      ! [M: nat,N: nat] :
% 9.10/5.20        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.20         != zero_zero_int )
% 9.10/5.20       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.10/5.20         != zero_zero_int ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2818_exp__add__not__zero__imp__right,axiom,
% 9.10/5.20      ! [M: nat,N: nat] :
% 9.10/5.20        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.20         != zero_zero_nat )
% 9.10/5.20       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.20         != zero_zero_nat ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2819_exp__add__not__zero__imp__right,axiom,
% 9.10/5.20      ! [M: nat,N: nat] :
% 9.10/5.20        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.20         != zero_zero_int )
% 9.10/5.20       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.10/5.20         != zero_zero_int ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2820_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.20         != zero_zero_nat )
% 9.10/5.20       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.10/5.20         != zero_zero_nat ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2821_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.10/5.20         != zero_zero_int )
% 9.10/5.20       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.10/5.20         != zero_zero_int ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2822_power__diff__power__eq,axiom,
% 9.10/5.20      ! [A: nat,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_nat )
% 9.10/5.20       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.10/5.20              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.10/5.20          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.10/5.20           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 9.10/5.20              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2823_power__diff__power__eq,axiom,
% 9.10/5.20      ! [A: int,N: nat,M: nat] :
% 9.10/5.20        ( ( A != zero_zero_int )
% 9.10/5.20       => ( ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.10/5.20              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.10/5.20          & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.10/5.20           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 9.10/5.20              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2824_less__2__cases,axiom,
% 9.10/5.20      ! [N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20       => ( ( N = zero_zero_nat )
% 9.10/5.20          | ( N
% 9.10/5.20            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2825_less__2__cases__iff,axiom,
% 9.10/5.20      ! [N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.20        = ( ( N = zero_zero_nat )
% 9.10/5.20          | ( N
% 9.10/5.20            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2826_power__eq__if,axiom,
% 9.10/5.20      ( power_power_complex
% 9.10/5.20      = ( ^ [P5: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2827_power__eq__if,axiom,
% 9.10/5.20      ( power_power_real
% 9.10/5.20      = ( ^ [P5: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2828_power__eq__if,axiom,
% 9.10/5.20      ( power_power_rat
% 9.10/5.20      = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2829_power__eq__if,axiom,
% 9.10/5.20      ( power_power_nat
% 9.10/5.20      = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2830_power__eq__if,axiom,
% 9.10/5.20      ( power_power_int
% 9.10/5.20      = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2831_power__minus__mult,axiom,
% 9.10/5.20      ! [N: nat,A: complex] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.10/5.20          = ( power_power_complex @ A @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2832_power__minus__mult,axiom,
% 9.10/5.20      ! [N: nat,A: real] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.10/5.20          = ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2833_power__minus__mult,axiom,
% 9.10/5.20      ! [N: nat,A: rat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.10/5.20          = ( power_power_rat @ A @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2834_power__minus__mult,axiom,
% 9.10/5.20      ! [N: nat,A: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.10/5.20          = ( power_power_nat @ A @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2835_power__minus__mult,axiom,
% 9.10/5.20      ! [N: nat,A: int] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 9.10/5.20          = ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2836_le__div__geq,axiom,
% 9.10/5.20      ! [N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.20         => ( ( divide_divide_nat @ M @ N )
% 9.10/5.20            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2837_split__div_H,axiom,
% 9.10/5.20      ! [P: nat > $o,M: nat,N: nat] :
% 9.10/5.20        ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.20        = ( ( ( N = zero_zero_nat )
% 9.10/5.20            & ( P @ zero_zero_nat ) )
% 9.10/5.20          | ? [Q4: nat] :
% 9.10/5.20              ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
% 9.10/5.20              & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
% 9.10/5.20              & ( P @ Q4 ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2838_div__exp__mod__exp__eq,axiom,
% 9.10/5.20      ! [A: nat,N: nat,M: nat] :
% 9.10/5.20        ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20        = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2839_div__exp__mod__exp__eq,axiom,
% 9.10/5.20      ! [A: int,N: nat,M: nat] :
% 9.10/5.20        ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20        = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2840_div__exp__mod__exp__eq,axiom,
% 9.10/5.20      ! [A: code_integer,N: nat,M: nat] :
% 9.10/5.20        ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.10/5.20        = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2841_vebt__pred_Osimps_I6_J,axiom,
% 9.10/5.20      ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 9.10/5.20        ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
% 9.10/5.20        = none_nat ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2842_power2__less__imp__less,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.20         => ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2843_power2__less__imp__less,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.10/5.20         => ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2844_power2__less__imp__less,axiom,
% 9.10/5.20      ! [X3: nat,Y3: nat] :
% 9.10/5.20        ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.10/5.20         => ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2845_power2__less__imp__less,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.20       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.20         => ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2846_sum__power2__ge__zero,axiom,
% 9.10/5.20      ! [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 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2847_sum__power2__ge__zero,axiom,
% 9.10/5.20      ! [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 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2848_sum__power2__ge__zero,axiom,
% 9.10/5.20      ! [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 ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2849_sum__power2__le__zero__iff,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( 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 )
% 9.10/5.20        = ( ( X3 = zero_zero_real )
% 9.10/5.20          & ( Y3 = zero_zero_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2850_sum__power2__le__zero__iff,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( 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 )
% 9.10/5.20        = ( ( X3 = zero_zero_rat )
% 9.10/5.20          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2851_sum__power2__le__zero__iff,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( 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 )
% 9.10/5.20        = ( ( X3 = zero_zero_int )
% 9.10/5.20          & ( Y3 = zero_zero_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2852_not__sum__power2__lt__zero,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ~ ( 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 ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2853_not__sum__power2__lt__zero,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ~ ( 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 ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2854_not__sum__power2__lt__zero,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ~ ( 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 ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2855_sum__power2__gt__zero__iff,axiom,
% 9.10/5.20      ! [X3: real,Y3: real] :
% 9.10/5.20        ( ( 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 ) ) ) ) )
% 9.10/5.20        = ( ( X3 != zero_zero_real )
% 9.10/5.20          | ( Y3 != zero_zero_real ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2856_sum__power2__gt__zero__iff,axiom,
% 9.10/5.20      ! [X3: rat,Y3: rat] :
% 9.10/5.20        ( ( 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 ) ) ) ) )
% 9.10/5.20        = ( ( X3 != zero_zero_rat )
% 9.10/5.20          | ( Y3 != zero_zero_rat ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2857_sum__power2__gt__zero__iff,axiom,
% 9.10/5.20      ! [X3: int,Y3: int] :
% 9.10/5.20        ( ( 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 ) ) ) ) )
% 9.10/5.20        = ( ( X3 != zero_zero_int )
% 9.10/5.20          | ( Y3 != zero_zero_int ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2858_zero__le__even__power_H,axiom,
% 9.10/5.20      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2859_zero__le__even__power_H,axiom,
% 9.10/5.20      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2860_zero__le__even__power_H,axiom,
% 9.10/5.20      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2861_nat__bit__induct,axiom,
% 9.10/5.20      ! [P: nat > $o,N: nat] :
% 9.10/5.20        ( ( P @ zero_zero_nat )
% 9.10/5.20       => ( ! [N3: nat] :
% 9.10/5.20              ( ( P @ N3 )
% 9.10/5.20             => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.10/5.20               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.10/5.20         => ( ! [N3: nat] :
% 9.10/5.20                ( ( P @ N3 )
% 9.10/5.20               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.10/5.20           => ( P @ N ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2862_div__2__gt__zero,axiom,
% 9.10/5.20      ! [N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2863_Suc__n__div__2__gt__zero,axiom,
% 9.10/5.20      ! [N: nat] :
% 9.10/5.20        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2864_mult__exp__mod__exp__eq,axiom,
% 9.10/5.20      ! [M: nat,N: nat,A: nat] :
% 9.10/5.20        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.20       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.20          = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2865_mult__exp__mod__exp__eq,axiom,
% 9.10/5.20      ! [M: nat,N: nat,A: int] :
% 9.10/5.20        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.20       => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.20          = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2866_mult__exp__mod__exp__eq,axiom,
% 9.10/5.20      ! [M: nat,N: nat,A: code_integer] :
% 9.10/5.20        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.20       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.20          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2867_vebt__member_Oelims_I2_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.20        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.20                   => A3 )
% 9.10/5.20                  & ( ( Xa2 != zero_zero_nat )
% 9.10/5.20                   => ( ( ( Xa2 = one_one_nat )
% 9.10/5.20                       => B3 )
% 9.10/5.20                      & ( Xa2 = one_one_nat ) ) ) ) )
% 9.10/5.20         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                ( ? [Summary2: vEBT_VEBT] :
% 9.10/5.20                    ( X3
% 9.10/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.20               => ~ ( ( Xa2 != Mi2 )
% 9.10/5.20                   => ( ( Xa2 != Ma2 )
% 9.10/5.20                     => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.20                        & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.20                         => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.20                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.20                             => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                                 => ( 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 ) ) ) ) ) )
% 9.10/5.20                                & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2868_vebt__member_Oelims_I1_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.20        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.10/5.20          = Y3 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ( Y3
% 9.10/5.20                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.20                       => A3 )
% 9.10/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.10/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.10/5.20                           => B3 )
% 9.10/5.20                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.10/5.20         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20                  ( X3
% 9.10/5.20                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.20             => Y3 )
% 9.10/5.20           => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.20                    ( X3
% 9.10/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.10/5.20               => Y3 )
% 9.10/5.20             => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.20                      ( X3
% 9.10/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.10/5.20                 => Y3 )
% 9.10/5.20               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                      ( ? [Summary2: vEBT_VEBT] :
% 9.10/5.20                          ( X3
% 9.10/5.20                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.20                     => ( Y3
% 9.10/5.20                        = ( ~ ( ( Xa2 != Mi2 )
% 9.10/5.20                             => ( ( Xa2 != Ma2 )
% 9.10/5.20                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.20                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.20                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.20                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.20                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                                           => ( 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 ) ) ) ) ) )
% 9.10/5.20                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2869_vebt__member_Oelims_I3_J,axiom,
% 9.10/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.20        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.10/5.20       => ( ! [A3: $o,B3: $o] :
% 9.10/5.20              ( ( X3
% 9.10/5.20                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.20             => ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.20                 => A3 )
% 9.10/5.20                & ( ( Xa2 != zero_zero_nat )
% 9.10/5.20                 => ( ( ( Xa2 = one_one_nat )
% 9.10/5.20                     => B3 )
% 9.10/5.20                    & ( Xa2 = one_one_nat ) ) ) ) )
% 9.10/5.20         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.20                ( X3
% 9.10/5.20               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.20           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.20                  ( X3
% 9.10/5.20                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.10/5.20             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.20                    ( X3
% 9.10/5.20                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.10/5.20               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.20                      ( ? [Summary2: vEBT_VEBT] :
% 9.10/5.20                          ( X3
% 9.10/5.20                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.20                     => ( ( Xa2 != Mi2 )
% 9.10/5.20                       => ( ( Xa2 != Ma2 )
% 9.10/5.20                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.20                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.20                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.20                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.20                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.20                                     => ( 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 ) ) ) ) ) )
% 9.10/5.20                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2870_odd__0__le__power__imp__0__le,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.20       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2871_odd__0__le__power__imp__0__le,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.20       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2872_odd__0__le__power__imp__0__le,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.20       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2873_odd__power__less__zero,axiom,
% 9.10/5.20      ! [A: real,N: nat] :
% 9.10/5.20        ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.20       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2874_odd__power__less__zero,axiom,
% 9.10/5.20      ! [A: rat,N: nat] :
% 9.10/5.20        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.20       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2875_odd__power__less__zero,axiom,
% 9.10/5.20      ! [A: int,N: nat] :
% 9.10/5.20        ( ( ord_less_int @ A @ zero_zero_int )
% 9.10/5.20       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2876_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 9.10/5.20      ! [X3: nat,N: nat,M: nat] :
% 9.10/5.20        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.10/5.20       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.20         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.20           => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 9.10/5.20  
% 9.10/5.20  thf(fact_2877_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 9.10/5.20      ! [X3: nat,N: nat,M: nat] :
% 9.10/5.21        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.10/5.21       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.21         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.21           => ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2878_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.10/5.21       => ( ! [Mi2: nat,Ma2: nat] :
% 9.10/5.21              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.10/5.21                  ( X3
% 9.10/5.21                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.10/5.21             => ~ ( ( Xa2 = Mi2 )
% 9.10/5.21                  | ( Xa2 = Ma2 ) ) )
% 9.10/5.21         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.21                ( ? [Vc2: vEBT_VEBT] :
% 9.10/5.21                    ( X3
% 9.10/5.21                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.10/5.21               => ~ ( ( Xa2 = Mi2 )
% 9.10/5.21                    | ( Xa2 = Ma2 )
% 9.10/5.21                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                       => ( 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 ) ) ) ) ) )
% 9.10/5.21                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.10/5.21           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.10/5.21                  ( ? [Vd2: vEBT_VEBT] :
% 9.10/5.21                      ( X3
% 9.10/5.21                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.10/5.21                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                       => ( 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 ) ) ) ) ) )
% 9.10/5.21                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2879_arith__geo__mean,axiom,
% 9.10/5.21      ! [U: real,X3: real,Y3: real] :
% 9.10/5.21        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.21          = ( times_times_real @ X3 @ Y3 ) )
% 9.10/5.21       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.21         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.21           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2880_arith__geo__mean,axiom,
% 9.10/5.21      ! [U: rat,X3: rat,Y3: rat] :
% 9.10/5.21        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.21          = ( times_times_rat @ X3 @ Y3 ) )
% 9.10/5.21       => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.10/5.21         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.10/5.21           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2881_invar__vebt_Osimps,axiom,
% 9.10/5.21      ( vEBT_invar_vebt
% 9.10/5.21      = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 9.10/5.21            ( ( ? [A5: $o,B4: $o] :
% 9.10/5.21                  ( A1
% 9.10/5.21                  = ( vEBT_Leaf @ A5 @ B4 ) )
% 9.10/5.21              & ( A22
% 9.10/5.21                = ( suc @ zero_zero_nat ) ) )
% 9.10/5.21            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 9.10/5.21                ( ( A1
% 9.10/5.21                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 9.10/5.21                & ! [X: vEBT_VEBT] :
% 9.10/5.21                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.10/5.21                & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 9.10/5.21                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.10/5.21                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.10/5.21                & ( A22
% 9.10/5.21                  = ( plus_plus_nat @ N2 @ N2 ) )
% 9.10/5.21                & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
% 9.10/5.21                & ! [X: vEBT_VEBT] :
% 9.10/5.21                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.21            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 9.10/5.21                ( ( A1
% 9.10/5.21                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 9.10/5.21                & ! [X: vEBT_VEBT] :
% 9.10/5.21                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.10/5.21                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 9.10/5.21                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.10/5.21                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.10/5.21                & ( A22
% 9.10/5.21                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 9.10/5.21                & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
% 9.10/5.21                & ! [X: vEBT_VEBT] :
% 9.10/5.21                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.21            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.10/5.21                ( ( A1
% 9.10/5.21                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 9.10/5.21                & ! [X: vEBT_VEBT] :
% 9.10/5.21                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.10/5.21                & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 9.10/5.21                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.10/5.21                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.10/5.21                & ( A22
% 9.10/5.21                  = ( plus_plus_nat @ N2 @ N2 ) )
% 9.10/5.21                & ! [I4: nat] :
% 9.10/5.21                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.10/5.21                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
% 9.10/5.21                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.10/5.21                & ( ( Mi3 = Ma3 )
% 9.10/5.21                 => ! [X: vEBT_VEBT] :
% 9.10/5.21                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.21                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.21                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 9.10/5.21                & ( ( Mi3 != Ma3 )
% 9.10/5.21                 => ! [I4: nat] :
% 9.10/5.21                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.10/5.21                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 9.10/5.21                            = I4 )
% 9.10/5.21                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 9.10/5.21                        & ! [X: nat] :
% 9.10/5.21                            ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 9.10/5.21                                = I4 )
% 9.10/5.21                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 9.10/5.21                           => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.21                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 9.10/5.21            | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.10/5.21                ( ( A1
% 9.10/5.21                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 9.10/5.21                & ! [X: vEBT_VEBT] :
% 9.10/5.21                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                   => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.10/5.21                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 9.10/5.21                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.10/5.21                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.10/5.21                & ( A22
% 9.10/5.21                  = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 9.10/5.21                & ! [I4: nat] :
% 9.10/5.21                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.10/5.21                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
% 9.10/5.21                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 9.10/5.21                & ( ( Mi3 = Ma3 )
% 9.10/5.21                 => ! [X: vEBT_VEBT] :
% 9.10/5.21                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.10/5.21                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.21                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.21                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 9.10/5.21                & ( ( Mi3 != Ma3 )
% 9.10/5.21                 => ! [I4: nat] :
% 9.10/5.21                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.10/5.21                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 9.10/5.21                            = I4 )
% 9.10/5.21                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 9.10/5.21                        & ! [X: nat] :
% 9.10/5.21                            ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 9.10/5.21                                = I4 )
% 9.10/5.21                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 9.10/5.21                           => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.21                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2882_invar__vebt_Ocases,axiom,
% 9.10/5.21      ! [A12: vEBT_VEBT,A23: nat] :
% 9.10/5.21        ( ( vEBT_invar_vebt @ A12 @ A23 )
% 9.10/5.21       => ( ( ? [A3: $o,B3: $o] :
% 9.10/5.21                ( A12
% 9.10/5.21                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21           => ( A23
% 9.10/5.21             != ( suc @ zero_zero_nat ) ) )
% 9.10/5.21         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 9.10/5.21                ( ( A12
% 9.10/5.21                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.21               => ( ( A23 = Deg2 )
% 9.10/5.21                 => ( ! [X6: vEBT_VEBT] :
% 9.10/5.21                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                       => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 9.10/5.21                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.10/5.21                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.21                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                       => ( ( M4 = N3 )
% 9.10/5.21                         => ( ( Deg2
% 9.10/5.21                              = ( plus_plus_nat @ N3 @ M4 ) )
% 9.10/5.21                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 9.10/5.21                             => ~ ! [X6: vEBT_VEBT] :
% 9.10/5.21                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.10/5.21           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 9.10/5.21                  ( ( A12
% 9.10/5.21                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.21                 => ( ( A23 = Deg2 )
% 9.10/5.21                   => ( ! [X6: vEBT_VEBT] :
% 9.10/5.21                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                         => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 9.10/5.21                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.10/5.21                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.21                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                         => ( ( M4
% 9.10/5.21                              = ( suc @ N3 ) )
% 9.10/5.21                           => ( ( Deg2
% 9.10/5.21                                = ( plus_plus_nat @ N3 @ M4 ) )
% 9.10/5.21                             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 9.10/5.21                               => ~ ! [X6: vEBT_VEBT] :
% 9.10/5.21                                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.10/5.21             => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 9.10/5.21                    ( ( A12
% 9.10/5.21                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.21                   => ( ( A23 = Deg2 )
% 9.10/5.21                     => ( ! [X6: vEBT_VEBT] :
% 9.10/5.21                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                           => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 9.10/5.21                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.10/5.21                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.21                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                           => ( ( M4 = N3 )
% 9.10/5.21                             => ( ( Deg2
% 9.10/5.21                                  = ( plus_plus_nat @ N3 @ M4 ) )
% 9.10/5.21                               => ( ! [I2: nat] :
% 9.10/5.21                                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
% 9.10/5.21                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.10/5.21                                 => ( ( ( Mi2 = Ma2 )
% 9.10/5.21                                     => ! [X6: vEBT_VEBT] :
% 9.10/5.21                                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
% 9.10/5.21                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.10/5.21                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.21                                       => ~ ( ( Mi2 != Ma2 )
% 9.10/5.21                                           => ! [I2: nat] :
% 9.10/5.21                                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 9.10/5.21                                                      = I2 )
% 9.10/5.21                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 9.10/5.21                                                  & ! [X6: nat] :
% 9.10/5.21                                                      ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
% 9.10/5.21                                                          = I2 )
% 9.10/5.21                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
% 9.10/5.21                                                     => ( ( ord_less_nat @ Mi2 @ X6 )
% 9.10/5.21                                                        & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.10/5.21               => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 9.10/5.21                      ( ( A12
% 9.10/5.21                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.21                     => ( ( A23 = Deg2 )
% 9.10/5.21                       => ( ! [X6: vEBT_VEBT] :
% 9.10/5.21                              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                             => ( vEBT_invar_vebt @ X6 @ N3 ) )
% 9.10/5.21                         => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.10/5.21                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.21                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                             => ( ( M4
% 9.10/5.21                                  = ( suc @ N3 ) )
% 9.10/5.21                               => ( ( Deg2
% 9.10/5.21                                    = ( plus_plus_nat @ N3 @ M4 ) )
% 9.10/5.21                                 => ( ! [I2: nat] :
% 9.10/5.21                                        ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                                       => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
% 9.10/5.21                                          = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.10/5.21                                   => ( ( ( Mi2 = Ma2 )
% 9.10/5.21                                       => ! [X6: vEBT_VEBT] :
% 9.10/5.21                                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.21                                           => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_12 ) ) )
% 9.10/5.21                                     => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.10/5.21                                       => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.21                                         => ~ ( ( Mi2 != Ma2 )
% 9.10/5.21                                             => ! [I2: nat] :
% 9.10/5.21                                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.10/5.21                                                 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 9.10/5.21                                                        = I2 )
% 9.10/5.21                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 9.10/5.21                                                    & ! [X6: nat] :
% 9.10/5.21                                                        ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
% 9.10/5.21                                                            = I2 )
% 9.10/5.21                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
% 9.10/5.21                                                       => ( ( ord_less_nat @ Mi2 @ X6 )
% 9.10/5.21                                                          & ( ord_less_eq_nat @ X6 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2883_vebt__member_Osimps_I2_J,axiom,
% 9.10/5.21      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
% 9.10/5.21        ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2884_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 9.10/5.21      ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.10/5.21        ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2885_inrange,axiom,
% 9.10/5.21      ! [T: vEBT_VEBT,N: nat] :
% 9.10/5.21        ( ( vEBT_invar_vebt @ T @ N )
% 9.10/5.21       => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2886_set__bit__0,axiom,
% 9.10/5.21      ! [A: int] :
% 9.10/5.21        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 9.10/5.21        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2887_set__bit__0,axiom,
% 9.10/5.21      ! [A: nat] :
% 9.10/5.21        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 9.10/5.21        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2888_vebt__pred_Opelims,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 9.10/5.21        ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
% 9.10/5.21          = Y3 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.21               => ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                 => ( ( Y3 = none_nat )
% 9.10/5.21                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 9.10/5.21           => ( ! [A3: $o,Uw2: $o] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 9.10/5.21                 => ( ( Xa2
% 9.10/5.21                      = ( suc @ zero_zero_nat ) )
% 9.10/5.21                   => ( ( ( A3
% 9.10/5.21                         => ( Y3
% 9.10/5.21                            = ( some_nat @ zero_zero_nat ) ) )
% 9.10/5.21                        & ( ~ A3
% 9.10/5.21                         => ( Y3 = none_nat ) ) )
% 9.10/5.21                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 9.10/5.21             => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21                   => ! [Va2: nat] :
% 9.10/5.21                        ( ( Xa2
% 9.10/5.21                          = ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.21                       => ( ( ( B3
% 9.10/5.21                             => ( Y3
% 9.10/5.21                                = ( some_nat @ one_one_nat ) ) )
% 9.10/5.21                            & ( ~ B3
% 9.10/5.21                             => ( ( A3
% 9.10/5.21                                 => ( Y3
% 9.10/5.21                                    = ( some_nat @ zero_zero_nat ) ) )
% 9.10/5.21                                & ( ~ A3
% 9.10/5.21                                 => ( Y3 = none_nat ) ) ) ) )
% 9.10/5.21                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 9.10/5.21               => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 9.10/5.21                      ( ( X3
% 9.10/5.21                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 9.10/5.21                     => ( ( Y3 = none_nat )
% 9.10/5.21                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 9.10/5.21                 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 9.10/5.21                        ( ( X3
% 9.10/5.21                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 9.10/5.21                       => ( ( Y3 = none_nat )
% 9.10/5.21                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
% 9.10/5.21                   => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 9.10/5.21                          ( ( X3
% 9.10/5.21                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 9.10/5.21                         => ( ( Y3 = none_nat )
% 9.10/5.21                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
% 9.10/5.21                     => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.21                            ( ( X3
% 9.10/5.21                              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.21                           => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                 => ( Y3
% 9.10/5.21                                    = ( some_nat @ Ma2 ) ) )
% 9.10/5.21                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                 => ( Y3
% 9.10/5.21                                    = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                                      @ ( if_option_nat
% 9.10/5.21                                        @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.21                                           != none_nat )
% 9.10/5.21                                          & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.10/5.21                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.21                                        @ ( if_option_nat
% 9.10/5.21                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.21                                            = none_nat )
% 9.10/5.21                                          @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 9.10/5.21                                          @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.10/5.21                                      @ none_nat ) ) ) )
% 9.10/5.21                             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2889_Diff__eq__empty__iff,axiom,
% 9.10/5.21      ! [A4: set_nat,B5: set_nat] :
% 9.10/5.21        ( ( ( minus_minus_set_nat @ A4 @ B5 )
% 9.10/5.21          = bot_bot_set_nat )
% 9.10/5.21        = ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2890_Diff__eq__empty__iff,axiom,
% 9.10/5.21      ! [A4: set_real,B5: set_real] :
% 9.10/5.21        ( ( ( minus_minus_set_real @ A4 @ B5 )
% 9.10/5.21          = bot_bot_set_real )
% 9.10/5.21        = ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2891_Diff__eq__empty__iff,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ( minus_minus_set_int @ A4 @ B5 )
% 9.10/5.21          = bot_bot_set_int )
% 9.10/5.21        = ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2892_empty__subsetI,axiom,
% 9.10/5.21      ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2893_empty__subsetI,axiom,
% 9.10/5.21      ! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2894_empty__subsetI,axiom,
% 9.10/5.21      ! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2895_subset__empty,axiom,
% 9.10/5.21      ! [A4: set_nat] :
% 9.10/5.21        ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
% 9.10/5.21        = ( A4 = bot_bot_set_nat ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2896_subset__empty,axiom,
% 9.10/5.21      ! [A4: set_real] :
% 9.10/5.21        ( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
% 9.10/5.21        = ( A4 = bot_bot_set_real ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2897_subset__empty,axiom,
% 9.10/5.21      ! [A4: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ bot_bot_set_int )
% 9.10/5.21        = ( A4 = bot_bot_set_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2898_double__eq__0__iff,axiom,
% 9.10/5.21      ! [A: real] :
% 9.10/5.21        ( ( ( plus_plus_real @ A @ A )
% 9.10/5.21          = zero_zero_real )
% 9.10/5.21        = ( A = zero_zero_real ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2899_double__eq__0__iff,axiom,
% 9.10/5.21      ! [A: rat] :
% 9.10/5.21        ( ( ( plus_plus_rat @ A @ A )
% 9.10/5.21          = zero_zero_rat )
% 9.10/5.21        = ( A = zero_zero_rat ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2900_double__eq__0__iff,axiom,
% 9.10/5.21      ! [A: int] :
% 9.10/5.21        ( ( ( plus_plus_int @ A @ A )
% 9.10/5.21          = zero_zero_int )
% 9.10/5.21        = ( A = zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2901_unset__bit__0,axiom,
% 9.10/5.21      ! [A: int] :
% 9.10/5.21        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 9.10/5.21        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2902_unset__bit__0,axiom,
% 9.10/5.21      ! [A: nat] :
% 9.10/5.21        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 9.10/5.21        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2903_set__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: code_integer] :
% 9.10/5.21        ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2904_set__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: int] :
% 9.10/5.21        ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2905_set__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: nat] :
% 9.10/5.21        ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2906_flip__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: code_integer] :
% 9.10/5.21        ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2907_flip__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: int] :
% 9.10/5.21        ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2908_flip__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: nat] :
% 9.10/5.21        ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2909_subsetI,axiom,
% 9.10/5.21      ! [A4: set_nat,B5: set_nat] :
% 9.10/5.21        ( ! [X5: nat] :
% 9.10/5.21            ( ( member_nat @ X5 @ A4 )
% 9.10/5.21           => ( member_nat @ X5 @ B5 ) )
% 9.10/5.21       => ( ord_less_eq_set_nat @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2910_subsetI,axiom,
% 9.10/5.21      ! [A4: set_real,B5: set_real] :
% 9.10/5.21        ( ! [X5: real] :
% 9.10/5.21            ( ( member_real @ X5 @ A4 )
% 9.10/5.21           => ( member_real @ X5 @ B5 ) )
% 9.10/5.21       => ( ord_less_eq_set_real @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2911_subsetI,axiom,
% 9.10/5.21      ! [A4: set_complex,B5: set_complex] :
% 9.10/5.21        ( ! [X5: complex] :
% 9.10/5.21            ( ( member_complex @ X5 @ A4 )
% 9.10/5.21           => ( member_complex @ X5 @ B5 ) )
% 9.10/5.21       => ( ord_le211207098394363844omplex @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2912_subsetI,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ! [X5: int] :
% 9.10/5.21            ( ( member_int @ X5 @ A4 )
% 9.10/5.21           => ( member_int @ X5 @ B5 ) )
% 9.10/5.21       => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2913_subset__antisym,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.21         => ( A4 = B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2914_psubsetI,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( A4 != B5 )
% 9.10/5.21         => ( ord_less_set_int @ A4 @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2915_div__pos__pos__trivial,axiom,
% 9.10/5.21      ! [K: int,L: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.21       => ( ( ord_less_int @ K @ L )
% 9.10/5.21         => ( ( divide_divide_int @ K @ L )
% 9.10/5.21            = zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2916_div__neg__neg__trivial,axiom,
% 9.10/5.21      ! [K: int,L: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_int @ L @ K )
% 9.10/5.21         => ( ( divide_divide_int @ K @ L )
% 9.10/5.21            = zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2917_i0__less,axiom,
% 9.10/5.21      ! [N: extended_enat] :
% 9.10/5.21        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.10/5.21        = ( N != zero_z5237406670263579293d_enat ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2918_idiff__0__right,axiom,
% 9.10/5.21      ! [N: extended_enat] :
% 9.10/5.21        ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 9.10/5.21        = N ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2919_idiff__0,axiom,
% 9.10/5.21      ! [N: extended_enat] :
% 9.10/5.21        ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.10/5.21        = zero_z5237406670263579293d_enat ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2920_not__real__square__gt__zero,axiom,
% 9.10/5.21      ! [X3: real] :
% 9.10/5.21        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
% 9.10/5.21        = ( X3 = zero_zero_real ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2921_zmod__numeral__Bit0,axiom,
% 9.10/5.21      ! [V: num,W: num] :
% 9.10/5.21        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.10/5.21        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2922_half__nonnegative__int__iff,axiom,
% 9.10/5.21      ! [K: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.10/5.21        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2923_half__negative__int__iff,axiom,
% 9.10/5.21      ! [K: int] :
% 9.10/5.21        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.10/5.21        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2924_psubset__imp__ex__mem,axiom,
% 9.10/5.21      ! [A4: set_nat,B5: set_nat] :
% 9.10/5.21        ( ( ord_less_set_nat @ A4 @ B5 )
% 9.10/5.21       => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2925_psubset__imp__ex__mem,axiom,
% 9.10/5.21      ! [A4: set_real,B5: set_real] :
% 9.10/5.21        ( ( ord_less_set_real @ A4 @ B5 )
% 9.10/5.21       => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2926_psubset__imp__ex__mem,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_set_int @ A4 @ B5 )
% 9.10/5.21       => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2927_psubset__imp__ex__mem,axiom,
% 9.10/5.21      ! [A4: set_complex,B5: set_complex] :
% 9.10/5.21        ( ( ord_less_set_complex @ A4 @ B5 )
% 9.10/5.21       => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2928_zdiv__mono__strict,axiom,
% 9.10/5.21      ! [A4: int,B5: int,N: int] :
% 9.10/5.21        ( ( ord_less_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.10/5.21         => ( ( ( modulo_modulo_int @ A4 @ N )
% 9.10/5.21              = zero_zero_int )
% 9.10/5.21           => ( ( ( modulo_modulo_int @ B5 @ N )
% 9.10/5.21                = zero_zero_int )
% 9.10/5.21             => ( ord_less_int @ ( divide_divide_int @ A4 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2929_verit__le__mono__div__int,axiom,
% 9.10/5.21      ! [A4: int,B5: int,N: int] :
% 9.10/5.21        ( ( ord_less_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.10/5.21         => ( ord_less_eq_int
% 9.10/5.21            @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N )
% 9.10/5.21              @ ( if_int
% 9.10/5.21                @ ( ( modulo_modulo_int @ B5 @ N )
% 9.10/5.21                  = zero_zero_int )
% 9.10/5.21                @ one_one_int
% 9.10/5.21                @ zero_zero_int ) )
% 9.10/5.21            @ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2930_zmod__zmult2__eq,axiom,
% 9.10/5.21      ! [C: int,A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.21       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2931_split__pos__lemma,axiom,
% 9.10/5.21      ! [K: int,P: int > int > $o,N: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.10/5.21          = ( ! [I4: int,J3: int] :
% 9.10/5.21                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.10/5.21                  & ( ord_less_int @ J3 @ K )
% 9.10/5.21                  & ( N
% 9.10/5.21                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.10/5.21               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2932_split__neg__lemma,axiom,
% 9.10/5.21      ! [K: int,P: int > int > $o,N: int] :
% 9.10/5.21        ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.21       => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.10/5.21          = ( ! [I4: int,J3: int] :
% 9.10/5.21                ( ( ( ord_less_int @ K @ J3 )
% 9.10/5.21                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.10/5.21                  & ( N
% 9.10/5.21                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.10/5.21               => ( P @ I4 @ J3 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2933_div__mod__decomp__int,axiom,
% 9.10/5.21      ! [A4: int,N: int] :
% 9.10/5.21        ( A4
% 9.10/5.21        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2934_div__neg__pos__less0,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_int @ A @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.21         => ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2935_neg__imp__zdiv__neg__iff,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 9.10/5.21          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2936_pos__imp__zdiv__neg__iff,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.21       => ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
% 9.10/5.21          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2937_div__pos__geq,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.21       => ( ( ord_less_eq_int @ L @ K )
% 9.10/5.21         => ( ( divide_divide_int @ K @ L )
% 9.10/5.21            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2938_nonneg1__imp__zdiv__pos__iff,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 9.10/5.21          = ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.21            & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2939_pos__imp__zdiv__nonneg__iff,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 9.10/5.21          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2940_neg__imp__zdiv__nonneg__iff,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
% 9.10/5.21          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2941_pos__imp__zdiv__pos__iff,axiom,
% 9.10/5.21      ! [K: int,I: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 9.10/5.21          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2942_div__nonpos__pos__le0,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.21         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2943_div__nonneg__neg__le0,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.21       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.10/5.21         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2944_int__div__less__self,axiom,
% 9.10/5.21      ! [X3: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ X3 )
% 9.10/5.21       => ( ( ord_less_int @ one_one_int @ K )
% 9.10/5.21         => ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2945_div__positive__int,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ L @ K )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.21         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2946_div__int__pos__iff,axiom,
% 9.10/5.21      ! [K: int,L: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 9.10/5.21        = ( ( K = zero_zero_int )
% 9.10/5.21          | ( L = zero_zero_int )
% 9.10/5.21          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.21            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 9.10/5.21          | ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.21            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2947_zdiv__mono2__neg,axiom,
% 9.10/5.21      ! [A: int,B: int,B2: int] :
% 9.10/5.21        ( ( ord_less_int @ A @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.10/5.21         => ( ( ord_less_eq_int @ B @ B2 )
% 9.10/5.21           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2948_zdiv__mono1__neg,axiom,
% 9.10/5.21      ! [A: int,A2: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ A @ A2 )
% 9.10/5.21       => ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.10/5.21         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2949_zdiv__eq__0__iff,axiom,
% 9.10/5.21      ! [I: int,K: int] :
% 9.10/5.21        ( ( ( divide_divide_int @ I @ K )
% 9.10/5.21          = zero_zero_int )
% 9.10/5.21        = ( ( K = zero_zero_int )
% 9.10/5.21          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.10/5.21            & ( ord_less_int @ I @ K ) )
% 9.10/5.21          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.10/5.21            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2950_zdiv__mono2,axiom,
% 9.10/5.21      ! [A: int,B: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.10/5.21         => ( ( ord_less_eq_int @ B @ B2 )
% 9.10/5.21           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2951_zdiv__mono1,axiom,
% 9.10/5.21      ! [A: int,A2: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ A @ A2 )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.21         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2952_zdiv__zmult2__eq,axiom,
% 9.10/5.21      ! [C: int,A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.10/5.21       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.21          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2953_int__div__pos__eq,axiom,
% 9.10/5.21      ! [A: int,B2: int,Q2: int,R: int] :
% 9.10/5.21        ( ( A
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 9.10/5.21         => ( ( ord_less_int @ R @ B2 )
% 9.10/5.21           => ( ( divide_divide_int @ A @ B2 )
% 9.10/5.21              = Q2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2954_int__div__neg__eq,axiom,
% 9.10/5.21      ! [A: int,B2: int,Q2: int,R: int] :
% 9.10/5.21        ( ( A
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 9.10/5.21         => ( ( ord_less_int @ B2 @ R )
% 9.10/5.21           => ( ( divide_divide_int @ A @ B2 )
% 9.10/5.21              = Q2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2955_split__zdiv,axiom,
% 9.10/5.21      ! [P: int > $o,N: int,K: int] :
% 9.10/5.21        ( ( P @ ( divide_divide_int @ N @ K ) )
% 9.10/5.21        = ( ( ( K = zero_zero_int )
% 9.10/5.21           => ( P @ zero_zero_int ) )
% 9.10/5.21          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21           => ! [I4: int,J3: int] :
% 9.10/5.21                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.10/5.21                  & ( ord_less_int @ J3 @ K )
% 9.10/5.21                  & ( N
% 9.10/5.21                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.10/5.21               => ( P @ I4 ) ) )
% 9.10/5.21          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.21           => ! [I4: int,J3: int] :
% 9.10/5.21                ( ( ( ord_less_int @ K @ J3 )
% 9.10/5.21                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.10/5.21                  & ( N
% 9.10/5.21                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.10/5.21               => ( P @ I4 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2956_neg__zmod__mult__2,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.10/5.21       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.21          = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2957_pos__zmod__mult__2,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.21       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.21          = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2958_enat__0__less__mult__iff,axiom,
% 9.10/5.21      ! [M: extended_enat,N: extended_enat] :
% 9.10/5.21        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 9.10/5.21        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 9.10/5.21          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2959_not__iless0,axiom,
% 9.10/5.21      ! [N: extended_enat] :
% 9.10/5.21        ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2960_iadd__is__0,axiom,
% 9.10/5.21      ! [M: extended_enat,N: extended_enat] :
% 9.10/5.21        ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 9.10/5.21          = zero_z5237406670263579293d_enat )
% 9.10/5.21        = ( ( M = zero_z5237406670263579293d_enat )
% 9.10/5.21          & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2961_i0__lb,axiom,
% 9.10/5.21      ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2962_ile0__eq,axiom,
% 9.10/5.21      ! [N: extended_enat] :
% 9.10/5.21        ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 9.10/5.21        = ( N = zero_z5237406670263579293d_enat ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2963_ex__nat__less,axiom,
% 9.10/5.21      ! [N: nat,P: nat > $o] :
% 9.10/5.21        ( ( ? [M3: nat] :
% 9.10/5.21              ( ( ord_less_eq_nat @ M3 @ N )
% 9.10/5.21              & ( P @ M3 ) ) )
% 9.10/5.21        = ( ? [X: nat] :
% 9.10/5.21              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.21              & ( P @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2964_all__nat__less,axiom,
% 9.10/5.21      ! [N: nat,P: nat > $o] :
% 9.10/5.21        ( ( ! [M3: nat] :
% 9.10/5.21              ( ( ord_less_eq_nat @ M3 @ N )
% 9.10/5.21             => ( P @ M3 ) ) )
% 9.10/5.21        = ( ! [X: nat] :
% 9.10/5.21              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.21             => ( P @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2965_not__psubset__empty,axiom,
% 9.10/5.21      ! [A4: set_nat] :
% 9.10/5.21        ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2966_not__psubset__empty,axiom,
% 9.10/5.21      ! [A4: set_int] :
% 9.10/5.21        ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2967_not__psubset__empty,axiom,
% 9.10/5.21      ! [A4: set_real] :
% 9.10/5.21        ~ ( ord_less_set_real @ A4 @ bot_bot_set_real ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2968_in__mono,axiom,
% 9.10/5.21      ! [A4: set_nat,B5: set_nat,X3: nat] :
% 9.10/5.21        ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.21       => ( ( member_nat @ X3 @ A4 )
% 9.10/5.21         => ( member_nat @ X3 @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2969_in__mono,axiom,
% 9.10/5.21      ! [A4: set_real,B5: set_real,X3: real] :
% 9.10/5.21        ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.21       => ( ( member_real @ X3 @ A4 )
% 9.10/5.21         => ( member_real @ X3 @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2970_in__mono,axiom,
% 9.10/5.21      ! [A4: set_complex,B5: set_complex,X3: complex] :
% 9.10/5.21        ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.21       => ( ( member_complex @ X3 @ A4 )
% 9.10/5.21         => ( member_complex @ X3 @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2971_in__mono,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,X3: int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( member_int @ X3 @ A4 )
% 9.10/5.21         => ( member_int @ X3 @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2972_subsetD,axiom,
% 9.10/5.21      ! [A4: set_nat,B5: set_nat,C: nat] :
% 9.10/5.21        ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.21       => ( ( member_nat @ C @ A4 )
% 9.10/5.21         => ( member_nat @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2973_subsetD,axiom,
% 9.10/5.21      ! [A4: set_real,B5: set_real,C: real] :
% 9.10/5.21        ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.21       => ( ( member_real @ C @ A4 )
% 9.10/5.21         => ( member_real @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2974_subsetD,axiom,
% 9.10/5.21      ! [A4: set_complex,B5: set_complex,C: complex] :
% 9.10/5.21        ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.21       => ( ( member_complex @ C @ A4 )
% 9.10/5.21         => ( member_complex @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2975_subsetD,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,C: int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( member_int @ C @ A4 )
% 9.10/5.21         => ( member_int @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2976_equalityE,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( A4 = B5 )
% 9.10/5.21       => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21           => ~ ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2977_subset__eq,axiom,
% 9.10/5.21      ( ord_less_eq_set_nat
% 9.10/5.21      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.10/5.21          ! [X: nat] :
% 9.10/5.21            ( ( member_nat @ X @ A6 )
% 9.10/5.21           => ( member_nat @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2978_subset__eq,axiom,
% 9.10/5.21      ( ord_less_eq_set_real
% 9.10/5.21      = ( ^ [A6: set_real,B6: set_real] :
% 9.10/5.21          ! [X: real] :
% 9.10/5.21            ( ( member_real @ X @ A6 )
% 9.10/5.21           => ( member_real @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2979_subset__eq,axiom,
% 9.10/5.21      ( ord_le211207098394363844omplex
% 9.10/5.21      = ( ^ [A6: set_complex,B6: set_complex] :
% 9.10/5.21          ! [X: complex] :
% 9.10/5.21            ( ( member_complex @ X @ A6 )
% 9.10/5.21           => ( member_complex @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2980_subset__eq,axiom,
% 9.10/5.21      ( ord_less_eq_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21          ! [X: int] :
% 9.10/5.21            ( ( member_int @ X @ A6 )
% 9.10/5.21           => ( member_int @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2981_equalityD1,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( A4 = B5 )
% 9.10/5.21       => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2982_equalityD2,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( A4 = B5 )
% 9.10/5.21       => ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2983_subset__iff,axiom,
% 9.10/5.21      ( ord_less_eq_set_nat
% 9.10/5.21      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.10/5.21          ! [T2: nat] :
% 9.10/5.21            ( ( member_nat @ T2 @ A6 )
% 9.10/5.21           => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2984_subset__iff,axiom,
% 9.10/5.21      ( ord_less_eq_set_real
% 9.10/5.21      = ( ^ [A6: set_real,B6: set_real] :
% 9.10/5.21          ! [T2: real] :
% 9.10/5.21            ( ( member_real @ T2 @ A6 )
% 9.10/5.21           => ( member_real @ T2 @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2985_subset__iff,axiom,
% 9.10/5.21      ( ord_le211207098394363844omplex
% 9.10/5.21      = ( ^ [A6: set_complex,B6: set_complex] :
% 9.10/5.21          ! [T2: complex] :
% 9.10/5.21            ( ( member_complex @ T2 @ A6 )
% 9.10/5.21           => ( member_complex @ T2 @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2986_subset__iff,axiom,
% 9.10/5.21      ( ord_less_eq_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21          ! [T2: int] :
% 9.10/5.21            ( ( member_int @ T2 @ A6 )
% 9.10/5.21           => ( member_int @ T2 @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2987_subset__refl,axiom,
% 9.10/5.21      ! [A4: set_int] : ( ord_less_eq_set_int @ A4 @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2988_Collect__mono,axiom,
% 9.10/5.21      ! [P: complex > $o,Q: complex > $o] :
% 9.10/5.21        ( ! [X5: complex] :
% 9.10/5.21            ( ( P @ X5 )
% 9.10/5.21           => ( Q @ X5 ) )
% 9.10/5.21       => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2989_Collect__mono,axiom,
% 9.10/5.21      ! [P: real > $o,Q: real > $o] :
% 9.10/5.21        ( ! [X5: real] :
% 9.10/5.21            ( ( P @ X5 )
% 9.10/5.21           => ( Q @ X5 ) )
% 9.10/5.21       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2990_Collect__mono,axiom,
% 9.10/5.21      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.10/5.21        ( ! [X5: list_nat] :
% 9.10/5.21            ( ( P @ X5 )
% 9.10/5.21           => ( Q @ X5 ) )
% 9.10/5.21       => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2991_Collect__mono,axiom,
% 9.10/5.21      ! [P: nat > $o,Q: nat > $o] :
% 9.10/5.21        ( ! [X5: nat] :
% 9.10/5.21            ( ( P @ X5 )
% 9.10/5.21           => ( Q @ X5 ) )
% 9.10/5.21       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2992_Collect__mono,axiom,
% 9.10/5.21      ! [P: int > $o,Q: int > $o] :
% 9.10/5.21        ( ! [X5: int] :
% 9.10/5.21            ( ( P @ X5 )
% 9.10/5.21           => ( Q @ X5 ) )
% 9.10/5.21       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2993_subset__trans,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,C4: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 9.10/5.21         => ( ord_less_eq_set_int @ A4 @ C4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2994_set__eq__subset,axiom,
% 9.10/5.21      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.10/5.21            & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2995_Collect__mono__iff,axiom,
% 9.10/5.21      ! [P: complex > $o,Q: complex > $o] :
% 9.10/5.21        ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 9.10/5.21        = ( ! [X: complex] :
% 9.10/5.21              ( ( P @ X )
% 9.10/5.21             => ( Q @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2996_Collect__mono__iff,axiom,
% 9.10/5.21      ! [P: real > $o,Q: real > $o] :
% 9.10/5.21        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 9.10/5.21        = ( ! [X: real] :
% 9.10/5.21              ( ( P @ X )
% 9.10/5.21             => ( Q @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2997_Collect__mono__iff,axiom,
% 9.10/5.21      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.10/5.21        ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 9.10/5.21        = ( ! [X: list_nat] :
% 9.10/5.21              ( ( P @ X )
% 9.10/5.21             => ( Q @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2998_Collect__mono__iff,axiom,
% 9.10/5.21      ! [P: nat > $o,Q: nat > $o] :
% 9.10/5.21        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 9.10/5.21        = ( ! [X: nat] :
% 9.10/5.21              ( ( P @ X )
% 9.10/5.21             => ( Q @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_2999_Collect__mono__iff,axiom,
% 9.10/5.21      ! [P: int > $o,Q: int > $o] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 9.10/5.21        = ( ! [X: int] :
% 9.10/5.21              ( ( P @ X )
% 9.10/5.21             => ( Q @ X ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3000_Diff__mono,axiom,
% 9.10/5.21      ! [A4: set_int,C4: set_int,D4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ C4 )
% 9.10/5.21       => ( ( ord_less_eq_set_int @ D4 @ B5 )
% 9.10/5.21         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3001_Diff__subset,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ B5 ) @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3002_double__diff,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,C4: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 9.10/5.21         => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A4 ) )
% 9.10/5.21            = A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3003_subset__iff__psubset__eq,axiom,
% 9.10/5.21      ( ord_less_eq_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21            ( ( ord_less_set_int @ A6 @ B6 )
% 9.10/5.21            | ( A6 = B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3004_subset__psubset__trans,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,C4: set_int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_set_int @ B5 @ C4 )
% 9.10/5.21         => ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3005_subset__not__subset__eq,axiom,
% 9.10/5.21      ( ord_less_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.10/5.21            & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3006_psubset__subset__trans,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,C4: set_int] :
% 9.10/5.21        ( ( ord_less_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 9.10/5.21         => ( ord_less_set_int @ A4 @ C4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3007_psubset__imp__subset,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ord_less_eq_set_int @ A4 @ B5 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3008_psubset__eq,axiom,
% 9.10/5.21      ( ord_less_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21            ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.10/5.21            & ( A6 != B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3009_psubsetE,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_set_int @ A4 @ B5 )
% 9.10/5.21       => ~ ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.21           => ( ord_less_eq_set_int @ B5 @ A4 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3010_not__exp__less__eq__0__int,axiom,
% 9.10/5.21      ! [N: nat] :
% 9.10/5.21        ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3011_realpow__pos__nth2,axiom,
% 9.10/5.21      ! [A: real,N: nat] :
% 9.10/5.21        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.21       => ? [R3: real] :
% 9.10/5.21            ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.10/5.21            & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 9.10/5.21              = A ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3012_real__arch__pow__inv,axiom,
% 9.10/5.21      ! [Y3: real,X3: real] :
% 9.10/5.21        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.21       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.21         => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y3 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3013_realpow__pos__nth__unique,axiom,
% 9.10/5.21      ! [N: nat,A: real] :
% 9.10/5.21        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.21       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.21         => ? [X5: real] :
% 9.10/5.21              ( ( ord_less_real @ zero_zero_real @ X5 )
% 9.10/5.21              & ( ( power_power_real @ X5 @ N )
% 9.10/5.21                = A )
% 9.10/5.21              & ! [Y5: real] :
% 9.10/5.21                  ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 9.10/5.21                    & ( ( power_power_real @ Y5 @ N )
% 9.10/5.21                      = A ) )
% 9.10/5.21                 => ( Y5 = X5 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3014_realpow__pos__nth,axiom,
% 9.10/5.21      ! [N: nat,A: real] :
% 9.10/5.21        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.21       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.21         => ? [R3: real] :
% 9.10/5.21              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.10/5.21              & ( ( power_power_real @ R3 @ N )
% 9.10/5.21                = A ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3015_neg__zdiv__mult__2,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.10/5.21       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.21          = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3016_pos__zdiv__mult__2,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.21       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.21          = ( divide_divide_int @ B2 @ A ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3017_less__eq__set__def,axiom,
% 9.10/5.21      ( ord_less_eq_set_nat
% 9.10/5.21      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.10/5.21            ( ord_less_eq_nat_o
% 9.10/5.21            @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.10/5.21            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3018_less__eq__set__def,axiom,
% 9.10/5.21      ( ord_less_eq_set_real
% 9.10/5.21      = ( ^ [A6: set_real,B6: set_real] :
% 9.10/5.21            ( ord_less_eq_real_o
% 9.10/5.21            @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.10/5.21            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3019_less__eq__set__def,axiom,
% 9.10/5.21      ( ord_le211207098394363844omplex
% 9.10/5.21      = ( ^ [A6: set_complex,B6: set_complex] :
% 9.10/5.21            ( ord_le4573692005234683329plex_o
% 9.10/5.21            @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 9.10/5.21            @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3020_less__eq__set__def,axiom,
% 9.10/5.21      ( ord_less_eq_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21            ( ord_less_eq_int_o
% 9.10/5.21            @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.10/5.21            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3021_Collect__subset,axiom,
% 9.10/5.21      ! [A4: set_complex,P: complex > $o] :
% 9.10/5.21        ( ord_le211207098394363844omplex
% 9.10/5.21        @ ( collect_complex
% 9.10/5.21          @ ^ [X: complex] :
% 9.10/5.21              ( ( member_complex @ X @ A4 )
% 9.10/5.21              & ( P @ X ) ) )
% 9.10/5.21        @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3022_Collect__subset,axiom,
% 9.10/5.21      ! [A4: set_real,P: real > $o] :
% 9.10/5.21        ( ord_less_eq_set_real
% 9.10/5.21        @ ( collect_real
% 9.10/5.21          @ ^ [X: real] :
% 9.10/5.21              ( ( member_real @ X @ A4 )
% 9.10/5.21              & ( P @ X ) ) )
% 9.10/5.21        @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3023_Collect__subset,axiom,
% 9.10/5.21      ! [A4: set_list_nat,P: list_nat > $o] :
% 9.10/5.21        ( ord_le6045566169113846134st_nat
% 9.10/5.21        @ ( collect_list_nat
% 9.10/5.21          @ ^ [X: list_nat] :
% 9.10/5.21              ( ( member_list_nat @ X @ A4 )
% 9.10/5.21              & ( P @ X ) ) )
% 9.10/5.21        @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3024_Collect__subset,axiom,
% 9.10/5.21      ! [A4: set_nat,P: nat > $o] :
% 9.10/5.21        ( ord_less_eq_set_nat
% 9.10/5.21        @ ( collect_nat
% 9.10/5.21          @ ^ [X: nat] :
% 9.10/5.21              ( ( member_nat @ X @ A4 )
% 9.10/5.21              & ( P @ X ) ) )
% 9.10/5.21        @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3025_Collect__subset,axiom,
% 9.10/5.21      ! [A4: set_int,P: int > $o] :
% 9.10/5.21        ( ord_less_eq_set_int
% 9.10/5.21        @ ( collect_int
% 9.10/5.21          @ ^ [X: int] :
% 9.10/5.21              ( ( member_int @ X @ A4 )
% 9.10/5.21              & ( P @ X ) ) )
% 9.10/5.21        @ A4 ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3026_int__power__div__base,axiom,
% 9.10/5.21      ! [M: nat,K: int] :
% 9.10/5.21        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 9.10/5.21            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3027_unset__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: code_integer] :
% 9.10/5.21        ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3028_unset__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: int] :
% 9.10/5.21        ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3029_unset__bit__Suc,axiom,
% 9.10/5.21      ! [N: nat,A: nat] :
% 9.10/5.21        ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 9.10/5.21        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3030_vebt__member_Opelims_I3_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.10/5.21                 => ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                     => A3 )
% 9.10/5.21                    & ( ( Xa2 != zero_zero_nat )
% 9.10/5.21                     => ( ( ( Xa2 = one_one_nat )
% 9.10/5.21                         => B3 )
% 9.10/5.21                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.10/5.21           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.21                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.10/5.21             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.10/5.21                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 9.10/5.21               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.21                      ( ( X3
% 9.10/5.21                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.10/5.21                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 9.10/5.21                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.21                        ( ( X3
% 9.10/5.21                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.21                       => ( ( 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 ) )
% 9.10/5.21                         => ( ( Xa2 != Mi2 )
% 9.10/5.21                           => ( ( Xa2 != Ma2 )
% 9.10/5.21                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.21                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.21                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                                         => ( 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 ) ) ) ) ) )
% 9.10/5.21                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3031_vebt__member_Opelims_I1_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.21        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.10/5.21          = Y3 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21               => ( ( Y3
% 9.10/5.21                    = ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                       => A3 )
% 9.10/5.21                      & ( ( Xa2 != zero_zero_nat )
% 9.10/5.21                       => ( ( ( Xa2 = one_one_nat )
% 9.10/5.21                           => B3 )
% 9.10/5.21                          & ( Xa2 = one_one_nat ) ) ) ) )
% 9.10/5.21                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 9.10/5.21           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.21                 => ( ~ Y3
% 9.10/5.21                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.10/5.21             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.10/5.21                   => ( ~ Y3
% 9.10/5.21                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 9.10/5.21               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.21                      ( ( X3
% 9.10/5.21                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.10/5.21                     => ( ~ Y3
% 9.10/5.21                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 9.10/5.21                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.21                        ( ( X3
% 9.10/5.21                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.21                       => ( ( Y3
% 9.10/5.21                            = ( ( Xa2 != Mi2 )
% 9.10/5.21                             => ( ( Xa2 != Ma2 )
% 9.10/5.21                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.21                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.21                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                                           => ( 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 ) ) ) ) ) )
% 9.10/5.21                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 9.10/5.21                         => ~ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3032_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.10/5.21                 => ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                     => A3 )
% 9.10/5.21                    & ( ( Xa2 != zero_zero_nat )
% 9.10/5.21                     => ( ( ( Xa2 = one_one_nat )
% 9.10/5.21                         => B3 )
% 9.10/5.21                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.10/5.21           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.10/5.21                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.10/5.21             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 9.10/5.21                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
% 9.10/5.21                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                         => ( 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 ) ) ) ) ) )
% 9.10/5.21                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3033_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.10/5.21                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                       => A3 )
% 9.10/5.21                      & ( ( Xa2 != zero_zero_nat )
% 9.10/5.21                       => ( ( ( Xa2 = one_one_nat )
% 9.10/5.21                           => B3 )
% 9.10/5.21                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.10/5.21           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 9.10/5.21                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) )
% 9.10/5.21                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                         => ( 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 ) ) ) ) ) )
% 9.10/5.21                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3034_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.21        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.10/5.21          = Y3 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21               => ( ( Y3
% 9.10/5.21                    = ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                       => A3 )
% 9.10/5.21                      & ( ( Xa2 != zero_zero_nat )
% 9.10/5.21                       => ( ( ( Xa2 = one_one_nat )
% 9.10/5.21                           => B3 )
% 9.10/5.21                          & ( Xa2 = one_one_nat ) ) ) ) )
% 9.10/5.21                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 9.10/5.21           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.10/5.21                 => ( ~ Y3
% 9.10/5.21                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.10/5.21             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 9.10/5.21                   => ( ( Y3
% 9.10/5.21                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                           => ( 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 ) ) ) ) ) )
% 9.10/5.21                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.10/5.21                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3035_vebt__member_Opelims_I2_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [A3: $o,B3: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.21               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
% 9.10/5.21                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.21                       => A3 )
% 9.10/5.21                      & ( ( Xa2 != zero_zero_nat )
% 9.10/5.21                       => ( ( ( Xa2 = one_one_nat )
% 9.10/5.21                           => B3 )
% 9.10/5.21                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.10/5.21           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.21                 => ( ( 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 ) )
% 9.10/5.21                   => ~ ( ( Xa2 != Mi2 )
% 9.10/5.21                       => ( ( Xa2 != Ma2 )
% 9.10/5.21                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.21                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.10/5.21                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.10/5.21                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                                     => ( 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 ) ) ) ) ) )
% 9.10/5.21                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3036_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.21               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 9.10/5.21           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.10/5.21                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 9.10/5.21             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.10/5.21                   => ( ( 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 ) )
% 9.10/5.21                     => ( ( Xa2 = Mi2 )
% 9.10/5.21                        | ( Xa2 = Ma2 ) ) ) )
% 9.10/5.21               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.21                      ( ( X3
% 9.10/5.21                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.10/5.21                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 9.10/5.21                       => ( ( Xa2 = Mi2 )
% 9.10/5.21                          | ( Xa2 = Ma2 )
% 9.10/5.21                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                             => ( 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 ) ) ) ) ) )
% 9.10/5.21                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.10/5.21                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.10/5.21                        ( ( X3
% 9.10/5.21                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.10/5.21                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 9.10/5.21                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                             => ( 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 ) ) ) ) ) )
% 9.10/5.21                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3037_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.21        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.10/5.21          = Y3 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.21               => ( ~ Y3
% 9.10/5.21                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.10/5.21           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.10/5.21                 => ( ~ Y3
% 9.10/5.21                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 9.10/5.21             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.10/5.21                   => ( ( Y3
% 9.10/5.21                        = ( ( Xa2 = Mi2 )
% 9.10/5.21                          | ( Xa2 = Ma2 ) ) )
% 9.10/5.21                     => ~ ( 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 ) ) ) )
% 9.10/5.21               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.21                      ( ( X3
% 9.10/5.21                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.10/5.21                     => ( ( Y3
% 9.10/5.21                          = ( ( Xa2 = Mi2 )
% 9.10/5.21                            | ( Xa2 = Ma2 )
% 9.10/5.21                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                               => ( 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 ) ) ) ) ) )
% 9.10/5.21                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.10/5.21                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 9.10/5.21                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.10/5.21                        ( ( X3
% 9.10/5.21                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.10/5.21                       => ( ( Y3
% 9.10/5.21                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                               => ( 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 ) ) ) ) ) )
% 9.10/5.21                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.10/5.21                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3038_set__bit__nonnegative__int__iff,axiom,
% 9.10/5.21      ! [N: nat,K: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 9.10/5.21        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3039_flip__bit__nonnegative__int__iff,axiom,
% 9.10/5.21      ! [N: nat,K: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 9.10/5.21        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3040_unset__bit__nonnegative__int__iff,axiom,
% 9.10/5.21      ! [N: nat,K: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 9.10/5.21        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3041_set__bit__negative__int__iff,axiom,
% 9.10/5.21      ! [N: nat,K: int] :
% 9.10/5.21        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 9.10/5.21        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3042_flip__bit__negative__int__iff,axiom,
% 9.10/5.21      ! [N: nat,K: int] :
% 9.10/5.21        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 9.10/5.21        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3043_unset__bit__negative__int__iff,axiom,
% 9.10/5.21      ! [N: nat,K: int] :
% 9.10/5.21        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 9.10/5.21        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3044_zle__add1__eq__le,axiom,
% 9.10/5.21      ! [W: int,Z: int] :
% 9.10/5.21        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 9.10/5.21        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3045_zle__diff1__eq,axiom,
% 9.10/5.21      ! [W: int,Z: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 9.10/5.21        = ( ord_less_int @ W @ Z ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3046_mod__neg__neg__trivial,axiom,
% 9.10/5.21      ! [K: int,L: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_int @ L @ K )
% 9.10/5.21         => ( ( modulo_modulo_int @ K @ L )
% 9.10/5.21            = K ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3047_mod__pos__pos__trivial,axiom,
% 9.10/5.21      ! [K: int,L: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.21       => ( ( ord_less_int @ K @ L )
% 9.10/5.21         => ( ( modulo_modulo_int @ K @ L )
% 9.10/5.21            = K ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3048_split__zmod,axiom,
% 9.10/5.21      ! [P: int > $o,N: int,K: int] :
% 9.10/5.21        ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 9.10/5.21        = ( ( ( K = zero_zero_int )
% 9.10/5.21           => ( P @ N ) )
% 9.10/5.21          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21           => ! [I4: int,J3: int] :
% 9.10/5.21                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.10/5.21                  & ( ord_less_int @ J3 @ K )
% 9.10/5.21                  & ( N
% 9.10/5.21                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.10/5.21               => ( P @ J3 ) ) )
% 9.10/5.21          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.21           => ! [I4: int,J3: int] :
% 9.10/5.21                ( ( ( ord_less_int @ K @ J3 )
% 9.10/5.21                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.10/5.21                  & ( N
% 9.10/5.21                    = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.10/5.21               => ( P @ J3 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3049_q__pos__lemma,axiom,
% 9.10/5.21      ! [B: int,Q5: int,R4: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 9.10/5.21       => ( ( ord_less_int @ R4 @ B )
% 9.10/5.21         => ( ( ord_less_int @ zero_zero_int @ B )
% 9.10/5.21           => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3050_int__mod__neg__eq,axiom,
% 9.10/5.21      ! [A: int,B2: int,Q2: int,R: int] :
% 9.10/5.21        ( ( A
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 9.10/5.21         => ( ( ord_less_int @ B2 @ R )
% 9.10/5.21           => ( ( modulo_modulo_int @ A @ B2 )
% 9.10/5.21              = R ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3051_int__mod__pos__eq,axiom,
% 9.10/5.21      ! [A: int,B2: int,Q2: int,R: int] :
% 9.10/5.21        ( ( A
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 9.10/5.21         => ( ( ord_less_int @ R @ B2 )
% 9.10/5.21           => ( ( modulo_modulo_int @ A @ B2 )
% 9.10/5.21              = R ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3052_zdiv__mono2__lemma,axiom,
% 9.10/5.21      ! [B2: int,Q2: int,R: int,B: int,Q5: int,R4: int] :
% 9.10/5.21        ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R )
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 9.10/5.21         => ( ( ord_less_int @ R4 @ B )
% 9.10/5.21           => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 9.10/5.21             => ( ( ord_less_int @ zero_zero_int @ B )
% 9.10/5.21               => ( ( ord_less_eq_int @ B @ B2 )
% 9.10/5.21                 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3053_zdiv__mono2__neg__lemma,axiom,
% 9.10/5.21      ! [B2: int,Q2: int,R: int,B: int,Q5: int,R4: int] :
% 9.10/5.21        ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R )
% 9.10/5.21          = ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) )
% 9.10/5.21       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ zero_zero_int )
% 9.10/5.21         => ( ( ord_less_int @ R @ B2 )
% 9.10/5.21           => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.10/5.21             => ( ( ord_less_int @ zero_zero_int @ B )
% 9.10/5.21               => ( ( ord_less_eq_int @ B @ B2 )
% 9.10/5.21                 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3054_unique__quotient__lemma,axiom,
% 9.10/5.21      ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.10/5.21         => ( ( ord_less_int @ R4 @ B2 )
% 9.10/5.21           => ( ( ord_less_int @ R @ B2 )
% 9.10/5.21             => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3055_unique__quotient__lemma__neg,axiom,
% 9.10/5.21      ! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q2 ) @ R ) )
% 9.10/5.21       => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 9.10/5.21         => ( ( ord_less_int @ B2 @ R )
% 9.10/5.21           => ( ( ord_less_int @ B2 @ R4 )
% 9.10/5.21             => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3056_zmod__eq__0__iff,axiom,
% 9.10/5.21      ! [M: int,D: int] :
% 9.10/5.21        ( ( ( modulo_modulo_int @ M @ D )
% 9.10/5.21          = zero_zero_int )
% 9.10/5.21        = ( ? [Q4: int] :
% 9.10/5.21              ( M
% 9.10/5.21              = ( times_times_int @ D @ Q4 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3057_zmod__eq__0D,axiom,
% 9.10/5.21      ! [M: int,D: int] :
% 9.10/5.21        ( ( ( modulo_modulo_int @ M @ D )
% 9.10/5.21          = zero_zero_int )
% 9.10/5.21       => ? [Q3: int] :
% 9.10/5.21            ( M
% 9.10/5.21            = ( times_times_int @ D @ Q3 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3058_pos__zmult__eq__1__iff,axiom,
% 9.10/5.21      ! [M: int,N: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ M )
% 9.10/5.21       => ( ( ( times_times_int @ M @ N )
% 9.10/5.21            = one_one_int )
% 9.10/5.21          = ( ( M = one_one_int )
% 9.10/5.21            & ( N = one_one_int ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3059_zmult__zless__mono2,axiom,
% 9.10/5.21      ! [I: int,J: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ I @ J )
% 9.10/5.21       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3060_neg__mod__conj,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ B2 @ zero_zero_int )
% 9.10/5.21       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B2 ) @ zero_zero_int )
% 9.10/5.21          & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A @ B2 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3061_pos__mod__conj,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.21       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B2 ) )
% 9.10/5.21          & ( ord_less_int @ ( modulo_modulo_int @ A @ B2 ) @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3062_zmod__trivial__iff,axiom,
% 9.10/5.21      ! [I: int,K: int] :
% 9.10/5.21        ( ( ( modulo_modulo_int @ I @ K )
% 9.10/5.21          = I )
% 9.10/5.21        = ( ( K = zero_zero_int )
% 9.10/5.21          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.10/5.21            & ( ord_less_int @ I @ K ) )
% 9.10/5.21          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.10/5.21            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3063_int__one__le__iff__zero__less,axiom,
% 9.10/5.21      ! [Z: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ one_one_int @ Z )
% 9.10/5.21        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3064_zmod__le__nonneg__dividend,axiom,
% 9.10/5.21      ! [M: int,K: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.10/5.21       => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3065_neg__mod__sign,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ L @ zero_zero_int )
% 9.10/5.21       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3066_Euclidean__Division_Opos__mod__sign,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.21       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3067_zero__one__enat__neq_I1_J,axiom,
% 9.10/5.21      zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3068_mod__pos__neg__trivial,axiom,
% 9.10/5.21      ! [K: int,L: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.21       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.10/5.21         => ( ( modulo_modulo_int @ K @ L )
% 9.10/5.21            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3069_zless__imp__add1__zle,axiom,
% 9.10/5.21      ! [W: int,Z: int] :
% 9.10/5.21        ( ( ord_less_int @ W @ Z )
% 9.10/5.21       => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3070_int__less__induct,axiom,
% 9.10/5.21      ! [I: int,K: int,P: int > $o] :
% 9.10/5.21        ( ( ord_less_int @ I @ K )
% 9.10/5.21       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 9.10/5.21         => ( ! [I3: int] :
% 9.10/5.21                ( ( ord_less_int @ I3 @ K )
% 9.10/5.21               => ( ( P @ I3 )
% 9.10/5.21                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 9.10/5.21           => ( P @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3071_mod__pos__geq,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.21       => ( ( ord_less_eq_int @ L @ K )
% 9.10/5.21         => ( ( modulo_modulo_int @ K @ L )
% 9.10/5.21            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3072_odd__less__0__iff,axiom,
% 9.10/5.21      ! [Z: int] :
% 9.10/5.21        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 9.10/5.21        = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3073_zless__add1__eq,axiom,
% 9.10/5.21      ! [W: int,Z: int] :
% 9.10/5.21        ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 9.10/5.21        = ( ( ord_less_int @ W @ Z )
% 9.10/5.21          | ( W = Z ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3074_le__imp__0__less,axiom,
% 9.10/5.21      ! [Z: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.21       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3075_int__le__induct,axiom,
% 9.10/5.21      ! [I: int,K: int,P: int > $o] :
% 9.10/5.21        ( ( ord_less_eq_int @ I @ K )
% 9.10/5.21       => ( ( P @ K )
% 9.10/5.21         => ( ! [I3: int] :
% 9.10/5.21                ( ( ord_less_eq_int @ I3 @ K )
% 9.10/5.21               => ( ( P @ I3 )
% 9.10/5.21                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 9.10/5.21           => ( P @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3076_int__gr__induct,axiom,
% 9.10/5.21      ! [K: int,I: int,P: int > $o] :
% 9.10/5.21        ( ( ord_less_int @ K @ I )
% 9.10/5.21       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 9.10/5.21         => ( ! [I3: int] :
% 9.10/5.21                ( ( ord_less_int @ K @ I3 )
% 9.10/5.21               => ( ( P @ I3 )
% 9.10/5.21                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 9.10/5.21           => ( P @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3077_int__ge__induct,axiom,
% 9.10/5.21      ! [K: int,I: int,P: int > $o] :
% 9.10/5.21        ( ( ord_less_eq_int @ K @ I )
% 9.10/5.21       => ( ( P @ K )
% 9.10/5.21         => ( ! [I3: int] :
% 9.10/5.21                ( ( ord_less_eq_int @ K @ I3 )
% 9.10/5.21               => ( ( P @ I3 )
% 9.10/5.21                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 9.10/5.21           => ( P @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3078_add1__zle__eq,axiom,
% 9.10/5.21      ! [W: int,Z: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 9.10/5.21        = ( ord_less_int @ W @ Z ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3079_int__induct,axiom,
% 9.10/5.21      ! [P: int > $o,K: int,I: int] :
% 9.10/5.21        ( ( P @ K )
% 9.10/5.21       => ( ! [I3: int] :
% 9.10/5.21              ( ( ord_less_eq_int @ K @ I3 )
% 9.10/5.21             => ( ( P @ I3 )
% 9.10/5.21               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 9.10/5.21         => ( ! [I3: int] :
% 9.10/5.21                ( ( ord_less_eq_int @ I3 @ K )
% 9.10/5.21               => ( ( P @ I3 )
% 9.10/5.21                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 9.10/5.21           => ( P @ I ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3080_Euclidean__Division_Opos__mod__bound,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.21       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3081_neg__mod__bound,axiom,
% 9.10/5.21      ! [L: int,K: int] :
% 9.10/5.21        ( ( ord_less_int @ L @ zero_zero_int )
% 9.10/5.21       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3082_less__int__code_I1_J,axiom,
% 9.10/5.21      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3083_unset__bit__less__eq,axiom,
% 9.10/5.21      ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3084_set__bit__greater__eq,axiom,
% 9.10/5.21      ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3085_verit__la__generic,axiom,
% 9.10/5.21      ! [A: int,X3: int] :
% 9.10/5.21        ( ( ord_less_eq_int @ A @ X3 )
% 9.10/5.21        | ( A = X3 )
% 9.10/5.21        | ( ord_less_eq_int @ X3 @ A ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3086_less__eq__int__code_I1_J,axiom,
% 9.10/5.21      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3087_times__int__code_I2_J,axiom,
% 9.10/5.21      ! [L: int] :
% 9.10/5.21        ( ( times_times_int @ zero_zero_int @ L )
% 9.10/5.21        = zero_zero_int ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3088_times__int__code_I1_J,axiom,
% 9.10/5.21      ! [K: int] :
% 9.10/5.21        ( ( times_times_int @ K @ zero_zero_int )
% 9.10/5.21        = zero_zero_int ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3089_imult__is__0,axiom,
% 9.10/5.21      ! [M: extended_enat,N: extended_enat] :
% 9.10/5.21        ( ( ( times_7803423173614009249d_enat @ M @ N )
% 9.10/5.21          = zero_z5237406670263579293d_enat )
% 9.10/5.21        = ( ( M = zero_z5237406670263579293d_enat )
% 9.10/5.21          | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3090_int__distrib_I1_J,axiom,
% 9.10/5.21      ! [Z1: int,Z22: int,W: int] :
% 9.10/5.21        ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 9.10/5.21        = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3091_int__distrib_I2_J,axiom,
% 9.10/5.21      ! [W: int,Z1: int,Z22: int] :
% 9.10/5.21        ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 9.10/5.21        = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3092_psubsetD,axiom,
% 9.10/5.21      ! [A4: set_nat,B5: set_nat,C: nat] :
% 9.10/5.21        ( ( ord_less_set_nat @ A4 @ B5 )
% 9.10/5.21       => ( ( member_nat @ C @ A4 )
% 9.10/5.21         => ( member_nat @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3093_psubsetD,axiom,
% 9.10/5.21      ! [A4: set_real,B5: set_real,C: real] :
% 9.10/5.21        ( ( ord_less_set_real @ A4 @ B5 )
% 9.10/5.21       => ( ( member_real @ C @ A4 )
% 9.10/5.21         => ( member_real @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3094_psubsetD,axiom,
% 9.10/5.21      ! [A4: set_int,B5: set_int,C: int] :
% 9.10/5.21        ( ( ord_less_set_int @ A4 @ B5 )
% 9.10/5.21       => ( ( member_int @ C @ A4 )
% 9.10/5.21         => ( member_int @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3095_psubsetD,axiom,
% 9.10/5.21      ! [A4: set_complex,B5: set_complex,C: complex] :
% 9.10/5.21        ( ( ord_less_set_complex @ A4 @ B5 )
% 9.10/5.21       => ( ( member_complex @ C @ A4 )
% 9.10/5.21         => ( member_complex @ C @ B5 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3096_less__set__def,axiom,
% 9.10/5.21      ( ord_less_set_nat
% 9.10/5.21      = ( ^ [A6: set_nat,B6: set_nat] :
% 9.10/5.21            ( ord_less_nat_o
% 9.10/5.21            @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.10/5.21            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3097_less__set__def,axiom,
% 9.10/5.21      ( ord_less_set_real
% 9.10/5.21      = ( ^ [A6: set_real,B6: set_real] :
% 9.10/5.21            ( ord_less_real_o
% 9.10/5.21            @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.10/5.21            @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3098_less__set__def,axiom,
% 9.10/5.21      ( ord_less_set_int
% 9.10/5.21      = ( ^ [A6: set_int,B6: set_int] :
% 9.10/5.21            ( ord_less_int_o
% 9.10/5.21            @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.10/5.21            @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3099_less__set__def,axiom,
% 9.10/5.21      ( ord_less_set_complex
% 9.10/5.21      = ( ^ [A6: set_complex,B6: set_complex] :
% 9.10/5.21            ( ord_less_complex_o
% 9.10/5.21            @ ^ [X: complex] : ( member_complex @ X @ A6 )
% 9.10/5.21            @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3100_int__distrib_I4_J,axiom,
% 9.10/5.21      ! [W: int,Z1: int,Z22: int] :
% 9.10/5.21        ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 9.10/5.21        = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3101_int__distrib_I3_J,axiom,
% 9.10/5.21      ! [Z1: int,Z22: int,W: int] :
% 9.10/5.21        ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 9.10/5.21        = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3102_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 9.10/5.21      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.21        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.10/5.21       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.21         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.10/5.21                ( ( X3
% 9.10/5.21                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.10/5.21               => ( ( 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 ) )
% 9.10/5.21                 => ~ ( ( Xa2 = Mi2 )
% 9.10/5.21                      | ( Xa2 = Ma2 ) ) ) )
% 9.10/5.21           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.21                  ( ( X3
% 9.10/5.21                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.10/5.21                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 9.10/5.21                   => ~ ( ( Xa2 = Mi2 )
% 9.10/5.21                        | ( Xa2 = Ma2 )
% 9.10/5.21                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                           => ( 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 ) ) ) ) ) )
% 9.10/5.21                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.10/5.21             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.10/5.21                    ( ( X3
% 9.10/5.21                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 9.10/5.21                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 9.10/5.21                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.21                           => ( 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 ) ) ) ) ) )
% 9.10/5.21                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3103_atLeastatMost__empty,axiom,
% 9.10/5.21      ! [B2: rat,A: rat] :
% 9.10/5.21        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.21       => ( ( set_or633870826150836451st_rat @ A @ B2 )
% 9.10/5.21          = bot_bot_set_rat ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3104_atLeastatMost__empty,axiom,
% 9.10/5.21      ! [B2: num,A: num] :
% 9.10/5.21        ( ( ord_less_num @ B2 @ A )
% 9.10/5.21       => ( ( set_or7049704709247886629st_num @ A @ B2 )
% 9.10/5.21          = bot_bot_set_num ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3105_atLeastatMost__empty,axiom,
% 9.10/5.21      ! [B2: nat,A: nat] :
% 9.10/5.21        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.21       => ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 9.10/5.21          = bot_bot_set_nat ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3106_atLeastatMost__empty,axiom,
% 9.10/5.21      ! [B2: int,A: int] :
% 9.10/5.21        ( ( ord_less_int @ B2 @ A )
% 9.10/5.21       => ( ( set_or1266510415728281911st_int @ A @ B2 )
% 9.10/5.21          = bot_bot_set_int ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3107_atLeastatMost__empty,axiom,
% 9.10/5.21      ! [B2: real,A: real] :
% 9.10/5.21        ( ( ord_less_real @ B2 @ A )
% 9.10/5.21       => ( ( set_or1222579329274155063t_real @ A @ B2 )
% 9.10/5.21          = bot_bot_set_real ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3108_atLeastatMost__subset__iff,axiom,
% 9.10/5.21      ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
% 9.10/5.21        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.21          | ( ( ord_less_eq_set_int @ C @ A )
% 9.10/5.21            & ( ord_less_eq_set_int @ B2 @ D ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3109_atLeastatMost__subset__iff,axiom,
% 9.10/5.21      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.10/5.21        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.21          | ( ( ord_less_eq_rat @ C @ A )
% 9.10/5.21            & ( ord_less_eq_rat @ B2 @ D ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3110_atLeastatMost__subset__iff,axiom,
% 9.10/5.21      ! [A: num,B2: num,C: num,D: num] :
% 9.10/5.21        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_num @ A @ B2 )
% 9.10/5.21          | ( ( ord_less_eq_num @ C @ A )
% 9.10/5.21            & ( ord_less_eq_num @ B2 @ D ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3111_atLeastatMost__subset__iff,axiom,
% 9.10/5.21      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.10/5.21        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.21          | ( ( ord_less_eq_nat @ C @ A )
% 9.10/5.21            & ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3112_atLeastatMost__subset__iff,axiom,
% 9.10/5.21      ! [A: int,B2: int,C: int,D: int] :
% 9.10/5.21        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_int @ A @ B2 )
% 9.10/5.21          | ( ( ord_less_eq_int @ C @ A )
% 9.10/5.21            & ( ord_less_eq_int @ B2 @ D ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3113_atLeastatMost__subset__iff,axiom,
% 9.10/5.21      ! [A: real,B2: real,C: real,D: real] :
% 9.10/5.21        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_real @ A @ B2 )
% 9.10/5.21          | ( ( ord_less_eq_real @ C @ A )
% 9.10/5.21            & ( ord_less_eq_real @ B2 @ D ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3114_atLeastatMost__empty__iff2,axiom,
% 9.10/5.21      ! [A: set_int,B2: set_int] :
% 9.10/5.21        ( ( bot_bot_set_set_int
% 9.10/5.21          = ( set_or370866239135849197et_int @ A @ B2 ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3115_atLeastatMost__empty__iff2,axiom,
% 9.10/5.21      ! [A: rat,B2: rat] :
% 9.10/5.21        ( ( bot_bot_set_rat
% 9.10/5.21          = ( set_or633870826150836451st_rat @ A @ B2 ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3116_atLeastatMost__empty__iff2,axiom,
% 9.10/5.21      ! [A: num,B2: num] :
% 9.10/5.21        ( ( bot_bot_set_num
% 9.10/5.21          = ( set_or7049704709247886629st_num @ A @ B2 ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3117_atLeastatMost__empty__iff2,axiom,
% 9.10/5.21      ! [A: nat,B2: nat] :
% 9.10/5.21        ( ( bot_bot_set_nat
% 9.10/5.21          = ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3118_atLeastatMost__empty__iff2,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( bot_bot_set_int
% 9.10/5.21          = ( set_or1266510415728281911st_int @ A @ B2 ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3119_atLeastatMost__empty__iff2,axiom,
% 9.10/5.21      ! [A: real,B2: real] :
% 9.10/5.21        ( ( bot_bot_set_real
% 9.10/5.21          = ( set_or1222579329274155063t_real @ A @ B2 ) )
% 9.10/5.21        = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3120_atLeastatMost__empty__iff,axiom,
% 9.10/5.21      ! [A: set_int,B2: set_int] :
% 9.10/5.21        ( ( ( set_or370866239135849197et_int @ A @ B2 )
% 9.10/5.21          = bot_bot_set_set_int )
% 9.10/5.21        = ( ~ ( ord_less_eq_set_int @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3121_atLeastatMost__empty__iff,axiom,
% 9.10/5.21      ! [A: rat,B2: rat] :
% 9.10/5.21        ( ( ( set_or633870826150836451st_rat @ A @ B2 )
% 9.10/5.21          = bot_bot_set_rat )
% 9.10/5.21        = ( ~ ( ord_less_eq_rat @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3122_atLeastatMost__empty__iff,axiom,
% 9.10/5.21      ! [A: num,B2: num] :
% 9.10/5.21        ( ( ( set_or7049704709247886629st_num @ A @ B2 )
% 9.10/5.21          = bot_bot_set_num )
% 9.10/5.21        = ( ~ ( ord_less_eq_num @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3123_atLeastatMost__empty__iff,axiom,
% 9.10/5.21      ! [A: nat,B2: nat] :
% 9.10/5.21        ( ( ( set_or1269000886237332187st_nat @ A @ B2 )
% 9.10/5.21          = bot_bot_set_nat )
% 9.10/5.21        = ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3124_atLeastatMost__empty__iff,axiom,
% 9.10/5.21      ! [A: int,B2: int] :
% 9.10/5.21        ( ( ( set_or1266510415728281911st_int @ A @ B2 )
% 9.10/5.21          = bot_bot_set_int )
% 9.10/5.21        = ( ~ ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3125_atLeastatMost__empty__iff,axiom,
% 9.10/5.21      ! [A: real,B2: real] :
% 9.10/5.21        ( ( ( set_or1222579329274155063t_real @ A @ B2 )
% 9.10/5.21          = bot_bot_set_real )
% 9.10/5.21        = ( ~ ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3126_decr__mult__lemma,axiom,
% 9.10/5.21      ! [D: int,P: int > $o,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.21       => ( ! [X5: int] :
% 9.10/5.21              ( ( P @ X5 )
% 9.10/5.21             => ( P @ ( minus_minus_int @ X5 @ D ) ) )
% 9.10/5.21         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.21           => ! [X6: int] :
% 9.10/5.21                ( ( P @ X6 )
% 9.10/5.21               => ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3127_incr__mult__lemma,axiom,
% 9.10/5.21      ! [D: int,P: int > $o,K: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.21       => ( ! [X5: int] :
% 9.10/5.21              ( ( P @ X5 )
% 9.10/5.21             => ( P @ ( plus_plus_int @ X5 @ D ) ) )
% 9.10/5.21         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.21           => ! [X6: int] :
% 9.10/5.21                ( ( P @ X6 )
% 9.10/5.21               => ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3128_atLeastAtMost__iff,axiom,
% 9.10/5.21      ! [I: set_int,L: set_int,U: set_int] :
% 9.10/5.21        ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 9.10/5.21        = ( ( ord_less_eq_set_int @ L @ I )
% 9.10/5.21          & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3129_atLeastAtMost__iff,axiom,
% 9.10/5.21      ! [I: rat,L: rat,U: rat] :
% 9.10/5.21        ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 9.10/5.21        = ( ( ord_less_eq_rat @ L @ I )
% 9.10/5.21          & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3130_atLeastAtMost__iff,axiom,
% 9.10/5.21      ! [I: num,L: num,U: num] :
% 9.10/5.21        ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 9.10/5.21        = ( ( ord_less_eq_num @ L @ I )
% 9.10/5.21          & ( ord_less_eq_num @ I @ U ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3131_atLeastAtMost__iff,axiom,
% 9.10/5.21      ! [I: nat,L: nat,U: nat] :
% 9.10/5.21        ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.10/5.21        = ( ( ord_less_eq_nat @ L @ I )
% 9.10/5.21          & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3132_atLeastAtMost__iff,axiom,
% 9.10/5.21      ! [I: int,L: int,U: int] :
% 9.10/5.21        ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.10/5.21        = ( ( ord_less_eq_int @ L @ I )
% 9.10/5.21          & ( ord_less_eq_int @ I @ U ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3133_atLeastAtMost__iff,axiom,
% 9.10/5.21      ! [I: real,L: real,U: real] :
% 9.10/5.21        ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 9.10/5.21        = ( ( ord_less_eq_real @ L @ I )
% 9.10/5.21          & ( ord_less_eq_real @ I @ U ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3134_Icc__eq__Icc,axiom,
% 9.10/5.21      ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 9.10/5.21        ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 9.10/5.21          = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 9.10/5.21        = ( ( ( L = L3 )
% 9.10/5.21            & ( H2 = H3 ) )
% 9.10/5.21          | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.10/5.21            & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3135_Icc__eq__Icc,axiom,
% 9.10/5.21      ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 9.10/5.21        ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 9.10/5.21          = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 9.10/5.21        = ( ( ( L = L3 )
% 9.10/5.21            & ( H2 = H3 ) )
% 9.10/5.21          | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.10/5.21            & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3136_Icc__eq__Icc,axiom,
% 9.10/5.21      ! [L: num,H2: num,L3: num,H3: num] :
% 9.10/5.21        ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 9.10/5.21          = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 9.10/5.21        = ( ( ( L = L3 )
% 9.10/5.21            & ( H2 = H3 ) )
% 9.10/5.21          | ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.10/5.21            & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3137_Icc__eq__Icc,axiom,
% 9.10/5.21      ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 9.10/5.21        ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 9.10/5.21          = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 9.10/5.21        = ( ( ( L = L3 )
% 9.10/5.21            & ( H2 = H3 ) )
% 9.10/5.21          | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.10/5.21            & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3138_Icc__eq__Icc,axiom,
% 9.10/5.21      ! [L: int,H2: int,L3: int,H3: int] :
% 9.10/5.21        ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 9.10/5.21          = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 9.10/5.21        = ( ( ( L = L3 )
% 9.10/5.21            & ( H2 = H3 ) )
% 9.10/5.21          | ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.10/5.21            & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3139_Icc__eq__Icc,axiom,
% 9.10/5.21      ! [L: real,H2: real,L3: real,H3: real] :
% 9.10/5.21        ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 9.10/5.21          = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 9.10/5.21        = ( ( ( L = L3 )
% 9.10/5.21            & ( H2 = H3 ) )
% 9.10/5.21          | ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.10/5.21            & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3140_minf_I7_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.21       => ~ ( ord_less_real @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3141_minf_I7_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.21       => ~ ( ord_less_rat @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3142_minf_I7_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.21       => ~ ( ord_less_num @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3143_minf_I7_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.21       => ~ ( ord_less_nat @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3144_minf_I7_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.21       => ~ ( ord_less_int @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3145_minf_I5_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.21       => ( ord_less_real @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3146_minf_I5_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.21       => ( ord_less_rat @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3147_minf_I5_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.21       => ( ord_less_num @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3148_minf_I5_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.21       => ( ord_less_nat @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3149_minf_I5_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.21       => ( ord_less_int @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3150_minf_I4_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3151_minf_I4_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3152_minf_I4_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3153_minf_I4_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3154_minf_I4_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3155_minf_I3_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3156_minf_I3_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3157_minf_I3_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3158_minf_I3_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3159_minf_I3_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3160_minf_I2_J,axiom,
% 9.10/5.21      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.10/5.21        ( ? [Z3: real] :
% 9.10/5.21          ! [X5: real] :
% 9.10/5.21            ( ( ord_less_real @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: real] :
% 9.10/5.21            ! [X5: real] :
% 9.10/5.21              ( ( ord_less_real @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: real] :
% 9.10/5.21            ! [X6: real] :
% 9.10/5.21              ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3161_minf_I2_J,axiom,
% 9.10/5.21      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.10/5.21        ( ? [Z3: rat] :
% 9.10/5.21          ! [X5: rat] :
% 9.10/5.21            ( ( ord_less_rat @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: rat] :
% 9.10/5.21            ! [X5: rat] :
% 9.10/5.21              ( ( ord_less_rat @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: rat] :
% 9.10/5.21            ! [X6: rat] :
% 9.10/5.21              ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3162_minf_I2_J,axiom,
% 9.10/5.21      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.10/5.21        ( ? [Z3: num] :
% 9.10/5.21          ! [X5: num] :
% 9.10/5.21            ( ( ord_less_num @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: num] :
% 9.10/5.21            ! [X5: num] :
% 9.10/5.21              ( ( ord_less_num @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: num] :
% 9.10/5.21            ! [X6: num] :
% 9.10/5.21              ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3163_minf_I2_J,axiom,
% 9.10/5.21      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.10/5.21        ( ? [Z3: nat] :
% 9.10/5.21          ! [X5: nat] :
% 9.10/5.21            ( ( ord_less_nat @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: nat] :
% 9.10/5.21            ! [X5: nat] :
% 9.10/5.21              ( ( ord_less_nat @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: nat] :
% 9.10/5.21            ! [X6: nat] :
% 9.10/5.21              ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3164_minf_I2_J,axiom,
% 9.10/5.21      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.10/5.21        ( ? [Z3: int] :
% 9.10/5.21          ! [X5: int] :
% 9.10/5.21            ( ( ord_less_int @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: int] :
% 9.10/5.21            ! [X5: int] :
% 9.10/5.21              ( ( ord_less_int @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: int] :
% 9.10/5.21            ! [X6: int] :
% 9.10/5.21              ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3165_minf_I1_J,axiom,
% 9.10/5.21      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.10/5.21        ( ? [Z3: real] :
% 9.10/5.21          ! [X5: real] :
% 9.10/5.21            ( ( ord_less_real @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: real] :
% 9.10/5.21            ! [X5: real] :
% 9.10/5.21              ( ( ord_less_real @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: real] :
% 9.10/5.21            ! [X6: real] :
% 9.10/5.21              ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3166_minf_I1_J,axiom,
% 9.10/5.21      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.10/5.21        ( ? [Z3: rat] :
% 9.10/5.21          ! [X5: rat] :
% 9.10/5.21            ( ( ord_less_rat @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: rat] :
% 9.10/5.21            ! [X5: rat] :
% 9.10/5.21              ( ( ord_less_rat @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: rat] :
% 9.10/5.21            ! [X6: rat] :
% 9.10/5.21              ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3167_minf_I1_J,axiom,
% 9.10/5.21      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.10/5.21        ( ? [Z3: num] :
% 9.10/5.21          ! [X5: num] :
% 9.10/5.21            ( ( ord_less_num @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: num] :
% 9.10/5.21            ! [X5: num] :
% 9.10/5.21              ( ( ord_less_num @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: num] :
% 9.10/5.21            ! [X6: num] :
% 9.10/5.21              ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3168_minf_I1_J,axiom,
% 9.10/5.21      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.10/5.21        ( ? [Z3: nat] :
% 9.10/5.21          ! [X5: nat] :
% 9.10/5.21            ( ( ord_less_nat @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: nat] :
% 9.10/5.21            ! [X5: nat] :
% 9.10/5.21              ( ( ord_less_nat @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: nat] :
% 9.10/5.21            ! [X6: nat] :
% 9.10/5.21              ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3169_minf_I1_J,axiom,
% 9.10/5.21      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.10/5.21        ( ? [Z3: int] :
% 9.10/5.21          ! [X5: int] :
% 9.10/5.21            ( ( ord_less_int @ X5 @ Z3 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: int] :
% 9.10/5.21            ! [X5: int] :
% 9.10/5.21              ( ( ord_less_int @ X5 @ Z3 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: int] :
% 9.10/5.21            ! [X6: int] :
% 9.10/5.21              ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3170_pinf_I7_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.21       => ( ord_less_real @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3171_pinf_I7_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.21       => ( ord_less_rat @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3172_pinf_I7_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.21       => ( ord_less_num @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3173_pinf_I7_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.21       => ( ord_less_nat @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3174_pinf_I7_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.21       => ( ord_less_int @ T @ X6 ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3175_pinf_I5_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.21       => ~ ( ord_less_real @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3176_pinf_I5_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.21       => ~ ( ord_less_rat @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3177_pinf_I5_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.21       => ~ ( ord_less_num @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3178_pinf_I5_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.21       => ~ ( ord_less_nat @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3179_pinf_I5_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.21       => ~ ( ord_less_int @ X6 @ T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3180_pinf_I4_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3181_pinf_I4_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3182_pinf_I4_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3183_pinf_I4_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3184_pinf_I4_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3185_pinf_I3_J,axiom,
% 9.10/5.21      ! [T: real] :
% 9.10/5.21      ? [Z4: real] :
% 9.10/5.21      ! [X6: real] :
% 9.10/5.21        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3186_pinf_I3_J,axiom,
% 9.10/5.21      ! [T: rat] :
% 9.10/5.21      ? [Z4: rat] :
% 9.10/5.21      ! [X6: rat] :
% 9.10/5.21        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3187_pinf_I3_J,axiom,
% 9.10/5.21      ! [T: num] :
% 9.10/5.21      ? [Z4: num] :
% 9.10/5.21      ! [X6: num] :
% 9.10/5.21        ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3188_pinf_I3_J,axiom,
% 9.10/5.21      ! [T: nat] :
% 9.10/5.21      ? [Z4: nat] :
% 9.10/5.21      ! [X6: nat] :
% 9.10/5.21        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3189_pinf_I3_J,axiom,
% 9.10/5.21      ! [T: int] :
% 9.10/5.21      ? [Z4: int] :
% 9.10/5.21      ! [X6: int] :
% 9.10/5.21        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.21       => ( X6 != T ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3190_pinf_I2_J,axiom,
% 9.10/5.21      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.10/5.21        ( ? [Z3: real] :
% 9.10/5.21          ! [X5: real] :
% 9.10/5.21            ( ( ord_less_real @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: real] :
% 9.10/5.21            ! [X5: real] :
% 9.10/5.21              ( ( ord_less_real @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: real] :
% 9.10/5.21            ! [X6: real] :
% 9.10/5.21              ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3191_pinf_I2_J,axiom,
% 9.10/5.21      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.10/5.21        ( ? [Z3: rat] :
% 9.10/5.21          ! [X5: rat] :
% 9.10/5.21            ( ( ord_less_rat @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: rat] :
% 9.10/5.21            ! [X5: rat] :
% 9.10/5.21              ( ( ord_less_rat @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: rat] :
% 9.10/5.21            ! [X6: rat] :
% 9.10/5.21              ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3192_pinf_I2_J,axiom,
% 9.10/5.21      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.10/5.21        ( ? [Z3: num] :
% 9.10/5.21          ! [X5: num] :
% 9.10/5.21            ( ( ord_less_num @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: num] :
% 9.10/5.21            ! [X5: num] :
% 9.10/5.21              ( ( ord_less_num @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: num] :
% 9.10/5.21            ! [X6: num] :
% 9.10/5.21              ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3193_pinf_I2_J,axiom,
% 9.10/5.21      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.10/5.21        ( ? [Z3: nat] :
% 9.10/5.21          ! [X5: nat] :
% 9.10/5.21            ( ( ord_less_nat @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: nat] :
% 9.10/5.21            ! [X5: nat] :
% 9.10/5.21              ( ( ord_less_nat @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: nat] :
% 9.10/5.21            ! [X6: nat] :
% 9.10/5.21              ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3194_pinf_I2_J,axiom,
% 9.10/5.21      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.10/5.21        ( ? [Z3: int] :
% 9.10/5.21          ! [X5: int] :
% 9.10/5.21            ( ( ord_less_int @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: int] :
% 9.10/5.21            ! [X5: int] :
% 9.10/5.21              ( ( ord_less_int @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: int] :
% 9.10/5.21            ! [X6: int] :
% 9.10/5.21              ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  | ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  | ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3195_pinf_I1_J,axiom,
% 9.10/5.21      ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 9.10/5.21        ( ? [Z3: real] :
% 9.10/5.21          ! [X5: real] :
% 9.10/5.21            ( ( ord_less_real @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: real] :
% 9.10/5.21            ! [X5: real] :
% 9.10/5.21              ( ( ord_less_real @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: real] :
% 9.10/5.21            ! [X6: real] :
% 9.10/5.21              ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3196_pinf_I1_J,axiom,
% 9.10/5.21      ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 9.10/5.21        ( ? [Z3: rat] :
% 9.10/5.21          ! [X5: rat] :
% 9.10/5.21            ( ( ord_less_rat @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: rat] :
% 9.10/5.21            ! [X5: rat] :
% 9.10/5.21              ( ( ord_less_rat @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: rat] :
% 9.10/5.21            ! [X6: rat] :
% 9.10/5.21              ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3197_pinf_I1_J,axiom,
% 9.10/5.21      ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 9.10/5.21        ( ? [Z3: num] :
% 9.10/5.21          ! [X5: num] :
% 9.10/5.21            ( ( ord_less_num @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: num] :
% 9.10/5.21            ! [X5: num] :
% 9.10/5.21              ( ( ord_less_num @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: num] :
% 9.10/5.21            ! [X6: num] :
% 9.10/5.21              ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3198_pinf_I1_J,axiom,
% 9.10/5.21      ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 9.10/5.21        ( ? [Z3: nat] :
% 9.10/5.21          ! [X5: nat] :
% 9.10/5.21            ( ( ord_less_nat @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: nat] :
% 9.10/5.21            ! [X5: nat] :
% 9.10/5.21              ( ( ord_less_nat @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: nat] :
% 9.10/5.21            ! [X6: nat] :
% 9.10/5.21              ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3199_pinf_I1_J,axiom,
% 9.10/5.21      ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 9.10/5.21        ( ? [Z3: int] :
% 9.10/5.21          ! [X5: int] :
% 9.10/5.21            ( ( ord_less_int @ Z3 @ X5 )
% 9.10/5.21           => ( ( P @ X5 )
% 9.10/5.21              = ( P6 @ X5 ) ) )
% 9.10/5.21       => ( ? [Z3: int] :
% 9.10/5.21            ! [X5: int] :
% 9.10/5.21              ( ( ord_less_int @ Z3 @ X5 )
% 9.10/5.21             => ( ( Q @ X5 )
% 9.10/5.21                = ( Q6 @ X5 ) ) )
% 9.10/5.21         => ? [Z4: int] :
% 9.10/5.21            ! [X6: int] :
% 9.10/5.21              ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.21             => ( ( ( P @ X6 )
% 9.10/5.21                  & ( Q @ X6 ) )
% 9.10/5.21                = ( ( P6 @ X6 )
% 9.10/5.21                  & ( Q6 @ X6 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3200_bounded__Max__nat,axiom,
% 9.10/5.21      ! [P: nat > $o,X3: nat,M7: nat] :
% 9.10/5.21        ( ( P @ X3 )
% 9.10/5.21       => ( ! [X5: nat] :
% 9.10/5.21              ( ( P @ X5 )
% 9.10/5.21             => ( ord_less_eq_nat @ X5 @ M7 ) )
% 9.10/5.21         => ~ ! [M4: nat] :
% 9.10/5.21                ( ( P @ M4 )
% 9.10/5.21               => ~ ! [X6: nat] :
% 9.10/5.21                      ( ( P @ X6 )
% 9.10/5.21                     => ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3201_periodic__finite__ex,axiom,
% 9.10/5.21      ! [D: int,P: int > $o] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.21       => ( ! [X5: int,K3: int] :
% 9.10/5.21              ( ( P @ X5 )
% 9.10/5.21              = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 9.10/5.21         => ( ( ? [X7: int] : ( P @ X7 ) )
% 9.10/5.21            = ( ? [X: int] :
% 9.10/5.21                  ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 9.10/5.21                  & ( P @ X ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3202_aset_I7_J,axiom,
% 9.10/5.21      ! [D4: int,A4: set_int,T: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ! [X6: int] :
% 9.10/5.21            ( ! [Xa3: int] :
% 9.10/5.21                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21               => ! [Xb3: int] :
% 9.10/5.21                    ( ( member_int @ Xb3 @ A4 )
% 9.10/5.21                   => ( X6
% 9.10/5.21                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21           => ( ( ord_less_int @ T @ X6 )
% 9.10/5.21             => ( ord_less_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3203_aset_I5_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,A4: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ T @ A4 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ A4 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( ord_less_int @ X6 @ T )
% 9.10/5.21               => ( ord_less_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3204_aset_I4_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,A4: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ T @ A4 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ A4 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( X6 != T )
% 9.10/5.21               => ( ( plus_plus_int @ X6 @ D4 )
% 9.10/5.21                 != T ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3205_aset_I3_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,A4: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ A4 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( X6 = T )
% 9.10/5.21               => ( ( plus_plus_int @ X6 @ D4 )
% 9.10/5.21                  = T ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3206_bset_I7_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ T @ B5 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ B5 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( ord_less_int @ T @ X6 )
% 9.10/5.21               => ( ord_less_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3207_bset_I5_J,axiom,
% 9.10/5.21      ! [D4: int,B5: set_int,T: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ! [X6: int] :
% 9.10/5.21            ( ! [Xa3: int] :
% 9.10/5.21                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21               => ! [Xb3: int] :
% 9.10/5.21                    ( ( member_int @ Xb3 @ B5 )
% 9.10/5.21                   => ( X6
% 9.10/5.21                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21           => ( ( ord_less_int @ X6 @ T )
% 9.10/5.21             => ( ord_less_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3208_bset_I4_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ T @ B5 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ B5 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( X6 != T )
% 9.10/5.21               => ( ( minus_minus_int @ X6 @ D4 )
% 9.10/5.21                 != T ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3209_bset_I3_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,B5: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ B5 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( X6 = T )
% 9.10/5.21               => ( ( minus_minus_int @ X6 @ D4 )
% 9.10/5.21                  = T ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3210_aset_I8_J,axiom,
% 9.10/5.21      ! [D4: int,A4: set_int,T: int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ! [X6: int] :
% 9.10/5.21            ( ! [Xa3: int] :
% 9.10/5.21                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21               => ! [Xb3: int] :
% 9.10/5.21                    ( ( member_int @ Xb3 @ A4 )
% 9.10/5.21                   => ( X6
% 9.10/5.21                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21           => ( ( ord_less_eq_int @ T @ X6 )
% 9.10/5.21             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X6 @ D4 ) ) ) ) ) ).
% 9.10/5.21  
% 9.10/5.21  thf(fact_3211_aset_I6_J,axiom,
% 9.10/5.21      ! [D4: int,T: int,A4: set_int] :
% 9.10/5.21        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.21       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A4 )
% 9.10/5.21         => ! [X6: int] :
% 9.10/5.21              ( ! [Xa3: int] :
% 9.10/5.21                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.21                 => ! [Xb3: int] :
% 9.10/5.21                      ( ( member_int @ Xb3 @ A4 )
% 9.10/5.21                     => ( X6
% 9.10/5.21                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.21             => ( ( ord_less_eq_int @ X6 @ T )
% 9.10/5.22               => ( ord_less_eq_int @ ( plus_plus_int @ X6 @ D4 ) @ T ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3212_bset_I8_J,axiom,
% 9.10/5.22      ! [D4: int,T: int,B5: set_int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.22       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 9.10/5.22         => ! [X6: int] :
% 9.10/5.22              ( ! [Xa3: int] :
% 9.10/5.22                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                 => ! [Xb3: int] :
% 9.10/5.22                      ( ( member_int @ Xb3 @ B5 )
% 9.10/5.22                     => ( X6
% 9.10/5.22                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.22             => ( ( ord_less_eq_int @ T @ X6 )
% 9.10/5.22               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X6 @ D4 ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3213_bset_I6_J,axiom,
% 9.10/5.22      ! [D4: int,B5: set_int,T: int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.22       => ! [X6: int] :
% 9.10/5.22            ( ! [Xa3: int] :
% 9.10/5.22                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22               => ! [Xb3: int] :
% 9.10/5.22                    ( ( member_int @ Xb3 @ B5 )
% 9.10/5.22                   => ( X6
% 9.10/5.22                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 9.10/5.22           => ( ( ord_less_eq_int @ X6 @ T )
% 9.10/5.22             => ( ord_less_eq_int @ ( minus_minus_int @ X6 @ D4 ) @ T ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3214_cppi,axiom,
% 9.10/5.22      ! [D4: int,P: int > $o,P6: int > $o,A4: set_int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.22       => ( ? [Z3: int] :
% 9.10/5.22            ! [X5: int] :
% 9.10/5.22              ( ( ord_less_int @ Z3 @ X5 )
% 9.10/5.22             => ( ( P @ X5 )
% 9.10/5.22                = ( P6 @ X5 ) ) )
% 9.10/5.22         => ( ! [X5: int] :
% 9.10/5.22                ( ! [Xa: int] :
% 9.10/5.22                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                   => ! [Xb: int] :
% 9.10/5.22                        ( ( member_int @ Xb @ A4 )
% 9.10/5.22                       => ( X5
% 9.10/5.22                         != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 9.10/5.22               => ( ( P @ X5 )
% 9.10/5.22                 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 9.10/5.22           => ( ! [X5: int,K3: int] :
% 9.10/5.22                  ( ( P6 @ X5 )
% 9.10/5.22                  = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 9.10/5.22             => ( ( ? [X7: int] : ( P @ X7 ) )
% 9.10/5.22                = ( ? [X: int] :
% 9.10/5.22                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                      & ( P6 @ X ) )
% 9.10/5.22                  | ? [X: int] :
% 9.10/5.22                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                      & ? [Y: int] :
% 9.10/5.22                          ( ( member_int @ Y @ A4 )
% 9.10/5.22                          & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3215_cpmi,axiom,
% 9.10/5.22      ! [D4: int,P: int > $o,P6: int > $o,B5: set_int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.10/5.22       => ( ? [Z3: int] :
% 9.10/5.22            ! [X5: int] :
% 9.10/5.22              ( ( ord_less_int @ X5 @ Z3 )
% 9.10/5.22             => ( ( P @ X5 )
% 9.10/5.22                = ( P6 @ X5 ) ) )
% 9.10/5.22         => ( ! [X5: int] :
% 9.10/5.22                ( ! [Xa: int] :
% 9.10/5.22                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                   => ! [Xb: int] :
% 9.10/5.22                        ( ( member_int @ Xb @ B5 )
% 9.10/5.22                       => ( X5
% 9.10/5.22                         != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 9.10/5.22               => ( ( P @ X5 )
% 9.10/5.22                 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 9.10/5.22           => ( ! [X5: int,K3: int] :
% 9.10/5.22                  ( ( P6 @ X5 )
% 9.10/5.22                  = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 9.10/5.22             => ( ( ? [X7: int] : ( P @ X7 ) )
% 9.10/5.22                = ( ? [X: int] :
% 9.10/5.22                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                      & ( P6 @ X ) )
% 9.10/5.22                  | ? [X: int] :
% 9.10/5.22                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.10/5.22                      & ? [Y: int] :
% 9.10/5.22                          ( ( member_int @ Y @ B5 )
% 9.10/5.22                          & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3216_minf_I8_J,axiom,
% 9.10/5.22      ! [T: real] :
% 9.10/5.22      ? [Z4: real] :
% 9.10/5.22      ! [X6: real] :
% 9.10/5.22        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.22       => ~ ( ord_less_eq_real @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3217_minf_I8_J,axiom,
% 9.10/5.22      ! [T: rat] :
% 9.10/5.22      ? [Z4: rat] :
% 9.10/5.22      ! [X6: rat] :
% 9.10/5.22        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.22       => ~ ( ord_less_eq_rat @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3218_minf_I8_J,axiom,
% 9.10/5.22      ! [T: num] :
% 9.10/5.22      ? [Z4: num] :
% 9.10/5.22      ! [X6: num] :
% 9.10/5.22        ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.22       => ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3219_minf_I8_J,axiom,
% 9.10/5.22      ! [T: nat] :
% 9.10/5.22      ? [Z4: nat] :
% 9.10/5.22      ! [X6: nat] :
% 9.10/5.22        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.22       => ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3220_minf_I8_J,axiom,
% 9.10/5.22      ! [T: int] :
% 9.10/5.22      ? [Z4: int] :
% 9.10/5.22      ! [X6: int] :
% 9.10/5.22        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.22       => ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3221_minf_I6_J,axiom,
% 9.10/5.22      ! [T: real] :
% 9.10/5.22      ? [Z4: real] :
% 9.10/5.22      ! [X6: real] :
% 9.10/5.22        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.22       => ( ord_less_eq_real @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3222_minf_I6_J,axiom,
% 9.10/5.22      ! [T: rat] :
% 9.10/5.22      ? [Z4: rat] :
% 9.10/5.22      ! [X6: rat] :
% 9.10/5.22        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.22       => ( ord_less_eq_rat @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3223_minf_I6_J,axiom,
% 9.10/5.22      ! [T: num] :
% 9.10/5.22      ? [Z4: num] :
% 9.10/5.22      ! [X6: num] :
% 9.10/5.22        ( ( ord_less_num @ X6 @ Z4 )
% 9.10/5.22       => ( ord_less_eq_num @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3224_minf_I6_J,axiom,
% 9.10/5.22      ! [T: nat] :
% 9.10/5.22      ? [Z4: nat] :
% 9.10/5.22      ! [X6: nat] :
% 9.10/5.22        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.22       => ( ord_less_eq_nat @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3225_minf_I6_J,axiom,
% 9.10/5.22      ! [T: int] :
% 9.10/5.22      ? [Z4: int] :
% 9.10/5.22      ! [X6: int] :
% 9.10/5.22        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.22       => ( ord_less_eq_int @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3226_pinf_I8_J,axiom,
% 9.10/5.22      ! [T: real] :
% 9.10/5.22      ? [Z4: real] :
% 9.10/5.22      ! [X6: real] :
% 9.10/5.22        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.22       => ( ord_less_eq_real @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3227_pinf_I8_J,axiom,
% 9.10/5.22      ! [T: rat] :
% 9.10/5.22      ? [Z4: rat] :
% 9.10/5.22      ! [X6: rat] :
% 9.10/5.22        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.22       => ( ord_less_eq_rat @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3228_pinf_I8_J,axiom,
% 9.10/5.22      ! [T: num] :
% 9.10/5.22      ? [Z4: num] :
% 9.10/5.22      ! [X6: num] :
% 9.10/5.22        ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.22       => ( ord_less_eq_num @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3229_pinf_I8_J,axiom,
% 9.10/5.22      ! [T: nat] :
% 9.10/5.22      ? [Z4: nat] :
% 9.10/5.22      ! [X6: nat] :
% 9.10/5.22        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.22       => ( ord_less_eq_nat @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3230_pinf_I8_J,axiom,
% 9.10/5.22      ! [T: int] :
% 9.10/5.22      ? [Z4: int] :
% 9.10/5.22      ! [X6: int] :
% 9.10/5.22        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.22       => ( ord_less_eq_int @ T @ X6 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3231_pinf_I6_J,axiom,
% 9.10/5.22      ! [T: real] :
% 9.10/5.22      ? [Z4: real] :
% 9.10/5.22      ! [X6: real] :
% 9.10/5.22        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.22       => ~ ( ord_less_eq_real @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3232_pinf_I6_J,axiom,
% 9.10/5.22      ! [T: rat] :
% 9.10/5.22      ? [Z4: rat] :
% 9.10/5.22      ! [X6: rat] :
% 9.10/5.22        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.22       => ~ ( ord_less_eq_rat @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3233_pinf_I6_J,axiom,
% 9.10/5.22      ! [T: num] :
% 9.10/5.22      ? [Z4: num] :
% 9.10/5.22      ! [X6: num] :
% 9.10/5.22        ( ( ord_less_num @ Z4 @ X6 )
% 9.10/5.22       => ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3234_pinf_I6_J,axiom,
% 9.10/5.22      ! [T: nat] :
% 9.10/5.22      ? [Z4: nat] :
% 9.10/5.22      ! [X6: nat] :
% 9.10/5.22        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.22       => ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3235_pinf_I6_J,axiom,
% 9.10/5.22      ! [T: int] :
% 9.10/5.22      ? [Z4: int] :
% 9.10/5.22      ! [X6: int] :
% 9.10/5.22        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.22       => ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3236_inf__period_I2_J,axiom,
% 9.10/5.22      ! [P: complex > $o,D4: complex,Q: complex > $o] :
% 9.10/5.22        ( ! [X5: complex,K3: complex] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: complex,K3: complex] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: complex,K4: complex] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                | ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
% 9.10/5.22                | ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3237_inf__period_I2_J,axiom,
% 9.10/5.22      ! [P: real > $o,D4: real,Q: real > $o] :
% 9.10/5.22        ( ! [X5: real,K3: real] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: real,K3: real] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: real,K4: real] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                | ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
% 9.10/5.22                | ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3238_inf__period_I2_J,axiom,
% 9.10/5.22      ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 9.10/5.22        ( ! [X5: rat,K3: rat] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: rat,K3: rat] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: rat,K4: rat] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                | ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
% 9.10/5.22                | ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3239_inf__period_I2_J,axiom,
% 9.10/5.22      ! [P: int > $o,D4: int,Q: int > $o] :
% 9.10/5.22        ( ! [X5: int,K3: int] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: int,K3: int] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: int,K4: int] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                | ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
% 9.10/5.22                | ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3240_inf__period_I1_J,axiom,
% 9.10/5.22      ! [P: complex > $o,D4: complex,Q: complex > $o] :
% 9.10/5.22        ( ! [X5: complex,K3: complex] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: complex,K3: complex] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_complex @ X5 @ ( times_times_complex @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: complex,K4: complex] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                & ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) )
% 9.10/5.22                & ( Q @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3241_inf__period_I1_J,axiom,
% 9.10/5.22      ! [P: real > $o,D4: real,Q: real > $o] :
% 9.10/5.22        ( ! [X5: real,K3: real] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: real,K3: real] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: real,K4: real] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                & ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) )
% 9.10/5.22                & ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3242_inf__period_I1_J,axiom,
% 9.10/5.22      ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 9.10/5.22        ( ! [X5: rat,K3: rat] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: rat,K3: rat] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: rat,K4: rat] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                & ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) )
% 9.10/5.22                & ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3243_inf__period_I1_J,axiom,
% 9.10/5.22      ! [P: int > $o,D4: int,Q: int > $o] :
% 9.10/5.22        ( ! [X5: int,K3: int] :
% 9.10/5.22            ( ( P @ X5 )
% 9.10/5.22            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 9.10/5.22       => ( ! [X5: int,K3: int] :
% 9.10/5.22              ( ( Q @ X5 )
% 9.10/5.22              = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 9.10/5.22         => ! [X6: int,K4: int] :
% 9.10/5.22              ( ( ( P @ X6 )
% 9.10/5.22                & ( Q @ X6 ) )
% 9.10/5.22              = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) )
% 9.10/5.22                & ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3244_imp__le__cong,axiom,
% 9.10/5.22      ! [X3: int,X8: int,P: $o,P6: $o] :
% 9.10/5.22        ( ( X3 = X8 )
% 9.10/5.22       => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 9.10/5.22           => ( P = P6 ) )
% 9.10/5.22         => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.22             => P )
% 9.10/5.22            = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 9.10/5.22             => P6 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3245_conj__le__cong,axiom,
% 9.10/5.22      ! [X3: int,X8: int,P: $o,P6: $o] :
% 9.10/5.22        ( ( X3 = X8 )
% 9.10/5.22       => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 9.10/5.22           => ( P = P6 ) )
% 9.10/5.22         => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.22              & P )
% 9.10/5.22            = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 9.10/5.22              & P6 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3246_atLeastatMost__psubset__iff,axiom,
% 9.10/5.22      ! [A: set_int,B2: set_int,C: set_int,D: set_int] :
% 9.10/5.22        ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B2 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.10/5.22        = ( ( ~ ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.22            | ( ( ord_less_eq_set_int @ C @ A )
% 9.10/5.22              & ( ord_less_eq_set_int @ B2 @ D )
% 9.10/5.22              & ( ( ord_less_set_int @ C @ A )
% 9.10/5.22                | ( ord_less_set_int @ B2 @ D ) ) ) )
% 9.10/5.22          & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3247_atLeastatMost__psubset__iff,axiom,
% 9.10/5.22      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.10/5.22        ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.10/5.22        = ( ( ~ ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.22            | ( ( ord_less_eq_rat @ C @ A )
% 9.10/5.22              & ( ord_less_eq_rat @ B2 @ D )
% 9.10/5.22              & ( ( ord_less_rat @ C @ A )
% 9.10/5.22                | ( ord_less_rat @ B2 @ D ) ) ) )
% 9.10/5.22          & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3248_atLeastatMost__psubset__iff,axiom,
% 9.10/5.22      ! [A: num,B2: num,C: num,D: num] :
% 9.10/5.22        ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B2 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.10/5.22        = ( ( ~ ( ord_less_eq_num @ A @ B2 )
% 9.10/5.22            | ( ( ord_less_eq_num @ C @ A )
% 9.10/5.22              & ( ord_less_eq_num @ B2 @ D )
% 9.10/5.22              & ( ( ord_less_num @ C @ A )
% 9.10/5.22                | ( ord_less_num @ B2 @ D ) ) ) )
% 9.10/5.22          & ( ord_less_eq_num @ C @ D ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3249_atLeastatMost__psubset__iff,axiom,
% 9.10/5.22      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.10/5.22        ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.10/5.22        = ( ( ~ ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.22            | ( ( ord_less_eq_nat @ C @ A )
% 9.10/5.22              & ( ord_less_eq_nat @ B2 @ D )
% 9.10/5.22              & ( ( ord_less_nat @ C @ A )
% 9.10/5.22                | ( ord_less_nat @ B2 @ D ) ) ) )
% 9.10/5.22          & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3250_atLeastatMost__psubset__iff,axiom,
% 9.10/5.22      ! [A: int,B2: int,C: int,D: int] :
% 9.10/5.22        ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B2 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.10/5.22        = ( ( ~ ( ord_less_eq_int @ A @ B2 )
% 9.10/5.22            | ( ( ord_less_eq_int @ C @ A )
% 9.10/5.22              & ( ord_less_eq_int @ B2 @ D )
% 9.10/5.22              & ( ( ord_less_int @ C @ A )
% 9.10/5.22                | ( ord_less_int @ B2 @ D ) ) ) )
% 9.10/5.22          & ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3251_atLeastatMost__psubset__iff,axiom,
% 9.10/5.22      ! [A: real,B2: real,C: real,D: real] :
% 9.10/5.22        ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.10/5.22        = ( ( ~ ( ord_less_eq_real @ A @ B2 )
% 9.10/5.22            | ( ( ord_less_eq_real @ C @ A )
% 9.10/5.22              & ( ord_less_eq_real @ B2 @ D )
% 9.10/5.22              & ( ( ord_less_real @ C @ A )
% 9.10/5.22                | ( ord_less_real @ B2 @ D ) ) ) )
% 9.10/5.22          & ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3252_minusinfinity,axiom,
% 9.10/5.22      ! [D: int,P1: int > $o,P: int > $o] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.22       => ( ! [X5: int,K3: int] :
% 9.10/5.22              ( ( P1 @ X5 )
% 9.10/5.22              = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 9.10/5.22         => ( ? [Z3: int] :
% 9.10/5.22              ! [X5: int] :
% 9.10/5.22                ( ( ord_less_int @ X5 @ Z3 )
% 9.10/5.22               => ( ( P @ X5 )
% 9.10/5.22                  = ( P1 @ X5 ) ) )
% 9.10/5.22           => ( ? [X_12: int] : ( P1 @ X_12 )
% 9.10/5.22             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3253_plusinfinity,axiom,
% 9.10/5.22      ! [D: int,P6: int > $o,P: int > $o] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.22       => ( ! [X5: int,K3: int] :
% 9.10/5.22              ( ( P6 @ X5 )
% 9.10/5.22              = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K3 @ D ) ) ) )
% 9.10/5.22         => ( ? [Z3: int] :
% 9.10/5.22              ! [X5: int] :
% 9.10/5.22                ( ( ord_less_int @ Z3 @ X5 )
% 9.10/5.22               => ( ( P @ X5 )
% 9.10/5.22                  = ( P6 @ X5 ) ) )
% 9.10/5.22           => ( ? [X_12: int] : ( P6 @ X_12 )
% 9.10/5.22             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3254_Bolzano,axiom,
% 9.10/5.22      ! [A: real,B2: real,P: real > real > $o] :
% 9.10/5.22        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.22       => ( ! [A3: real,B3: real,C2: real] :
% 9.10/5.22              ( ( P @ A3 @ B3 )
% 9.10/5.22             => ( ( P @ B3 @ C2 )
% 9.10/5.22               => ( ( ord_less_eq_real @ A3 @ B3 )
% 9.10/5.22                 => ( ( ord_less_eq_real @ B3 @ C2 )
% 9.10/5.22                   => ( P @ A3 @ C2 ) ) ) ) )
% 9.10/5.22         => ( ! [X5: real] :
% 9.10/5.22                ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.22               => ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.22                 => ? [D5: real] :
% 9.10/5.22                      ( ( ord_less_real @ zero_zero_real @ D5 )
% 9.10/5.22                      & ! [A3: real,B3: real] :
% 9.10/5.22                          ( ( ( ord_less_eq_real @ A3 @ X5 )
% 9.10/5.22                            & ( ord_less_eq_real @ X5 @ B3 )
% 9.10/5.22                            & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D5 ) )
% 9.10/5.22                         => ( P @ A3 @ B3 ) ) ) ) )
% 9.10/5.22           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3255_mult__le__cancel__iff1,axiom,
% 9.10/5.22      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.22        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.10/5.22       => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 9.10/5.22          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3256_mult__le__cancel__iff1,axiom,
% 9.10/5.22      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.22        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.10/5.22       => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 9.10/5.22          = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3257_mult__le__cancel__iff1,axiom,
% 9.10/5.22      ! [Z: int,X3: int,Y3: int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.10/5.22       => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 9.10/5.22          = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3258_mult__le__cancel__iff2,axiom,
% 9.10/5.22      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.22        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.10/5.22       => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X3 ) @ ( times_times_real @ Z @ Y3 ) )
% 9.10/5.22          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3259_mult__le__cancel__iff2,axiom,
% 9.10/5.22      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.22        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.10/5.22       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X3 ) @ ( times_times_rat @ Z @ Y3 ) )
% 9.10/5.22          = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3260_mult__le__cancel__iff2,axiom,
% 9.10/5.22      ! [Z: int,X3: int,Y3: int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.10/5.22       => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X3 ) @ ( times_times_int @ Z @ Y3 ) )
% 9.10/5.22          = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3261_divides__aux__eq,axiom,
% 9.10/5.22      ! [Q2: nat,R: nat] :
% 9.10/5.22        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 9.10/5.22        = ( R = zero_zero_nat ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3262_divides__aux__eq,axiom,
% 9.10/5.22      ! [Q2: int,R: int] :
% 9.10/5.22        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22        = ( R = zero_zero_int ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3263_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_int,Ys: list_int] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( product_Pair_int_int @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3264_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_int,Ys: list_VEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3265_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_int,Ys: list_o] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( product_Pair_int_o @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3266_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_int,Ys: list_nat] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( product_Pair_int_nat @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3267_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3268_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3269_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3270_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3271_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_o,Ys: list_int] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3272_product__nth,axiom,
% 9.10/5.22      ! [N: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.10/5.22       => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) @ N )
% 9.10/5.22          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3273_neg__eucl__rel__int__mult__2,axiom,
% 9.10/5.22      ! [B2: int,A: int,Q2: int,R: int] :
% 9.10/5.22        ( ( ord_less_eq_int @ B2 @ zero_zero_int )
% 9.10/5.22       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22         => ( 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 ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3274_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.10/5.22        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3275_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
% 9.10/5.22        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3276_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
% 9.10/5.22        ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3277_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_o,Ys: list_VEBT_VEBT] :
% 9.10/5.22        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3278_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_o,Ys: list_o] :
% 9.10/5.22        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3279_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_o,Ys: list_nat] :
% 9.10/5.22        ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3280_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_nat,Ys: list_VEBT_VEBT] :
% 9.10/5.22        ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3281_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_nat,Ys: list_o] :
% 9.10/5.22        ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3282_length__product,axiom,
% 9.10/5.22      ! [Xs2: list_nat,Ys: list_nat] :
% 9.10/5.22        ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs2 @ Ys ) )
% 9.10/5.22        = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3283_unique__remainder,axiom,
% 9.10/5.22      ! [A: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
% 9.10/5.22        ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22       => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 9.10/5.22         => ( R = R4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3284_unique__quotient,axiom,
% 9.10/5.22      ! [A: int,B2: int,Q2: int,R: int,Q5: int,R4: int] :
% 9.10/5.22        ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22       => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q5 @ R4 ) )
% 9.10/5.22         => ( Q2 = Q5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3285_eucl__rel__int__by0,axiom,
% 9.10/5.22      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3286_div__int__unique,axiom,
% 9.10/5.22      ! [K: int,L: int,Q2: int,R: int] :
% 9.10/5.22        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22       => ( ( divide_divide_int @ K @ L )
% 9.10/5.22          = Q2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3287_mod__int__unique,axiom,
% 9.10/5.22      ! [K: int,L: int,Q2: int,R: int] :
% 9.10/5.22        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22       => ( ( modulo_modulo_int @ K @ L )
% 9.10/5.22          = R ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3288_eucl__rel__int__dividesI,axiom,
% 9.10/5.22      ! [L: int,K: int,Q2: int] :
% 9.10/5.22        ( ( L != zero_zero_int )
% 9.10/5.22       => ( ( K
% 9.10/5.22            = ( times_times_int @ Q2 @ L ) )
% 9.10/5.22         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3289_eucl__rel__int,axiom,
% 9.10/5.22      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3290_eucl__rel__int__iff,axiom,
% 9.10/5.22      ! [K: int,L: int,Q2: int,R: int] :
% 9.10/5.22        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22        = ( ( K
% 9.10/5.22            = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R ) )
% 9.10/5.22          & ( ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.22           => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 9.10/5.22              & ( ord_less_int @ R @ L ) ) )
% 9.10/5.22          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 9.10/5.22           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 9.10/5.22               => ( ( ord_less_int @ L @ R )
% 9.10/5.22                  & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 9.10/5.22              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 9.10/5.22               => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3291_mult__less__iff1,axiom,
% 9.10/5.22      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.22        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.10/5.22       => ( ( ord_less_real @ ( times_times_real @ X3 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 9.10/5.22          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3292_mult__less__iff1,axiom,
% 9.10/5.22      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.22        ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.10/5.22       => ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 9.10/5.22          = ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3293_mult__less__iff1,axiom,
% 9.10/5.22      ! [Z: int,X3: int,Y3: int] :
% 9.10/5.22        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.10/5.22       => ( ( ord_less_int @ ( times_times_int @ X3 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 9.10/5.22          = ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3294_pos__eucl__rel__int__mult__2,axiom,
% 9.10/5.22      ! [B2: int,A: int,Q2: int,R: int] :
% 9.10/5.22        ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.22       => ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.22         => ( 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 ) ) @ B2 ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3295_obtain__set__pred,axiom,
% 9.10/5.22      ! [Z: nat,X3: nat,A4: set_nat] :
% 9.10/5.22        ( ( ord_less_nat @ Z @ X3 )
% 9.10/5.22       => ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
% 9.10/5.22         => ( ( finite_finite_nat @ A4 )
% 9.10/5.22           => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X3 @ X_1 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3296_set__vebt__finite,axiom,
% 9.10/5.22      ! [T: vEBT_VEBT,N: nat] :
% 9.10/5.22        ( ( vEBT_invar_vebt @ T @ N )
% 9.10/5.22       => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3297_pred__none__empty,axiom,
% 9.10/5.22      ! [Xs2: set_nat,A: nat] :
% 9.10/5.22        ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_1 )
% 9.10/5.22       => ( ( finite_finite_nat @ Xs2 )
% 9.10/5.22         => ~ ? [X6: nat] :
% 9.10/5.22                ( ( member_nat @ X6 @ Xs2 )
% 9.10/5.22                & ( ord_less_nat @ X6 @ A ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3298_prod_Oinject,axiom,
% 9.10/5.22      ! [X1: int,X2: int,Y1: int,Y2: int] :
% 9.10/5.22        ( ( ( product_Pair_int_int @ X1 @ X2 )
% 9.10/5.22          = ( product_Pair_int_int @ Y1 @ Y2 ) )
% 9.10/5.22        = ( ( X1 = Y1 )
% 9.10/5.22          & ( X2 = Y2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3299_prod_Oinject,axiom,
% 9.10/5.22      ! [X1: code_integer > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 9.10/5.22        ( ( ( produc6137756002093451184nteger @ X1 @ X2 )
% 9.10/5.22          = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
% 9.10/5.22        = ( ( X1 = Y1 )
% 9.10/5.22          & ( X2 = Y2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3300_prod_Oinject,axiom,
% 9.10/5.22      ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X2: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 9.10/5.22        ( ( ( produc8603105652947943368nteger @ X1 @ X2 )
% 9.10/5.22          = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
% 9.10/5.22        = ( ( X1 = Y1 )
% 9.10/5.22          & ( X2 = Y2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3301_prod_Oinject,axiom,
% 9.10/5.22      ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X2: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 9.10/5.22        ( ( ( produc5700946648718959541nt_int @ X1 @ X2 )
% 9.10/5.22          = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
% 9.10/5.22        = ( ( X1 = Y1 )
% 9.10/5.22          & ( X2 = Y2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3302_prod_Oinject,axiom,
% 9.10/5.22      ! [X1: int > option6357759511663192854e_term,X2: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 9.10/5.22        ( ( ( produc4305682042979456191nt_int @ X1 @ X2 )
% 9.10/5.22          = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
% 9.10/5.22        = ( ( X1 = Y1 )
% 9.10/5.22          & ( X2 = Y2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3303_old_Oprod_Oinject,axiom,
% 9.10/5.22      ! [A: int,B2: int,A2: int,B: int] :
% 9.10/5.22        ( ( ( product_Pair_int_int @ A @ B2 )
% 9.10/5.22          = ( product_Pair_int_int @ A2 @ B ) )
% 9.10/5.22        = ( ( A = A2 )
% 9.10/5.22          & ( B2 = B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3304_old_Oprod_Oinject,axiom,
% 9.10/5.22      ! [A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.10/5.22        ( ( ( produc6137756002093451184nteger @ A @ B2 )
% 9.10/5.22          = ( produc6137756002093451184nteger @ A2 @ B ) )
% 9.10/5.22        = ( ( A = A2 )
% 9.10/5.22          & ( B2 = B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3305_old_Oprod_Oinject,axiom,
% 9.10/5.22      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.10/5.22        ( ( ( produc8603105652947943368nteger @ A @ B2 )
% 9.10/5.22          = ( produc8603105652947943368nteger @ A2 @ B ) )
% 9.10/5.22        = ( ( A = A2 )
% 9.10/5.22          & ( B2 = B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3306_old_Oprod_Oinject,axiom,
% 9.10/5.22      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.10/5.22        ( ( ( produc5700946648718959541nt_int @ A @ B2 )
% 9.10/5.22          = ( produc5700946648718959541nt_int @ A2 @ B ) )
% 9.10/5.22        = ( ( A = A2 )
% 9.10/5.22          & ( B2 = B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3307_old_Oprod_Oinject,axiom,
% 9.10/5.22      ! [A: int > option6357759511663192854e_term,B2: product_prod_int_int,A2: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.10/5.22        ( ( ( produc4305682042979456191nt_int @ A @ B2 )
% 9.10/5.22          = ( produc4305682042979456191nt_int @ A2 @ B ) )
% 9.10/5.22        = ( ( A = A2 )
% 9.10/5.22          & ( B2 = B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3308_List_Ofinite__set,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3309_List_Ofinite__set,axiom,
% 9.10/5.22      ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3310_List_Ofinite__set,axiom,
% 9.10/5.22      ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3311_List_Ofinite__set,axiom,
% 9.10/5.22      ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3312_infinite__Icc__iff,axiom,
% 9.10/5.22      ! [A: rat,B2: rat] :
% 9.10/5.22        ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) ) )
% 9.10/5.22        = ( ord_less_rat @ A @ B2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3313_infinite__Icc__iff,axiom,
% 9.10/5.22      ! [A: real,B2: real] :
% 9.10/5.22        ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) )
% 9.10/5.22        = ( ord_less_real @ A @ B2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3314_bounded__nat__set__is__finite,axiom,
% 9.10/5.22      ! [N4: set_nat,N: nat] :
% 9.10/5.22        ( ! [X5: nat] :
% 9.10/5.22            ( ( member_nat @ X5 @ N4 )
% 9.10/5.22           => ( ord_less_nat @ X5 @ N ) )
% 9.10/5.22       => ( finite_finite_nat @ N4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3315_finite__nat__set__iff__bounded,axiom,
% 9.10/5.22      ( finite_finite_nat
% 9.10/5.22      = ( ^ [N6: set_nat] :
% 9.10/5.22          ? [M3: nat] :
% 9.10/5.22          ! [X: nat] :
% 9.10/5.22            ( ( member_nat @ X @ N6 )
% 9.10/5.22           => ( ord_less_nat @ X @ M3 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3316_finite__nat__set__iff__bounded__le,axiom,
% 9.10/5.22      ( finite_finite_nat
% 9.10/5.22      = ( ^ [N6: set_nat] :
% 9.10/5.22          ? [M3: nat] :
% 9.10/5.22          ! [X: nat] :
% 9.10/5.22            ( ( member_nat @ X @ N6 )
% 9.10/5.22           => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3317_finite__list,axiom,
% 9.10/5.22      ! [A4: set_VEBT_VEBT] :
% 9.10/5.22        ( ( finite5795047828879050333T_VEBT @ A4 )
% 9.10/5.22       => ? [Xs3: list_VEBT_VEBT] :
% 9.10/5.22            ( ( set_VEBT_VEBT2 @ Xs3 )
% 9.10/5.22            = A4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3318_finite__list,axiom,
% 9.10/5.22      ! [A4: set_nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ? [Xs3: list_nat] :
% 9.10/5.22            ( ( set_nat2 @ Xs3 )
% 9.10/5.22            = A4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3319_finite__list,axiom,
% 9.10/5.22      ! [A4: set_int] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ? [Xs3: list_int] :
% 9.10/5.22            ( ( set_int2 @ Xs3 )
% 9.10/5.22            = A4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3320_finite__list,axiom,
% 9.10/5.22      ! [A4: set_complex] :
% 9.10/5.22        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.22       => ? [Xs3: list_complex] :
% 9.10/5.22            ( ( set_complex2 @ Xs3 )
% 9.10/5.22            = A4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3321_finite__M__bounded__by__nat,axiom,
% 9.10/5.22      ! [P: nat > $o,I: nat] :
% 9.10/5.22        ( finite_finite_nat
% 9.10/5.22        @ ( collect_nat
% 9.10/5.22          @ ^ [K2: nat] :
% 9.10/5.22              ( ( P @ K2 )
% 9.10/5.22              & ( ord_less_nat @ K2 @ I ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3322_finite__less__ub,axiom,
% 9.10/5.22      ! [F: nat > nat,U: nat] :
% 9.10/5.22        ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 9.10/5.22       => ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3323_finite__lists__length__eq,axiom,
% 9.10/5.22      ! [A4: set_complex,N: nat] :
% 9.10/5.22        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.22       => ( finite8712137658972009173omplex
% 9.10/5.22          @ ( collect_list_complex
% 9.10/5.22            @ ^ [Xs: list_complex] :
% 9.10/5.22                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ( size_s3451745648224563538omplex @ Xs )
% 9.10/5.22                  = N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3324_finite__lists__length__eq,axiom,
% 9.10/5.22      ! [A4: set_VEBT_VEBT,N: nat] :
% 9.10/5.22        ( ( finite5795047828879050333T_VEBT @ A4 )
% 9.10/5.22       => ( finite3004134309566078307T_VEBT
% 9.10/5.22          @ ( collec5608196760682091941T_VEBT
% 9.10/5.22            @ ^ [Xs: list_VEBT_VEBT] :
% 9.10/5.22                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.10/5.22                  = N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3325_finite__lists__length__eq,axiom,
% 9.10/5.22      ! [A4: set_o,N: nat] :
% 9.10/5.22        ( ( finite_finite_o @ A4 )
% 9.10/5.22       => ( finite_finite_list_o
% 9.10/5.22          @ ( collect_list_o
% 9.10/5.22            @ ^ [Xs: list_o] :
% 9.10/5.22                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ( size_size_list_o @ Xs )
% 9.10/5.22                  = N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3326_finite__lists__length__eq,axiom,
% 9.10/5.22      ! [A4: set_nat,N: nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( finite8100373058378681591st_nat
% 9.10/5.22          @ ( collect_list_nat
% 9.10/5.22            @ ^ [Xs: list_nat] :
% 9.10/5.22                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ( size_size_list_nat @ Xs )
% 9.10/5.22                  = N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3327_finite__lists__length__eq,axiom,
% 9.10/5.22      ! [A4: set_int,N: nat] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( finite3922522038869484883st_int
% 9.10/5.22          @ ( collect_list_int
% 9.10/5.22            @ ^ [Xs: list_int] :
% 9.10/5.22                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ( size_size_list_int @ Xs )
% 9.10/5.22                  = N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3328_infinite__Icc,axiom,
% 9.10/5.22      ! [A: rat,B2: rat] :
% 9.10/5.22        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.22       => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3329_infinite__Icc,axiom,
% 9.10/5.22      ! [A: real,B2: real] :
% 9.10/5.22        ( ( ord_less_real @ A @ B2 )
% 9.10/5.22       => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3330_finite__lists__length__le,axiom,
% 9.10/5.22      ! [A4: set_complex,N: nat] :
% 9.10/5.22        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.22       => ( finite8712137658972009173omplex
% 9.10/5.22          @ ( collect_list_complex
% 9.10/5.22            @ ^ [Xs: list_complex] :
% 9.10/5.22                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3331_finite__lists__length__le,axiom,
% 9.10/5.22      ! [A4: set_VEBT_VEBT,N: nat] :
% 9.10/5.22        ( ( finite5795047828879050333T_VEBT @ A4 )
% 9.10/5.22       => ( finite3004134309566078307T_VEBT
% 9.10/5.22          @ ( collec5608196760682091941T_VEBT
% 9.10/5.22            @ ^ [Xs: list_VEBT_VEBT] :
% 9.10/5.22                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3332_finite__lists__length__le,axiom,
% 9.10/5.22      ! [A4: set_o,N: nat] :
% 9.10/5.22        ( ( finite_finite_o @ A4 )
% 9.10/5.22       => ( finite_finite_list_o
% 9.10/5.22          @ ( collect_list_o
% 9.10/5.22            @ ^ [Xs: list_o] :
% 9.10/5.22                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3333_finite__lists__length__le,axiom,
% 9.10/5.22      ! [A4: set_nat,N: nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( finite8100373058378681591st_nat
% 9.10/5.22          @ ( collect_list_nat
% 9.10/5.22            @ ^ [Xs: list_nat] :
% 9.10/5.22                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3334_finite__lists__length__le,axiom,
% 9.10/5.22      ! [A4: set_int,N: nat] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( finite3922522038869484883st_int
% 9.10/5.22          @ ( collect_list_int
% 9.10/5.22            @ ^ [Xs: list_int] :
% 9.10/5.22                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
% 9.10/5.22                & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3335_old_Oprod_Oexhaust,axiom,
% 9.10/5.22      ! [Y3: product_prod_int_int] :
% 9.10/5.22        ~ ! [A3: int,B3: int] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( product_Pair_int_int @ A3 @ B3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3336_old_Oprod_Oexhaust,axiom,
% 9.10/5.22      ! [Y3: produc8763457246119570046nteger] :
% 9.10/5.22        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc6137756002093451184nteger @ A3 @ B3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3337_old_Oprod_Oexhaust,axiom,
% 9.10/5.22      ! [Y3: produc1908205239877642774nteger] :
% 9.10/5.22        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc8603105652947943368nteger @ A3 @ B3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3338_old_Oprod_Oexhaust,axiom,
% 9.10/5.22      ! [Y3: produc2285326912895808259nt_int] :
% 9.10/5.22        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc5700946648718959541nt_int @ A3 @ B3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3339_old_Oprod_Oexhaust,axiom,
% 9.10/5.22      ! [Y3: produc7773217078559923341nt_int] :
% 9.10/5.22        ~ ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc4305682042979456191nt_int @ A3 @ B3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3340_surj__pair,axiom,
% 9.10/5.22      ! [P4: product_prod_int_int] :
% 9.10/5.22      ? [X5: int,Y4: int] :
% 9.10/5.22        ( P4
% 9.10/5.22        = ( product_Pair_int_int @ X5 @ Y4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3341_surj__pair,axiom,
% 9.10/5.22      ! [P4: produc8763457246119570046nteger] :
% 9.10/5.22      ? [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.22        ( P4
% 9.10/5.22        = ( produc6137756002093451184nteger @ X5 @ Y4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3342_surj__pair,axiom,
% 9.10/5.22      ! [P4: produc1908205239877642774nteger] :
% 9.10/5.22      ? [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.22        ( P4
% 9.10/5.22        = ( produc8603105652947943368nteger @ X5 @ Y4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3343_surj__pair,axiom,
% 9.10/5.22      ! [P4: produc2285326912895808259nt_int] :
% 9.10/5.22      ? [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.22        ( P4
% 9.10/5.22        = ( produc5700946648718959541nt_int @ X5 @ Y4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3344_surj__pair,axiom,
% 9.10/5.22      ! [P4: produc7773217078559923341nt_int] :
% 9.10/5.22      ? [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.22        ( P4
% 9.10/5.22        = ( produc4305682042979456191nt_int @ X5 @ Y4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3345_prod__cases,axiom,
% 9.10/5.22      ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 9.10/5.22        ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
% 9.10/5.22       => ( P @ P4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3346_prod__cases,axiom,
% 9.10/5.22      ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
% 9.10/5.22        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.10/5.22       => ( P @ P4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3347_prod__cases,axiom,
% 9.10/5.22      ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
% 9.10/5.22        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 9.10/5.22       => ( P @ P4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3348_prod__cases,axiom,
% 9.10/5.22      ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
% 9.10/5.22        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 9.10/5.22       => ( P @ P4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3349_prod__cases,axiom,
% 9.10/5.22      ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
% 9.10/5.22        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.10/5.22       => ( P @ P4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3350_Pair__inject,axiom,
% 9.10/5.22      ! [A: int,B2: int,A2: int,B: int] :
% 9.10/5.22        ( ( ( product_Pair_int_int @ A @ B2 )
% 9.10/5.22          = ( product_Pair_int_int @ A2 @ B ) )
% 9.10/5.22       => ~ ( ( A = A2 )
% 9.10/5.22           => ( B2 != B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3351_Pair__inject,axiom,
% 9.10/5.22      ! [A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.10/5.22        ( ( ( produc6137756002093451184nteger @ A @ B2 )
% 9.10/5.22          = ( produc6137756002093451184nteger @ A2 @ B ) )
% 9.10/5.22       => ~ ( ( A = A2 )
% 9.10/5.22           => ( B2 != B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3352_Pair__inject,axiom,
% 9.10/5.22      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger,A2: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.10/5.22        ( ( ( produc8603105652947943368nteger @ A @ B2 )
% 9.10/5.22          = ( produc8603105652947943368nteger @ A2 @ B ) )
% 9.10/5.22       => ~ ( ( A = A2 )
% 9.10/5.22           => ( B2 != B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3353_Pair__inject,axiom,
% 9.10/5.22      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int,A2: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.10/5.22        ( ( ( produc5700946648718959541nt_int @ A @ B2 )
% 9.10/5.22          = ( produc5700946648718959541nt_int @ A2 @ B ) )
% 9.10/5.22       => ~ ( ( A = A2 )
% 9.10/5.22           => ( B2 != B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3354_Pair__inject,axiom,
% 9.10/5.22      ! [A: int > option6357759511663192854e_term,B2: product_prod_int_int,A2: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.10/5.22        ( ( ( produc4305682042979456191nt_int @ A @ B2 )
% 9.10/5.22          = ( produc4305682042979456191nt_int @ A2 @ B ) )
% 9.10/5.22       => ~ ( ( A = A2 )
% 9.10/5.22           => ( B2 != B ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3355_subset__eq__atLeast0__atMost__finite,axiom,
% 9.10/5.22      ! [N4: set_nat,N: nat] :
% 9.10/5.22        ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.22       => ( finite_finite_nat @ N4 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3356_prod__cases3,axiom,
% 9.10/5.22      ! [Y3: produc8763457246119570046nteger] :
% 9.10/5.22        ~ ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C2: code_integer] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3357_prod__cases3,axiom,
% 9.10/5.22      ! [Y3: produc1908205239877642774nteger] :
% 9.10/5.22        ~ ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C2: code_integer] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3358_prod__cases3,axiom,
% 9.10/5.22      ! [Y3: produc2285326912895808259nt_int] :
% 9.10/5.22        ~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3359_prod__cases3,axiom,
% 9.10/5.22      ! [Y3: produc7773217078559923341nt_int] :
% 9.10/5.22        ~ ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] :
% 9.10/5.22            ( Y3
% 9.10/5.22           != ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3360_prod__induct3,axiom,
% 9.10/5.22      ! [P: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
% 9.10/5.22        ( ! [A3: code_integer > option6357759511663192854e_term,B3: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) )
% 9.10/5.22       => ( P @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3361_prod__induct3,axiom,
% 9.10/5.22      ! [P: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
% 9.10/5.22        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A3 @ ( produc1086072967326762835nteger @ B3 @ C2 ) ) )
% 9.10/5.22       => ( P @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3362_prod__induct3,axiom,
% 9.10/5.22      ! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
% 9.10/5.22        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
% 9.10/5.22       => ( P @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3363_prod__induct3,axiom,
% 9.10/5.22      ! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
% 9.10/5.22        ( ! [A3: int > option6357759511663192854e_term,B3: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B3 @ C2 ) ) )
% 9.10/5.22       => ( P @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3364_finite__Collect__le__nat,axiom,
% 9.10/5.22      ! [K: nat] :
% 9.10/5.22        ( finite_finite_nat
% 9.10/5.22        @ ( collect_nat
% 9.10/5.22          @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3365_finite__Collect__less__nat,axiom,
% 9.10/5.22      ! [K: nat] :
% 9.10/5.22        ( finite_finite_nat
% 9.10/5.22        @ ( collect_nat
% 9.10/5.22          @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3366_finite__Collect__subsets,axiom,
% 9.10/5.22      ! [A4: set_nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( finite1152437895449049373et_nat
% 9.10/5.22          @ ( collect_set_nat
% 9.10/5.22            @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3367_finite__Collect__subsets,axiom,
% 9.10/5.22      ! [A4: set_complex] :
% 9.10/5.22        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.22       => ( finite6551019134538273531omplex
% 9.10/5.22          @ ( collect_set_complex
% 9.10/5.22            @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A4 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3368_finite__Collect__subsets,axiom,
% 9.10/5.22      ! [A4: set_int] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( finite6197958912794628473et_int
% 9.10/5.22          @ ( collect_set_int
% 9.10/5.22            @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A4 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3369_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: real > $o,Q: real > real > $o] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [X: real] :
% 9.10/5.22                ? [Y: real] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: real] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_real
% 9.10/5.22                  @ ( collect_real
% 9.10/5.22                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3370_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: real > $o,Q: nat > real > $o] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [X: nat] :
% 9.10/5.22                ? [Y: real] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: real] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_nat
% 9.10/5.22                  @ ( collect_nat
% 9.10/5.22                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3371_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: real > $o,Q: int > real > $o] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [X: int] :
% 9.10/5.22                ? [Y: real] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: real] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_int
% 9.10/5.22                  @ ( collect_int
% 9.10/5.22                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3372_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: real > $o,Q: complex > real > $o] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [X: complex] :
% 9.10/5.22                ? [Y: real] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: real] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite3207457112153483333omplex
% 9.10/5.22                  @ ( collect_complex
% 9.10/5.22                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3373_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: nat > $o,Q: real > nat > $o] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [X: real] :
% 9.10/5.22                ? [Y: nat] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: nat] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_real
% 9.10/5.22                  @ ( collect_real
% 9.10/5.22                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3374_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: nat > $o,Q: nat > nat > $o] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [X: nat] :
% 9.10/5.22                ? [Y: nat] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: nat] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_nat
% 9.10/5.22                  @ ( collect_nat
% 9.10/5.22                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3375_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: nat > $o,Q: int > nat > $o] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [X: int] :
% 9.10/5.22                ? [Y: nat] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: nat] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_int
% 9.10/5.22                  @ ( collect_int
% 9.10/5.22                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3376_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: nat > $o,Q: complex > nat > $o] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [X: complex] :
% 9.10/5.22                ? [Y: nat] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: nat] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite3207457112153483333omplex
% 9.10/5.22                  @ ( collect_complex
% 9.10/5.22                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3377_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: int > $o,Q: real > int > $o] :
% 9.10/5.22        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [X: real] :
% 9.10/5.22                ? [Y: int] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: int] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_real
% 9.10/5.22                  @ ( collect_real
% 9.10/5.22                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3378_finite__Collect__bounded__ex,axiom,
% 9.10/5.22      ! [P: int > $o,Q: nat > int > $o] :
% 9.10/5.22        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [X: nat] :
% 9.10/5.22                ? [Y: int] :
% 9.10/5.22                  ( ( P @ Y )
% 9.10/5.22                  & ( Q @ X @ Y ) ) ) )
% 9.10/5.22          = ( ! [Y: int] :
% 9.10/5.22                ( ( P @ Y )
% 9.10/5.22               => ( finite_finite_nat
% 9.10/5.22                  @ ( collect_nat
% 9.10/5.22                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3379_finite__roots__unity,axiom,
% 9.10/5.22      ! [N: nat] :
% 9.10/5.22        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.22       => ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [Z2: real] :
% 9.10/5.22                ( ( power_power_real @ Z2 @ N )
% 9.10/5.22                = one_one_real ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3380_finite__roots__unity,axiom,
% 9.10/5.22      ! [N: nat] :
% 9.10/5.22        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.22       => ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [Z2: complex] :
% 9.10/5.22                ( ( power_power_complex @ Z2 @ N )
% 9.10/5.22                = one_one_complex ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3381_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 9.10/5.22        ( ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [I4: real] :
% 9.10/5.22                ( ( member_real @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_complex ) ) ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_complex ) ) ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3382_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 9.10/5.22        ( ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [I4: nat] :
% 9.10/5.22                ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_complex ) ) ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_complex ) ) ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3383_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 9.10/5.22        ( ( finite_finite_int
% 9.10/5.22          @ ( collect_int
% 9.10/5.22            @ ^ [I4: int] :
% 9.10/5.22                ( ( member_int @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_complex ) ) ) )
% 9.10/5.22       => ( ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_complex ) ) ) )
% 9.10/5.22         => ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3384_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 9.10/5.22        ( ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [I4: complex] :
% 9.10/5.22                ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_complex ) ) ) )
% 9.10/5.22       => ( ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_complex ) ) ) )
% 9.10/5.22         => ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3385_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 9.10/5.22        ( ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [I4: real] :
% 9.10/5.22                ( ( member_real @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_real ) ) ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_real ) ) ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3386_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 9.10/5.22        ( ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [I4: nat] :
% 9.10/5.22                ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_real ) ) ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_real ) ) ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3387_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 9.10/5.22        ( ( finite_finite_int
% 9.10/5.22          @ ( collect_int
% 9.10/5.22            @ ^ [I4: int] :
% 9.10/5.22                ( ( member_int @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_real ) ) ) )
% 9.10/5.22       => ( ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_real ) ) ) )
% 9.10/5.22         => ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3388_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 9.10/5.22        ( ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [I4: complex] :
% 9.10/5.22                ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_real ) ) ) )
% 9.10/5.22       => ( ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_real ) ) ) )
% 9.10/5.22         => ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3389_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 9.10/5.22        ( ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [I4: real] :
% 9.10/5.22                ( ( member_real @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_rat ) ) ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_rat ) ) ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_rat ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3390_prod_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 9.10/5.22        ( ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [I4: nat] :
% 9.10/5.22                ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != one_one_rat ) ) ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != one_one_rat ) ) ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( times_times_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != one_one_rat ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3391_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 9.10/5.22        ( ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [I4: real] :
% 9.10/5.22                ( ( member_real @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_complex ) ) ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_complex ) ) ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3392_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 9.10/5.22        ( ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [I4: nat] :
% 9.10/5.22                ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_complex ) ) ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_complex ) ) ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3393_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 9.10/5.22        ( ( finite_finite_int
% 9.10/5.22          @ ( collect_int
% 9.10/5.22            @ ^ [I4: int] :
% 9.10/5.22                ( ( member_int @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_complex ) ) ) )
% 9.10/5.22       => ( ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_complex ) ) ) )
% 9.10/5.22         => ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3394_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 9.10/5.22        ( ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [I4: complex] :
% 9.10/5.22                ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_complex ) ) ) )
% 9.10/5.22       => ( ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_complex ) ) ) )
% 9.10/5.22         => ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_complex @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3395_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 9.10/5.22        ( ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [I4: real] :
% 9.10/5.22                ( ( member_real @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_real ) ) ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_real ) ) ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3396_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 9.10/5.22        ( ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [I4: nat] :
% 9.10/5.22                ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_real ) ) ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_real ) ) ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3397_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 9.10/5.22        ( ( finite_finite_int
% 9.10/5.22          @ ( collect_int
% 9.10/5.22            @ ^ [I4: int] :
% 9.10/5.22                ( ( member_int @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_real ) ) ) )
% 9.10/5.22       => ( ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_real ) ) ) )
% 9.10/5.22         => ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [I4: int] :
% 9.10/5.22                  ( ( member_int @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3398_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 9.10/5.22        ( ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [I4: complex] :
% 9.10/5.22                ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_real ) ) ) )
% 9.10/5.22       => ( ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_real ) ) ) )
% 9.10/5.22         => ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [I4: complex] :
% 9.10/5.22                  ( ( member_complex @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3399_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 9.10/5.22        ( ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [I4: real] :
% 9.10/5.22                ( ( member_real @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_rat ) ) ) )
% 9.10/5.22       => ( ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_rat ) ) ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [I4: real] :
% 9.10/5.22                  ( ( member_real @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3400_sum_Ofinite__Collect__op,axiom,
% 9.10/5.22      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 9.10/5.22        ( ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [I4: nat] :
% 9.10/5.22                ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                & ( ( X3 @ I4 )
% 9.10/5.22                 != zero_zero_rat ) ) ) )
% 9.10/5.22       => ( ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( Y3 @ I4 )
% 9.10/5.22                   != zero_zero_rat ) ) ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [I4: nat] :
% 9.10/5.22                  ( ( member_nat @ I4 @ I5 )
% 9.10/5.22                  & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y3 @ I4 ) )
% 9.10/5.22                   != zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3401_finite__interval__int1,axiom,
% 9.10/5.22      ! [A: int,B2: int] :
% 9.10/5.22        ( finite_finite_int
% 9.10/5.22        @ ( collect_int
% 9.10/5.22          @ ^ [I4: int] :
% 9.10/5.22              ( ( ord_less_eq_int @ A @ I4 )
% 9.10/5.22              & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3402_finite__interval__int4,axiom,
% 9.10/5.22      ! [A: int,B2: int] :
% 9.10/5.22        ( finite_finite_int
% 9.10/5.22        @ ( collect_int
% 9.10/5.22          @ ^ [I4: int] :
% 9.10/5.22              ( ( ord_less_int @ A @ I4 )
% 9.10/5.22              & ( ord_less_int @ I4 @ B2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3403_finite__interval__int3,axiom,
% 9.10/5.22      ! [A: int,B2: int] :
% 9.10/5.22        ( finite_finite_int
% 9.10/5.22        @ ( collect_int
% 9.10/5.22          @ ^ [I4: int] :
% 9.10/5.22              ( ( ord_less_int @ A @ I4 )
% 9.10/5.22              & ( ord_less_eq_int @ I4 @ B2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3404_finite__interval__int2,axiom,
% 9.10/5.22      ! [A: int,B2: int] :
% 9.10/5.22        ( finite_finite_int
% 9.10/5.22        @ ( collect_int
% 9.10/5.22          @ ^ [I4: int] :
% 9.10/5.22              ( ( ord_less_eq_int @ A @ I4 )
% 9.10/5.22              & ( ord_less_int @ I4 @ B2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3405_finite__maxlen,axiom,
% 9.10/5.22      ! [M7: set_list_VEBT_VEBT] :
% 9.10/5.22        ( ( finite3004134309566078307T_VEBT @ M7 )
% 9.10/5.22       => ? [N3: nat] :
% 9.10/5.22          ! [X6: list_VEBT_VEBT] :
% 9.10/5.22            ( ( member2936631157270082147T_VEBT @ X6 @ M7 )
% 9.10/5.22           => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X6 ) @ N3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3406_finite__maxlen,axiom,
% 9.10/5.22      ! [M7: set_list_o] :
% 9.10/5.22        ( ( finite_finite_list_o @ M7 )
% 9.10/5.22       => ? [N3: nat] :
% 9.10/5.22          ! [X6: list_o] :
% 9.10/5.22            ( ( member_list_o @ X6 @ M7 )
% 9.10/5.22           => ( ord_less_nat @ ( size_size_list_o @ X6 ) @ N3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3407_finite__maxlen,axiom,
% 9.10/5.22      ! [M7: set_list_nat] :
% 9.10/5.22        ( ( finite8100373058378681591st_nat @ M7 )
% 9.10/5.22       => ? [N3: nat] :
% 9.10/5.22          ! [X6: list_nat] :
% 9.10/5.22            ( ( member_list_nat @ X6 @ M7 )
% 9.10/5.22           => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3408_finite__has__minimal2,axiom,
% 9.10/5.22      ! [A4: set_real,A: real] :
% 9.10/5.22        ( ( finite_finite_real @ A4 )
% 9.10/5.22       => ( ( member_real @ A @ A4 )
% 9.10/5.22         => ? [X5: real] :
% 9.10/5.22              ( ( member_real @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_real @ X5 @ A )
% 9.10/5.22              & ! [Xa: real] :
% 9.10/5.22                  ( ( member_real @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_real @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3409_finite__has__minimal2,axiom,
% 9.10/5.22      ! [A4: set_set_int,A: set_int] :
% 9.10/5.22        ( ( finite6197958912794628473et_int @ A4 )
% 9.10/5.22       => ( ( member_set_int @ A @ A4 )
% 9.10/5.22         => ? [X5: set_int] :
% 9.10/5.22              ( ( member_set_int @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_set_int @ X5 @ A )
% 9.10/5.22              & ! [Xa: set_int] :
% 9.10/5.22                  ( ( member_set_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3410_finite__has__minimal2,axiom,
% 9.10/5.22      ! [A4: set_rat,A: rat] :
% 9.10/5.22        ( ( finite_finite_rat @ A4 )
% 9.10/5.22       => ( ( member_rat @ A @ A4 )
% 9.10/5.22         => ? [X5: rat] :
% 9.10/5.22              ( ( member_rat @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_rat @ X5 @ A )
% 9.10/5.22              & ! [Xa: rat] :
% 9.10/5.22                  ( ( member_rat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3411_finite__has__minimal2,axiom,
% 9.10/5.22      ! [A4: set_num,A: num] :
% 9.10/5.22        ( ( finite_finite_num @ A4 )
% 9.10/5.22       => ( ( member_num @ A @ A4 )
% 9.10/5.22         => ? [X5: num] :
% 9.10/5.22              ( ( member_num @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_num @ X5 @ A )
% 9.10/5.22              & ! [Xa: num] :
% 9.10/5.22                  ( ( member_num @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_num @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3412_finite__has__minimal2,axiom,
% 9.10/5.22      ! [A4: set_nat,A: nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( ( member_nat @ A @ A4 )
% 9.10/5.22         => ? [X5: nat] :
% 9.10/5.22              ( ( member_nat @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_nat @ X5 @ A )
% 9.10/5.22              & ! [Xa: nat] :
% 9.10/5.22                  ( ( member_nat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3413_finite__has__minimal2,axiom,
% 9.10/5.22      ! [A4: set_int,A: int] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( ( member_int @ A @ A4 )
% 9.10/5.22         => ? [X5: int] :
% 9.10/5.22              ( ( member_int @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_int @ X5 @ A )
% 9.10/5.22              & ! [Xa: int] :
% 9.10/5.22                  ( ( member_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_int @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3414_finite__has__maximal2,axiom,
% 9.10/5.22      ! [A4: set_real,A: real] :
% 9.10/5.22        ( ( finite_finite_real @ A4 )
% 9.10/5.22       => ( ( member_real @ A @ A4 )
% 9.10/5.22         => ? [X5: real] :
% 9.10/5.22              ( ( member_real @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_real @ A @ X5 )
% 9.10/5.22              & ! [Xa: real] :
% 9.10/5.22                  ( ( member_real @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_real @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3415_finite__has__maximal2,axiom,
% 9.10/5.22      ! [A4: set_set_int,A: set_int] :
% 9.10/5.22        ( ( finite6197958912794628473et_int @ A4 )
% 9.10/5.22       => ( ( member_set_int @ A @ A4 )
% 9.10/5.22         => ? [X5: set_int] :
% 9.10/5.22              ( ( member_set_int @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_set_int @ A @ X5 )
% 9.10/5.22              & ! [Xa: set_int] :
% 9.10/5.22                  ( ( member_set_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3416_finite__has__maximal2,axiom,
% 9.10/5.22      ! [A4: set_rat,A: rat] :
% 9.10/5.22        ( ( finite_finite_rat @ A4 )
% 9.10/5.22       => ( ( member_rat @ A @ A4 )
% 9.10/5.22         => ? [X5: rat] :
% 9.10/5.22              ( ( member_rat @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_rat @ A @ X5 )
% 9.10/5.22              & ! [Xa: rat] :
% 9.10/5.22                  ( ( member_rat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3417_finite__has__maximal2,axiom,
% 9.10/5.22      ! [A4: set_num,A: num] :
% 9.10/5.22        ( ( finite_finite_num @ A4 )
% 9.10/5.22       => ( ( member_num @ A @ A4 )
% 9.10/5.22         => ? [X5: num] :
% 9.10/5.22              ( ( member_num @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_num @ A @ X5 )
% 9.10/5.22              & ! [Xa: num] :
% 9.10/5.22                  ( ( member_num @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_num @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3418_finite__has__maximal2,axiom,
% 9.10/5.22      ! [A4: set_nat,A: nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( ( member_nat @ A @ A4 )
% 9.10/5.22         => ? [X5: nat] :
% 9.10/5.22              ( ( member_nat @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_nat @ A @ X5 )
% 9.10/5.22              & ! [Xa: nat] :
% 9.10/5.22                  ( ( member_nat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3419_finite__has__maximal2,axiom,
% 9.10/5.22      ! [A4: set_int,A: int] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( ( member_int @ A @ A4 )
% 9.10/5.22         => ? [X5: int] :
% 9.10/5.22              ( ( member_int @ X5 @ A4 )
% 9.10/5.22              & ( ord_less_eq_int @ A @ X5 )
% 9.10/5.22              & ! [Xa: int] :
% 9.10/5.22                  ( ( member_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_int @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3420_rev__finite__subset,axiom,
% 9.10/5.22      ! [B5: set_nat,A4: set_nat] :
% 9.10/5.22        ( ( finite_finite_nat @ B5 )
% 9.10/5.22       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.22         => ( finite_finite_nat @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3421_rev__finite__subset,axiom,
% 9.10/5.22      ! [B5: set_complex,A4: set_complex] :
% 9.10/5.22        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.22       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.22         => ( finite3207457112153483333omplex @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3422_rev__finite__subset,axiom,
% 9.10/5.22      ! [B5: set_int,A4: set_int] :
% 9.10/5.22        ( ( finite_finite_int @ B5 )
% 9.10/5.22       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.22         => ( finite_finite_int @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3423_infinite__super,axiom,
% 9.10/5.22      ! [S2: set_nat,T3: set_nat] :
% 9.10/5.22        ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.22       => ( ~ ( finite_finite_nat @ S2 )
% 9.10/5.22         => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3424_infinite__super,axiom,
% 9.10/5.22      ! [S2: set_complex,T3: set_complex] :
% 9.10/5.22        ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.22       => ( ~ ( finite3207457112153483333omplex @ S2 )
% 9.10/5.22         => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3425_infinite__super,axiom,
% 9.10/5.22      ! [S2: set_int,T3: set_int] :
% 9.10/5.22        ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.22       => ( ~ ( finite_finite_int @ S2 )
% 9.10/5.22         => ~ ( finite_finite_int @ T3 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3426_finite__subset,axiom,
% 9.10/5.22      ! [A4: set_nat,B5: set_nat] :
% 9.10/5.22        ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.22       => ( ( finite_finite_nat @ B5 )
% 9.10/5.22         => ( finite_finite_nat @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3427_finite__subset,axiom,
% 9.10/5.22      ! [A4: set_complex,B5: set_complex] :
% 9.10/5.22        ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.22       => ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.22         => ( finite3207457112153483333omplex @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3428_finite__subset,axiom,
% 9.10/5.22      ! [A4: set_int,B5: set_int] :
% 9.10/5.22        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.22       => ( ( finite_finite_int @ B5 )
% 9.10/5.22         => ( finite_finite_int @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3429_finite__psubset__induct,axiom,
% 9.10/5.22      ! [A4: set_nat,P: set_nat > $o] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( ! [A7: set_nat] :
% 9.10/5.22              ( ( finite_finite_nat @ A7 )
% 9.10/5.22             => ( ! [B7: set_nat] :
% 9.10/5.22                    ( ( ord_less_set_nat @ B7 @ A7 )
% 9.10/5.22                   => ( P @ B7 ) )
% 9.10/5.22               => ( P @ A7 ) ) )
% 9.10/5.22         => ( P @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3430_finite__psubset__induct,axiom,
% 9.10/5.22      ! [A4: set_int,P: set_int > $o] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( ! [A7: set_int] :
% 9.10/5.22              ( ( finite_finite_int @ A7 )
% 9.10/5.22             => ( ! [B7: set_int] :
% 9.10/5.22                    ( ( ord_less_set_int @ B7 @ A7 )
% 9.10/5.22                   => ( P @ B7 ) )
% 9.10/5.22               => ( P @ A7 ) ) )
% 9.10/5.22         => ( P @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3431_finite__psubset__induct,axiom,
% 9.10/5.22      ! [A4: set_complex,P: set_complex > $o] :
% 9.10/5.22        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.22       => ( ! [A7: set_complex] :
% 9.10/5.22              ( ( finite3207457112153483333omplex @ A7 )
% 9.10/5.22             => ( ! [B7: set_complex] :
% 9.10/5.22                    ( ( ord_less_set_complex @ B7 @ A7 )
% 9.10/5.22                   => ( P @ B7 ) )
% 9.10/5.22               => ( P @ A7 ) ) )
% 9.10/5.22         => ( P @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3432_finite__image__set,axiom,
% 9.10/5.22      ! [P: real > $o,F: real > real] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [Uu3: real] :
% 9.10/5.22              ? [X: real] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3433_finite__image__set,axiom,
% 9.10/5.22      ! [P: real > $o,F: real > nat] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [Uu3: nat] :
% 9.10/5.22              ? [X: real] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3434_finite__image__set,axiom,
% 9.10/5.22      ! [P: real > $o,F: real > int] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( finite_finite_int
% 9.10/5.22          @ ( collect_int
% 9.10/5.22            @ ^ [Uu3: int] :
% 9.10/5.22              ? [X: real] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3435_finite__image__set,axiom,
% 9.10/5.22      ! [P: real > $o,F: real > complex] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [Uu3: complex] :
% 9.10/5.22              ? [X: real] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3436_finite__image__set,axiom,
% 9.10/5.22      ! [P: nat > $o,F: nat > real] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [Uu3: real] :
% 9.10/5.22              ? [X: nat] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3437_finite__image__set,axiom,
% 9.10/5.22      ! [P: nat > $o,F: nat > nat] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [Uu3: nat] :
% 9.10/5.22              ? [X: nat] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3438_finite__image__set,axiom,
% 9.10/5.22      ! [P: nat > $o,F: nat > int] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( finite_finite_int
% 9.10/5.22          @ ( collect_int
% 9.10/5.22            @ ^ [Uu3: int] :
% 9.10/5.22              ? [X: nat] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3439_finite__image__set,axiom,
% 9.10/5.22      ! [P: nat > $o,F: nat > complex] :
% 9.10/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.10/5.22       => ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [Uu3: complex] :
% 9.10/5.22              ? [X: nat] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3440_finite__image__set,axiom,
% 9.10/5.22      ! [P: int > $o,F: int > real] :
% 9.10/5.22        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.10/5.22       => ( finite_finite_real
% 9.10/5.22          @ ( collect_real
% 9.10/5.22            @ ^ [Uu3: real] :
% 9.10/5.22              ? [X: int] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3441_finite__image__set,axiom,
% 9.10/5.22      ! [P: int > $o,F: int > nat] :
% 9.10/5.22        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.10/5.22       => ( finite_finite_nat
% 9.10/5.22          @ ( collect_nat
% 9.10/5.22            @ ^ [Uu3: nat] :
% 9.10/5.22              ? [X: int] :
% 9.10/5.22                ( ( Uu3
% 9.10/5.22                  = ( F @ X ) )
% 9.10/5.22                & ( P @ X ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3442_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: real > $o,F: real > real > real] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [Uu3: real] :
% 9.10/5.22                ? [X: real,Y: real] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3443_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: real > $o,F: real > real > nat] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [Uu3: nat] :
% 9.10/5.22                ? [X: real,Y: real] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3444_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: real > $o,F: real > real > int] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.10/5.22         => ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [Uu3: int] :
% 9.10/5.22                ? [X: real,Y: real] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3445_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: real > $o,F: real > real > complex] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.10/5.22         => ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [Uu3: complex] :
% 9.10/5.22                ? [X: real,Y: real] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3446_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [Uu3: real] :
% 9.10/5.22                ? [X: real,Y: nat] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3447_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [Uu3: nat] :
% 9.10/5.22                ? [X: real,Y: nat] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3448_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: nat > $o,F: real > nat > int] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.10/5.22         => ( finite_finite_int
% 9.10/5.22            @ ( collect_int
% 9.10/5.22              @ ^ [Uu3: int] :
% 9.10/5.22                ? [X: real,Y: nat] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3449_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.10/5.22         => ( finite3207457112153483333omplex
% 9.10/5.22            @ ( collect_complex
% 9.10/5.22              @ ^ [Uu3: complex] :
% 9.10/5.22                ? [X: real,Y: nat] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3450_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: int > $o,F: real > int > real] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 9.10/5.22         => ( finite_finite_real
% 9.10/5.22            @ ( collect_real
% 9.10/5.22              @ ^ [Uu3: real] :
% 9.10/5.22                ? [X: real,Y: int] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3451_finite__image__set2,axiom,
% 9.10/5.22      ! [P: real > $o,Q: int > $o,F: real > int > nat] :
% 9.10/5.22        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.10/5.22       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 9.10/5.22         => ( finite_finite_nat
% 9.10/5.22            @ ( collect_nat
% 9.10/5.22              @ ^ [Uu3: nat] :
% 9.10/5.22                ? [X: real,Y: int] :
% 9.10/5.22                  ( ( Uu3
% 9.10/5.22                    = ( F @ X @ Y ) )
% 9.10/5.22                  & ( P @ X )
% 9.10/5.22                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3452_finite__has__minimal,axiom,
% 9.10/5.22      ! [A4: set_real] :
% 9.10/5.22        ( ( finite_finite_real @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_real )
% 9.10/5.22         => ? [X5: real] :
% 9.10/5.22              ( ( member_real @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: real] :
% 9.10/5.22                  ( ( member_real @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_real @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3453_finite__has__minimal,axiom,
% 9.10/5.22      ! [A4: set_set_int] :
% 9.10/5.22        ( ( finite6197958912794628473et_int @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_set_int )
% 9.10/5.22         => ? [X5: set_int] :
% 9.10/5.22              ( ( member_set_int @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: set_int] :
% 9.10/5.22                  ( ( member_set_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3454_finite__has__minimal,axiom,
% 9.10/5.22      ! [A4: set_rat] :
% 9.10/5.22        ( ( finite_finite_rat @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_rat )
% 9.10/5.22         => ? [X5: rat] :
% 9.10/5.22              ( ( member_rat @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: rat] :
% 9.10/5.22                  ( ( member_rat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3455_finite__has__minimal,axiom,
% 9.10/5.22      ! [A4: set_num] :
% 9.10/5.22        ( ( finite_finite_num @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_num )
% 9.10/5.22         => ? [X5: num] :
% 9.10/5.22              ( ( member_num @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: num] :
% 9.10/5.22                  ( ( member_num @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_num @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3456_finite__has__minimal,axiom,
% 9.10/5.22      ! [A4: set_nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_nat )
% 9.10/5.22         => ? [X5: nat] :
% 9.10/5.22              ( ( member_nat @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: nat] :
% 9.10/5.22                  ( ( member_nat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3457_finite__has__minimal,axiom,
% 9.10/5.22      ! [A4: set_int] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_int )
% 9.10/5.22         => ? [X5: int] :
% 9.10/5.22              ( ( member_int @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: int] :
% 9.10/5.22                  ( ( member_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_int @ Xa @ X5 )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3458_finite__has__maximal,axiom,
% 9.10/5.22      ! [A4: set_real] :
% 9.10/5.22        ( ( finite_finite_real @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_real )
% 9.10/5.22         => ? [X5: real] :
% 9.10/5.22              ( ( member_real @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: real] :
% 9.10/5.22                  ( ( member_real @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_real @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3459_finite__has__maximal,axiom,
% 9.10/5.22      ! [A4: set_set_int] :
% 9.10/5.22        ( ( finite6197958912794628473et_int @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_set_int )
% 9.10/5.22         => ? [X5: set_int] :
% 9.10/5.22              ( ( member_set_int @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: set_int] :
% 9.10/5.22                  ( ( member_set_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3460_finite__has__maximal,axiom,
% 9.10/5.22      ! [A4: set_rat] :
% 9.10/5.22        ( ( finite_finite_rat @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_rat )
% 9.10/5.22         => ? [X5: rat] :
% 9.10/5.22              ( ( member_rat @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: rat] :
% 9.10/5.22                  ( ( member_rat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3461_finite__has__maximal,axiom,
% 9.10/5.22      ! [A4: set_num] :
% 9.10/5.22        ( ( finite_finite_num @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_num )
% 9.10/5.22         => ? [X5: num] :
% 9.10/5.22              ( ( member_num @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: num] :
% 9.10/5.22                  ( ( member_num @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_num @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3462_finite__has__maximal,axiom,
% 9.10/5.22      ! [A4: set_nat] :
% 9.10/5.22        ( ( finite_finite_nat @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_nat )
% 9.10/5.22         => ? [X5: nat] :
% 9.10/5.22              ( ( member_nat @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: nat] :
% 9.10/5.22                  ( ( member_nat @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3463_finite__has__maximal,axiom,
% 9.10/5.22      ! [A4: set_int] :
% 9.10/5.22        ( ( finite_finite_int @ A4 )
% 9.10/5.22       => ( ( A4 != bot_bot_set_int )
% 9.10/5.22         => ? [X5: int] :
% 9.10/5.22              ( ( member_int @ X5 @ A4 )
% 9.10/5.22              & ! [Xa: int] :
% 9.10/5.22                  ( ( member_int @ Xa @ A4 )
% 9.10/5.22                 => ( ( ord_less_eq_int @ X5 @ Xa )
% 9.10/5.22                   => ( X5 = Xa ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3464_finite__nth__roots,axiom,
% 9.10/5.22      ! [N: nat,C: complex] :
% 9.10/5.22        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.22       => ( finite3207457112153483333omplex
% 9.10/5.22          @ ( collect_complex
% 9.10/5.22            @ ^ [Z2: complex] :
% 9.10/5.22                ( ( power_power_complex @ Z2 @ N )
% 9.10/5.22                = C ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3465_insert__simp__excp,axiom,
% 9.10/5.22      ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X3: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.10/5.22       => ( ( ord_less_nat @ X3 @ Mi )
% 9.10/5.22         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.10/5.22           => ( ( X3 != Ma )
% 9.10/5.22             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.10/5.22                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3466_insert__simp__norm,axiom,
% 9.10/5.22      ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.10/5.22       => ( ( ord_less_nat @ Mi @ X3 )
% 9.10/5.22         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.10/5.22           => ( ( X3 != Ma )
% 9.10/5.22             => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.10/5.22                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X3 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( 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 ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3467_gcd__nat__induct,axiom,
% 9.10/5.22      ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.10/5.22        ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 9.10/5.22       => ( ! [M4: nat,N3: nat] :
% 9.10/5.22              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.10/5.22             => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
% 9.10/5.22               => ( P @ M4 @ N3 ) ) )
% 9.10/5.22         => ( P @ M @ N ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3468_concat__bit__Suc,axiom,
% 9.10/5.22      ! [N: nat,K: int,L: int] :
% 9.10/5.22        ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 9.10/5.22        = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3469_order__refl,axiom,
% 9.10/5.22      ! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3470_order__refl,axiom,
% 9.10/5.22      ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3471_order__refl,axiom,
% 9.10/5.22      ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3472_order__refl,axiom,
% 9.10/5.22      ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3473_order__refl,axiom,
% 9.10/5.22      ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3474_dual__order_Orefl,axiom,
% 9.10/5.22      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3475_dual__order_Orefl,axiom,
% 9.10/5.22      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3476_dual__order_Orefl,axiom,
% 9.10/5.22      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3477_dual__order_Orefl,axiom,
% 9.10/5.22      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3478_dual__order_Orefl,axiom,
% 9.10/5.22      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3479_length__list__update,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 9.10/5.22        ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) )
% 9.10/5.22        = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3480_length__list__update,axiom,
% 9.10/5.22      ! [Xs2: list_o,I: nat,X3: $o] :
% 9.10/5.22        ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X3 ) )
% 9.10/5.22        = ( size_size_list_o @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3481_length__list__update,axiom,
% 9.10/5.22      ! [Xs2: list_nat,I: nat,X3: nat] :
% 9.10/5.22        ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) )
% 9.10/5.22        = ( size_size_list_nat @ Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3482_max__Suc__Suc,axiom,
% 9.10/5.22      ! [M: nat,N: nat] :
% 9.10/5.22        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.10/5.22        = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3483_max__0R,axiom,
% 9.10/5.22      ! [N: nat] :
% 9.10/5.22        ( ( ord_max_nat @ N @ zero_zero_nat )
% 9.10/5.22        = N ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3484_max__0L,axiom,
% 9.10/5.22      ! [N: nat] :
% 9.10/5.22        ( ( ord_max_nat @ zero_zero_nat @ N )
% 9.10/5.22        = N ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3485_max__nat_Oright__neutral,axiom,
% 9.10/5.22      ! [A: nat] :
% 9.10/5.22        ( ( ord_max_nat @ A @ zero_zero_nat )
% 9.10/5.22        = A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3486_max__nat_Oneutr__eq__iff,axiom,
% 9.10/5.22      ! [A: nat,B2: nat] :
% 9.10/5.22        ( ( zero_zero_nat
% 9.10/5.22          = ( ord_max_nat @ A @ B2 ) )
% 9.10/5.22        = ( ( A = zero_zero_nat )
% 9.10/5.22          & ( B2 = zero_zero_nat ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3487_max__nat_Oleft__neutral,axiom,
% 9.10/5.22      ! [A: nat] :
% 9.10/5.22        ( ( ord_max_nat @ zero_zero_nat @ A )
% 9.10/5.22        = A ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3488_max__nat_Oeq__neutr__iff,axiom,
% 9.10/5.22      ! [A: nat,B2: nat] :
% 9.10/5.22        ( ( ( ord_max_nat @ A @ B2 )
% 9.10/5.22          = zero_zero_nat )
% 9.10/5.22        = ( ( A = zero_zero_nat )
% 9.10/5.22          & ( B2 = zero_zero_nat ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3489_nth__list__update__neq,axiom,
% 9.10/5.22      ! [I: nat,J: nat,Xs2: list_int,X3: int] :
% 9.10/5.22        ( ( I != J )
% 9.10/5.22       => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.22          = ( nth_int @ Xs2 @ J ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3490_nth__list__update__neq,axiom,
% 9.10/5.22      ! [I: nat,J: nat,Xs2: list_nat,X3: nat] :
% 9.10/5.22        ( ( I != J )
% 9.10/5.22       => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.22          = ( nth_nat @ Xs2 @ J ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3491_nth__list__update__neq,axiom,
% 9.10/5.22      ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.10/5.22        ( ( I != J )
% 9.10/5.22       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.22          = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3492_list__update__id,axiom,
% 9.10/5.22      ! [Xs2: list_int,I: nat] :
% 9.10/5.22        ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
% 9.10/5.22        = Xs2 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3493_list__update__id,axiom,
% 9.10/5.22      ! [Xs2: list_nat,I: nat] :
% 9.10/5.22        ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
% 9.10/5.22        = Xs2 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3494_list__update__id,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,I: nat] :
% 9.10/5.22        ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
% 9.10/5.22        = Xs2 ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3495_concat__bit__0,axiom,
% 9.10/5.22      ! [K: int,L: int] :
% 9.10/5.22        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 9.10/5.22        = L ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3496_max__number__of_I1_J,axiom,
% 9.10/5.22      ! [U: num,V: num] :
% 9.10/5.22        ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.10/5.22         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.10/5.22            = ( numera1916890842035813515d_enat @ V ) ) )
% 9.10/5.22        & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.10/5.22         => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.10/5.22            = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3497_max__number__of_I1_J,axiom,
% 9.10/5.22      ! [U: num,V: num] :
% 9.10/5.22        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.22         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.22            = ( numeral_numeral_real @ V ) ) )
% 9.10/5.22        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.22         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.22            = ( numeral_numeral_real @ U ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3498_max__number__of_I1_J,axiom,
% 9.10/5.22      ! [U: num,V: num] :
% 9.10/5.22        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.22         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.22            = ( numeral_numeral_rat @ V ) ) )
% 9.10/5.22        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.22         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.22            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3499_max__number__of_I1_J,axiom,
% 9.10/5.22      ! [U: num,V: num] :
% 9.10/5.22        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.10/5.22         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.10/5.22            = ( numeral_numeral_nat @ V ) ) )
% 9.10/5.22        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.10/5.22         => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.10/5.22            = ( numeral_numeral_nat @ U ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3500_max__number__of_I1_J,axiom,
% 9.10/5.22      ! [U: num,V: num] :
% 9.10/5.22        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.22         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.22            = ( numeral_numeral_int @ V ) ) )
% 9.10/5.22        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.22         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.22            = ( numeral_numeral_int @ U ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3501_max__0__1_I4_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ zero_z5237406670263579293d_enat )
% 9.10/5.22        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3502_max__0__1_I4_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ zero_zero_real )
% 9.10/5.22        = ( numeral_numeral_real @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3503_max__0__1_I4_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ zero_zero_rat )
% 9.10/5.22        = ( numeral_numeral_rat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3504_max__0__1_I4_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ zero_zero_nat )
% 9.10/5.22        = ( numeral_numeral_nat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3505_max__0__1_I4_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ zero_zero_int )
% 9.10/5.22        = ( numeral_numeral_int @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3506_max__0__1_I3_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
% 9.10/5.22        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3507_max__0__1_I3_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_real @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3508_max__0__1_I3_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_rat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3509_max__0__1_I3_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_nat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3510_max__0__1_I3_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_int @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3511_max__0__1_I1_J,axiom,
% 9.10/5.22      ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 9.10/5.22      = one_one_real ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3512_max__0__1_I1_J,axiom,
% 9.10/5.22      ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 9.10/5.22      = one_one_rat ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3513_max__0__1_I1_J,axiom,
% 9.10/5.22      ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 9.10/5.22      = one_one_nat ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3514_max__0__1_I1_J,axiom,
% 9.10/5.22      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 9.10/5.22      = one_on7984719198319812577d_enat ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3515_max__0__1_I1_J,axiom,
% 9.10/5.22      ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 9.10/5.22      = one_one_int ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3516_max__0__1_I2_J,axiom,
% 9.10/5.22      ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 9.10/5.22      = one_one_real ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3517_max__0__1_I2_J,axiom,
% 9.10/5.22      ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 9.10/5.22      = one_one_rat ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3518_max__0__1_I2_J,axiom,
% 9.10/5.22      ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 9.10/5.22      = one_one_nat ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3519_max__0__1_I2_J,axiom,
% 9.10/5.22      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 9.10/5.22      = one_on7984719198319812577d_enat ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3520_max__0__1_I2_J,axiom,
% 9.10/5.22      ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 9.10/5.22      = one_one_int ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3521_max__0__1_I6_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat )
% 9.10/5.22        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3522_max__0__1_I6_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_real @ ( numeral_numeral_real @ X3 ) @ one_one_real )
% 9.10/5.22        = ( numeral_numeral_real @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3523_max__0__1_I6_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat )
% 9.10/5.22        = ( numeral_numeral_rat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3524_max__0__1_I6_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat )
% 9.10/5.22        = ( numeral_numeral_nat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3525_max__0__1_I6_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_int @ ( numeral_numeral_int @ X3 ) @ one_one_int )
% 9.10/5.22        = ( numeral_numeral_int @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3526_max__0__1_I5_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
% 9.10/5.22        = ( numera1916890842035813515d_enat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3527_max__0__1_I5_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_real @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3528_max__0__1_I5_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_rat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3529_max__0__1_I5_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_nat @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3530_max__0__1_I5_J,axiom,
% 9.10/5.22      ! [X3: num] :
% 9.10/5.22        ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
% 9.10/5.22        = ( numeral_numeral_int @ X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3531_list__update__beyond,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
% 9.10/5.22       => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 )
% 9.10/5.22          = Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3532_list__update__beyond,axiom,
% 9.10/5.22      ! [Xs2: list_o,I: nat,X3: $o] :
% 9.10/5.22        ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
% 9.10/5.22       => ( ( list_update_o @ Xs2 @ I @ X3 )
% 9.10/5.22          = Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3533_list__update__beyond,axiom,
% 9.10/5.22      ! [Xs2: list_nat,I: nat,X3: nat] :
% 9.10/5.22        ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
% 9.10/5.22       => ( ( list_update_nat @ Xs2 @ I @ X3 )
% 9.10/5.22          = Xs2 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3534_concat__bit__nonnegative__iff,axiom,
% 9.10/5.22      ! [N: nat,K: int,L: int] :
% 9.10/5.22        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 9.10/5.22        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3535_concat__bit__negative__iff,axiom,
% 9.10/5.22      ! [N: nat,K: int,L: int] :
% 9.10/5.22        ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 9.10/5.22        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3536_nth__list__update__eq,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_int,X3: int] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 9.10/5.22       => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ I )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3537_nth__list__update__eq,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.10/5.22       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ I )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3538_nth__list__update__eq,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_o,X3: $o] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 9.10/5.22       => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X3 ) @ I )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3539_nth__list__update__eq,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_nat,X3: nat] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 9.10/5.22       => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ I )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3540_set__swap,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_int,J: nat] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 9.10/5.22       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
% 9.10/5.22         => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
% 9.10/5.22            = ( set_int2 @ Xs2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3541_set__swap,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.10/5.22       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.10/5.22         => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
% 9.10/5.22            = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3542_set__swap,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_o,J: nat] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 9.10/5.22       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
% 9.10/5.22         => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
% 9.10/5.22            = ( set_o2 @ Xs2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3543_set__swap,axiom,
% 9.10/5.22      ! [I: nat,Xs2: list_nat,J: nat] :
% 9.10/5.22        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 9.10/5.22       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
% 9.10/5.22         => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
% 9.10/5.22            = ( set_nat2 @ Xs2 ) ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3544_max__def,axiom,
% 9.10/5.22      ( ord_ma741700101516333627d_enat
% 9.10/5.22      = ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3545_max__def,axiom,
% 9.10/5.22      ( ord_max_set_int
% 9.10/5.22      = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3546_max__def,axiom,
% 9.10/5.22      ( ord_max_rat
% 9.10/5.22      = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3547_max__def,axiom,
% 9.10/5.22      ( ord_max_num
% 9.10/5.22      = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3548_max__def,axiom,
% 9.10/5.22      ( ord_max_nat
% 9.10/5.22      = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3549_max__def,axiom,
% 9.10/5.22      ( ord_max_int
% 9.10/5.22      = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3550_max__absorb1,axiom,
% 9.10/5.22      ! [Y3: extended_enat,X3: extended_enat] :
% 9.10/5.22        ( ( ord_le2932123472753598470d_enat @ Y3 @ X3 )
% 9.10/5.22       => ( ( ord_ma741700101516333627d_enat @ X3 @ Y3 )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3551_max__absorb1,axiom,
% 9.10/5.22      ! [Y3: set_int,X3: set_int] :
% 9.10/5.22        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.10/5.22       => ( ( ord_max_set_int @ X3 @ Y3 )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3552_max__absorb1,axiom,
% 9.10/5.22      ! [Y3: rat,X3: rat] :
% 9.10/5.22        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.10/5.22       => ( ( ord_max_rat @ X3 @ Y3 )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3553_max__absorb1,axiom,
% 9.10/5.22      ! [Y3: num,X3: num] :
% 9.10/5.22        ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.10/5.22       => ( ( ord_max_num @ X3 @ Y3 )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3554_max__absorb1,axiom,
% 9.10/5.22      ! [Y3: nat,X3: nat] :
% 9.10/5.22        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.22       => ( ( ord_max_nat @ X3 @ Y3 )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3555_max__absorb1,axiom,
% 9.10/5.22      ! [Y3: int,X3: int] :
% 9.10/5.22        ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.10/5.22       => ( ( ord_max_int @ X3 @ Y3 )
% 9.10/5.22          = X3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3556_max__absorb2,axiom,
% 9.10/5.22      ! [X3: extended_enat,Y3: extended_enat] :
% 9.10/5.22        ( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
% 9.10/5.22       => ( ( ord_ma741700101516333627d_enat @ X3 @ Y3 )
% 9.10/5.22          = Y3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3557_max__absorb2,axiom,
% 9.10/5.22      ! [X3: set_int,Y3: set_int] :
% 9.10/5.22        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.22       => ( ( ord_max_set_int @ X3 @ Y3 )
% 9.10/5.22          = Y3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3558_max__absorb2,axiom,
% 9.10/5.22      ! [X3: rat,Y3: rat] :
% 9.10/5.22        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.22       => ( ( ord_max_rat @ X3 @ Y3 )
% 9.10/5.22          = Y3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3559_max__absorb2,axiom,
% 9.10/5.22      ! [X3: num,Y3: num] :
% 9.10/5.22        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.22       => ( ( ord_max_num @ X3 @ Y3 )
% 9.10/5.22          = Y3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3560_max__absorb2,axiom,
% 9.10/5.22      ! [X3: nat,Y3: nat] :
% 9.10/5.22        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.22       => ( ( ord_max_nat @ X3 @ Y3 )
% 9.10/5.22          = Y3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3561_max__absorb2,axiom,
% 9.10/5.22      ! [X3: int,Y3: int] :
% 9.10/5.22        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.22       => ( ( ord_max_int @ X3 @ Y3 )
% 9.10/5.22          = Y3 ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3562_max__add__distrib__left,axiom,
% 9.10/5.22      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.22        ( ( plus_plus_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_real @ ( plus_plus_real @ X3 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3563_max__add__distrib__left,axiom,
% 9.10/5.22      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.22        ( ( plus_plus_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3564_max__add__distrib__left,axiom,
% 9.10/5.22      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.22        ( ( plus_plus_nat @ ( ord_max_nat @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3565_max__add__distrib__left,axiom,
% 9.10/5.22      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.22        ( ( plus_plus_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_int @ ( plus_plus_int @ X3 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3566_max__add__distrib__right,axiom,
% 9.10/5.22      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.22        ( ( plus_plus_real @ X3 @ ( ord_max_real @ Y3 @ Z ) )
% 9.10/5.22        = ( ord_max_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( plus_plus_real @ X3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3567_max__add__distrib__right,axiom,
% 9.10/5.22      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.22        ( ( plus_plus_rat @ X3 @ ( ord_max_rat @ Y3 @ Z ) )
% 9.10/5.22        = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( plus_plus_rat @ X3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3568_max__add__distrib__right,axiom,
% 9.10/5.22      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.22        ( ( plus_plus_nat @ X3 @ ( ord_max_nat @ Y3 @ Z ) )
% 9.10/5.22        = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( plus_plus_nat @ X3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3569_max__add__distrib__right,axiom,
% 9.10/5.22      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.22        ( ( plus_plus_int @ X3 @ ( ord_max_int @ Y3 @ Z ) )
% 9.10/5.22        = ( ord_max_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( plus_plus_int @ X3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3570_max__diff__distrib__left,axiom,
% 9.10/5.22      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.22        ( ( minus_minus_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_real @ ( minus_minus_real @ X3 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3571_max__diff__distrib__left,axiom,
% 9.10/5.22      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.22        ( ( minus_minus_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_rat @ ( minus_minus_rat @ X3 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3572_max__diff__distrib__left,axiom,
% 9.10/5.22      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.22        ( ( minus_minus_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
% 9.10/5.22        = ( ord_max_int @ ( minus_minus_int @ X3 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3573_nat__add__max__left,axiom,
% 9.10/5.22      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.22        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 9.10/5.22        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3574_nat__add__max__right,axiom,
% 9.10/5.22      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.22        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 9.10/5.22        = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3575_nat__mult__max__right,axiom,
% 9.10/5.22      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.22        ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 9.10/5.22        = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3576_nat__mult__max__left,axiom,
% 9.10/5.22      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.22        ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 9.10/5.22        = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3577_max__def__raw,axiom,
% 9.10/5.22      ( ord_ma741700101516333627d_enat
% 9.10/5.22      = ( ^ [A5: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3578_max__def__raw,axiom,
% 9.10/5.22      ( ord_max_set_int
% 9.10/5.22      = ( ^ [A5: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3579_max__def__raw,axiom,
% 9.10/5.22      ( ord_max_rat
% 9.10/5.22      = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3580_max__def__raw,axiom,
% 9.10/5.22      ( ord_max_num
% 9.10/5.22      = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3581_max__def__raw,axiom,
% 9.10/5.22      ( ord_max_nat
% 9.10/5.22      = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3582_max__def__raw,axiom,
% 9.10/5.22      ( ord_max_int
% 9.10/5.22      = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3583_concat__bit__assoc,axiom,
% 9.10/5.22      ! [N: nat,K: int,M: nat,L: int,R: int] :
% 9.10/5.22        ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R ) )
% 9.10/5.22        = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3584_nat__minus__add__max,axiom,
% 9.10/5.22      ! [N: nat,M: nat] :
% 9.10/5.22        ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 9.10/5.22        = ( ord_max_nat @ N @ M ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3585_set__update__subsetI,axiom,
% 9.10/5.22      ! [Xs2: list_real,A4: set_real,X3: real,I: nat] :
% 9.10/5.22        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A4 )
% 9.10/5.22       => ( ( member_real @ X3 @ A4 )
% 9.10/5.22         => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3586_set__update__subsetI,axiom,
% 9.10/5.22      ! [Xs2: list_complex,A4: set_complex,X3: complex,I: nat] :
% 9.10/5.22        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A4 )
% 9.10/5.22       => ( ( member_complex @ X3 @ A4 )
% 9.10/5.22         => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3587_set__update__subsetI,axiom,
% 9.10/5.22      ! [Xs2: list_nat,A4: set_nat,X3: nat,I: nat] :
% 9.10/5.22        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
% 9.10/5.22       => ( ( member_nat @ X3 @ A4 )
% 9.10/5.22         => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 9.10/5.22  
% 9.10/5.22  thf(fact_3588_set__update__subsetI,axiom,
% 9.10/5.22      ! [Xs2: list_VEBT_VEBT,A4: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
% 9.10/5.23        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A4 )
% 9.10/5.23       => ( ( member_VEBT_VEBT @ X3 @ A4 )
% 9.10/5.23         => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3589_set__update__subsetI,axiom,
% 9.10/5.23      ! [Xs2: list_int,A4: set_int,X3: int,I: nat] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A4 )
% 9.10/5.23       => ( ( member_int @ X3 @ A4 )
% 9.10/5.23         => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3590_set__update__memI,axiom,
% 9.10/5.23      ! [N: nat,Xs2: list_real,X3: real] :
% 9.10/5.23        ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 9.10/5.23       => ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X3 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3591_set__update__memI,axiom,
% 9.10/5.23      ! [N: nat,Xs2: list_complex,X3: complex] :
% 9.10/5.23        ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 9.10/5.23       => ( member_complex @ X3 @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X3 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3592_set__update__memI,axiom,
% 9.10/5.23      ! [N: nat,Xs2: list_int,X3: int] :
% 9.10/5.23        ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 9.10/5.23       => ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X3 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3593_set__update__memI,axiom,
% 9.10/5.23      ! [N: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.10/5.23        ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.10/5.23       => ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X3 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3594_set__update__memI,axiom,
% 9.10/5.23      ! [N: nat,Xs2: list_o,X3: $o] :
% 9.10/5.23        ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 9.10/5.23       => ( member_o @ X3 @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X3 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3595_set__update__memI,axiom,
% 9.10/5.23      ! [N: nat,Xs2: list_nat,X3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 9.10/5.23       => ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X3 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3596_nle__le,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_rat @ A @ B2 ) )
% 9.10/5.23        = ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.23          & ( B2 != A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3597_nle__le,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_num @ A @ B2 ) )
% 9.10/5.23        = ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.23          & ( B2 != A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3598_nle__le,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
% 9.10/5.23        = ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.23          & ( B2 != A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3599_nle__le,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_int @ A @ B2 ) )
% 9.10/5.23        = ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.23          & ( B2 != A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3600_le__cases3,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.23        ( ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23         => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 9.10/5.23       => ( ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.10/5.23           => ~ ( ord_less_eq_rat @ X3 @ Z ) )
% 9.10/5.23         => ( ( ( ord_less_eq_rat @ X3 @ Z )
% 9.10/5.23             => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 9.10/5.23           => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 9.10/5.23               => ~ ( ord_less_eq_rat @ Y3 @ X3 ) )
% 9.10/5.23             => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 9.10/5.23                 => ~ ( ord_less_eq_rat @ Z @ X3 ) )
% 9.10/5.23               => ~ ( ( ord_less_eq_rat @ Z @ X3 )
% 9.10/5.23                   => ~ ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3601_le__cases3,axiom,
% 9.10/5.23      ! [X3: num,Y3: num,Z: num] :
% 9.10/5.23        ( ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23         => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 9.10/5.23       => ( ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.10/5.23           => ~ ( ord_less_eq_num @ X3 @ Z ) )
% 9.10/5.23         => ( ( ( ord_less_eq_num @ X3 @ Z )
% 9.10/5.23             => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 9.10/5.23           => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 9.10/5.23               => ~ ( ord_less_eq_num @ Y3 @ X3 ) )
% 9.10/5.23             => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 9.10/5.23                 => ~ ( ord_less_eq_num @ Z @ X3 ) )
% 9.10/5.23               => ~ ( ( ord_less_eq_num @ Z @ X3 )
% 9.10/5.23                   => ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3602_le__cases3,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.23        ( ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23         => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 9.10/5.23       => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.23           => ~ ( ord_less_eq_nat @ X3 @ Z ) )
% 9.10/5.23         => ( ( ( ord_less_eq_nat @ X3 @ Z )
% 9.10/5.23             => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 9.10/5.23           => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 9.10/5.23               => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
% 9.10/5.23             => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 9.10/5.23                 => ~ ( ord_less_eq_nat @ Z @ X3 ) )
% 9.10/5.23               => ~ ( ( ord_less_eq_nat @ Z @ X3 )
% 9.10/5.23                   => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3603_le__cases3,axiom,
% 9.10/5.23      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.23        ( ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23         => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 9.10/5.23       => ( ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.10/5.23           => ~ ( ord_less_eq_int @ X3 @ Z ) )
% 9.10/5.23         => ( ( ( ord_less_eq_int @ X3 @ Z )
% 9.10/5.23             => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 9.10/5.23           => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 9.10/5.23               => ~ ( ord_less_eq_int @ Y3 @ X3 ) )
% 9.10/5.23             => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 9.10/5.23                 => ~ ( ord_less_eq_int @ Z @ X3 ) )
% 9.10/5.23               => ~ ( ( ord_less_eq_int @ Z @ X3 )
% 9.10/5.23                   => ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3604_order__class_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [X: set_int,Y: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ X @ Y )
% 9.10/5.23            & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3605_order__class_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [X: rat,Y: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ X @ Y )
% 9.10/5.23            & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3606_order__class_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [X: num,Y: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ X @ Y )
% 9.10/5.23            & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3607_order__class_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [X: nat,Y: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ X @ Y )
% 9.10/5.23            & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3608_order__class_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [X: int,Y: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ X @ Y )
% 9.10/5.23            & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3609_ord__eq__le__trans,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int,C: set_int] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3610_ord__eq__le__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3611_ord__eq__le__trans,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3612_ord__eq__le__trans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3613_ord__eq__le__trans,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_eq_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3614_ord__le__eq__trans,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3615_ord__le__eq__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3616_ord__le__eq__trans,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3617_ord__le__eq__trans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3618_ord__le__eq__trans,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3619_order__antisym,axiom,
% 9.10/5.23      ! [X3: set_int,Y3: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.10/5.23         => ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3620_order__antisym,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.10/5.23         => ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3621_order__antisym,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.10/5.23         => ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3622_order__antisym,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.23         => ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3623_order__antisym,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.10/5.23         => ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3624_order_Otrans,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3625_order_Otrans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3626_order_Otrans,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3627_order_Otrans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3628_order_Otrans,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3629_order__trans,axiom,
% 9.10/5.23      ! [X3: set_int,Y3: set_int,Z: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_eq_set_int @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3630_order__trans,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_eq_rat @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3631_order__trans,axiom,
% 9.10/5.23      ! [X3: num,Y3: num,Z: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_num @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_eq_num @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3632_order__trans,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3633_order__trans,axiom,
% 9.10/5.23      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_eq_int @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3634_linorder__wlog,axiom,
% 9.10/5.23      ! [P: rat > rat > $o,A: rat,B2: rat] :
% 9.10/5.23        ( ! [A3: rat,B3: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: rat,B3: rat] :
% 9.10/5.23              ( ( P @ B3 @ A3 )
% 9.10/5.23             => ( P @ A3 @ B3 ) )
% 9.10/5.23         => ( P @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3635_linorder__wlog,axiom,
% 9.10/5.23      ! [P: num > num > $o,A: num,B2: num] :
% 9.10/5.23        ( ! [A3: num,B3: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: num,B3: num] :
% 9.10/5.23              ( ( P @ B3 @ A3 )
% 9.10/5.23             => ( P @ A3 @ B3 ) )
% 9.10/5.23         => ( P @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3636_linorder__wlog,axiom,
% 9.10/5.23      ! [P: nat > nat > $o,A: nat,B2: nat] :
% 9.10/5.23        ( ! [A3: nat,B3: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: nat,B3: nat] :
% 9.10/5.23              ( ( P @ B3 @ A3 )
% 9.10/5.23             => ( P @ A3 @ B3 ) )
% 9.10/5.23         => ( P @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3637_linorder__wlog,axiom,
% 9.10/5.23      ! [P: int > int > $o,A: int,B2: int] :
% 9.10/5.23        ( ! [A3: int,B3: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: int,B3: int] :
% 9.10/5.23              ( ( P @ B3 @ A3 )
% 9.10/5.23             => ( P @ A3 @ B3 ) )
% 9.10/5.23         => ( P @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3638_dual__order_Oeq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: set_int,B4: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ B4 @ A5 )
% 9.10/5.23            & ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3639_dual__order_Oeq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: rat,B4: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ B4 @ A5 )
% 9.10/5.23            & ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3640_dual__order_Oeq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: num,B4: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ B4 @ A5 )
% 9.10/5.23            & ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3641_dual__order_Oeq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ B4 @ A5 )
% 9.10/5.23            & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3642_dual__order_Oeq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: int,B4: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ B4 @ A5 )
% 9.10/5.23            & ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3643_dual__order_Oantisym,axiom,
% 9.10/5.23      ! [B2: set_int,A: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3644_dual__order_Oantisym,axiom,
% 9.10/5.23      ! [B2: rat,A: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3645_dual__order_Oantisym,axiom,
% 9.10/5.23      ! [B2: num,A: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3646_dual__order_Oantisym,axiom,
% 9.10/5.23      ! [B2: nat,A: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3647_dual__order_Oantisym,axiom,
% 9.10/5.23      ! [B2: int,A: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3648_dual__order_Otrans,axiom,
% 9.10/5.23      ! [B2: set_int,A: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3649_dual__order_Otrans,axiom,
% 9.10/5.23      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_rat @ C @ B2 )
% 9.10/5.23         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3650_dual__order_Otrans,axiom,
% 9.10/5.23      ! [B2: num,A: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_num @ C @ B2 )
% 9.10/5.23         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3651_dual__order_Otrans,axiom,
% 9.10/5.23      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_nat @ C @ B2 )
% 9.10/5.23         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3652_dual__order_Otrans,axiom,
% 9.10/5.23      ! [B2: int,A: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3653_antisym,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ B2 @ A )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3654_antisym,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3655_antisym,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3656_antisym,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3657_antisym,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.23         => ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3658_Orderings_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: set_int,Z5: set_int] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: set_int,B4: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ A5 @ B4 )
% 9.10/5.23            & ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3659_Orderings_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: rat,Z5: rat] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: rat,B4: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ A5 @ B4 )
% 9.10/5.23            & ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3660_Orderings_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: num,Z5: num] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: num,B4: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ A5 @ B4 )
% 9.10/5.23            & ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3661_Orderings_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ A5 @ B4 )
% 9.10/5.23            & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3662_Orderings_Oorder__eq__iff,axiom,
% 9.10/5.23      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.10/5.23      = ( ^ [A5: int,B4: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ A5 @ B4 )
% 9.10/5.23            & ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3663_order__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3664_order__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: num > rat,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3665_order__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3666_order__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: int > rat,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_int @ B2 @ C )
% 9.10/5.23         => ( ! [X5: int,Y4: int] :
% 9.10/5.23                ( ( ord_less_eq_int @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3667_order__subst1,axiom,
% 9.10/5.23      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3668_order__subst1,axiom,
% 9.10/5.23      ! [A: num,F: num > num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3669_order__subst1,axiom,
% 9.10/5.23      ! [A: num,F: nat > num,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3670_order__subst1,axiom,
% 9.10/5.23      ! [A: num,F: int > num,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_int @ B2 @ C )
% 9.10/5.23         => ( ! [X5: int,Y4: int] :
% 9.10/5.23                ( ( ord_less_eq_int @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3671_order__subst1,axiom,
% 9.10/5.23      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3672_order__subst1,axiom,
% 9.10/5.23      ! [A: nat,F: num > nat,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3673_order__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3674_order__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3675_order__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3676_order__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3677_order__subst2,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3678_order__subst2,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3679_order__subst2,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3680_order__subst2,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3681_order__subst2,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3682_order__subst2,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,F: nat > num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3683_order__eq__refl,axiom,
% 9.10/5.23      ! [X3: set_int,Y3: set_int] :
% 9.10/5.23        ( ( X3 = Y3 )
% 9.10/5.23       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3684_order__eq__refl,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( X3 = Y3 )
% 9.10/5.23       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3685_order__eq__refl,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( X3 = Y3 )
% 9.10/5.23       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3686_order__eq__refl,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( X3 = Y3 )
% 9.10/5.23       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3687_order__eq__refl,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( X3 = Y3 )
% 9.10/5.23       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3688_linorder__linear,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23        | ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3689_linorder__linear,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23        | ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3690_linorder__linear,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23        | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3691_linorder__linear,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23        | ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3692_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3693_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3694_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3695_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: int,F: rat > int,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3696_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: rat,F: num > rat,B2: num,C: num] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3697_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: num,F: num > num,B2: num,C: num] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3698_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: nat,F: num > nat,B2: num,C: num] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3699_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: int,F: num > int,B2: num,C: num] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3700_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3701_ord__eq__le__subst,axiom,
% 9.10/5.23      ! [A: num,F: nat > num,B2: nat,C: nat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3702_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3703_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3704_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3705_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3706_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3707_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3708_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3709_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: num,B2: num,F: num > int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3710_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3711_ord__le__eq__subst,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,F: nat > num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3712_linorder__le__cases,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ~ ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3713_linorder__le__cases,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ~ ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3714_linorder__le__cases,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3715_linorder__le__cases,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ~ ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3716_order__antisym__conv,axiom,
% 9.10/5.23      ! [Y3: set_int,X3: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3717_order__antisym__conv,axiom,
% 9.10/5.23      ! [Y3: rat,X3: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3718_order__antisym__conv,axiom,
% 9.10/5.23      ! [Y3: num,X3: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.10/5.23       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3719_order__antisym__conv,axiom,
% 9.10/5.23      ! [Y3: nat,X3: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3720_order__antisym__conv,axiom,
% 9.10/5.23      ! [Y3: int,X3: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.10/5.23       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3721_nth__list__update,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_int,J: nat,X3: int] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 9.10/5.23       => ( ( ( I = J )
% 9.10/5.23           => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23              = X3 ) )
% 9.10/5.23          & ( ( I != J )
% 9.10/5.23           => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23              = ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3722_nth__list__update,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.10/5.23       => ( ( ( I = J )
% 9.10/5.23           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23              = X3 ) )
% 9.10/5.23          & ( ( I != J )
% 9.10/5.23           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23              = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3723_nth__list__update,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_o,X3: $o,J: nat] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 9.10/5.23       => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23          = ( ( ( I = J )
% 9.10/5.23             => X3 )
% 9.10/5.23            & ( ( I != J )
% 9.10/5.23             => ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3724_nth__list__update,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_nat,J: nat,X3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 9.10/5.23       => ( ( ( I = J )
% 9.10/5.23           => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23              = X3 ) )
% 9.10/5.23          & ( ( I != J )
% 9.10/5.23           => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X3 ) @ J )
% 9.10/5.23              = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3725_list__update__same__conv,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_int,X3: int] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 9.10/5.23       => ( ( ( list_update_int @ Xs2 @ I @ X3 )
% 9.10/5.23            = Xs2 )
% 9.10/5.23          = ( ( nth_int @ Xs2 @ I )
% 9.10/5.23            = X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3726_list__update__same__conv,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.10/5.23       => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 )
% 9.10/5.23            = Xs2 )
% 9.10/5.23          = ( ( nth_VEBT_VEBT @ Xs2 @ I )
% 9.10/5.23            = X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3727_list__update__same__conv,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_o,X3: $o] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 9.10/5.23       => ( ( ( list_update_o @ Xs2 @ I @ X3 )
% 9.10/5.23            = Xs2 )
% 9.10/5.23          = ( ( nth_o @ Xs2 @ I )
% 9.10/5.23            = X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3728_list__update__same__conv,axiom,
% 9.10/5.23      ! [I: nat,Xs2: list_nat,X3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 9.10/5.23       => ( ( ( list_update_nat @ Xs2 @ I @ X3 )
% 9.10/5.23            = Xs2 )
% 9.10/5.23          = ( ( nth_nat @ Xs2 @ I )
% 9.10/5.23            = X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3729_lt__ex,axiom,
% 9.10/5.23      ! [X3: real] :
% 9.10/5.23      ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3730_lt__ex,axiom,
% 9.10/5.23      ! [X3: rat] :
% 9.10/5.23      ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3731_lt__ex,axiom,
% 9.10/5.23      ! [X3: int] :
% 9.10/5.23      ? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3732_gt__ex,axiom,
% 9.10/5.23      ! [X3: real] :
% 9.10/5.23      ? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3733_gt__ex,axiom,
% 9.10/5.23      ! [X3: rat] :
% 9.10/5.23      ? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3734_gt__ex,axiom,
% 9.10/5.23      ! [X3: nat] :
% 9.10/5.23      ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3735_gt__ex,axiom,
% 9.10/5.23      ! [X3: int] :
% 9.10/5.23      ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3736_dense,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ? [Z4: real] :
% 9.10/5.23            ( ( ord_less_real @ X3 @ Z4 )
% 9.10/5.23            & ( ord_less_real @ Z4 @ Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3737_dense,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ? [Z4: rat] :
% 9.10/5.23            ( ( ord_less_rat @ X3 @ Z4 )
% 9.10/5.23            & ( ord_less_rat @ Z4 @ Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3738_less__imp__neq,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3739_less__imp__neq,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3740_less__imp__neq,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3741_less__imp__neq,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3742_less__imp__neq,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3743_order_Oasym,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_real @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3744_order_Oasym,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3745_order_Oasym,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_num @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3746_order_Oasym,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3747_order_Oasym,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_int @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3748_ord__eq__less__trans,axiom,
% 9.10/5.23      ! [A: real,B2: real,C: real] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3749_ord__eq__less__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3750_ord__eq__less__trans,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3751_ord__eq__less__trans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3752_ord__eq__less__trans,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( A = B2 )
% 9.10/5.23       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3753_ord__less__eq__trans,axiom,
% 9.10/5.23      ! [A: real,B2: real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3754_ord__less__eq__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3755_ord__less__eq__trans,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3756_ord__less__eq__trans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3757_ord__less__eq__trans,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ( ( B2 = C )
% 9.10/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3758_less__induct,axiom,
% 9.10/5.23      ! [P: nat > $o,A: nat] :
% 9.10/5.23        ( ! [X5: nat] :
% 9.10/5.23            ( ! [Y5: nat] :
% 9.10/5.23                ( ( ord_less_nat @ Y5 @ X5 )
% 9.10/5.23               => ( P @ Y5 ) )
% 9.10/5.23           => ( P @ X5 ) )
% 9.10/5.23       => ( P @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3759_antisym__conv3,axiom,
% 9.10/5.23      ! [Y3: real,X3: real] :
% 9.10/5.23        ( ~ ( ord_less_real @ Y3 @ X3 )
% 9.10/5.23       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3760_antisym__conv3,axiom,
% 9.10/5.23      ! [Y3: rat,X3: rat] :
% 9.10/5.23        ( ~ ( ord_less_rat @ Y3 @ X3 )
% 9.10/5.23       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3761_antisym__conv3,axiom,
% 9.10/5.23      ! [Y3: num,X3: num] :
% 9.10/5.23        ( ~ ( ord_less_num @ Y3 @ X3 )
% 9.10/5.23       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3762_antisym__conv3,axiom,
% 9.10/5.23      ! [Y3: nat,X3: nat] :
% 9.10/5.23        ( ~ ( ord_less_nat @ Y3 @ X3 )
% 9.10/5.23       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3763_antisym__conv3,axiom,
% 9.10/5.23      ! [Y3: int,X3: int] :
% 9.10/5.23        ( ~ ( ord_less_int @ Y3 @ X3 )
% 9.10/5.23       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3764_linorder__cases,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ( X3 != Y3 )
% 9.10/5.23         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3765_linorder__cases,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( X3 != Y3 )
% 9.10/5.23         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3766_linorder__cases,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( X3 != Y3 )
% 9.10/5.23         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3767_linorder__cases,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( X3 != Y3 )
% 9.10/5.23         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3768_linorder__cases,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( X3 != Y3 )
% 9.10/5.23         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3769_dual__order_Oasym,axiom,
% 9.10/5.23      ! [B2: real,A: real] :
% 9.10/5.23        ( ( ord_less_real @ B2 @ A )
% 9.10/5.23       => ~ ( ord_less_real @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3770_dual__order_Oasym,axiom,
% 9.10/5.23      ! [B2: rat,A: rat] :
% 9.10/5.23        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.23       => ~ ( ord_less_rat @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3771_dual__order_Oasym,axiom,
% 9.10/5.23      ! [B2: num,A: num] :
% 9.10/5.23        ( ( ord_less_num @ B2 @ A )
% 9.10/5.23       => ~ ( ord_less_num @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3772_dual__order_Oasym,axiom,
% 9.10/5.23      ! [B2: nat,A: nat] :
% 9.10/5.23        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.23       => ~ ( ord_less_nat @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3773_dual__order_Oasym,axiom,
% 9.10/5.23      ! [B2: int,A: int] :
% 9.10/5.23        ( ( ord_less_int @ B2 @ A )
% 9.10/5.23       => ~ ( ord_less_int @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3774_dual__order_Oirrefl,axiom,
% 9.10/5.23      ! [A: real] :
% 9.10/5.23        ~ ( ord_less_real @ A @ A ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3775_dual__order_Oirrefl,axiom,
% 9.10/5.23      ! [A: rat] :
% 9.10/5.23        ~ ( ord_less_rat @ A @ A ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3776_dual__order_Oirrefl,axiom,
% 9.10/5.23      ! [A: num] :
% 9.10/5.23        ~ ( ord_less_num @ A @ A ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3777_dual__order_Oirrefl,axiom,
% 9.10/5.23      ! [A: nat] :
% 9.10/5.23        ~ ( ord_less_nat @ A @ A ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3778_dual__order_Oirrefl,axiom,
% 9.10/5.23      ! [A: int] :
% 9.10/5.23        ~ ( ord_less_int @ A @ A ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3779_exists__least__iff,axiom,
% 9.10/5.23      ( ( ^ [P2: nat > $o] :
% 9.10/5.23          ? [X4: nat] : ( P2 @ X4 ) )
% 9.10/5.23      = ( ^ [P3: nat > $o] :
% 9.10/5.23          ? [N2: nat] :
% 9.10/5.23            ( ( P3 @ N2 )
% 9.10/5.23            & ! [M3: nat] :
% 9.10/5.23                ( ( ord_less_nat @ M3 @ N2 )
% 9.10/5.23               => ~ ( P3 @ M3 ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3780_linorder__less__wlog,axiom,
% 9.10/5.23      ! [P: real > real > $o,A: real,B2: real] :
% 9.10/5.23        ( ! [A3: real,B3: real] :
% 9.10/5.23            ( ( ord_less_real @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: real] : ( P @ A3 @ A3 )
% 9.10/5.23         => ( ! [A3: real,B3: real] :
% 9.10/5.23                ( ( P @ B3 @ A3 )
% 9.10/5.23               => ( P @ A3 @ B3 ) )
% 9.10/5.23           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3781_linorder__less__wlog,axiom,
% 9.10/5.23      ! [P: rat > rat > $o,A: rat,B2: rat] :
% 9.10/5.23        ( ! [A3: rat,B3: rat] :
% 9.10/5.23            ( ( ord_less_rat @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: rat] : ( P @ A3 @ A3 )
% 9.10/5.23         => ( ! [A3: rat,B3: rat] :
% 9.10/5.23                ( ( P @ B3 @ A3 )
% 9.10/5.23               => ( P @ A3 @ B3 ) )
% 9.10/5.23           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3782_linorder__less__wlog,axiom,
% 9.10/5.23      ! [P: num > num > $o,A: num,B2: num] :
% 9.10/5.23        ( ! [A3: num,B3: num] :
% 9.10/5.23            ( ( ord_less_num @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: num] : ( P @ A3 @ A3 )
% 9.10/5.23         => ( ! [A3: num,B3: num] :
% 9.10/5.23                ( ( P @ B3 @ A3 )
% 9.10/5.23               => ( P @ A3 @ B3 ) )
% 9.10/5.23           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3783_linorder__less__wlog,axiom,
% 9.10/5.23      ! [P: nat > nat > $o,A: nat,B2: nat] :
% 9.10/5.23        ( ! [A3: nat,B3: nat] :
% 9.10/5.23            ( ( ord_less_nat @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 9.10/5.23         => ( ! [A3: nat,B3: nat] :
% 9.10/5.23                ( ( P @ B3 @ A3 )
% 9.10/5.23               => ( P @ A3 @ B3 ) )
% 9.10/5.23           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3784_linorder__less__wlog,axiom,
% 9.10/5.23      ! [P: int > int > $o,A: int,B2: int] :
% 9.10/5.23        ( ! [A3: int,B3: int] :
% 9.10/5.23            ( ( ord_less_int @ A3 @ B3 )
% 9.10/5.23           => ( P @ A3 @ B3 ) )
% 9.10/5.23       => ( ! [A3: int] : ( P @ A3 @ A3 )
% 9.10/5.23         => ( ! [A3: int,B3: int] :
% 9.10/5.23                ( ( P @ B3 @ A3 )
% 9.10/5.23               => ( P @ A3 @ B3 ) )
% 9.10/5.23           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3785_order_Ostrict__trans,axiom,
% 9.10/5.23      ! [A: real,B2: real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3786_order_Ostrict__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3787_order_Ostrict__trans,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3788_order_Ostrict__trans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3789_order_Ostrict__trans,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3790_not__less__iff__gr__or__eq,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.10/5.23        = ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.23          | ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3791_not__less__iff__gr__or__eq,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.10/5.23        = ( ( ord_less_rat @ Y3 @ X3 )
% 9.10/5.23          | ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3792_not__less__iff__gr__or__eq,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.10/5.23        = ( ( ord_less_num @ Y3 @ X3 )
% 9.10/5.23          | ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3793_not__less__iff__gr__or__eq,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.10/5.23        = ( ( ord_less_nat @ Y3 @ X3 )
% 9.10/5.23          | ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3794_not__less__iff__gr__or__eq,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.10/5.23        = ( ( ord_less_int @ Y3 @ X3 )
% 9.10/5.23          | ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3795_dual__order_Ostrict__trans,axiom,
% 9.10/5.23      ! [B2: real,A: real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_real @ C @ B2 )
% 9.10/5.23         => ( ord_less_real @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3796_dual__order_Ostrict__trans,axiom,
% 9.10/5.23      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_rat @ C @ B2 )
% 9.10/5.23         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3797_dual__order_Ostrict__trans,axiom,
% 9.10/5.23      ! [B2: num,A: num,C: num] :
% 9.10/5.23        ( ( ord_less_num @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_num @ C @ B2 )
% 9.10/5.23         => ( ord_less_num @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3798_dual__order_Ostrict__trans,axiom,
% 9.10/5.23      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_nat @ C @ B2 )
% 9.10/5.23         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3799_dual__order_Ostrict__trans,axiom,
% 9.10/5.23      ! [B2: int,A: int,C: int] :
% 9.10/5.23        ( ( ord_less_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3800_order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3801_order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3802_order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3803_order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3804_order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3805_dual__order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [B2: real,A: real] :
% 9.10/5.23        ( ( ord_less_real @ B2 @ A )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3806_dual__order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [B2: rat,A: rat] :
% 9.10/5.23        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3807_dual__order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [B2: num,A: num] :
% 9.10/5.23        ( ( ord_less_num @ B2 @ A )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3808_dual__order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [B2: nat,A: nat] :
% 9.10/5.23        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3809_dual__order_Ostrict__implies__not__eq,axiom,
% 9.10/5.23      ! [B2: int,A: int] :
% 9.10/5.23        ( ( ord_less_int @ B2 @ A )
% 9.10/5.23       => ( A != B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3810_linorder__neqE,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3811_linorder__neqE,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3812_linorder__neqE,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23       => ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3813_linorder__neqE,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3814_linorder__neqE,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3815_order__less__asym,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3816_order__less__asym,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3817_order__less__asym,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3818_order__less__asym,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3819_order__less__asym,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3820_linorder__neq__iff,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23        = ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23          | ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3821_linorder__neq__iff,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23        = ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23          | ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3822_linorder__neq__iff,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23        = ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23          | ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3823_linorder__neq__iff,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23        = ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23          | ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3824_linorder__neq__iff,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( X3 != Y3 )
% 9.10/5.23        = ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23          | ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3825_order__less__asym_H,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_real @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3826_order__less__asym_H,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_rat @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3827_order__less__asym_H,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_num @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3828_order__less__asym_H,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_nat @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3829_order__less__asym_H,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ~ ( ord_less_int @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3830_order__less__trans,axiom,
% 9.10/5.23      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_real @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_real @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3831_order__less__trans,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_rat @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_rat @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3832_order__less__trans,axiom,
% 9.10/5.23      ! [X3: num,Y3: num,Z: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_num @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_num @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3833_order__less__trans,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_nat @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_nat @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3834_order__less__trans,axiom,
% 9.10/5.23      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_int @ Y3 @ Z )
% 9.10/5.23         => ( ord_less_int @ X3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3835_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: real,F: real > real,B2: real,C: real] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3836_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: rat,F: real > rat,B2: real,C: real] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3837_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: num,F: real > num,B2: real,C: real] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3838_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: nat,F: real > nat,B2: real,C: real] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3839_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: int,F: real > int,B2: real,C: real] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3840_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3841_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3842_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3843_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3844_ord__eq__less__subst,axiom,
% 9.10/5.23      ! [A: int,F: rat > int,B2: rat,C: rat] :
% 9.10/5.23        ( ( A
% 9.10/5.23          = ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3845_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3846_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3847_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > num,C: num] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3848_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3849_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > int,C: int] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3850_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3851_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3852_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3853_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3854_ord__less__eq__subst,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ( F @ B2 )
% 9.10/5.23            = C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3855_order__less__irrefl,axiom,
% 9.10/5.23      ! [X3: real] :
% 9.10/5.23        ~ ( ord_less_real @ X3 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3856_order__less__irrefl,axiom,
% 9.10/5.23      ! [X3: rat] :
% 9.10/5.23        ~ ( ord_less_rat @ X3 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3857_order__less__irrefl,axiom,
% 9.10/5.23      ! [X3: num] :
% 9.10/5.23        ~ ( ord_less_num @ X3 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3858_order__less__irrefl,axiom,
% 9.10/5.23      ! [X3: nat] :
% 9.10/5.23        ~ ( ord_less_nat @ X3 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3859_order__less__irrefl,axiom,
% 9.10/5.23      ! [X3: int] :
% 9.10/5.23        ~ ( ord_less_int @ X3 @ X3 ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3860_order__less__subst1,axiom,
% 9.10/5.23      ! [A: real,F: real > real,B2: real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3861_order__less__subst1,axiom,
% 9.10/5.23      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3862_order__less__subst1,axiom,
% 9.10/5.23      ! [A: real,F: num > real,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3863_order__less__subst1,axiom,
% 9.10/5.23      ! [A: real,F: nat > real,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3864_order__less__subst1,axiom,
% 9.10/5.23      ! [A: real,F: int > real,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.23         => ( ! [X5: int,Y4: int] :
% 9.10/5.23                ( ( ord_less_int @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3865_order__less__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: real > rat,B2: real,C: real] :
% 9.10/5.23        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3866_order__less__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3867_order__less__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: num > rat,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.23         => ( ! [X5: num,Y4: num] :
% 9.10/5.23                ( ( ord_less_num @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3868_order__less__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.23         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.23                ( ( ord_less_nat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3869_order__less__subst1,axiom,
% 9.10/5.23      ! [A: rat,F: int > rat,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.23       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.23         => ( ! [X5: int,Y4: int] :
% 9.10/5.23                ( ( ord_less_int @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3870_order__less__subst2,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3871_order__less__subst2,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3872_order__less__subst2,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > num,C: num] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3873_order__less__subst2,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3874_order__less__subst2,axiom,
% 9.10/5.23      ! [A: real,B2: real,F: real > int,C: int] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: real,Y4: real] :
% 9.10/5.23                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3875_order__less__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3876_order__less__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3877_order__less__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3878_order__less__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3879_order__less__subst2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 9.10/5.23         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.23                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.23               => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.23           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3880_order__less__not__sym,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3881_order__less__not__sym,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3882_order__less__not__sym,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3883_order__less__not__sym,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3884_order__less__not__sym,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3885_order__less__imp__triv,axiom,
% 9.10/5.23      ! [X3: real,Y3: real,P: $o] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.23         => P ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3886_order__less__imp__triv,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat,P: $o] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_rat @ Y3 @ X3 )
% 9.10/5.23         => P ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3887_order__less__imp__triv,axiom,
% 9.10/5.23      ! [X3: num,Y3: num,P: $o] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_num @ Y3 @ X3 )
% 9.10/5.23         => P ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3888_order__less__imp__triv,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat,P: $o] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_nat @ Y3 @ X3 )
% 9.10/5.23         => P ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3889_order__less__imp__triv,axiom,
% 9.10/5.23      ! [X3: int,Y3: int,P: $o] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_int @ Y3 @ X3 )
% 9.10/5.23         => P ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3890_linorder__less__linear,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23        | ( X3 = Y3 )
% 9.10/5.23        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3891_linorder__less__linear,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23        | ( X3 = Y3 )
% 9.10/5.23        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3892_linorder__less__linear,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23        | ( X3 = Y3 )
% 9.10/5.23        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3893_linorder__less__linear,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23        | ( X3 = Y3 )
% 9.10/5.23        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3894_linorder__less__linear,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23        | ( X3 = Y3 )
% 9.10/5.23        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3895_order__less__imp__not__eq,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3896_order__less__imp__not__eq,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3897_order__less__imp__not__eq,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3898_order__less__imp__not__eq,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3899_order__less__imp__not__eq,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( X3 != Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3900_order__less__imp__not__eq2,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( Y3 != X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3901_order__less__imp__not__eq2,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( Y3 != X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3902_order__less__imp__not__eq2,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( Y3 != X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3903_order__less__imp__not__eq2,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( Y3 != X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3904_order__less__imp__not__eq2,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( Y3 != X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3905_order__less__imp__not__less,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3906_order__less__imp__not__less,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3907_order__less__imp__not__less,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3908_order__less__imp__not__less,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3909_order__less__imp__not__less,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3910_small__lazy_H_Ocases,axiom,
% 9.10/5.23      ! [X3: product_prod_int_int] :
% 9.10/5.23        ~ ! [D3: int,I3: int] :
% 9.10/5.23            ( X3
% 9.10/5.23           != ( product_Pair_int_int @ D3 @ I3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3911_exhaustive__int_H_Ocases,axiom,
% 9.10/5.23      ! [X3: produc7773217078559923341nt_int] :
% 9.10/5.23        ~ ! [F2: int > option6357759511663192854e_term,D3: int,I3: int] :
% 9.10/5.23            ( X3
% 9.10/5.23           != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3912_full__exhaustive__int_H_Ocases,axiom,
% 9.10/5.23      ! [X3: produc2285326912895808259nt_int] :
% 9.10/5.23        ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I3: int] :
% 9.10/5.23            ( X3
% 9.10/5.23           != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3913_leD,axiom,
% 9.10/5.23      ! [Y3: real,X3: real] :
% 9.10/5.23        ( ( ord_less_eq_real @ Y3 @ X3 )
% 9.10/5.23       => ~ ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3914_leD,axiom,
% 9.10/5.23      ! [Y3: set_int,X3: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.10/5.23       => ~ ( ord_less_set_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3915_leD,axiom,
% 9.10/5.23      ! [Y3: rat,X3: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.10/5.23       => ~ ( ord_less_rat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3916_leD,axiom,
% 9.10/5.23      ! [Y3: num,X3: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.10/5.23       => ~ ( ord_less_num @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3917_leD,axiom,
% 9.10/5.23      ! [Y3: nat,X3: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.23       => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3918_leD,axiom,
% 9.10/5.23      ! [Y3: int,X3: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.10/5.23       => ~ ( ord_less_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3919_leI,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3920_leI,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3921_leI,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3922_leI,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3923_leI,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3924_nless__le,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( ~ ( ord_less_real @ A @ B2 ) )
% 9.10/5.23        = ( ~ ( ord_less_eq_real @ A @ B2 )
% 9.10/5.23          | ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3925_nless__le,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int] :
% 9.10/5.23        ( ( ~ ( ord_less_set_int @ A @ B2 ) )
% 9.10/5.23        = ( ~ ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23          | ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3926_nless__le,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ~ ( ord_less_rat @ A @ B2 ) )
% 9.10/5.23        = ( ~ ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23          | ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3927_nless__le,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ~ ( ord_less_num @ A @ B2 ) )
% 9.10/5.23        = ( ~ ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23          | ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3928_nless__le,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ~ ( ord_less_nat @ A @ B2 ) )
% 9.10/5.23        = ( ~ ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23          | ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3929_nless__le,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ~ ( ord_less_int @ A @ B2 ) )
% 9.10/5.23        = ( ~ ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23          | ( A = B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3930_antisym__conv1,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3931_antisym__conv1,axiom,
% 9.10/5.23      ! [X3: set_int,Y3: set_int] :
% 9.10/5.23        ( ~ ( ord_less_set_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3932_antisym__conv1,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3933_antisym__conv1,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3934_antisym__conv1,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3935_antisym__conv1,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3936_antisym__conv2,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.23       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3937_antisym__conv2,axiom,
% 9.10/5.23      ! [X3: set_int,Y3: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ~ ( ord_less_set_int @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3938_antisym__conv2,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3939_antisym__conv2,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.23       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3940_antisym__conv2,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3941_antisym__conv2,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.23       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.10/5.23          = ( X3 = Y3 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3942_dense__ge,axiom,
% 9.10/5.23      ! [Z: real,Y3: real] :
% 9.10/5.23        ( ! [X5: real] :
% 9.10/5.23            ( ( ord_less_real @ Z @ X5 )
% 9.10/5.23           => ( ord_less_eq_real @ Y3 @ X5 ) )
% 9.10/5.23       => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3943_dense__ge,axiom,
% 9.10/5.23      ! [Z: rat,Y3: rat] :
% 9.10/5.23        ( ! [X5: rat] :
% 9.10/5.23            ( ( ord_less_rat @ Z @ X5 )
% 9.10/5.23           => ( ord_less_eq_rat @ Y3 @ X5 ) )
% 9.10/5.23       => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3944_dense__le,axiom,
% 9.10/5.23      ! [Y3: real,Z: real] :
% 9.10/5.23        ( ! [X5: real] :
% 9.10/5.23            ( ( ord_less_real @ X5 @ Y3 )
% 9.10/5.23           => ( ord_less_eq_real @ X5 @ Z ) )
% 9.10/5.23       => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3945_dense__le,axiom,
% 9.10/5.23      ! [Y3: rat,Z: rat] :
% 9.10/5.23        ( ! [X5: rat] :
% 9.10/5.23            ( ( ord_less_rat @ X5 @ Y3 )
% 9.10/5.23           => ( ord_less_eq_rat @ X5 @ Z ) )
% 9.10/5.23       => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3946_less__le__not__le,axiom,
% 9.10/5.23      ( ord_less_real
% 9.10/5.23      = ( ^ [X: real,Y: real] :
% 9.10/5.23            ( ( ord_less_eq_real @ X @ Y )
% 9.10/5.23            & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3947_less__le__not__le,axiom,
% 9.10/5.23      ( ord_less_set_int
% 9.10/5.23      = ( ^ [X: set_int,Y: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ X @ Y )
% 9.10/5.23            & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3948_less__le__not__le,axiom,
% 9.10/5.23      ( ord_less_rat
% 9.10/5.23      = ( ^ [X: rat,Y: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ X @ Y )
% 9.10/5.23            & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3949_less__le__not__le,axiom,
% 9.10/5.23      ( ord_less_num
% 9.10/5.23      = ( ^ [X: num,Y: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ X @ Y )
% 9.10/5.23            & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3950_less__le__not__le,axiom,
% 9.10/5.23      ( ord_less_nat
% 9.10/5.23      = ( ^ [X: nat,Y: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ X @ Y )
% 9.10/5.23            & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3951_less__le__not__le,axiom,
% 9.10/5.23      ( ord_less_int
% 9.10/5.23      = ( ^ [X: int,Y: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ X @ Y )
% 9.10/5.23            & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3952_not__le__imp__less,axiom,
% 9.10/5.23      ! [Y3: real,X3: real] :
% 9.10/5.23        ( ~ ( ord_less_eq_real @ Y3 @ X3 )
% 9.10/5.23       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3953_not__le__imp__less,axiom,
% 9.10/5.23      ! [Y3: rat,X3: rat] :
% 9.10/5.23        ( ~ ( ord_less_eq_rat @ Y3 @ X3 )
% 9.10/5.23       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3954_not__le__imp__less,axiom,
% 9.10/5.23      ! [Y3: num,X3: num] :
% 9.10/5.23        ( ~ ( ord_less_eq_num @ Y3 @ X3 )
% 9.10/5.23       => ( ord_less_num @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3955_not__le__imp__less,axiom,
% 9.10/5.23      ! [Y3: nat,X3: nat] :
% 9.10/5.23        ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.23       => ( ord_less_nat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3956_not__le__imp__less,axiom,
% 9.10/5.23      ! [Y3: int,X3: int] :
% 9.10/5.23        ( ~ ( ord_less_eq_int @ Y3 @ X3 )
% 9.10/5.23       => ( ord_less_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3957_order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_real
% 9.10/5.23      = ( ^ [A5: real,B4: real] :
% 9.10/5.23            ( ( ord_less_real @ A5 @ B4 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3958_order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_set_int
% 9.10/5.23      = ( ^ [A5: set_int,B4: set_int] :
% 9.10/5.23            ( ( ord_less_set_int @ A5 @ B4 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3959_order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_rat
% 9.10/5.23      = ( ^ [A5: rat,B4: rat] :
% 9.10/5.23            ( ( ord_less_rat @ A5 @ B4 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3960_order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_num
% 9.10/5.23      = ( ^ [A5: num,B4: num] :
% 9.10/5.23            ( ( ord_less_num @ A5 @ B4 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3961_order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_nat
% 9.10/5.23      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.23            ( ( ord_less_nat @ A5 @ B4 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3962_order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_int
% 9.10/5.23      = ( ^ [A5: int,B4: int] :
% 9.10/5.23            ( ( ord_less_int @ A5 @ B4 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3963_order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_real
% 9.10/5.23      = ( ^ [A5: real,B4: real] :
% 9.10/5.23            ( ( ord_less_eq_real @ A5 @ B4 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3964_order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_set_int
% 9.10/5.23      = ( ^ [A5: set_int,B4: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ A5 @ B4 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3965_order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_rat
% 9.10/5.23      = ( ^ [A5: rat,B4: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ A5 @ B4 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3966_order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_num
% 9.10/5.23      = ( ^ [A5: num,B4: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ A5 @ B4 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3967_order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_nat
% 9.10/5.23      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ A5 @ B4 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3968_order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_int
% 9.10/5.23      = ( ^ [A5: int,B4: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ A5 @ B4 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3969_order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [A: real,B2: real,C: real] :
% 9.10/5.23        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3970_order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_set_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3971_order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3972_order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3973_order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3974_order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3975_order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [A: real,B2: real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_real @ B2 @ C )
% 9.10/5.23         => ( ord_less_real @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3976_order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_set_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3977_order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.23         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3978_order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [A: num,B2: num,C: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.23         => ( ord_less_num @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3979_order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_nat @ B2 @ C )
% 9.10/5.23         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3980_order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [A: int,B2: int,C: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ( ( ord_less_eq_int @ B2 @ C )
% 9.10/5.23         => ( ord_less_int @ A @ C ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3981_order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_real
% 9.10/5.23      = ( ^ [A5: real,B4: real] :
% 9.10/5.23            ( ( ord_less_eq_real @ A5 @ B4 )
% 9.10/5.23            & ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3982_order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_set_int
% 9.10/5.23      = ( ^ [A5: set_int,B4: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ A5 @ B4 )
% 9.10/5.23            & ~ ( ord_less_eq_set_int @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3983_order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_rat
% 9.10/5.23      = ( ^ [A5: rat,B4: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ A5 @ B4 )
% 9.10/5.23            & ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3984_order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_num
% 9.10/5.23      = ( ^ [A5: num,B4: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ A5 @ B4 )
% 9.10/5.23            & ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3985_order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_nat
% 9.10/5.23      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ A5 @ B4 )
% 9.10/5.23            & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3986_order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_int
% 9.10/5.23      = ( ^ [A5: int,B4: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ A5 @ B4 )
% 9.10/5.23            & ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3987_dense__ge__bounded,axiom,
% 9.10/5.23      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ Z @ X3 )
% 9.10/5.23       => ( ! [W2: real] :
% 9.10/5.23              ( ( ord_less_real @ Z @ W2 )
% 9.10/5.23             => ( ( ord_less_real @ W2 @ X3 )
% 9.10/5.23               => ( ord_less_eq_real @ Y3 @ W2 ) ) )
% 9.10/5.23         => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3988_dense__ge__bounded,axiom,
% 9.10/5.23      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ Z @ X3 )
% 9.10/5.23       => ( ! [W2: rat] :
% 9.10/5.23              ( ( ord_less_rat @ Z @ W2 )
% 9.10/5.23             => ( ( ord_less_rat @ W2 @ X3 )
% 9.10/5.23               => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
% 9.10/5.23         => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3989_dense__le__bounded,axiom,
% 9.10/5.23      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ! [W2: real] :
% 9.10/5.23              ( ( ord_less_real @ X3 @ W2 )
% 9.10/5.23             => ( ( ord_less_real @ W2 @ Y3 )
% 9.10/5.23               => ( ord_less_eq_real @ W2 @ Z ) ) )
% 9.10/5.23         => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3990_dense__le__bounded,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ! [W2: rat] :
% 9.10/5.23              ( ( ord_less_rat @ X3 @ W2 )
% 9.10/5.23             => ( ( ord_less_rat @ W2 @ Y3 )
% 9.10/5.23               => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 9.10/5.23         => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3991_dual__order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_real
% 9.10/5.23      = ( ^ [B4: real,A5: real] :
% 9.10/5.23            ( ( ord_less_real @ B4 @ A5 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3992_dual__order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_set_int
% 9.10/5.23      = ( ^ [B4: set_int,A5: set_int] :
% 9.10/5.23            ( ( ord_less_set_int @ B4 @ A5 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3993_dual__order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_rat
% 9.10/5.23      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.23            ( ( ord_less_rat @ B4 @ A5 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3994_dual__order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_num
% 9.10/5.23      = ( ^ [B4: num,A5: num] :
% 9.10/5.23            ( ( ord_less_num @ B4 @ A5 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3995_dual__order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_nat
% 9.10/5.23      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.23            ( ( ord_less_nat @ B4 @ A5 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3996_dual__order_Oorder__iff__strict,axiom,
% 9.10/5.23      ( ord_less_eq_int
% 9.10/5.23      = ( ^ [B4: int,A5: int] :
% 9.10/5.23            ( ( ord_less_int @ B4 @ A5 )
% 9.10/5.23            | ( A5 = B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3997_dual__order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_real
% 9.10/5.23      = ( ^ [B4: real,A5: real] :
% 9.10/5.23            ( ( ord_less_eq_real @ B4 @ A5 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3998_dual__order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_set_int
% 9.10/5.23      = ( ^ [B4: set_int,A5: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ B4 @ A5 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_3999_dual__order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_rat
% 9.10/5.23      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ B4 @ A5 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4000_dual__order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_num
% 9.10/5.23      = ( ^ [B4: num,A5: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ B4 @ A5 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4001_dual__order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_nat
% 9.10/5.23      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ B4 @ A5 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4002_dual__order_Ostrict__iff__order,axiom,
% 9.10/5.23      ( ord_less_int
% 9.10/5.23      = ( ^ [B4: int,A5: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ B4 @ A5 )
% 9.10/5.23            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4003_dual__order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [B2: real,A: real,C: real] :
% 9.10/5.23        ( ( ord_less_eq_real @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_real @ C @ B2 )
% 9.10/5.23         => ( ord_less_real @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4004_dual__order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [B2: set_int,A: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_set_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4005_dual__order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_rat @ C @ B2 )
% 9.10/5.23         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4006_dual__order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [B2: num,A: num,C: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_num @ C @ B2 )
% 9.10/5.23         => ( ord_less_num @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4007_dual__order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_nat @ C @ B2 )
% 9.10/5.23         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4008_dual__order_Ostrict__trans1,axiom,
% 9.10/5.23      ! [B2: int,A: int,C: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4009_dual__order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [B2: real,A: real,C: real] :
% 9.10/5.23        ( ( ord_less_real @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_real @ C @ B2 )
% 9.10/5.23         => ( ord_less_real @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4010_dual__order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [B2: set_int,A: set_int,C: set_int] :
% 9.10/5.23        ( ( ord_less_set_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4011_dual__order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.23        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_rat @ C @ B2 )
% 9.10/5.23         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4012_dual__order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [B2: num,A: num,C: num] :
% 9.10/5.23        ( ( ord_less_num @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_num @ C @ B2 )
% 9.10/5.23         => ( ord_less_num @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4013_dual__order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.23        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_nat @ C @ B2 )
% 9.10/5.23         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4014_dual__order_Ostrict__trans2,axiom,
% 9.10/5.23      ! [B2: int,A: int,C: int] :
% 9.10/5.23        ( ( ord_less_int @ B2 @ A )
% 9.10/5.23       => ( ( ord_less_eq_int @ C @ B2 )
% 9.10/5.23         => ( ord_less_int @ C @ A ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4015_dual__order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_real
% 9.10/5.23      = ( ^ [B4: real,A5: real] :
% 9.10/5.23            ( ( ord_less_eq_real @ B4 @ A5 )
% 9.10/5.23            & ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4016_dual__order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_set_int
% 9.10/5.23      = ( ^ [B4: set_int,A5: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ B4 @ A5 )
% 9.10/5.23            & ~ ( ord_less_eq_set_int @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4017_dual__order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_rat
% 9.10/5.23      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ B4 @ A5 )
% 9.10/5.23            & ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4018_dual__order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_num
% 9.10/5.23      = ( ^ [B4: num,A5: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ B4 @ A5 )
% 9.10/5.23            & ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4019_dual__order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_nat
% 9.10/5.23      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ B4 @ A5 )
% 9.10/5.23            & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4020_dual__order_Ostrict__iff__not,axiom,
% 9.10/5.23      ( ord_less_int
% 9.10/5.23      = ( ^ [B4: int,A5: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ B4 @ A5 )
% 9.10/5.23            & ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4021_order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( ord_less_real @ A @ B2 )
% 9.10/5.23       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4022_order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int] :
% 9.10/5.23        ( ( ord_less_set_int @ A @ B2 )
% 9.10/5.23       => ( ord_less_eq_set_int @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4023_order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.23       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4024_order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ord_less_num @ A @ B2 )
% 9.10/5.23       => ( ord_less_eq_num @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4025_order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.23       => ( ord_less_eq_nat @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4026_order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ord_less_int @ A @ B2 )
% 9.10/5.23       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4027_dual__order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [B2: real,A: real] :
% 9.10/5.23        ( ( ord_less_real @ B2 @ A )
% 9.10/5.23       => ( ord_less_eq_real @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4028_dual__order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [B2: set_int,A: set_int] :
% 9.10/5.23        ( ( ord_less_set_int @ B2 @ A )
% 9.10/5.23       => ( ord_less_eq_set_int @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4029_dual__order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [B2: rat,A: rat] :
% 9.10/5.23        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.23       => ( ord_less_eq_rat @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4030_dual__order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [B2: num,A: num] :
% 9.10/5.23        ( ( ord_less_num @ B2 @ A )
% 9.10/5.23       => ( ord_less_eq_num @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4031_dual__order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [B2: nat,A: nat] :
% 9.10/5.23        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.23       => ( ord_less_eq_nat @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4032_dual__order_Ostrict__implies__order,axiom,
% 9.10/5.23      ! [B2: int,A: int] :
% 9.10/5.23        ( ( ord_less_int @ B2 @ A )
% 9.10/5.23       => ( ord_less_eq_int @ B2 @ A ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4033_order__le__less,axiom,
% 9.10/5.23      ( ord_less_eq_real
% 9.10/5.23      = ( ^ [X: real,Y: real] :
% 9.10/5.23            ( ( ord_less_real @ X @ Y )
% 9.10/5.23            | ( X = Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4034_order__le__less,axiom,
% 9.10/5.23      ( ord_less_eq_set_int
% 9.10/5.23      = ( ^ [X: set_int,Y: set_int] :
% 9.10/5.23            ( ( ord_less_set_int @ X @ Y )
% 9.10/5.23            | ( X = Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4035_order__le__less,axiom,
% 9.10/5.23      ( ord_less_eq_rat
% 9.10/5.23      = ( ^ [X: rat,Y: rat] :
% 9.10/5.23            ( ( ord_less_rat @ X @ Y )
% 9.10/5.23            | ( X = Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4036_order__le__less,axiom,
% 9.10/5.23      ( ord_less_eq_num
% 9.10/5.23      = ( ^ [X: num,Y: num] :
% 9.10/5.23            ( ( ord_less_num @ X @ Y )
% 9.10/5.23            | ( X = Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4037_order__le__less,axiom,
% 9.10/5.23      ( ord_less_eq_nat
% 9.10/5.23      = ( ^ [X: nat,Y: nat] :
% 9.10/5.23            ( ( ord_less_nat @ X @ Y )
% 9.10/5.23            | ( X = Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4038_order__le__less,axiom,
% 9.10/5.23      ( ord_less_eq_int
% 9.10/5.23      = ( ^ [X: int,Y: int] :
% 9.10/5.23            ( ( ord_less_int @ X @ Y )
% 9.10/5.23            | ( X = Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4039_order__less__le,axiom,
% 9.10/5.23      ( ord_less_real
% 9.10/5.23      = ( ^ [X: real,Y: real] :
% 9.10/5.23            ( ( ord_less_eq_real @ X @ Y )
% 9.10/5.23            & ( X != Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4040_order__less__le,axiom,
% 9.10/5.23      ( ord_less_set_int
% 9.10/5.23      = ( ^ [X: set_int,Y: set_int] :
% 9.10/5.23            ( ( ord_less_eq_set_int @ X @ Y )
% 9.10/5.23            & ( X != Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4041_order__less__le,axiom,
% 9.10/5.23      ( ord_less_rat
% 9.10/5.23      = ( ^ [X: rat,Y: rat] :
% 9.10/5.23            ( ( ord_less_eq_rat @ X @ Y )
% 9.10/5.23            & ( X != Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4042_order__less__le,axiom,
% 9.10/5.23      ( ord_less_num
% 9.10/5.23      = ( ^ [X: num,Y: num] :
% 9.10/5.23            ( ( ord_less_eq_num @ X @ Y )
% 9.10/5.23            & ( X != Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4043_order__less__le,axiom,
% 9.10/5.23      ( ord_less_nat
% 9.10/5.23      = ( ^ [X: nat,Y: nat] :
% 9.10/5.23            ( ( ord_less_eq_nat @ X @ Y )
% 9.10/5.23            & ( X != Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4044_order__less__le,axiom,
% 9.10/5.23      ( ord_less_int
% 9.10/5.23      = ( ^ [X: int,Y: int] :
% 9.10/5.23            ( ( ord_less_eq_int @ X @ Y )
% 9.10/5.23            & ( X != Y ) ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4045_linorder__not__le,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_real @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4046_linorder__not__le,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_rat @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4047_linorder__not__le,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4048_linorder__not__le,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4049_linorder__not__le,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4050_linorder__not__less,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4051_linorder__not__less,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4052_linorder__not__less,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4053_linorder__not__less,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4054_linorder__not__less,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.10/5.23        = ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4055_order__less__imp__le,axiom,
% 9.10/5.23      ! [X3: real,Y3: real] :
% 9.10/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4056_order__less__imp__le,axiom,
% 9.10/5.23      ! [X3: set_int,Y3: set_int] :
% 9.10/5.23        ( ( ord_less_set_int @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4057_order__less__imp__le,axiom,
% 9.10/5.23      ! [X3: rat,Y3: rat] :
% 9.10/5.23        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4058_order__less__imp__le,axiom,
% 9.10/5.23      ! [X3: num,Y3: num] :
% 9.10/5.23        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4059_order__less__imp__le,axiom,
% 9.10/5.23      ! [X3: nat,Y3: nat] :
% 9.10/5.23        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4060_order__less__imp__le,axiom,
% 9.10/5.23      ! [X3: int,Y3: int] :
% 9.10/5.23        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.23       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4061_order__le__neq__trans,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.23       => ( ( A != B2 )
% 9.10/5.23         => ( ord_less_real @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4062_order__le__neq__trans,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int] :
% 9.10/5.23        ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23       => ( ( A != B2 )
% 9.10/5.23         => ( ord_less_set_int @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4063_order__le__neq__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.23       => ( ( A != B2 )
% 9.10/5.23         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4064_order__le__neq__trans,axiom,
% 9.10/5.23      ! [A: num,B2: num] :
% 9.10/5.23        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.23       => ( ( A != B2 )
% 9.10/5.23         => ( ord_less_num @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4065_order__le__neq__trans,axiom,
% 9.10/5.23      ! [A: nat,B2: nat] :
% 9.10/5.23        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.23       => ( ( A != B2 )
% 9.10/5.23         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4066_order__le__neq__trans,axiom,
% 9.10/5.23      ! [A: int,B2: int] :
% 9.10/5.23        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.23       => ( ( A != B2 )
% 9.10/5.23         => ( ord_less_int @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4067_order__neq__le__trans,axiom,
% 9.10/5.23      ! [A: real,B2: real] :
% 9.10/5.23        ( ( A != B2 )
% 9.10/5.23       => ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.23         => ( ord_less_real @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4068_order__neq__le__trans,axiom,
% 9.10/5.23      ! [A: set_int,B2: set_int] :
% 9.10/5.23        ( ( A != B2 )
% 9.10/5.23       => ( ( ord_less_eq_set_int @ A @ B2 )
% 9.10/5.23         => ( ord_less_set_int @ A @ B2 ) ) ) ).
% 9.10/5.23  
% 9.10/5.23  thf(fact_4069_order__neq__le__trans,axiom,
% 9.10/5.23      ! [A: rat,B2: rat] :
% 9.10/5.23        ( ( A != B2 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24         => ( ord_less_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4070_order__neq__le__trans,axiom,
% 9.10/5.24      ! [A: num,B2: num] :
% 9.10/5.24        ( ( A != B2 )
% 9.10/5.24       => ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24         => ( ord_less_num @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4071_order__neq__le__trans,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( A != B2 )
% 9.10/5.24       => ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.24         => ( ord_less_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4072_order__neq__le__trans,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( A != B2 )
% 9.10/5.24       => ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.24         => ( ord_less_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4073_order__le__less__trans,axiom,
% 9.10/5.24      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.24        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_real @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_real @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4074_order__le__less__trans,axiom,
% 9.10/5.24      ! [X3: set_int,Y3: set_int,Z: set_int] :
% 9.10/5.24        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_set_int @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_set_int @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4075_order__le__less__trans,axiom,
% 9.10/5.24      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_rat @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_rat @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4076_order__le__less__trans,axiom,
% 9.10/5.24      ! [X3: num,Y3: num,Z: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_num @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_num @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4077_order__le__less__trans,axiom,
% 9.10/5.24      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_nat @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_nat @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4078_order__le__less__trans,axiom,
% 9.10/5.24      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_int @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_int @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4079_order__less__le__trans,axiom,
% 9.10/5.24      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.24        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_eq_real @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_real @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4080_order__less__le__trans,axiom,
% 9.10/5.24      ! [X3: set_int,Y3: set_int,Z: set_int] :
% 9.10/5.24        ( ( ord_less_set_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_set_int @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4081_order__less__le__trans,axiom,
% 9.10/5.24      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.24        ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_rat @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4082_order__less__le__trans,axiom,
% 9.10/5.24      ! [X3: num,Y3: num,Z: num] :
% 9.10/5.24        ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_eq_num @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_num @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4083_order__less__le__trans,axiom,
% 9.10/5.24      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.24        ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_eq_nat @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_nat @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4084_order__less__le__trans,axiom,
% 9.10/5.24      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.24        ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.10/5.24         => ( ord_less_int @ X3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4085_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: real,F: real > real,B2: real,C: real] :
% 9.10/5.24        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.24         => ( ! [X5: real,Y4: real] :
% 9.10/5.24                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4086_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4087_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: real,F: num > real,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4088_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: real,F: nat > real,B2: nat,C: nat] :
% 9.10/5.24        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.24                ( ( ord_less_nat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4089_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: real,F: int > real,B2: int,C: int] :
% 9.10/5.24        ( ( ord_less_eq_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.24         => ( ! [X5: int,Y4: int] :
% 9.10/5.24                ( ( ord_less_int @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4090_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: real > rat,B2: real,C: real] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_real @ B2 @ C )
% 9.10/5.24         => ( ! [X5: real,Y4: real] :
% 9.10/5.24                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4091_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4092_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: num > rat,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4093_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: nat > rat,B2: nat,C: nat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_nat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.24                ( ( ord_less_nat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4094_order__le__less__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: int > rat,B2: int,C: int] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_int @ B2 @ C )
% 9.10/5.24         => ( ! [X5: int,Y4: int] :
% 9.10/5.24                ( ( ord_less_int @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4095_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4096_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4097_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > num,C: num] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4098_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > nat,C: nat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4099_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > int,C: int] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4100_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > real,C: real] :
% 9.10/5.24        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4101_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4102_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > num,C: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_num @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4103_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > nat,C: nat] :
% 9.10/5.24        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_nat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4104_order__le__less__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > int,C: int] :
% 9.10/5.24        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_int @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4105_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: real,F: rat > real,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4106_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: rat > rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4107_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: num,F: rat > num,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_num @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4108_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: nat,F: rat > nat,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4109_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: int,F: rat > int,B2: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_int @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_rat @ B2 @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4110_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: real,F: num > real,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_real @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4111_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: rat,F: num > rat,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_rat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4112_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: num,F: num > num,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_num @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4113_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: nat,F: num > nat,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_nat @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4114_order__less__le__subst1,axiom,
% 9.10/5.24      ! [A: int,F: num > int,B2: num,C: num] :
% 9.10/5.24        ( ( ord_less_int @ A @ ( F @ B2 ) )
% 9.10/5.24       => ( ( ord_less_eq_num @ B2 @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4115_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: real,B2: real,F: real > real,C: real] :
% 9.10/5.24        ( ( ord_less_real @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: real,Y4: real] :
% 9.10/5.24                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4116_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > real,C: real] :
% 9.10/5.24        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4117_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > real,C: real] :
% 9.10/5.24        ( ( ord_less_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4118_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,F: nat > real,C: real] :
% 9.10/5.24        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.24                ( ( ord_less_nat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4119_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: int,B2: int,F: int > real,C: real] :
% 9.10/5.24        ( ( ord_less_int @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: int,Y4: int] :
% 9.10/5.24                ( ( ord_less_int @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4120_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: real,B2: real,F: real > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_real @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: real,Y4: real] :
% 9.10/5.24                ( ( ord_less_real @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4121_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,F: rat > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: rat,Y4: rat] :
% 9.10/5.24                ( ( ord_less_rat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4122_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: num,B2: num,F: num > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: num,Y4: num] :
% 9.10/5.24                ( ( ord_less_num @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4123_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,F: nat > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: nat,Y4: nat] :
% 9.10/5.24                ( ( ord_less_nat @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4124_order__less__le__subst2,axiom,
% 9.10/5.24      ! [A: int,B2: int,F: int > rat,C: rat] :
% 9.10/5.24        ( ( ord_less_int @ A @ B2 )
% 9.10/5.24       => ( ( ord_less_eq_rat @ ( F @ B2 ) @ C )
% 9.10/5.24         => ( ! [X5: int,Y4: int] :
% 9.10/5.24                ( ( ord_less_int @ X5 @ Y4 )
% 9.10/5.24               => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.10/5.24           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4125_linorder__le__less__linear,axiom,
% 9.10/5.24      ! [X3: real,Y3: real] :
% 9.10/5.24        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.24        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4126_linorder__le__less__linear,axiom,
% 9.10/5.24      ! [X3: rat,Y3: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.24        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4127_linorder__le__less__linear,axiom,
% 9.10/5.24      ! [X3: num,Y3: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.24        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4128_linorder__le__less__linear,axiom,
% 9.10/5.24      ! [X3: nat,Y3: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.24        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4129_linorder__le__less__linear,axiom,
% 9.10/5.24      ! [X3: int,Y3: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.24        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4130_order__le__imp__less__or__eq,axiom,
% 9.10/5.24      ! [X3: real,Y3: real] :
% 9.10/5.24        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.24          | ( X3 = Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4131_order__le__imp__less__or__eq,axiom,
% 9.10/5.24      ! [X3: set_int,Y3: set_int] :
% 9.10/5.24        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_set_int @ X3 @ Y3 )
% 9.10/5.24          | ( X3 = Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4132_order__le__imp__less__or__eq,axiom,
% 9.10/5.24      ! [X3: rat,Y3: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_rat @ X3 @ Y3 )
% 9.10/5.24          | ( X3 = Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4133_order__le__imp__less__or__eq,axiom,
% 9.10/5.24      ! [X3: num,Y3: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_num @ X3 @ Y3 )
% 9.10/5.24          | ( X3 = Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4134_order__le__imp__less__or__eq,axiom,
% 9.10/5.24      ! [X3: nat,Y3: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.24          | ( X3 = Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4135_order__le__imp__less__or__eq,axiom,
% 9.10/5.24      ! [X3: int,Y3: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( ord_less_int @ X3 @ Y3 )
% 9.10/5.24          | ( X3 = Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4136_bot_Oextremum,axiom,
% 9.10/5.24      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4137_bot_Oextremum,axiom,
% 9.10/5.24      ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4138_bot_Oextremum,axiom,
% 9.10/5.24      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4139_bot_Oextremum,axiom,
% 9.10/5.24      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4140_bot_Oextremum,axiom,
% 9.10/5.24      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4141_bot_Oextremum__unique,axiom,
% 9.10/5.24      ! [A: set_nat] :
% 9.10/5.24        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.10/5.24        = ( A = bot_bot_set_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4142_bot_Oextremum__unique,axiom,
% 9.10/5.24      ! [A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 9.10/5.24        = ( A = bot_bo4199563552545308370d_enat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4143_bot_Oextremum__unique,axiom,
% 9.10/5.24      ! [A: set_real] :
% 9.10/5.24        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.10/5.24        = ( A = bot_bot_set_real ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4144_bot_Oextremum__unique,axiom,
% 9.10/5.24      ! [A: set_int] :
% 9.10/5.24        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.10/5.24        = ( A = bot_bot_set_int ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4145_bot_Oextremum__unique,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.10/5.24        = ( A = bot_bot_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4146_bot_Oextremum__uniqueI,axiom,
% 9.10/5.24      ! [A: set_nat] :
% 9.10/5.24        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.10/5.24       => ( A = bot_bot_set_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4147_bot_Oextremum__uniqueI,axiom,
% 9.10/5.24      ! [A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 9.10/5.24       => ( A = bot_bo4199563552545308370d_enat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4148_bot_Oextremum__uniqueI,axiom,
% 9.10/5.24      ! [A: set_real] :
% 9.10/5.24        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.10/5.24       => ( A = bot_bot_set_real ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4149_bot_Oextremum__uniqueI,axiom,
% 9.10/5.24      ! [A: set_int] :
% 9.10/5.24        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.10/5.24       => ( A = bot_bot_set_int ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4150_bot_Oextremum__uniqueI,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.10/5.24       => ( A = bot_bot_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4151_bot_Oextremum__strict,axiom,
% 9.10/5.24      ! [A: set_nat] :
% 9.10/5.24        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4152_bot_Oextremum__strict,axiom,
% 9.10/5.24      ! [A: extended_enat] :
% 9.10/5.24        ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4153_bot_Oextremum__strict,axiom,
% 9.10/5.24      ! [A: set_int] :
% 9.10/5.24        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4154_bot_Oextremum__strict,axiom,
% 9.10/5.24      ! [A: set_real] :
% 9.10/5.24        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4155_bot_Oextremum__strict,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4156_bot_Onot__eq__extremum,axiom,
% 9.10/5.24      ! [A: set_nat] :
% 9.10/5.24        ( ( A != bot_bot_set_nat )
% 9.10/5.24        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4157_bot_Onot__eq__extremum,axiom,
% 9.10/5.24      ! [A: extended_enat] :
% 9.10/5.24        ( ( A != bot_bo4199563552545308370d_enat )
% 9.10/5.24        = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4158_bot_Onot__eq__extremum,axiom,
% 9.10/5.24      ! [A: set_int] :
% 9.10/5.24        ( ( A != bot_bot_set_int )
% 9.10/5.24        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4159_bot_Onot__eq__extremum,axiom,
% 9.10/5.24      ! [A: set_real] :
% 9.10/5.24        ( ( A != bot_bot_set_real )
% 9.10/5.24        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4160_bot_Onot__eq__extremum,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( A != bot_bot_nat )
% 9.10/5.24        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4161_Euclid__induct,axiom,
% 9.10/5.24      ! [P: nat > nat > $o,A: nat,B2: nat] :
% 9.10/5.24        ( ! [A3: nat,B3: nat] :
% 9.10/5.24            ( ( P @ A3 @ B3 )
% 9.10/5.24            = ( P @ B3 @ A3 ) )
% 9.10/5.24       => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
% 9.10/5.24         => ( ! [A3: nat,B3: nat] :
% 9.10/5.24                ( ( P @ A3 @ B3 )
% 9.10/5.24               => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
% 9.10/5.24           => ( P @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4162_vebt__insert_Osimps_I5_J,axiom,
% 9.10/5.24      ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.10/5.24        ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 9.10/5.24        = ( if_VEBT_VEBT
% 9.10/5.24          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.10/5.24            & ~ ( ( X3 = Mi )
% 9.10/5.24                | ( X3 = Ma ) ) )
% 9.10/5.24          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ X3 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi ) @ Mi @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 9.10/5.24          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4163_vebt__insert_Oelims,axiom,
% 9.10/5.24      ! [X3: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.10/5.24        ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
% 9.10/5.24          = Y3 )
% 9.10/5.24       => ( ! [A3: $o,B3: $o] :
% 9.10/5.24              ( ( X3
% 9.10/5.24                = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.24             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.24                   => ( Y3
% 9.10/5.24                      = ( vEBT_Leaf @ $true @ B3 ) ) )
% 9.10/5.24                  & ( ( Xa2 != zero_zero_nat )
% 9.10/5.24                   => ( ( ( Xa2 = one_one_nat )
% 9.10/5.24                       => ( Y3
% 9.10/5.24                          = ( vEBT_Leaf @ A3 @ $true ) ) )
% 9.10/5.24                      & ( ( Xa2 != one_one_nat )
% 9.10/5.24                       => ( Y3
% 9.10/5.24                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
% 9.10/5.24         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.24                ( ( X3
% 9.10/5.24                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 9.10/5.24               => ( Y3
% 9.10/5.24                 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
% 9.10/5.24           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.24                  ( ( X3
% 9.10/5.24                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 9.10/5.24                 => ( Y3
% 9.10/5.24                   != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
% 9.10/5.24             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.24                    ( ( X3
% 9.10/5.24                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.24                   => ( Y3
% 9.10/5.24                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
% 9.10/5.24               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.24                      ( ( X3
% 9.10/5.24                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.24                     => ( Y3
% 9.10/5.24                       != ( if_VEBT_VEBT
% 9.10/5.24                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.24                            & ~ ( ( Xa2 = Mi2 )
% 9.10/5.24                                | ( Xa2 = Ma2 ) ) )
% 9.10/5.24                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 9.10/5.24                          @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4164_vebt__insert_Opelims,axiom,
% 9.10/5.24      ! [X3: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.10/5.24        ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
% 9.10/5.24          = Y3 )
% 9.10/5.24       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.24         => ( ! [A3: $o,B3: $o] :
% 9.10/5.24                ( ( X3
% 9.10/5.24                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.24               => ( ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.24                     => ( Y3
% 9.10/5.24                        = ( vEBT_Leaf @ $true @ B3 ) ) )
% 9.10/5.24                    & ( ( Xa2 != zero_zero_nat )
% 9.10/5.24                     => ( ( ( Xa2 = one_one_nat )
% 9.10/5.24                         => ( Y3
% 9.10/5.24                            = ( vEBT_Leaf @ A3 @ $true ) ) )
% 9.10/5.24                        & ( ( Xa2 != one_one_nat )
% 9.10/5.24                         => ( Y3
% 9.10/5.24                            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
% 9.10/5.24                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
% 9.10/5.24           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.24                  ( ( X3
% 9.10/5.24                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 9.10/5.24                 => ( ( Y3
% 9.10/5.24                      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 9.10/5.24                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa2 ) ) ) )
% 9.10/5.24             => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.10/5.24                    ( ( X3
% 9.10/5.24                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 9.10/5.24                   => ( ( Y3
% 9.10/5.24                        = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 9.10/5.24                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
% 9.10/5.24               => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.24                      ( ( X3
% 9.10/5.24                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.24                     => ( ( Y3
% 9.10/5.24                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.24                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
% 9.10/5.24                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.24                        ( ( X3
% 9.10/5.24                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.10/5.24                       => ( ( Y3
% 9.10/5.24                            = ( if_VEBT_VEBT
% 9.10/5.24                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.10/5.24                                & ~ ( ( Xa2 = Mi2 )
% 9.10/5.24                                    | ( Xa2 = Ma2 ) ) )
% 9.10/5.24                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 9.10/5.24                              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 9.10/5.24                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4165_max__less__iff__conj,axiom,
% 9.10/5.24      ! [X3: extended_enat,Y3: extended_enat,Z: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) @ Z )
% 9.10/5.24        = ( ( ord_le72135733267957522d_enat @ X3 @ Z )
% 9.10/5.24          & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4166_max__less__iff__conj,axiom,
% 9.10/5.24      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.24        ( ( ord_less_real @ ( ord_max_real @ X3 @ Y3 ) @ Z )
% 9.10/5.24        = ( ( ord_less_real @ X3 @ Z )
% 9.10/5.24          & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4167_max__less__iff__conj,axiom,
% 9.10/5.24      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.24        ( ( ord_less_rat @ ( ord_max_rat @ X3 @ Y3 ) @ Z )
% 9.10/5.24        = ( ( ord_less_rat @ X3 @ Z )
% 9.10/5.24          & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4168_max__less__iff__conj,axiom,
% 9.10/5.24      ! [X3: num,Y3: num,Z: num] :
% 9.10/5.24        ( ( ord_less_num @ ( ord_max_num @ X3 @ Y3 ) @ Z )
% 9.10/5.24        = ( ( ord_less_num @ X3 @ Z )
% 9.10/5.24          & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4169_max__less__iff__conj,axiom,
% 9.10/5.24      ! [X3: nat,Y3: nat,Z: nat] :
% 9.10/5.24        ( ( ord_less_nat @ ( ord_max_nat @ X3 @ Y3 ) @ Z )
% 9.10/5.24        = ( ( ord_less_nat @ X3 @ Z )
% 9.10/5.24          & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4170_max__less__iff__conj,axiom,
% 9.10/5.24      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.24        ( ( ord_less_int @ ( ord_max_int @ X3 @ Y3 ) @ Z )
% 9.10/5.24        = ( ( ord_less_int @ X3 @ Z )
% 9.10/5.24          & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4171_max_Oabsorb4,axiom,
% 9.10/5.24      ! [A: extended_enat,B2: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ A @ B2 )
% 9.10/5.24       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4172_max_Oabsorb4,axiom,
% 9.10/5.24      ! [A: real,B2: real] :
% 9.10/5.24        ( ( ord_less_real @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_real @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4173_max_Oabsorb4,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] :
% 9.10/5.24        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_rat @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4174_max_Oabsorb4,axiom,
% 9.10/5.24      ! [A: num,B2: num] :
% 9.10/5.24        ( ( ord_less_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_num @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4175_max_Oabsorb4,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( ord_less_nat @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_nat @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4176_max_Oabsorb4,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( ord_less_int @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_int @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4177_max_Oabsorb3,axiom,
% 9.10/5.24      ! [B2: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ B2 @ A )
% 9.10/5.24       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4178_max_Oabsorb3,axiom,
% 9.10/5.24      ! [B2: real,A: real] :
% 9.10/5.24        ( ( ord_less_real @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_real @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4179_max_Oabsorb3,axiom,
% 9.10/5.24      ! [B2: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_rat @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_rat @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4180_max_Oabsorb3,axiom,
% 9.10/5.24      ! [B2: num,A: num] :
% 9.10/5.24        ( ( ord_less_num @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_num @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4181_max_Oabsorb3,axiom,
% 9.10/5.24      ! [B2: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_nat @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_nat @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4182_max_Oabsorb3,axiom,
% 9.10/5.24      ! [B2: int,A: int] :
% 9.10/5.24        ( ( ord_less_int @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_int @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4183_max_Oabsorb1,axiom,
% 9.10/5.24      ! [B2: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 9.10/5.24       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4184_max_Oabsorb1,axiom,
% 9.10/5.24      ! [B2: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_rat @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4185_max_Oabsorb1,axiom,
% 9.10/5.24      ! [B2: num,A: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_num @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4186_max_Oabsorb1,axiom,
% 9.10/5.24      ! [B2: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_nat @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4187_max_Oabsorb1,axiom,
% 9.10/5.24      ! [B2: int,A: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.24       => ( ( ord_max_int @ A @ B2 )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4188_max_Oabsorb2,axiom,
% 9.10/5.24      ! [A: extended_enat,B2: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ A @ B2 )
% 9.10/5.24       => ( ( ord_ma741700101516333627d_enat @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4189_max_Oabsorb2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_rat @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4190_max_Oabsorb2,axiom,
% 9.10/5.24      ! [A: num,B2: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_num @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4191_max_Oabsorb2,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_nat @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4192_max_Oabsorb2,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.24       => ( ( ord_max_int @ A @ B2 )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4193_max_Obounded__iff,axiom,
% 9.10/5.24      ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
% 9.10/5.24        = ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 9.10/5.24          & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4194_max_Obounded__iff,axiom,
% 9.10/5.24      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A )
% 9.10/5.24        = ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.24          & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4195_max_Obounded__iff,axiom,
% 9.10/5.24      ! [B2: num,C: num,A: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A )
% 9.10/5.24        = ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.24          & ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4196_max_Obounded__iff,axiom,
% 9.10/5.24      ! [B2: nat,C: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
% 9.10/5.24        = ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.24          & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4197_max_Obounded__iff,axiom,
% 9.10/5.24      ! [B2: int,C: int,A: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
% 9.10/5.24        = ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.24          & ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4198_dbl__simps_I3_J,axiom,
% 9.10/5.24      ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 9.10/5.24      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4199_dbl__simps_I3_J,axiom,
% 9.10/5.24      ( ( neg_numeral_dbl_real @ one_one_real )
% 9.10/5.24      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4200_dbl__simps_I3_J,axiom,
% 9.10/5.24      ( ( neg_numeral_dbl_rat @ one_one_rat )
% 9.10/5.24      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4201_dbl__simps_I3_J,axiom,
% 9.10/5.24      ( ( neg_numeral_dbl_int @ one_one_int )
% 9.10/5.24      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4202_max__enat__simps_I3_J,axiom,
% 9.10/5.24      ! [Q2: extended_enat] :
% 9.10/5.24        ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 9.10/5.24        = Q2 ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4203_max__enat__simps_I2_J,axiom,
% 9.10/5.24      ! [Q2: extended_enat] :
% 9.10/5.24        ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 9.10/5.24        = Q2 ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4204_dbl__simps_I2_J,axiom,
% 9.10/5.24      ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 9.10/5.24      = zero_zero_complex ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4205_dbl__simps_I2_J,axiom,
% 9.10/5.24      ( ( neg_numeral_dbl_real @ zero_zero_real )
% 9.10/5.24      = zero_zero_real ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4206_dbl__simps_I2_J,axiom,
% 9.10/5.24      ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 9.10/5.24      = zero_zero_rat ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4207_dbl__simps_I2_J,axiom,
% 9.10/5.24      ( ( neg_numeral_dbl_int @ zero_zero_int )
% 9.10/5.24      = zero_zero_int ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4208_dbl__simps_I5_J,axiom,
% 9.10/5.24      ! [K: num] :
% 9.10/5.24        ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 9.10/5.24        = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4209_dbl__simps_I5_J,axiom,
% 9.10/5.24      ! [K: num] :
% 9.10/5.24        ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 9.10/5.24        = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4210_dbl__simps_I5_J,axiom,
% 9.10/5.24      ! [K: num] :
% 9.10/5.24        ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 9.10/5.24        = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4211_dbl__simps_I5_J,axiom,
% 9.10/5.24      ! [K: num] :
% 9.10/5.24        ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 9.10/5.24        = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4212_bot__nat__def,axiom,
% 9.10/5.24      bot_bot_nat = zero_zero_nat ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4213_bot__enat__def,axiom,
% 9.10/5.24      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4214_dbl__def,axiom,
% 9.10/5.24      ( neg_numeral_dbl_real
% 9.10/5.24      = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4215_dbl__def,axiom,
% 9.10/5.24      ( neg_numeral_dbl_rat
% 9.10/5.24      = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4216_dbl__def,axiom,
% 9.10/5.24      ( neg_numeral_dbl_int
% 9.10/5.24      = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4217_max_OcoboundedI2,axiom,
% 9.10/5.24      ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ C @ B2 )
% 9.10/5.24       => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4218_max_OcoboundedI2,axiom,
% 9.10/5.24      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ C @ B2 )
% 9.10/5.24       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4219_max_OcoboundedI2,axiom,
% 9.10/5.24      ! [C: num,B2: num,A: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ C @ B2 )
% 9.10/5.24       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4220_max_OcoboundedI2,axiom,
% 9.10/5.24      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ C @ B2 )
% 9.10/5.24       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4221_max_OcoboundedI2,axiom,
% 9.10/5.24      ! [C: int,B2: int,A: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ C @ B2 )
% 9.10/5.24       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4222_max_OcoboundedI1,axiom,
% 9.10/5.24      ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ C @ A )
% 9.10/5.24       => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4223_max_OcoboundedI1,axiom,
% 9.10/5.24      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ C @ A )
% 9.10/5.24       => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4224_max_OcoboundedI1,axiom,
% 9.10/5.24      ! [C: num,A: num,B2: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ C @ A )
% 9.10/5.24       => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4225_max_OcoboundedI1,axiom,
% 9.10/5.24      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ C @ A )
% 9.10/5.24       => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4226_max_OcoboundedI1,axiom,
% 9.10/5.24      ! [C: int,A: int,B2: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ C @ A )
% 9.10/5.24       => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4227_max_Oabsorb__iff2,axiom,
% 9.10/5.24      ( ord_le2932123472753598470d_enat
% 9.10/5.24      = ( ^ [A5: extended_enat,B4: extended_enat] :
% 9.10/5.24            ( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
% 9.10/5.24            = B4 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4228_max_Oabsorb__iff2,axiom,
% 9.10/5.24      ( ord_less_eq_rat
% 9.10/5.24      = ( ^ [A5: rat,B4: rat] :
% 9.10/5.24            ( ( ord_max_rat @ A5 @ B4 )
% 9.10/5.24            = B4 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4229_max_Oabsorb__iff2,axiom,
% 9.10/5.24      ( ord_less_eq_num
% 9.10/5.24      = ( ^ [A5: num,B4: num] :
% 9.10/5.24            ( ( ord_max_num @ A5 @ B4 )
% 9.10/5.24            = B4 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4230_max_Oabsorb__iff2,axiom,
% 9.10/5.24      ( ord_less_eq_nat
% 9.10/5.24      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.24            ( ( ord_max_nat @ A5 @ B4 )
% 9.10/5.24            = B4 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4231_max_Oabsorb__iff2,axiom,
% 9.10/5.24      ( ord_less_eq_int
% 9.10/5.24      = ( ^ [A5: int,B4: int] :
% 9.10/5.24            ( ( ord_max_int @ A5 @ B4 )
% 9.10/5.24            = B4 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4232_max_Oabsorb__iff1,axiom,
% 9.10/5.24      ( ord_le2932123472753598470d_enat
% 9.10/5.24      = ( ^ [B4: extended_enat,A5: extended_enat] :
% 9.10/5.24            ( ( ord_ma741700101516333627d_enat @ A5 @ B4 )
% 9.10/5.24            = A5 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4233_max_Oabsorb__iff1,axiom,
% 9.10/5.24      ( ord_less_eq_rat
% 9.10/5.24      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.24            ( ( ord_max_rat @ A5 @ B4 )
% 9.10/5.24            = A5 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4234_max_Oabsorb__iff1,axiom,
% 9.10/5.24      ( ord_less_eq_num
% 9.10/5.24      = ( ^ [B4: num,A5: num] :
% 9.10/5.24            ( ( ord_max_num @ A5 @ B4 )
% 9.10/5.24            = A5 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4235_max_Oabsorb__iff1,axiom,
% 9.10/5.24      ( ord_less_eq_nat
% 9.10/5.24      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.24            ( ( ord_max_nat @ A5 @ B4 )
% 9.10/5.24            = A5 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4236_max_Oabsorb__iff1,axiom,
% 9.10/5.24      ( ord_less_eq_int
% 9.10/5.24      = ( ^ [B4: int,A5: int] :
% 9.10/5.24            ( ( ord_max_int @ A5 @ B4 )
% 9.10/5.24            = A5 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4237_le__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: extended_enat,X3: extended_enat,Y3: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_le2932123472753598470d_enat @ Z @ X3 )
% 9.10/5.24          | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4238_le__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_eq_rat @ Z @ X3 )
% 9.10/5.24          | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4239_le__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: num,X3: num,Y3: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_eq_num @ Z @ X3 )
% 9.10/5.24          | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4240_le__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: nat,X3: nat,Y3: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_eq_nat @ Z @ X3 )
% 9.10/5.24          | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4241_le__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: int,X3: int,Y3: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_eq_int @ Z @ X3 )
% 9.10/5.24          | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4242_max_Ocobounded2,axiom,
% 9.10/5.24      ! [B2: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B2 @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4243_max_Ocobounded2,axiom,
% 9.10/5.24      ! [B2: rat,A: rat] : ( ord_less_eq_rat @ B2 @ ( ord_max_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4244_max_Ocobounded2,axiom,
% 9.10/5.24      ! [B2: num,A: num] : ( ord_less_eq_num @ B2 @ ( ord_max_num @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4245_max_Ocobounded2,axiom,
% 9.10/5.24      ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( ord_max_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4246_max_Ocobounded2,axiom,
% 9.10/5.24      ! [B2: int,A: int] : ( ord_less_eq_int @ B2 @ ( ord_max_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4247_max_Ocobounded1,axiom,
% 9.10/5.24      ! [A: extended_enat,B2: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4248_max_Ocobounded1,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4249_max_Ocobounded1,axiom,
% 9.10/5.24      ! [A: num,B2: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4250_max_Ocobounded1,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4251_max_Ocobounded1,axiom,
% 9.10/5.24      ! [A: int,B2: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4252_max_Oorder__iff,axiom,
% 9.10/5.24      ( ord_le2932123472753598470d_enat
% 9.10/5.24      = ( ^ [B4: extended_enat,A5: extended_enat] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( ord_ma741700101516333627d_enat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4253_max_Oorder__iff,axiom,
% 9.10/5.24      ( ord_less_eq_rat
% 9.10/5.24      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( ord_max_rat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4254_max_Oorder__iff,axiom,
% 9.10/5.24      ( ord_less_eq_num
% 9.10/5.24      = ( ^ [B4: num,A5: num] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( ord_max_num @ A5 @ B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4255_max_Oorder__iff,axiom,
% 9.10/5.24      ( ord_less_eq_nat
% 9.10/5.24      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( ord_max_nat @ A5 @ B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4256_max_Oorder__iff,axiom,
% 9.10/5.24      ( ord_less_eq_int
% 9.10/5.24      = ( ^ [B4: int,A5: int] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( ord_max_int @ A5 @ B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4257_max_OboundedI,axiom,
% 9.10/5.24      ! [B2: extended_enat,A: extended_enat,C: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 9.10/5.24       => ( ( ord_le2932123472753598470d_enat @ C @ A )
% 9.10/5.24         => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4258_max_OboundedI,axiom,
% 9.10/5.24      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.24       => ( ( ord_less_eq_rat @ C @ A )
% 9.10/5.24         => ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4259_max_OboundedI,axiom,
% 9.10/5.24      ! [B2: num,A: num,C: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.24       => ( ( ord_less_eq_num @ C @ A )
% 9.10/5.24         => ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4260_max_OboundedI,axiom,
% 9.10/5.24      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.24       => ( ( ord_less_eq_nat @ C @ A )
% 9.10/5.24         => ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4261_max_OboundedI,axiom,
% 9.10/5.24      ! [B2: int,A: int,C: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.24       => ( ( ord_less_eq_int @ C @ A )
% 9.10/5.24         => ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4262_max_OboundedE,axiom,
% 9.10/5.24      ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 9.10/5.24           => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4263_max_OboundedE,axiom,
% 9.10/5.24      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4264_max_OboundedE,axiom,
% 9.10/5.24      ! [B2: num,C: num,A: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4265_max_OboundedE,axiom,
% 9.10/5.24      ! [B2: nat,C: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4266_max_OboundedE,axiom,
% 9.10/5.24      ! [B2: int,C: int,A: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4267_max_OorderI,axiom,
% 9.10/5.24      ! [A: extended_enat,B2: extended_enat] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( ord_ma741700101516333627d_enat @ A @ B2 ) )
% 9.10/5.24       => ( ord_le2932123472753598470d_enat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4268_max_OorderI,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( ord_max_rat @ A @ B2 ) )
% 9.10/5.24       => ( ord_less_eq_rat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4269_max_OorderI,axiom,
% 9.10/5.24      ! [A: num,B2: num] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( ord_max_num @ A @ B2 ) )
% 9.10/5.24       => ( ord_less_eq_num @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4270_max_OorderI,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( ord_max_nat @ A @ B2 ) )
% 9.10/5.24       => ( ord_less_eq_nat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4271_max_OorderI,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( ord_max_int @ A @ B2 ) )
% 9.10/5.24       => ( ord_less_eq_int @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4272_max_OorderE,axiom,
% 9.10/5.24      ! [B2: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ B2 @ A )
% 9.10/5.24       => ( A
% 9.10/5.24          = ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4273_max_OorderE,axiom,
% 9.10/5.24      ! [B2: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ B2 @ A )
% 9.10/5.24       => ( A
% 9.10/5.24          = ( ord_max_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4274_max_OorderE,axiom,
% 9.10/5.24      ! [B2: num,A: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ B2 @ A )
% 9.10/5.24       => ( A
% 9.10/5.24          = ( ord_max_num @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4275_max_OorderE,axiom,
% 9.10/5.24      ! [B2: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ B2 @ A )
% 9.10/5.24       => ( A
% 9.10/5.24          = ( ord_max_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4276_max_OorderE,axiom,
% 9.10/5.24      ! [B2: int,A: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ B2 @ A )
% 9.10/5.24       => ( A
% 9.10/5.24          = ( ord_max_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4277_max_Omono,axiom,
% 9.10/5.24      ! [C: extended_enat,A: extended_enat,D: extended_enat,B2: extended_enat] :
% 9.10/5.24        ( ( ord_le2932123472753598470d_enat @ C @ A )
% 9.10/5.24       => ( ( ord_le2932123472753598470d_enat @ D @ B2 )
% 9.10/5.24         => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4278_max_Omono,axiom,
% 9.10/5.24      ! [C: rat,A: rat,D: rat,B2: rat] :
% 9.10/5.24        ( ( ord_less_eq_rat @ C @ A )
% 9.10/5.24       => ( ( ord_less_eq_rat @ D @ B2 )
% 9.10/5.24         => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4279_max_Omono,axiom,
% 9.10/5.24      ! [C: num,A: num,D: num,B2: num] :
% 9.10/5.24        ( ( ord_less_eq_num @ C @ A )
% 9.10/5.24       => ( ( ord_less_eq_num @ D @ B2 )
% 9.10/5.24         => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4280_max_Omono,axiom,
% 9.10/5.24      ! [C: nat,A: nat,D: nat,B2: nat] :
% 9.10/5.24        ( ( ord_less_eq_nat @ C @ A )
% 9.10/5.24       => ( ( ord_less_eq_nat @ D @ B2 )
% 9.10/5.24         => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4281_max_Omono,axiom,
% 9.10/5.24      ! [C: int,A: int,D: int,B2: int] :
% 9.10/5.24        ( ( ord_less_eq_int @ C @ A )
% 9.10/5.24       => ( ( ord_less_eq_int @ D @ B2 )
% 9.10/5.24         => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4282_less__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: extended_enat,X3: extended_enat,Y3: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_le72135733267957522d_enat @ Z @ X3 )
% 9.10/5.24          | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4283_less__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.24        ( ( ord_less_real @ Z @ ( ord_max_real @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_real @ Z @ X3 )
% 9.10/5.24          | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4284_less__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.24        ( ( ord_less_rat @ Z @ ( ord_max_rat @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_rat @ Z @ X3 )
% 9.10/5.24          | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4285_less__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: num,X3: num,Y3: num] :
% 9.10/5.24        ( ( ord_less_num @ Z @ ( ord_max_num @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_num @ Z @ X3 )
% 9.10/5.24          | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4286_less__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: nat,X3: nat,Y3: nat] :
% 9.10/5.24        ( ( ord_less_nat @ Z @ ( ord_max_nat @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_nat @ Z @ X3 )
% 9.10/5.24          | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4287_less__max__iff__disj,axiom,
% 9.10/5.24      ! [Z: int,X3: int,Y3: int] :
% 9.10/5.24        ( ( ord_less_int @ Z @ ( ord_max_int @ X3 @ Y3 ) )
% 9.10/5.24        = ( ( ord_less_int @ Z @ X3 )
% 9.10/5.24          | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4288_max_Ostrict__boundedE,axiom,
% 9.10/5.24      ! [B2: extended_enat,C: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_le72135733267957522d_enat @ B2 @ A )
% 9.10/5.24           => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4289_max_Ostrict__boundedE,axiom,
% 9.10/5.24      ! [B2: real,C: real,A: real] :
% 9.10/5.24        ( ( ord_less_real @ ( ord_max_real @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_real @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_real @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4290_max_Ostrict__boundedE,axiom,
% 9.10/5.24      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_rat @ ( ord_max_rat @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_rat @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4291_max_Ostrict__boundedE,axiom,
% 9.10/5.24      ! [B2: num,C: num,A: num] :
% 9.10/5.24        ( ( ord_less_num @ ( ord_max_num @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_num @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_num @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4292_max_Ostrict__boundedE,axiom,
% 9.10/5.24      ! [B2: nat,C: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_nat @ ( ord_max_nat @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_nat @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4293_max_Ostrict__boundedE,axiom,
% 9.10/5.24      ! [B2: int,C: int,A: int] :
% 9.10/5.24        ( ( ord_less_int @ ( ord_max_int @ B2 @ C ) @ A )
% 9.10/5.24       => ~ ( ( ord_less_int @ B2 @ A )
% 9.10/5.24           => ~ ( ord_less_int @ C @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4294_max_Ostrict__order__iff,axiom,
% 9.10/5.24      ( ord_le72135733267957522d_enat
% 9.10/5.24      = ( ^ [B4: extended_enat,A5: extended_enat] :
% 9.10/5.24            ( ( A5
% 9.10/5.24              = ( ord_ma741700101516333627d_enat @ A5 @ B4 ) )
% 9.10/5.24            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4295_max_Ostrict__order__iff,axiom,
% 9.10/5.24      ( ord_less_real
% 9.10/5.24      = ( ^ [B4: real,A5: real] :
% 9.10/5.24            ( ( A5
% 9.10/5.24              = ( ord_max_real @ A5 @ B4 ) )
% 9.10/5.24            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4296_max_Ostrict__order__iff,axiom,
% 9.10/5.24      ( ord_less_rat
% 9.10/5.24      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.24            ( ( A5
% 9.10/5.24              = ( ord_max_rat @ A5 @ B4 ) )
% 9.10/5.24            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4297_max_Ostrict__order__iff,axiom,
% 9.10/5.24      ( ord_less_num
% 9.10/5.24      = ( ^ [B4: num,A5: num] :
% 9.10/5.24            ( ( A5
% 9.10/5.24              = ( ord_max_num @ A5 @ B4 ) )
% 9.10/5.24            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4298_max_Ostrict__order__iff,axiom,
% 9.10/5.24      ( ord_less_nat
% 9.10/5.24      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.24            ( ( A5
% 9.10/5.24              = ( ord_max_nat @ A5 @ B4 ) )
% 9.10/5.24            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4299_max_Ostrict__order__iff,axiom,
% 9.10/5.24      ( ord_less_int
% 9.10/5.24      = ( ^ [B4: int,A5: int] :
% 9.10/5.24            ( ( A5
% 9.10/5.24              = ( ord_max_int @ A5 @ B4 ) )
% 9.10/5.24            & ( A5 != B4 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4300_max_Ostrict__coboundedI1,axiom,
% 9.10/5.24      ! [C: extended_enat,A: extended_enat,B2: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ C @ A )
% 9.10/5.24       => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4301_max_Ostrict__coboundedI1,axiom,
% 9.10/5.24      ! [C: real,A: real,B2: real] :
% 9.10/5.24        ( ( ord_less_real @ C @ A )
% 9.10/5.24       => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4302_max_Ostrict__coboundedI1,axiom,
% 9.10/5.24      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.24        ( ( ord_less_rat @ C @ A )
% 9.10/5.24       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4303_max_Ostrict__coboundedI1,axiom,
% 9.10/5.24      ! [C: num,A: num,B2: num] :
% 9.10/5.24        ( ( ord_less_num @ C @ A )
% 9.10/5.24       => ( ord_less_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4304_max_Ostrict__coboundedI1,axiom,
% 9.10/5.24      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.24        ( ( ord_less_nat @ C @ A )
% 9.10/5.24       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4305_max_Ostrict__coboundedI1,axiom,
% 9.10/5.24      ! [C: int,A: int,B2: int] :
% 9.10/5.24        ( ( ord_less_int @ C @ A )
% 9.10/5.24       => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4306_max_Ostrict__coboundedI2,axiom,
% 9.10/5.24      ! [C: extended_enat,B2: extended_enat,A: extended_enat] :
% 9.10/5.24        ( ( ord_le72135733267957522d_enat @ C @ B2 )
% 9.10/5.24       => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4307_max_Ostrict__coboundedI2,axiom,
% 9.10/5.24      ! [C: real,B2: real,A: real] :
% 9.10/5.24        ( ( ord_less_real @ C @ B2 )
% 9.10/5.24       => ( ord_less_real @ C @ ( ord_max_real @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4308_max_Ostrict__coboundedI2,axiom,
% 9.10/5.24      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.24        ( ( ord_less_rat @ C @ B2 )
% 9.10/5.24       => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4309_max_Ostrict__coboundedI2,axiom,
% 9.10/5.24      ! [C: num,B2: num,A: num] :
% 9.10/5.24        ( ( ord_less_num @ C @ B2 )
% 9.10/5.24       => ( ord_less_num @ C @ ( ord_max_num @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4310_max_Ostrict__coboundedI2,axiom,
% 9.10/5.24      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.24        ( ( ord_less_nat @ C @ B2 )
% 9.10/5.24       => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4311_max_Ostrict__coboundedI2,axiom,
% 9.10/5.24      ! [C: int,B2: int,A: int] :
% 9.10/5.24        ( ( ord_less_int @ C @ B2 )
% 9.10/5.24       => ( ord_less_int @ C @ ( ord_max_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4312_even__succ__mod__exp,axiom,
% 9.10/5.24      ! [A: nat,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.24            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4313_even__succ__mod__exp,axiom,
% 9.10/5.24      ! [A: int,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.24            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4314_even__succ__mod__exp,axiom,
% 9.10/5.24      ! [A: code_integer,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.24            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4315_option_Osize__gen_I2_J,axiom,
% 9.10/5.24      ! [X3: nat > nat,X2: nat] :
% 9.10/5.24        ( ( size_option_nat @ X3 @ ( some_nat @ X2 ) )
% 9.10/5.24        = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4316_option_Osize__gen_I2_J,axiom,
% 9.10/5.24      ! [X3: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
% 9.10/5.24        ( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X2 ) )
% 9.10/5.24        = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4317_option_Osize__gen_I2_J,axiom,
% 9.10/5.24      ! [X3: num > nat,X2: num] :
% 9.10/5.24        ( ( size_option_num @ X3 @ ( some_num @ X2 ) )
% 9.10/5.24        = ( plus_plus_nat @ ( X3 @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4318_even__succ__div__exp,axiom,
% 9.10/5.24      ! [A: code_integer,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.24            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4319_even__succ__div__exp,axiom,
% 9.10/5.24      ! [A: nat,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.24            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4320_even__succ__div__exp,axiom,
% 9.10/5.24      ! [A: int,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.24            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4321_signed__take__bit__Suc,axiom,
% 9.10/5.24      ! [N: nat,A: code_integer] :
% 9.10/5.24        ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 9.10/5.24        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4322_signed__take__bit__Suc,axiom,
% 9.10/5.24      ! [N: nat,A: int] :
% 9.10/5.24        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 9.10/5.24        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4323_infinite__growing,axiom,
% 9.10/5.24      ! [X9: set_real] :
% 9.10/5.24        ( ( X9 != bot_bot_set_real )
% 9.10/5.24       => ( ! [X5: real] :
% 9.10/5.24              ( ( member_real @ X5 @ X9 )
% 9.10/5.24             => ? [Xa: real] :
% 9.10/5.24                  ( ( member_real @ Xa @ X9 )
% 9.10/5.24                  & ( ord_less_real @ X5 @ Xa ) ) )
% 9.10/5.24         => ~ ( finite_finite_real @ X9 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4324_infinite__growing,axiom,
% 9.10/5.24      ! [X9: set_rat] :
% 9.10/5.24        ( ( X9 != bot_bot_set_rat )
% 9.10/5.24       => ( ! [X5: rat] :
% 9.10/5.24              ( ( member_rat @ X5 @ X9 )
% 9.10/5.24             => ? [Xa: rat] :
% 9.10/5.24                  ( ( member_rat @ Xa @ X9 )
% 9.10/5.24                  & ( ord_less_rat @ X5 @ Xa ) ) )
% 9.10/5.24         => ~ ( finite_finite_rat @ X9 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4325_infinite__growing,axiom,
% 9.10/5.24      ! [X9: set_num] :
% 9.10/5.24        ( ( X9 != bot_bot_set_num )
% 9.10/5.24       => ( ! [X5: num] :
% 9.10/5.24              ( ( member_num @ X5 @ X9 )
% 9.10/5.24             => ? [Xa: num] :
% 9.10/5.24                  ( ( member_num @ Xa @ X9 )
% 9.10/5.24                  & ( ord_less_num @ X5 @ Xa ) ) )
% 9.10/5.24         => ~ ( finite_finite_num @ X9 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4326_infinite__growing,axiom,
% 9.10/5.24      ! [X9: set_nat] :
% 9.10/5.24        ( ( X9 != bot_bot_set_nat )
% 9.10/5.24       => ( ! [X5: nat] :
% 9.10/5.24              ( ( member_nat @ X5 @ X9 )
% 9.10/5.24             => ? [Xa: nat] :
% 9.10/5.24                  ( ( member_nat @ Xa @ X9 )
% 9.10/5.24                  & ( ord_less_nat @ X5 @ Xa ) ) )
% 9.10/5.24         => ~ ( finite_finite_nat @ X9 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4327_infinite__growing,axiom,
% 9.10/5.24      ! [X9: set_int] :
% 9.10/5.24        ( ( X9 != bot_bot_set_int )
% 9.10/5.24       => ( ! [X5: int] :
% 9.10/5.24              ( ( member_int @ X5 @ X9 )
% 9.10/5.24             => ? [Xa: int] :
% 9.10/5.24                  ( ( member_int @ Xa @ X9 )
% 9.10/5.24                  & ( ord_less_int @ X5 @ Xa ) ) )
% 9.10/5.24         => ~ ( finite_finite_int @ X9 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4328_nat__dvd__1__iff__1,axiom,
% 9.10/5.24      ! [M: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 9.10/5.24        = ( M = one_one_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4329_dvd__0__left__iff,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.10/5.24        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4330_dvd__0__left__iff,axiom,
% 9.10/5.24      ! [A: complex] :
% 9.10/5.24        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.10/5.24        = ( A = zero_zero_complex ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4331_dvd__0__left__iff,axiom,
% 9.10/5.24      ! [A: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.10/5.24        = ( A = zero_zero_real ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4332_dvd__0__left__iff,axiom,
% 9.10/5.24      ! [A: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.10/5.24        = ( A = zero_zero_rat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4333_dvd__0__left__iff,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.10/5.24        = ( A = zero_zero_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4334_dvd__0__left__iff,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.10/5.24        = ( A = zero_zero_int ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4335_dvd__0__right,axiom,
% 9.10/5.24      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4336_dvd__0__right,axiom,
% 9.10/5.24      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4337_dvd__0__right,axiom,
% 9.10/5.24      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4338_dvd__0__right,axiom,
% 9.10/5.24      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4339_dvd__0__right,axiom,
% 9.10/5.24      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4340_dvd__0__right,axiom,
% 9.10/5.24      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4341_dvd__add__triv__left__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4342_dvd__add__triv__left__iff,axiom,
% 9.10/5.24      ! [A: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4343_dvd__add__triv__left__iff,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4344_dvd__add__triv__left__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4345_dvd__add__triv__left__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4346_dvd__add__triv__right__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ A ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4347_dvd__add__triv__right__iff,axiom,
% 9.10/5.24      ! [A: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ A ) )
% 9.10/5.24        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4348_dvd__add__triv__right__iff,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ A ) )
% 9.10/5.24        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4349_dvd__add__triv__right__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
% 9.10/5.24        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4350_dvd__add__triv__right__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ A ) )
% 9.10/5.24        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4351_dvd__1__iff__1,axiom,
% 9.10/5.24      ! [M: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.10/5.24        = ( M
% 9.10/5.24          = ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4352_dvd__1__left,axiom,
% 9.10/5.24      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4353_div__dvd__div,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ A @ C )
% 9.10/5.24         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B2 @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 9.10/5.24            = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4354_div__dvd__div,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ A @ C )
% 9.10/5.24         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A ) @ ( divide_divide_nat @ C @ A ) )
% 9.10/5.24            = ( dvd_dvd_nat @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4355_div__dvd__div,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ A @ C )
% 9.10/5.24         => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A ) @ ( divide_divide_int @ C @ A ) )
% 9.10/5.24            = ( dvd_dvd_int @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4356_nat__mult__dvd__cancel__disj,axiom,
% 9.10/5.24      ! [K: nat,M: nat,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.10/5.24        = ( ( K = zero_zero_nat )
% 9.10/5.24          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4357_signed__take__bit__of__0,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 9.10/5.24        = zero_zero_int ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4358_dvd__times__right__cancel__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 9.10/5.24          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4359_dvd__times__right__cancel__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( A != zero_zero_nat )
% 9.10/5.24       => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A ) @ ( times_times_nat @ C @ A ) )
% 9.10/5.24          = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4360_dvd__times__right__cancel__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( A != zero_zero_int )
% 9.10/5.24       => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A ) @ ( times_times_int @ C @ A ) )
% 9.10/5.24          = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4361_dvd__times__left__cancel__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 9.10/5.24          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4362_dvd__times__left__cancel__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( A != zero_zero_nat )
% 9.10/5.24       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ ( times_times_nat @ A @ C ) )
% 9.10/5.24          = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4363_dvd__times__left__cancel__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( A != zero_zero_int )
% 9.10/5.24       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ ( times_times_int @ A @ C ) )
% 9.10/5.24          = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4364_dvd__mult__cancel__right,axiom,
% 9.10/5.24      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.10/5.24        = ( ( C = zero_z3403309356797280102nteger )
% 9.10/5.24          | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4365_dvd__mult__cancel__right,axiom,
% 9.10/5.24      ! [A: complex,C: complex,B2: complex] :
% 9.10/5.24        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B2 @ C ) )
% 9.10/5.24        = ( ( C = zero_zero_complex )
% 9.10/5.24          | ( dvd_dvd_complex @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4366_dvd__mult__cancel__right,axiom,
% 9.10/5.24      ! [A: real,C: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
% 9.10/5.24        = ( ( C = zero_zero_real )
% 9.10/5.24          | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4367_dvd__mult__cancel__right,axiom,
% 9.10/5.24      ! [A: rat,C: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ C ) )
% 9.10/5.24        = ( ( C = zero_zero_rat )
% 9.10/5.24          | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4368_dvd__mult__cancel__right,axiom,
% 9.10/5.24      ! [A: int,C: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
% 9.10/5.24        = ( ( C = zero_zero_int )
% 9.10/5.24          | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4369_dvd__mult__cancel__left,axiom,
% 9.10/5.24      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 9.10/5.24        = ( ( C = zero_z3403309356797280102nteger )
% 9.10/5.24          | ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4370_dvd__mult__cancel__left,axiom,
% 9.10/5.24      ! [C: complex,A: complex,B2: complex] :
% 9.10/5.24        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B2 ) )
% 9.10/5.24        = ( ( C = zero_zero_complex )
% 9.10/5.24          | ( dvd_dvd_complex @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4371_dvd__mult__cancel__left,axiom,
% 9.10/5.24      ! [C: real,A: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
% 9.10/5.24        = ( ( C = zero_zero_real )
% 9.10/5.24          | ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4372_dvd__mult__cancel__left,axiom,
% 9.10/5.24      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B2 ) )
% 9.10/5.24        = ( ( C = zero_zero_rat )
% 9.10/5.24          | ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4373_dvd__mult__cancel__left,axiom,
% 9.10/5.24      ! [C: int,A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.10/5.24        = ( ( C = zero_zero_int )
% 9.10/5.24          | ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4374_unit__prod,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4375_unit__prod,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.24       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.24         => ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4376_unit__prod,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.24       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.24         => ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4377_dvd__add__times__triv__right__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4378_dvd__add__times__triv__right__iff,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ ( times_times_real @ C @ A ) ) )
% 9.10/5.24        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4379_dvd__add__times__triv__right__iff,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C @ A ) ) )
% 9.10/5.24        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4380_dvd__add__times__triv__right__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C @ A ) ) )
% 9.10/5.24        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4381_dvd__add__times__triv__right__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ ( times_times_int @ C @ A ) ) )
% 9.10/5.24        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4382_dvd__add__times__triv__left__iff,axiom,
% 9.10/5.24      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4383_dvd__add__times__triv__left__iff,axiom,
% 9.10/5.24      ! [A: real,C: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4384_dvd__add__times__triv__left__iff,axiom,
% 9.10/5.24      ! [A: rat,C: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4385_dvd__add__times__triv__left__iff,axiom,
% 9.10/5.24      ! [A: nat,C: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4386_dvd__add__times__triv__left__iff,axiom,
% 9.10/5.24      ! [A: int,C: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4387_dvd__mult__div__cancel,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ A ) )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4388_dvd__mult__div__cancel,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ A ) )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4389_dvd__mult__div__cancel,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ A ) )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4390_dvd__div__mult__self,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4391_dvd__div__mult__self,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4392_dvd__div__mult__self,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4393_unit__div__1__div__1,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.24       => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4394_unit__div__1__div__1,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.24       => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4395_unit__div__1__div__1,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.24       => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4396_unit__div__1__unit,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4397_unit__div__1__unit,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.24       => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4398_unit__div__1__unit,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.24       => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4399_unit__div,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ one_one_Code_integer ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4400_unit__div,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.24       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.24         => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ one_one_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4401_unit__div,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.24       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.24         => ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4402_div__add,axiom,
% 9.10/5.24      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.24         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 9.10/5.24            = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4403_div__add,axiom,
% 9.10/5.24      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ C @ A )
% 9.10/5.24       => ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.24         => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.10/5.24            = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4404_div__add,axiom,
% 9.10/5.24      ! [C: int,A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ C @ A )
% 9.10/5.24       => ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.24         => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.10/5.24            = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4405_div__diff,axiom,
% 9.10/5.24      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.24         => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) @ C )
% 9.10/5.24            = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4406_div__diff,axiom,
% 9.10/5.24      ! [C: int,A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ C @ A )
% 9.10/5.24       => ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.24         => ( ( divide_divide_int @ ( minus_minus_int @ A @ B2 ) @ C )
% 9.10/5.24            = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4407_dvd__imp__mod__0,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( modulo_modulo_nat @ B2 @ A )
% 9.10/5.24          = zero_zero_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4408_dvd__imp__mod__0,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( modulo_modulo_int @ B2 @ A )
% 9.10/5.24          = zero_zero_int ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4409_dvd__imp__mod__0,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( modulo364778990260209775nteger @ B2 @ A )
% 9.10/5.24          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4410_signed__take__bit__Suc__1,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 9.10/5.24        = one_one_int ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4411_signed__take__bit__numeral__of__1,axiom,
% 9.10/5.24      ! [K: num] :
% 9.10/5.24        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 9.10/5.24        = one_one_int ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4412_unit__mult__div__div,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.24       => ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 9.10/5.24          = ( divide6298287555418463151nteger @ B2 @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4413_unit__mult__div__div,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.24       => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A ) )
% 9.10/5.24          = ( divide_divide_nat @ B2 @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4414_unit__mult__div__div,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.24       => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A ) )
% 9.10/5.24          = ( divide_divide_int @ B2 @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4415_unit__div__mult__self,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.24       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ A ) @ A )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4416_unit__div__mult__self,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.24       => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A ) @ A )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4417_unit__div__mult__self,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.24       => ( ( times_times_int @ ( divide_divide_int @ B2 @ A ) @ A )
% 9.10/5.24          = B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4418_even__Suc__Suc__iff,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 9.10/5.24        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4419_even__Suc,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4420_pow__divides__pow__iff,axiom,
% 9.10/5.24      ! [N: nat,A: nat,B2: nat] :
% 9.10/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) )
% 9.10/5.24          = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4421_pow__divides__pow__iff,axiom,
% 9.10/5.24      ! [N: nat,A: int,B2: int] :
% 9.10/5.24        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.24       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) )
% 9.10/5.24          = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4422_even__mult__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 9.10/5.24        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4423_even__mult__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B2 ) )
% 9.10/5.24        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4424_even__mult__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B2 ) )
% 9.10/5.24        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4425_even__add,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 9.10/5.24        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4426_even__add,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) )
% 9.10/5.24        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4427_even__add,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) )
% 9.10/5.24        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4428_odd__add,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) )
% 9.10/5.24        = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.24         != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4429_odd__add,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) )
% 9.10/5.24        = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.24         != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4430_odd__add,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) )
% 9.10/5.24        = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.24         != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4431_even__mod__2__iff,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.24        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4432_even__mod__2__iff,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.10/5.24        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4433_even__mod__2__iff,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4434_even__Suc__div__two,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4435_odd__Suc__div__two,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24       => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4436_signed__take__bit__Suc__bit0,axiom,
% 9.10/5.24      ! [N: nat,K: num] :
% 9.10/5.24        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.10/5.24        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4437_zero__le__power__eq__numeral,axiom,
% 9.10/5.24      ! [A: real,W: num] :
% 9.10/5.24        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.10/5.24        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4438_zero__le__power__eq__numeral,axiom,
% 9.10/5.24      ! [A: rat,W: num] :
% 9.10/5.24        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.10/5.24        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4439_zero__le__power__eq__numeral,axiom,
% 9.10/5.24      ! [A: int,W: num] :
% 9.10/5.24        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.10/5.24        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4440_power__less__zero__eq__numeral,axiom,
% 9.10/5.24      ! [A: real,W: num] :
% 9.10/5.24        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4441_power__less__zero__eq__numeral,axiom,
% 9.10/5.24      ! [A: rat,W: num] :
% 9.10/5.24        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4442_power__less__zero__eq__numeral,axiom,
% 9.10/5.24      ! [A: int,W: num] :
% 9.10/5.24        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4443_power__less__zero__eq,axiom,
% 9.10/5.24      ! [A: real,N: nat] :
% 9.10/5.24        ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4444_power__less__zero__eq,axiom,
% 9.10/5.24      ! [A: rat,N: nat] :
% 9.10/5.24        ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4445_power__less__zero__eq,axiom,
% 9.10/5.24      ! [A: int,N: nat] :
% 9.10/5.24        ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4446_even__plus__one__iff,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 9.10/5.24        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4447_even__plus__one__iff,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 9.10/5.24        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4448_even__plus__one__iff,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 9.10/5.24        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4449_even__diff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4450_even__diff,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4451_odd__Suc__minus__one,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24       => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.10/5.24          = N ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4452_even__diff__nat,axiom,
% 9.10/5.24      ! [M: nat,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.24        = ( ( ord_less_nat @ M @ N )
% 9.10/5.24          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4453_zero__less__power__eq__numeral,axiom,
% 9.10/5.24      ! [A: real,W: num] :
% 9.10/5.24        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.10/5.24        = ( ( ( numeral_numeral_nat @ W )
% 9.10/5.24            = zero_zero_nat )
% 9.10/5.24          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( A != zero_zero_real ) )
% 9.10/5.24          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4454_zero__less__power__eq__numeral,axiom,
% 9.10/5.24      ! [A: rat,W: num] :
% 9.10/5.24        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.10/5.24        = ( ( ( numeral_numeral_nat @ W )
% 9.10/5.24            = zero_zero_nat )
% 9.10/5.24          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( A != zero_zero_rat ) )
% 9.10/5.24          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4455_zero__less__power__eq__numeral,axiom,
% 9.10/5.24      ! [A: int,W: num] :
% 9.10/5.24        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 9.10/5.24        = ( ( ( numeral_numeral_nat @ W )
% 9.10/5.24            = zero_zero_nat )
% 9.10/5.24          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( A != zero_zero_int ) )
% 9.10/5.24          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4456_even__succ__div__2,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4457_even__succ__div__2,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4458_even__succ__div__2,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4459_even__succ__div__two,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4460_even__succ__div__two,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4461_even__succ__div__two,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4462_odd__succ__div__two,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4463_odd__succ__div__two,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4464_odd__succ__div__two,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.24          = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4465_even__power,axiom,
% 9.10/5.24      ! [A: code_integer,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.10/5.24        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4466_even__power,axiom,
% 9.10/5.24      ! [A: nat,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 9.10/5.24        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4467_even__power,axiom,
% 9.10/5.24      ! [A: int,N: nat] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 9.10/5.24        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4468_odd__two__times__div__two__nat,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.24       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.24          = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4469_odd__two__times__div__two__succ,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4470_odd__two__times__div__two__succ,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4471_odd__two__times__div__two__succ,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.24       => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 9.10/5.24          = A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4472_power__le__zero__eq__numeral,axiom,
% 9.10/5.24      ! [A: real,W: num] :
% 9.10/5.24        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 9.10/5.24        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.10/5.24            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4473_power__le__zero__eq__numeral,axiom,
% 9.10/5.24      ! [A: rat,W: num] :
% 9.10/5.24        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 9.10/5.24        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.10/5.24            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4474_power__le__zero__eq__numeral,axiom,
% 9.10/5.24      ! [A: int,W: num] :
% 9.10/5.24        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 9.10/5.24        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.10/5.24            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.24              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4475_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 9.10/5.24        = ( N = zero_zero_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4476_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 9.10/5.24        = ( N = zero_zero_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4477_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.10/5.24      ! [N: nat] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 9.10/5.24        = ( N = zero_zero_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4478_dvd__trans,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ B2 @ C )
% 9.10/5.24         => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4479_dvd__trans,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ B2 @ C )
% 9.10/5.24         => ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4480_dvd__trans,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ B2 @ C )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4481_dvd__refl,axiom,
% 9.10/5.24      ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4482_dvd__refl,axiom,
% 9.10/5.24      ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4483_dvd__refl,axiom,
% 9.10/5.24      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4484_dvd__0__left,axiom,
% 9.10/5.24      ! [A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.10/5.24       => ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4485_dvd__0__left,axiom,
% 9.10/5.24      ! [A: complex] :
% 9.10/5.24        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.10/5.24       => ( A = zero_zero_complex ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4486_dvd__0__left,axiom,
% 9.10/5.24      ! [A: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.10/5.24       => ( A = zero_zero_real ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4487_dvd__0__left,axiom,
% 9.10/5.24      ! [A: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.10/5.24       => ( A = zero_zero_rat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4488_dvd__0__left,axiom,
% 9.10/5.24      ! [A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.10/5.24       => ( A = zero_zero_nat ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4489_dvd__0__left,axiom,
% 9.10/5.24      ! [A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.10/5.24       => ( A = zero_zero_int ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4490_dvd__productE,axiom,
% 9.10/5.24      ! [P4: nat,A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B2 ) )
% 9.10/5.24       => ~ ! [X5: nat,Y4: nat] :
% 9.10/5.24              ( ( P4
% 9.10/5.24                = ( times_times_nat @ X5 @ Y4 ) )
% 9.10/5.24             => ( ( dvd_dvd_nat @ X5 @ A )
% 9.10/5.24               => ~ ( dvd_dvd_nat @ Y4 @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4491_dvd__productE,axiom,
% 9.10/5.24      ! [P4: int,A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B2 ) )
% 9.10/5.24       => ~ ! [X5: int,Y4: int] :
% 9.10/5.24              ( ( P4
% 9.10/5.24                = ( times_times_int @ X5 @ Y4 ) )
% 9.10/5.24             => ( ( dvd_dvd_int @ X5 @ A )
% 9.10/5.24               => ~ ( dvd_dvd_int @ Y4 @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4492_division__decomp,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.10/5.24       => ? [B8: nat,C5: nat] :
% 9.10/5.24            ( ( A
% 9.10/5.24              = ( times_times_nat @ B8 @ C5 ) )
% 9.10/5.24            & ( dvd_dvd_nat @ B8 @ B2 )
% 9.10/5.24            & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4493_division__decomp,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.24       => ? [B8: int,C5: int] :
% 9.10/5.24            ( ( A
% 9.10/5.24              = ( times_times_int @ B8 @ C5 ) )
% 9.10/5.24            & ( dvd_dvd_int @ B8 @ B2 )
% 9.10/5.24            & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4494_dvd__triv__right,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4495_dvd__triv__right,axiom,
% 9.10/5.24      ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4496_dvd__triv__right,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4497_dvd__triv__right,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4498_dvd__triv__right,axiom,
% 9.10/5.24      ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4499_dvd__mult__right,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ B2 @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4500_dvd__mult__right,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_real @ B2 @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4501_dvd__mult__right,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_rat @ B2 @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4502_dvd__mult__right,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_nat @ B2 @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4503_dvd__mult__right,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_int @ B2 @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4504_mult__dvd__mono,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ C @ D )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4505_mult__dvd__mono,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real,D: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_real @ C @ D )
% 9.10/5.24         => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4506_mult__dvd__mono,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_rat @ C @ D )
% 9.10/5.24         => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4507_mult__dvd__mono,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ C @ D )
% 9.10/5.24         => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4508_mult__dvd__mono,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int,D: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ C @ D )
% 9.10/5.24         => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4509_dvd__triv__left,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4510_dvd__triv__left,axiom,
% 9.10/5.24      ! [A: real,B2: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4511_dvd__triv__left,axiom,
% 9.10/5.24      ! [A: rat,B2: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4512_dvd__triv__left,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4513_dvd__triv__left,axiom,
% 9.10/5.24      ! [A: int,B2: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B2 ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4514_dvd__mult__left,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4515_dvd__mult__left,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ ( times_times_real @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_real @ A @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4516_dvd__mult__left,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_rat @ A @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4517_dvd__mult__left,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_nat @ A @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4518_dvd__mult__left,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.10/5.24       => ( dvd_dvd_int @ A @ C ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4519_dvd__mult2,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4520_dvd__mult2,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ B2 )
% 9.10/5.24       => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4521_dvd__mult2,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ B2 )
% 9.10/5.24       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4522_dvd__mult2,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4523_dvd__mult2,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4524_dvd__mult,axiom,
% 9.10/5.24      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ C )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4525_dvd__mult,axiom,
% 9.10/5.24      ! [A: real,C: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ C )
% 9.10/5.24       => ( dvd_dvd_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4526_dvd__mult,axiom,
% 9.10/5.24      ! [A: rat,C: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ C )
% 9.10/5.24       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4527_dvd__mult,axiom,
% 9.10/5.24      ! [A: nat,C: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ C )
% 9.10/5.24       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4528_dvd__mult,axiom,
% 9.10/5.24      ! [A: int,C: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ C )
% 9.10/5.24       => ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4529_dvd__def,axiom,
% 9.10/5.24      ( dvd_dvd_Code_integer
% 9.10/5.24      = ( ^ [B4: code_integer,A5: code_integer] :
% 9.10/5.24          ? [K2: code_integer] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( times_3573771949741848930nteger @ B4 @ K2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4530_dvd__def,axiom,
% 9.10/5.24      ( dvd_dvd_real
% 9.10/5.24      = ( ^ [B4: real,A5: real] :
% 9.10/5.24          ? [K2: real] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( times_times_real @ B4 @ K2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4531_dvd__def,axiom,
% 9.10/5.24      ( dvd_dvd_rat
% 9.10/5.24      = ( ^ [B4: rat,A5: rat] :
% 9.10/5.24          ? [K2: rat] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( times_times_rat @ B4 @ K2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4532_dvd__def,axiom,
% 9.10/5.24      ( dvd_dvd_nat
% 9.10/5.24      = ( ^ [B4: nat,A5: nat] :
% 9.10/5.24          ? [K2: nat] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( times_times_nat @ B4 @ K2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4533_dvd__def,axiom,
% 9.10/5.24      ( dvd_dvd_int
% 9.10/5.24      = ( ^ [B4: int,A5: int] :
% 9.10/5.24          ? [K2: int] :
% 9.10/5.24            ( A5
% 9.10/5.24            = ( times_times_int @ B4 @ K2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4534_dvdI,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,K: code_integer] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( times_3573771949741848930nteger @ B2 @ K ) )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4535_dvdI,axiom,
% 9.10/5.24      ! [A: real,B2: real,K: real] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( times_times_real @ B2 @ K ) )
% 9.10/5.24       => ( dvd_dvd_real @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4536_dvdI,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,K: rat] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( times_times_rat @ B2 @ K ) )
% 9.10/5.24       => ( dvd_dvd_rat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4537_dvdI,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,K: nat] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( times_times_nat @ B2 @ K ) )
% 9.10/5.24       => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4538_dvdI,axiom,
% 9.10/5.24      ! [A: int,B2: int,K: int] :
% 9.10/5.24        ( ( A
% 9.10/5.24          = ( times_times_int @ B2 @ K ) )
% 9.10/5.24       => ( dvd_dvd_int @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4539_dvdE,axiom,
% 9.10/5.24      ! [B2: code_integer,A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.24       => ~ ! [K3: code_integer] :
% 9.10/5.24              ( A
% 9.10/5.24             != ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4540_dvdE,axiom,
% 9.10/5.24      ! [B2: real,A: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ B2 @ A )
% 9.10/5.24       => ~ ! [K3: real] :
% 9.10/5.24              ( A
% 9.10/5.24             != ( times_times_real @ B2 @ K3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4541_dvdE,axiom,
% 9.10/5.24      ! [B2: rat,A: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ B2 @ A )
% 9.10/5.24       => ~ ! [K3: rat] :
% 9.10/5.24              ( A
% 9.10/5.24             != ( times_times_rat @ B2 @ K3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4542_dvdE,axiom,
% 9.10/5.24      ! [B2: nat,A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.24       => ~ ! [K3: nat] :
% 9.10/5.24              ( A
% 9.10/5.24             != ( times_times_nat @ B2 @ K3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4543_dvdE,axiom,
% 9.10/5.24      ! [B2: int,A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.24       => ~ ! [K3: int] :
% 9.10/5.24              ( A
% 9.10/5.24             != ( times_times_int @ B2 @ K3 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4544_dvd__unit__imp__unit,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4545_dvd__unit__imp__unit,axiom,
% 9.10/5.24      ! [A: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.24         => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4546_dvd__unit__imp__unit,axiom,
% 9.10/5.24      ! [A: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.24         => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4547_unit__imp__dvd,axiom,
% 9.10/5.24      ! [B2: code_integer,A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.24       => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4548_unit__imp__dvd,axiom,
% 9.10/5.24      ! [B2: nat,A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.24       => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4549_unit__imp__dvd,axiom,
% 9.10/5.24      ! [B2: int,A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.24       => ( dvd_dvd_int @ B2 @ A ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4550_one__dvd,axiom,
% 9.10/5.24      ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4551_one__dvd,axiom,
% 9.10/5.24      ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4552_one__dvd,axiom,
% 9.10/5.24      ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4553_one__dvd,axiom,
% 9.10/5.24      ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4554_one__dvd,axiom,
% 9.10/5.24      ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4555_one__dvd,axiom,
% 9.10/5.24      ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4556_dvd__add,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ A @ C )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4557_dvd__add,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_real @ A @ C )
% 9.10/5.24         => ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4558_dvd__add,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_rat @ A @ C )
% 9.10/5.24         => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4559_dvd__add,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ A @ C )
% 9.10/5.24         => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4560_dvd__add,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ A @ C )
% 9.10/5.24         => ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4561_dvd__add__left__iff,axiom,
% 9.10/5.24      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ C )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_Code_integer @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4562_dvd__add__left__iff,axiom,
% 9.10/5.24      ! [A: real,C: real,B2: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ C )
% 9.10/5.24       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_real @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4563_dvd__add__left__iff,axiom,
% 9.10/5.24      ! [A: rat,C: rat,B2: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ C )
% 9.10/5.24       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_rat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4564_dvd__add__left__iff,axiom,
% 9.10/5.24      ! [A: nat,C: nat,B2: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ C )
% 9.10/5.24       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_nat @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4565_dvd__add__left__iff,axiom,
% 9.10/5.24      ! [A: int,C: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ C )
% 9.10/5.24       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_int @ A @ B2 ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4566_dvd__add__right__iff,axiom,
% 9.10/5.24      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4567_dvd__add__right__iff,axiom,
% 9.10/5.24      ! [A: real,B2: real,C: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_real @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4568_dvd__add__right__iff,axiom,
% 9.10/5.24      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4569_dvd__add__right__iff,axiom,
% 9.10/5.24      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4570_dvd__add__right__iff,axiom,
% 9.10/5.24      ! [A: int,B2: int,C: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B2 @ C ) )
% 9.10/5.24          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4571_dvd__diff,axiom,
% 9.10/5.24      ! [X3: code_integer,Y3: code_integer,Z: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ X3 @ Z )
% 9.10/5.24         => ( dvd_dvd_Code_integer @ X3 @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4572_dvd__diff,axiom,
% 9.10/5.24      ! [X3: complex,Y3: complex,Z: complex] :
% 9.10/5.24        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 9.10/5.24       => ( ( dvd_dvd_complex @ X3 @ Z )
% 9.10/5.24         => ( dvd_dvd_complex @ X3 @ ( minus_minus_complex @ Y3 @ Z ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4573_dvd__diff,axiom,
% 9.10/5.24      ! [X3: real,Y3: real,Z: real] :
% 9.10/5.24        ( ( dvd_dvd_real @ X3 @ Y3 )
% 9.10/5.24       => ( ( dvd_dvd_real @ X3 @ Z )
% 9.10/5.24         => ( dvd_dvd_real @ X3 @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4574_dvd__diff,axiom,
% 9.10/5.24      ! [X3: rat,Y3: rat,Z: rat] :
% 9.10/5.24        ( ( dvd_dvd_rat @ X3 @ Y3 )
% 9.10/5.24       => ( ( dvd_dvd_rat @ X3 @ Z )
% 9.10/5.24         => ( dvd_dvd_rat @ X3 @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4575_dvd__diff,axiom,
% 9.10/5.24      ! [X3: int,Y3: int,Z: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ X3 @ Y3 )
% 9.10/5.24       => ( ( dvd_dvd_int @ X3 @ Z )
% 9.10/5.24         => ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4576_dvd__diff__commute,axiom,
% 9.10/5.24      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4577_dvd__diff__commute,axiom,
% 9.10/5.24      ! [A: int,C: int,B2: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B2 ) )
% 9.10/5.24        = ( dvd_dvd_int @ A @ ( minus_minus_int @ B2 @ C ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4578_div__div__div__same,axiom,
% 9.10/5.24      ! [D: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.24        ( ( dvd_dvd_Code_integer @ D @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.24         => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B2 @ D ) )
% 9.10/5.24            = ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4579_div__div__div__same,axiom,
% 9.10/5.24      ! [D: nat,B2: nat,A: nat] :
% 9.10/5.24        ( ( dvd_dvd_nat @ D @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.24         => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B2 @ D ) )
% 9.10/5.24            = ( divide_divide_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.24  
% 9.10/5.24  thf(fact_4580_div__div__div__same,axiom,
% 9.10/5.24      ! [D: int,B2: int,A: int] :
% 9.10/5.24        ( ( dvd_dvd_int @ D @ B2 )
% 9.10/5.24       => ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.24         => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B2 @ D ) )
% 9.10/5.25            = ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4581_dvd__div__eq__cancel,axiom,
% 9.10/5.25      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ( divide6298287555418463151nteger @ A @ C )
% 9.10/5.25          = ( divide6298287555418463151nteger @ B2 @ C ) )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ C @ A )
% 9.10/5.25         => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25           => ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4582_dvd__div__eq__cancel,axiom,
% 9.10/5.25      ! [A: complex,C: complex,B2: complex] :
% 9.10/5.25        ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.10/5.25          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.10/5.25       => ( ( dvd_dvd_complex @ C @ A )
% 9.10/5.25         => ( ( dvd_dvd_complex @ C @ B2 )
% 9.10/5.25           => ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4583_dvd__div__eq__cancel,axiom,
% 9.10/5.25      ! [A: real,C: real,B2: real] :
% 9.10/5.25        ( ( ( divide_divide_real @ A @ C )
% 9.10/5.25          = ( divide_divide_real @ B2 @ C ) )
% 9.10/5.25       => ( ( dvd_dvd_real @ C @ A )
% 9.10/5.25         => ( ( dvd_dvd_real @ C @ B2 )
% 9.10/5.25           => ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4584_dvd__div__eq__cancel,axiom,
% 9.10/5.25      ! [A: rat,C: rat,B2: rat] :
% 9.10/5.25        ( ( ( divide_divide_rat @ A @ C )
% 9.10/5.25          = ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.25       => ( ( dvd_dvd_rat @ C @ A )
% 9.10/5.25         => ( ( dvd_dvd_rat @ C @ B2 )
% 9.10/5.25           => ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4585_dvd__div__eq__cancel,axiom,
% 9.10/5.25      ! [A: nat,C: nat,B2: nat] :
% 9.10/5.25        ( ( ( divide_divide_nat @ A @ C )
% 9.10/5.25          = ( divide_divide_nat @ B2 @ C ) )
% 9.10/5.25       => ( ( dvd_dvd_nat @ C @ A )
% 9.10/5.25         => ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25           => ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4586_dvd__div__eq__cancel,axiom,
% 9.10/5.25      ! [A: int,C: int,B2: int] :
% 9.10/5.25        ( ( ( divide_divide_int @ A @ C )
% 9.10/5.25          = ( divide_divide_int @ B2 @ C ) )
% 9.10/5.25       => ( ( dvd_dvd_int @ C @ A )
% 9.10/5.25         => ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25           => ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4587_dvd__div__eq__iff,axiom,
% 9.10/5.25      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25         => ( ( ( divide6298287555418463151nteger @ A @ C )
% 9.10/5.25              = ( divide6298287555418463151nteger @ B2 @ C ) )
% 9.10/5.25            = ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4588_dvd__div__eq__iff,axiom,
% 9.10/5.25      ! [C: complex,A: complex,B2: complex] :
% 9.10/5.25        ( ( dvd_dvd_complex @ C @ A )
% 9.10/5.25       => ( ( dvd_dvd_complex @ C @ B2 )
% 9.10/5.25         => ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.10/5.25              = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.10/5.25            = ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4589_dvd__div__eq__iff,axiom,
% 9.10/5.25      ! [C: real,A: real,B2: real] :
% 9.10/5.25        ( ( dvd_dvd_real @ C @ A )
% 9.10/5.25       => ( ( dvd_dvd_real @ C @ B2 )
% 9.10/5.25         => ( ( ( divide_divide_real @ A @ C )
% 9.10/5.25              = ( divide_divide_real @ B2 @ C ) )
% 9.10/5.25            = ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4590_dvd__div__eq__iff,axiom,
% 9.10/5.25      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.25        ( ( dvd_dvd_rat @ C @ A )
% 9.10/5.25       => ( ( dvd_dvd_rat @ C @ B2 )
% 9.10/5.25         => ( ( ( divide_divide_rat @ A @ C )
% 9.10/5.25              = ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.25            = ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4591_dvd__div__eq__iff,axiom,
% 9.10/5.25      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ A )
% 9.10/5.25       => ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25         => ( ( ( divide_divide_nat @ A @ C )
% 9.10/5.25              = ( divide_divide_nat @ B2 @ C ) )
% 9.10/5.25            = ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4592_dvd__div__eq__iff,axiom,
% 9.10/5.25      ! [C: int,A: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ A )
% 9.10/5.25       => ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25         => ( ( ( divide_divide_int @ A @ C )
% 9.10/5.25              = ( divide_divide_int @ B2 @ C ) )
% 9.10/5.25            = ( A = B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4593_dvd__power__same,axiom,
% 9.10/5.25      ! [X3: code_integer,Y3: code_integer,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 9.10/5.25       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4594_dvd__power__same,axiom,
% 9.10/5.25      ! [X3: nat,Y3: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 9.10/5.25       => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4595_dvd__power__same,axiom,
% 9.10/5.25      ! [X3: real,Y3: real,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_real @ X3 @ Y3 )
% 9.10/5.25       => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4596_dvd__power__same,axiom,
% 9.10/5.25      ! [X3: int,Y3: int,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ X3 @ Y3 )
% 9.10/5.25       => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4597_dvd__power__same,axiom,
% 9.10/5.25      ! [X3: complex,Y3: complex,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 9.10/5.25       => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4598_mod__mod__cancel,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25       => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4599_mod__mod__cancel,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25       => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( modulo_modulo_int @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4600_mod__mod__cancel,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25       => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4601_dvd__mod,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ K @ M )
% 9.10/5.25       => ( ( dvd_dvd_nat @ K @ N )
% 9.10/5.25         => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4602_dvd__mod,axiom,
% 9.10/5.25      ! [K: int,M: int,N: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ K @ M )
% 9.10/5.25       => ( ( dvd_dvd_int @ K @ N )
% 9.10/5.25         => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4603_dvd__mod,axiom,
% 9.10/5.25      ! [K: code_integer,M: code_integer,N: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ K @ M )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ K @ N )
% 9.10/5.25         => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4604_dvd__mod__imp__dvd,axiom,
% 9.10/5.25      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.10/5.25       => ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25         => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4605_dvd__mod__imp__dvd,axiom,
% 9.10/5.25      ! [C: int,A: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 9.10/5.25       => ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25         => ( dvd_dvd_int @ C @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4606_dvd__mod__imp__dvd,axiom,
% 9.10/5.25      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25         => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4607_dvd__mod__iff,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25       => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4608_dvd__mod__iff,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25       => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_int @ C @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4609_dvd__mod__iff,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4610_signed__take__bit__mult,axiom,
% 9.10/5.25      ! [N: nat,K: int,L: int] :
% 9.10/5.25        ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 9.10/5.25        = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4611_signed__take__bit__add,axiom,
% 9.10/5.25      ! [N: nat,K: int,L: int] :
% 9.10/5.25        ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 9.10/5.25        = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4612_dvd__diff__nat,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ K @ M )
% 9.10/5.25       => ( ( dvd_dvd_nat @ K @ N )
% 9.10/5.25         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4613_signed__take__bit__diff,axiom,
% 9.10/5.25      ! [N: nat,K: int,L: int] :
% 9.10/5.25        ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
% 9.10/5.25        = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4614_subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( ord_le211207098394363844omplex
% 9.10/5.25          @ ( collect_complex
% 9.10/5.25            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 9.10/5.25          @ ( collect_complex
% 9.10/5.25            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B2 ) ) )
% 9.10/5.25        = ( dvd_dvd_complex @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4615_subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( ord_less_eq_set_real
% 9.10/5.25          @ ( collect_real
% 9.10/5.25            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 9.10/5.25          @ ( collect_real
% 9.10/5.25            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
% 9.10/5.25        = ( dvd_dvd_real @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4616_subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( ord_less_eq_set_nat
% 9.10/5.25          @ ( collect_nat
% 9.10/5.25            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 9.10/5.25          @ ( collect_nat
% 9.10/5.25            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
% 9.10/5.25        = ( dvd_dvd_nat @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4617_subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ord_le7084787975880047091nteger
% 9.10/5.25          @ ( collect_Code_integer
% 9.10/5.25            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 9.10/5.25          @ ( collect_Code_integer
% 9.10/5.25            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B2 ) ) )
% 9.10/5.25        = ( dvd_dvd_Code_integer @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4618_subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( ord_less_eq_set_int
% 9.10/5.25          @ ( collect_int
% 9.10/5.25            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 9.10/5.25          @ ( collect_int
% 9.10/5.25            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
% 9.10/5.25        = ( dvd_dvd_int @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4619_strict__subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( ord_less_set_complex
% 9.10/5.25          @ ( collect_complex
% 9.10/5.25            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 9.10/5.25          @ ( collect_complex
% 9.10/5.25            @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B2 ) ) )
% 9.10/5.25        = ( ( dvd_dvd_complex @ A @ B2 )
% 9.10/5.25          & ~ ( dvd_dvd_complex @ B2 @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4620_strict__subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( ord_less_set_real
% 9.10/5.25          @ ( collect_real
% 9.10/5.25            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 9.10/5.25          @ ( collect_real
% 9.10/5.25            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B2 ) ) )
% 9.10/5.25        = ( ( dvd_dvd_real @ A @ B2 )
% 9.10/5.25          & ~ ( dvd_dvd_real @ B2 @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4621_strict__subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( ord_less_set_nat
% 9.10/5.25          @ ( collect_nat
% 9.10/5.25            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 9.10/5.25          @ ( collect_nat
% 9.10/5.25            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B2 ) ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.25          & ~ ( dvd_dvd_nat @ B2 @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4622_strict__subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( ord_less_set_int
% 9.10/5.25          @ ( collect_int
% 9.10/5.25            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 9.10/5.25          @ ( collect_int
% 9.10/5.25            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B2 ) ) )
% 9.10/5.25        = ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.25          & ~ ( dvd_dvd_int @ B2 @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4623_strict__subset__divisors__dvd,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ord_le1307284697595431911nteger
% 9.10/5.25          @ ( collect_Code_integer
% 9.10/5.25            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 9.10/5.25          @ ( collect_Code_integer
% 9.10/5.25            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B2 ) ) )
% 9.10/5.25        = ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.25          & ~ ( dvd_dvd_Code_integer @ B2 @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4624_even__signed__take__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 9.10/5.25        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4625_even__signed__take__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 9.10/5.25        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4626_not__is__unit__0,axiom,
% 9.10/5.25      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4627_not__is__unit__0,axiom,
% 9.10/5.25      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4628_not__is__unit__0,axiom,
% 9.10/5.25      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4629_pinf_I9_J,axiom,
% 9.10/5.25      ! [D: code_integer,S: code_integer] :
% 9.10/5.25      ? [Z4: code_integer] :
% 9.10/5.25      ! [X6: code_integer] :
% 9.10/5.25        ( ( ord_le6747313008572928689nteger @ Z4 @ X6 )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4630_pinf_I9_J,axiom,
% 9.10/5.25      ! [D: real,S: real] :
% 9.10/5.25      ? [Z4: real] :
% 9.10/5.25      ! [X6: real] :
% 9.10/5.25        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.25       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4631_pinf_I9_J,axiom,
% 9.10/5.25      ! [D: rat,S: rat] :
% 9.10/5.25      ? [Z4: rat] :
% 9.10/5.25      ! [X6: rat] :
% 9.10/5.25        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.25       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4632_pinf_I9_J,axiom,
% 9.10/5.25      ! [D: nat,S: nat] :
% 9.10/5.25      ? [Z4: nat] :
% 9.10/5.25      ! [X6: nat] :
% 9.10/5.25        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.25       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4633_pinf_I9_J,axiom,
% 9.10/5.25      ! [D: int,S: int] :
% 9.10/5.25      ? [Z4: int] :
% 9.10/5.25      ! [X6: int] :
% 9.10/5.25        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.25       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4634_pinf_I10_J,axiom,
% 9.10/5.25      ! [D: code_integer,S: code_integer] :
% 9.10/5.25      ? [Z4: code_integer] :
% 9.10/5.25      ! [X6: code_integer] :
% 9.10/5.25        ( ( ord_le6747313008572928689nteger @ Z4 @ X6 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4635_pinf_I10_J,axiom,
% 9.10/5.25      ! [D: real,S: real] :
% 9.10/5.25      ? [Z4: real] :
% 9.10/5.25      ! [X6: real] :
% 9.10/5.25        ( ( ord_less_real @ Z4 @ X6 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4636_pinf_I10_J,axiom,
% 9.10/5.25      ! [D: rat,S: rat] :
% 9.10/5.25      ? [Z4: rat] :
% 9.10/5.25      ! [X6: rat] :
% 9.10/5.25        ( ( ord_less_rat @ Z4 @ X6 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4637_pinf_I10_J,axiom,
% 9.10/5.25      ! [D: nat,S: nat] :
% 9.10/5.25      ? [Z4: nat] :
% 9.10/5.25      ! [X6: nat] :
% 9.10/5.25        ( ( ord_less_nat @ Z4 @ X6 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4638_pinf_I10_J,axiom,
% 9.10/5.25      ! [D: int,S: int] :
% 9.10/5.25      ? [Z4: int] :
% 9.10/5.25      ! [X6: int] :
% 9.10/5.25        ( ( ord_less_int @ Z4 @ X6 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4639_minf_I9_J,axiom,
% 9.10/5.25      ! [D: code_integer,S: code_integer] :
% 9.10/5.25      ? [Z4: code_integer] :
% 9.10/5.25      ! [X6: code_integer] :
% 9.10/5.25        ( ( ord_le6747313008572928689nteger @ X6 @ Z4 )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4640_minf_I9_J,axiom,
% 9.10/5.25      ! [D: real,S: real] :
% 9.10/5.25      ? [Z4: real] :
% 9.10/5.25      ! [X6: real] :
% 9.10/5.25        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.25       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4641_minf_I9_J,axiom,
% 9.10/5.25      ! [D: rat,S: rat] :
% 9.10/5.25      ? [Z4: rat] :
% 9.10/5.25      ! [X6: rat] :
% 9.10/5.25        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.25       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4642_minf_I9_J,axiom,
% 9.10/5.25      ! [D: nat,S: nat] :
% 9.10/5.25      ? [Z4: nat] :
% 9.10/5.25      ! [X6: nat] :
% 9.10/5.25        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.25       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4643_minf_I9_J,axiom,
% 9.10/5.25      ! [D: int,S: int] :
% 9.10/5.25      ? [Z4: int] :
% 9.10/5.25      ! [X6: int] :
% 9.10/5.25        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.25       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) )
% 9.10/5.25          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4644_minf_I10_J,axiom,
% 9.10/5.25      ! [D: code_integer,S: code_integer] :
% 9.10/5.25      ? [Z4: code_integer] :
% 9.10/5.25      ! [X6: code_integer] :
% 9.10/5.25        ( ( ord_le6747313008572928689nteger @ X6 @ Z4 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4645_minf_I10_J,axiom,
% 9.10/5.25      ! [D: real,S: real] :
% 9.10/5.25      ? [Z4: real] :
% 9.10/5.25      ! [X6: real] :
% 9.10/5.25        ( ( ord_less_real @ X6 @ Z4 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4646_minf_I10_J,axiom,
% 9.10/5.25      ! [D: rat,S: rat] :
% 9.10/5.25      ? [Z4: rat] :
% 9.10/5.25      ! [X6: rat] :
% 9.10/5.25        ( ( ord_less_rat @ X6 @ Z4 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4647_minf_I10_J,axiom,
% 9.10/5.25      ! [D: nat,S: nat] :
% 9.10/5.25      ? [Z4: nat] :
% 9.10/5.25      ! [X6: nat] :
% 9.10/5.25        ( ( ord_less_nat @ X6 @ Z4 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4648_minf_I10_J,axiom,
% 9.10/5.25      ! [D: int,S: int] :
% 9.10/5.25      ? [Z4: int] :
% 9.10/5.25      ! [X6: int] :
% 9.10/5.25        ( ( ord_less_int @ X6 @ Z4 )
% 9.10/5.25       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) )
% 9.10/5.25          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ S ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4649_dvd__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.25       => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 9.10/5.25            = zero_z3403309356797280102nteger )
% 9.10/5.25          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4650_dvd__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: complex,A: complex] :
% 9.10/5.25        ( ( dvd_dvd_complex @ B2 @ A )
% 9.10/5.25       => ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 9.10/5.25            = zero_zero_complex )
% 9.10/5.25          = ( A = zero_zero_complex ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4651_dvd__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: real,A: real] :
% 9.10/5.25        ( ( dvd_dvd_real @ B2 @ A )
% 9.10/5.25       => ( ( ( divide_divide_real @ A @ B2 )
% 9.10/5.25            = zero_zero_real )
% 9.10/5.25          = ( A = zero_zero_real ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4652_dvd__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: rat,A: rat] :
% 9.10/5.25        ( ( dvd_dvd_rat @ B2 @ A )
% 9.10/5.25       => ( ( ( divide_divide_rat @ A @ B2 )
% 9.10/5.25            = zero_zero_rat )
% 9.10/5.25          = ( A = zero_zero_rat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4653_dvd__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.25       => ( ( ( divide_divide_nat @ A @ B2 )
% 9.10/5.25            = zero_zero_nat )
% 9.10/5.25          = ( A = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4654_dvd__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.25       => ( ( ( divide_divide_int @ A @ B2 )
% 9.10/5.25            = zero_zero_int )
% 9.10/5.25          = ( A = zero_zero_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4655_unit__mult__right__cancel,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25       => ( ( ( times_3573771949741848930nteger @ B2 @ A )
% 9.10/5.25            = ( times_3573771949741848930nteger @ C @ A ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4656_unit__mult__right__cancel,axiom,
% 9.10/5.25      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25       => ( ( ( times_times_nat @ B2 @ A )
% 9.10/5.25            = ( times_times_nat @ C @ A ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4657_unit__mult__right__cancel,axiom,
% 9.10/5.25      ! [A: int,B2: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25       => ( ( ( times_times_int @ B2 @ A )
% 9.10/5.25            = ( times_times_int @ C @ A ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4658_unit__mult__left__cancel,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25       => ( ( ( times_3573771949741848930nteger @ A @ B2 )
% 9.10/5.25            = ( times_3573771949741848930nteger @ A @ C ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4659_unit__mult__left__cancel,axiom,
% 9.10/5.25      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25       => ( ( ( times_times_nat @ A @ B2 )
% 9.10/5.25            = ( times_times_nat @ A @ C ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4660_unit__mult__left__cancel,axiom,
% 9.10/5.25      ! [A: int,B2: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25       => ( ( ( times_times_int @ A @ B2 )
% 9.10/5.25            = ( times_times_int @ A @ C ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4661_mult__unit__dvd__iff_H,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ B2 @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4662_mult__unit__dvd__iff_H,axiom,
% 9.10/5.25      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_nat @ B2 @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4663_mult__unit__dvd__iff_H,axiom,
% 9.10/5.25      ! [A: int,B2: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_int @ B2 @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4664_dvd__mult__unit__iff_H,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4665_dvd__mult__unit__iff_H,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.10/5.25          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4666_dvd__mult__unit__iff_H,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.25          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4667_mult__unit__dvd__iff,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4668_mult__unit__dvd__iff,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4669_mult__unit__dvd__iff,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4670_dvd__mult__unit__iff,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4671_dvd__mult__unit__iff,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4672_dvd__mult__unit__iff,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4673_is__unit__mult__iff,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) @ one_one_Code_integer )
% 9.10/5.25        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25          & ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4674_is__unit__mult__iff,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B2 ) @ one_one_nat )
% 9.10/5.25        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25          & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4675_is__unit__mult__iff,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( times_times_int @ A @ B2 ) @ one_one_int )
% 9.10/5.25        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25          & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4676_div__mult__div__if__dvd,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ D @ C )
% 9.10/5.25         => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 9.10/5.25            = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B2 @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4677_div__mult__div__if__dvd,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,D: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.25       => ( ( dvd_dvd_nat @ D @ C )
% 9.10/5.25         => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ ( divide_divide_nat @ C @ D ) )
% 9.10/5.25            = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4678_div__mult__div__if__dvd,axiom,
% 9.10/5.25      ! [B2: int,A: int,D: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.25       => ( ( dvd_dvd_int @ D @ C )
% 9.10/5.25         => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ C @ D ) )
% 9.10/5.25            = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4679_dvd__mult__imp__div,axiom,
% 9.10/5.25      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 )
% 9.10/5.25       => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4680_dvd__mult__imp__div,axiom,
% 9.10/5.25      ! [A: nat,C: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 )
% 9.10/5.25       => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4681_dvd__mult__imp__div,axiom,
% 9.10/5.25      ! [A: int,C: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 )
% 9.10/5.25       => ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4682_dvd__div__mult2__eq,axiom,
% 9.10/5.25      ! [B2: code_integer,C: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B2 @ C ) @ A )
% 9.10/5.25       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.10/5.25          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4683_dvd__div__mult2__eq,axiom,
% 9.10/5.25      ! [B2: nat,C: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C ) @ A )
% 9.10/5.25       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.10/5.25          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4684_dvd__div__mult2__eq,axiom,
% 9.10/5.25      ! [B2: int,C: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C ) @ A )
% 9.10/5.25       => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.25          = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4685_div__div__eq__right,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.25         => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 9.10/5.25            = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4686_div__div__eq__right,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25       => ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.25         => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 9.10/5.25            = ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4687_div__div__eq__right,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25       => ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.25         => ( ( divide_divide_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 9.10/5.25            = ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4688_div__mult__swap,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 9.10/5.25          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4689_div__mult__swap,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 9.10/5.25          = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4690_div__mult__swap,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25       => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 9.10/5.25          = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4691_dvd__div__mult,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B2 @ C ) @ A )
% 9.10/5.25          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B2 @ A ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4692_dvd__div__mult,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25       => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C ) @ A )
% 9.10/5.25          = ( divide_divide_nat @ ( times_times_nat @ B2 @ A ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4693_dvd__div__mult,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25       => ( ( times_times_int @ ( divide_divide_int @ B2 @ C ) @ A )
% 9.10/5.25          = ( divide_divide_int @ ( times_times_int @ B2 @ A ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4694_dvd__div__unit__iff,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4695_dvd__div__unit__iff,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4696_dvd__div__unit__iff,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B2 ) )
% 9.10/5.25          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4697_div__unit__dvd__iff,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4698_div__unit__dvd__iff,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4699_div__unit__dvd__iff,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4700_unit__div__cancel,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25       => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 9.10/5.25            = ( divide6298287555418463151nteger @ C @ A ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4701_unit__div__cancel,axiom,
% 9.10/5.25      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25       => ( ( ( divide_divide_nat @ B2 @ A )
% 9.10/5.25            = ( divide_divide_nat @ C @ A ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4702_unit__div__cancel,axiom,
% 9.10/5.25      ! [A: int,B2: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25       => ( ( ( divide_divide_int @ B2 @ A )
% 9.10/5.25            = ( divide_divide_int @ C @ A ) )
% 9.10/5.25          = ( B2 = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4703_div__plus__div__distrib__dvd__left,axiom,
% 9.10/5.25      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.10/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4704_div__plus__div__distrib__dvd__left,axiom,
% 9.10/5.25      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ A )
% 9.10/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4705_div__plus__div__distrib__dvd__left,axiom,
% 9.10/5.25      ! [C: int,A: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ A )
% 9.10/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4706_div__plus__div__distrib__dvd__right,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B2 @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4707_div__plus__div__distrib__dvd__right,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B2 @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4708_div__plus__div__distrib__dvd__right,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B2 @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4709_div__power,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.25       => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ N )
% 9.10/5.25          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4710_div__power,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.25       => ( ( power_power_nat @ ( divide_divide_nat @ A @ B2 ) @ N )
% 9.10/5.25          = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B2 @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4711_div__power,axiom,
% 9.10/5.25      ! [B2: int,A: int,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.25       => ( ( power_power_int @ ( divide_divide_int @ A @ B2 ) @ N )
% 9.10/5.25          = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4712_mod__eq__0__iff__dvd,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( ( modulo_modulo_nat @ A @ B2 )
% 9.10/5.25          = zero_zero_nat )
% 9.10/5.25        = ( dvd_dvd_nat @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4713_mod__eq__0__iff__dvd,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( ( modulo_modulo_int @ A @ B2 )
% 9.10/5.25          = zero_zero_int )
% 9.10/5.25        = ( dvd_dvd_int @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4714_mod__eq__0__iff__dvd,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 9.10/5.25          = zero_z3403309356797280102nteger )
% 9.10/5.25        = ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4715_dvd__eq__mod__eq__0,axiom,
% 9.10/5.25      ( dvd_dvd_nat
% 9.10/5.25      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.25            ( ( modulo_modulo_nat @ B4 @ A5 )
% 9.10/5.25            = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4716_dvd__eq__mod__eq__0,axiom,
% 9.10/5.25      ( dvd_dvd_int
% 9.10/5.25      = ( ^ [A5: int,B4: int] :
% 9.10/5.25            ( ( modulo_modulo_int @ B4 @ A5 )
% 9.10/5.25            = zero_zero_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4717_dvd__eq__mod__eq__0,axiom,
% 9.10/5.25      ( dvd_dvd_Code_integer
% 9.10/5.25      = ( ^ [A5: code_integer,B4: code_integer] :
% 9.10/5.25            ( ( modulo364778990260209775nteger @ B4 @ A5 )
% 9.10/5.25            = zero_z3403309356797280102nteger ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4718_mod__0__imp__dvd,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( ( modulo_modulo_nat @ A @ B2 )
% 9.10/5.25          = zero_zero_nat )
% 9.10/5.25       => ( dvd_dvd_nat @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4719_mod__0__imp__dvd,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( ( modulo_modulo_int @ A @ B2 )
% 9.10/5.25          = zero_zero_int )
% 9.10/5.25       => ( dvd_dvd_int @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4720_mod__0__imp__dvd,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 9.10/5.25          = zero_z3403309356797280102nteger )
% 9.10/5.25       => ( dvd_dvd_Code_integer @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4721_dvd__power__le,axiom,
% 9.10/5.25      ! [X3: code_integer,Y3: code_integer,N: nat,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4722_dvd__power__le,axiom,
% 9.10/5.25      ! [X3: nat,Y3: nat,N: nat,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25         => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4723_dvd__power__le,axiom,
% 9.10/5.25      ! [X3: real,Y3: real,N: nat,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_real @ X3 @ Y3 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25         => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4724_dvd__power__le,axiom,
% 9.10/5.25      ! [X3: int,Y3: int,N: nat,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ X3 @ Y3 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25         => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4725_dvd__power__le,axiom,
% 9.10/5.25      ! [X3: complex,Y3: complex,N: nat,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25         => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4726_power__le__dvd,axiom,
% 9.10/5.25      ! [A: code_integer,N: nat,B2: code_integer,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B2 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4727_power__le__dvd,axiom,
% 9.10/5.25      ! [A: nat,N: nat,B2: nat,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B2 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4728_power__le__dvd,axiom,
% 9.10/5.25      ! [A: real,N: nat,B2: real,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B2 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4729_power__le__dvd,axiom,
% 9.10/5.25      ! [A: int,N: nat,B2: int,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B2 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4730_power__le__dvd,axiom,
% 9.10/5.25      ! [A: complex,N: nat,B2: complex,M: nat] :
% 9.10/5.25        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B2 )
% 9.10/5.25       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4731_le__imp__power__dvd,axiom,
% 9.10/5.25      ! [M: nat,N: nat,A: code_integer] :
% 9.10/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4732_le__imp__power__dvd,axiom,
% 9.10/5.25      ! [M: nat,N: nat,A: nat] :
% 9.10/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4733_le__imp__power__dvd,axiom,
% 9.10/5.25      ! [M: nat,N: nat,A: real] :
% 9.10/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4734_le__imp__power__dvd,axiom,
% 9.10/5.25      ! [M: nat,N: nat,A: int] :
% 9.10/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4735_le__imp__power__dvd,axiom,
% 9.10/5.25      ! [M: nat,N: nat,A: complex] :
% 9.10/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4736_mod__eq__dvd__iff,axiom,
% 9.10/5.25      ! [A: int,C: int,B2: int] :
% 9.10/5.25        ( ( ( modulo_modulo_int @ A @ C )
% 9.10/5.25          = ( modulo_modulo_int @ B2 @ C ) )
% 9.10/5.25        = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4737_mod__eq__dvd__iff,axiom,
% 9.10/5.25      ! [A: code_integer,C: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.10/5.25          = ( modulo364778990260209775nteger @ B2 @ C ) )
% 9.10/5.25        = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4738_dvd__minus__mod,axiom,
% 9.10/5.25      ! [B2: nat,A: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4739_dvd__minus__mod,axiom,
% 9.10/5.25      ! [B2: int,A: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4740_dvd__minus__mod,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4741_dvd__pos__nat,axiom,
% 9.10/5.25      ! [N: nat,M: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25       => ( ( dvd_dvd_nat @ M @ N )
% 9.10/5.25         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4742_nat__dvd__not__less,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.25       => ( ( ord_less_nat @ M @ N )
% 9.10/5.25         => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4743_dvd__minus__self,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 9.10/5.25        = ( ( ord_less_nat @ N @ M )
% 9.10/5.25          | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4744_zdvd__antisym__nonneg,axiom,
% 9.10/5.25      ! [M: int,N: int] :
% 9.10/5.25        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.10/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.10/5.25         => ( ( dvd_dvd_int @ M @ N )
% 9.10/5.25           => ( ( dvd_dvd_int @ N @ M )
% 9.10/5.25             => ( M = N ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4745_less__eq__dvd__minus,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25       => ( ( dvd_dvd_nat @ M @ N )
% 9.10/5.25          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4746_dvd__diffD1,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.25       => ( ( dvd_dvd_nat @ K @ M )
% 9.10/5.25         => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25           => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4747_dvd__diffD,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.25       => ( ( dvd_dvd_nat @ K @ N )
% 9.10/5.25         => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4748_zdvd__not__zless,axiom,
% 9.10/5.25      ! [M: int,N: int] :
% 9.10/5.25        ( ( ord_less_int @ zero_zero_int @ M )
% 9.10/5.25       => ( ( ord_less_int @ M @ N )
% 9.10/5.25         => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4749_zdvd__mono,axiom,
% 9.10/5.25      ! [K: int,M: int,T: int] :
% 9.10/5.25        ( ( K != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ M @ T )
% 9.10/5.25          = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4750_zdvd__mult__cancel,axiom,
% 9.10/5.25      ! [K: int,M: int,N: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 9.10/5.25       => ( ( K != zero_zero_int )
% 9.10/5.25         => ( dvd_dvd_int @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4751_bezout__add__nat,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25      ? [D3: nat,X5: nat,Y4: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ D3 @ A )
% 9.10/5.25        & ( dvd_dvd_nat @ D3 @ B2 )
% 9.10/5.25        & ( ( ( times_times_nat @ A @ X5 )
% 9.10/5.25            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) )
% 9.10/5.25          | ( ( times_times_nat @ B2 @ X5 )
% 9.10/5.25            = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4752_bezout__lemma__nat,axiom,
% 9.10/5.25      ! [D: nat,A: nat,B2: nat,X3: nat,Y3: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ D @ A )
% 9.10/5.25       => ( ( dvd_dvd_nat @ D @ B2 )
% 9.10/5.25         => ( ( ( ( times_times_nat @ A @ X3 )
% 9.10/5.25                = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D ) )
% 9.10/5.25              | ( ( times_times_nat @ B2 @ X3 )
% 9.10/5.25                = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 9.10/5.25           => ? [X5: nat,Y4: nat] :
% 9.10/5.25                ( ( dvd_dvd_nat @ D @ A )
% 9.10/5.25                & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B2 ) )
% 9.10/5.25                & ( ( ( times_times_nat @ A @ X5 )
% 9.10/5.25                    = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ Y4 ) @ D ) )
% 9.10/5.25                  | ( ( times_times_nat @ ( plus_plus_nat @ A @ B2 ) @ X5 )
% 9.10/5.25                    = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4753_bezout1__nat,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25      ? [D3: nat,X5: nat,Y4: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ D3 @ A )
% 9.10/5.25        & ( dvd_dvd_nat @ D3 @ B2 )
% 9.10/5.25        & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 9.10/5.25            = D3 )
% 9.10/5.25          | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 9.10/5.25            = D3 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4754_zdvd__period,axiom,
% 9.10/5.25      ! [A: int,D: int,X3: int,T: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ D )
% 9.10/5.25       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
% 9.10/5.25          = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4755_zdvd__reduce,axiom,
% 9.10/5.25      ! [K: int,N: int,M: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 9.10/5.25        = ( dvd_dvd_int @ K @ N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4756_div2__even__ext__nat,axiom,
% 9.10/5.25      ! [X3: nat,Y3: nat] :
% 9.10/5.25        ( ( ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25          = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.25       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 )
% 9.10/5.25            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 9.10/5.25         => ( X3 = Y3 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4757_unit__dvdE,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.25           => ! [C2: code_integer] :
% 9.10/5.25                ( B2
% 9.10/5.25               != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4758_unit__dvdE,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25       => ~ ( ( A != zero_zero_nat )
% 9.10/5.25           => ! [C2: nat] :
% 9.10/5.25                ( B2
% 9.10/5.25               != ( times_times_nat @ A @ C2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4759_unit__dvdE,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25       => ~ ( ( A != zero_zero_int )
% 9.10/5.25           => ! [C2: int] :
% 9.10/5.25                ( B2
% 9.10/5.25               != ( times_times_int @ A @ C2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4760_unity__coeff__ex,axiom,
% 9.10/5.25      ! [P: code_integer > $o,L: code_integer] :
% 9.10/5.25        ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
% 9.10/5.25        = ( ? [X: code_integer] :
% 9.10/5.25              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 9.10/5.25              & ( P @ X ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4761_unity__coeff__ex,axiom,
% 9.10/5.25      ! [P: complex > $o,L: complex] :
% 9.10/5.25        ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
% 9.10/5.25        = ( ? [X: complex] :
% 9.10/5.25              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
% 9.10/5.25              & ( P @ X ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4762_unity__coeff__ex,axiom,
% 9.10/5.25      ! [P: real > $o,L: real] :
% 9.10/5.25        ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
% 9.10/5.25        = ( ? [X: real] :
% 9.10/5.25              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
% 9.10/5.25              & ( P @ X ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4763_unity__coeff__ex,axiom,
% 9.10/5.25      ! [P: rat > $o,L: rat] :
% 9.10/5.25        ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
% 9.10/5.25        = ( ? [X: rat] :
% 9.10/5.25              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 9.10/5.25              & ( P @ X ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4764_unity__coeff__ex,axiom,
% 9.10/5.25      ! [P: nat > $o,L: nat] :
% 9.10/5.25        ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
% 9.10/5.25        = ( ? [X: nat] :
% 9.10/5.25              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 9.10/5.25              & ( P @ X ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4765_unity__coeff__ex,axiom,
% 9.10/5.25      ! [P: int > $o,L: int] :
% 9.10/5.25        ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
% 9.10/5.25        = ( ? [X: int] :
% 9.10/5.25              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
% 9.10/5.25              & ( P @ X ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4766_dvd__div__eq__mult,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer,C: code_integer] :
% 9.10/5.25        ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.25         => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 9.10/5.25              = C )
% 9.10/5.25            = ( B2
% 9.10/5.25              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4767_dvd__div__eq__mult,axiom,
% 9.10/5.25      ! [A: nat,B2: nat,C: nat] :
% 9.10/5.25        ( ( A != zero_zero_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.25         => ( ( ( divide_divide_nat @ B2 @ A )
% 9.10/5.25              = C )
% 9.10/5.25            = ( B2
% 9.10/5.25              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4768_dvd__div__eq__mult,axiom,
% 9.10/5.25      ! [A: int,B2: int,C: int] :
% 9.10/5.25        ( ( A != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.25         => ( ( ( divide_divide_int @ B2 @ A )
% 9.10/5.25              = C )
% 9.10/5.25            = ( B2
% 9.10/5.25              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4769_div__dvd__iff__mult,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( B2 != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.25         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 9.10/5.25            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4770_div__dvd__iff__mult,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( B2 != zero_zero_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.25         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 9.10/5.25            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B2 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4771_div__dvd__iff__mult,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( B2 != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.25         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 9.10/5.25            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4772_dvd__div__iff__mult,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( C != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ C @ B2 )
% 9.10/5.25         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 9.10/5.25            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4773_dvd__div__iff__mult,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( C != zero_zero_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ C @ B2 )
% 9.10/5.25         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 9.10/5.25            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4774_dvd__div__iff__mult,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( C != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ C @ B2 )
% 9.10/5.25         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 9.10/5.25            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4775_dvd__div__div__eq__mult,axiom,
% 9.10/5.25      ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 9.10/5.25        ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( C != zero_z3403309356797280102nteger )
% 9.10/5.25         => ( ( dvd_dvd_Code_integer @ A @ B2 )
% 9.10/5.25           => ( ( dvd_dvd_Code_integer @ C @ D )
% 9.10/5.25             => ( ( ( divide6298287555418463151nteger @ B2 @ A )
% 9.10/5.25                  = ( divide6298287555418463151nteger @ D @ C ) )
% 9.10/5.25                = ( ( times_3573771949741848930nteger @ B2 @ C )
% 9.10/5.25                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4776_dvd__div__div__eq__mult,axiom,
% 9.10/5.25      ! [A: nat,C: nat,B2: nat,D: nat] :
% 9.10/5.25        ( ( A != zero_zero_nat )
% 9.10/5.25       => ( ( C != zero_zero_nat )
% 9.10/5.25         => ( ( dvd_dvd_nat @ A @ B2 )
% 9.10/5.25           => ( ( dvd_dvd_nat @ C @ D )
% 9.10/5.25             => ( ( ( divide_divide_nat @ B2 @ A )
% 9.10/5.25                  = ( divide_divide_nat @ D @ C ) )
% 9.10/5.25                = ( ( times_times_nat @ B2 @ C )
% 9.10/5.25                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4777_dvd__div__div__eq__mult,axiom,
% 9.10/5.25      ! [A: int,C: int,B2: int,D: int] :
% 9.10/5.25        ( ( A != zero_zero_int )
% 9.10/5.25       => ( ( C != zero_zero_int )
% 9.10/5.25         => ( ( dvd_dvd_int @ A @ B2 )
% 9.10/5.25           => ( ( dvd_dvd_int @ C @ D )
% 9.10/5.25             => ( ( ( divide_divide_int @ B2 @ A )
% 9.10/5.25                  = ( divide_divide_int @ D @ C ) )
% 9.10/5.25                = ( ( times_times_int @ B2 @ C )
% 9.10/5.25                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4778_unit__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 9.10/5.25            = zero_z3403309356797280102nteger )
% 9.10/5.25          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4779_unit__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( ( divide_divide_nat @ A @ B2 )
% 9.10/5.25            = zero_zero_nat )
% 9.10/5.25          = ( A = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4780_unit__div__eq__0__iff,axiom,
% 9.10/5.25      ! [B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( ( divide_divide_int @ A @ B2 )
% 9.10/5.25            = zero_zero_int )
% 9.10/5.25          = ( A = zero_zero_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4781_even__numeral,axiom,
% 9.10/5.25      ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4782_even__numeral,axiom,
% 9.10/5.25      ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4783_even__numeral,axiom,
% 9.10/5.25      ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4784_inf__period_I4_J,axiom,
% 9.10/5.25      ! [D: code_integer,D4: code_integer,T: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ D @ D4 )
% 9.10/5.25       => ! [X6: code_integer,K4: code_integer] :
% 9.10/5.25            ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) ) )
% 9.10/5.25            = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4785_inf__period_I4_J,axiom,
% 9.10/5.25      ! [D: complex,D4: complex,T: complex] :
% 9.10/5.25        ( ( dvd_dvd_complex @ D @ D4 )
% 9.10/5.25       => ! [X6: complex,K4: complex] :
% 9.10/5.25            ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X6 @ T ) ) )
% 9.10/5.25            = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4786_inf__period_I4_J,axiom,
% 9.10/5.25      ! [D: real,D4: real,T: real] :
% 9.10/5.25        ( ( dvd_dvd_real @ D @ D4 )
% 9.10/5.25       => ! [X6: real,K4: real] :
% 9.10/5.25            ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) ) )
% 9.10/5.25            = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4787_inf__period_I4_J,axiom,
% 9.10/5.25      ! [D: rat,D4: rat,T: rat] :
% 9.10/5.25        ( ( dvd_dvd_rat @ D @ D4 )
% 9.10/5.25       => ! [X6: rat,K4: rat] :
% 9.10/5.25            ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) ) )
% 9.10/5.25            = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4788_inf__period_I4_J,axiom,
% 9.10/5.25      ! [D: int,D4: int,T: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ D @ D4 )
% 9.10/5.25       => ! [X6: int,K4: int] :
% 9.10/5.25            ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) ) )
% 9.10/5.25            = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4789_inf__period_I3_J,axiom,
% 9.10/5.25      ! [D: code_integer,D4: code_integer,T: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ D @ D4 )
% 9.10/5.25       => ! [X6: code_integer,K4: code_integer] :
% 9.10/5.25            ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X6 @ T ) )
% 9.10/5.25            = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X6 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4790_inf__period_I3_J,axiom,
% 9.10/5.25      ! [D: complex,D4: complex,T: complex] :
% 9.10/5.25        ( ( dvd_dvd_complex @ D @ D4 )
% 9.10/5.25       => ! [X6: complex,K4: complex] :
% 9.10/5.25            ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X6 @ T ) )
% 9.10/5.25            = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X6 @ ( times_times_complex @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4791_inf__period_I3_J,axiom,
% 9.10/5.25      ! [D: real,D4: real,T: real] :
% 9.10/5.25        ( ( dvd_dvd_real @ D @ D4 )
% 9.10/5.25       => ! [X6: real,K4: real] :
% 9.10/5.25            ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X6 @ T ) )
% 9.10/5.25            = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X6 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4792_inf__period_I3_J,axiom,
% 9.10/5.25      ! [D: rat,D4: rat,T: rat] :
% 9.10/5.25        ( ( dvd_dvd_rat @ D @ D4 )
% 9.10/5.25       => ! [X6: rat,K4: rat] :
% 9.10/5.25            ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X6 @ T ) )
% 9.10/5.25            = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4793_inf__period_I3_J,axiom,
% 9.10/5.25      ! [D: int,D4: int,T: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ D @ D4 )
% 9.10/5.25       => ! [X6: int,K4: int] :
% 9.10/5.25            ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X6 @ T ) )
% 9.10/5.25            = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X6 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4794_unit__eq__div1,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( ( divide6298287555418463151nteger @ A @ B2 )
% 9.10/5.25            = C )
% 9.10/5.25          = ( A
% 9.10/5.25            = ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4795_unit__eq__div1,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( ( divide_divide_nat @ A @ B2 )
% 9.10/5.25            = C )
% 9.10/5.25          = ( A
% 9.10/5.25            = ( times_times_nat @ C @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4796_unit__eq__div1,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( ( divide_divide_int @ A @ B2 )
% 9.10/5.25            = C )
% 9.10/5.25          = ( A
% 9.10/5.25            = ( times_times_int @ C @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4797_unit__eq__div2,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( A
% 9.10/5.25            = ( divide6298287555418463151nteger @ C @ B2 ) )
% 9.10/5.25          = ( ( times_3573771949741848930nteger @ A @ B2 )
% 9.10/5.25            = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4798_unit__eq__div2,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( A
% 9.10/5.25            = ( divide_divide_nat @ C @ B2 ) )
% 9.10/5.25          = ( ( times_times_nat @ A @ B2 )
% 9.10/5.25            = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4799_unit__eq__div2,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( A
% 9.10/5.25            = ( divide_divide_int @ C @ B2 ) )
% 9.10/5.25          = ( ( times_times_int @ A @ B2 )
% 9.10/5.25            = C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4800_div__mult__unit2,axiom,
% 9.10/5.25      ! [C: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.25         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.10/5.25            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4801_div__mult__unit2,axiom,
% 9.10/5.25      ! [C: nat,B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ B2 @ A )
% 9.10/5.25         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.10/5.25            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4802_div__mult__unit2,axiom,
% 9.10/5.25      ! [C: int,B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.25         => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.25            = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4803_unit__div__commute,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C )
% 9.10/5.25          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4804_unit__div__commute,axiom,
% 9.10/5.25      ! [B2: nat,A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B2 ) @ C )
% 9.10/5.25          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4805_unit__div__commute,axiom,
% 9.10/5.25      ! [B2: int,A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( times_times_int @ ( divide_divide_int @ A @ B2 ) @ C )
% 9.10/5.25          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4806_unit__div__mult__swap,axiom,
% 9.10/5.25      ! [C: code_integer,A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.10/5.25       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B2 @ C ) )
% 9.10/5.25          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4807_unit__div__mult__swap,axiom,
% 9.10/5.25      ! [C: nat,A: nat,B2: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.10/5.25       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B2 @ C ) )
% 9.10/5.25          = ( divide_divide_nat @ ( times_times_nat @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4808_unit__div__mult__swap,axiom,
% 9.10/5.25      ! [C: int,A: int,B2: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ C @ one_one_int )
% 9.10/5.25       => ( ( times_times_int @ A @ ( divide_divide_int @ B2 @ C ) )
% 9.10/5.25          = ( divide_divide_int @ ( times_times_int @ A @ B2 ) @ C ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4809_is__unit__div__mult2__eq,axiom,
% 9.10/5.25      ! [B2: code_integer,C: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.10/5.25         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ C ) )
% 9.10/5.25            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4810_is__unit__div__mult2__eq,axiom,
% 9.10/5.25      ! [B2: nat,C: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.10/5.25         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ C ) )
% 9.10/5.25            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4811_is__unit__div__mult2__eq,axiom,
% 9.10/5.25      ! [B2: int,C: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ C @ one_one_int )
% 9.10/5.25         => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
% 9.10/5.25            = ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4812_unit__imp__mod__eq__0,axiom,
% 9.10/5.25      ! [B2: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25       => ( ( modulo_modulo_nat @ A @ B2 )
% 9.10/5.25          = zero_zero_nat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4813_unit__imp__mod__eq__0,axiom,
% 9.10/5.25      ! [B2: int,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25       => ( ( modulo_modulo_int @ A @ B2 )
% 9.10/5.25          = zero_zero_int ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4814_unit__imp__mod__eq__0,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25       => ( ( modulo364778990260209775nteger @ A @ B2 )
% 9.10/5.25          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4815_is__unit__power__iff,axiom,
% 9.10/5.25      ! [A: code_integer,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 9.10/5.25        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4816_is__unit__power__iff,axiom,
% 9.10/5.25      ! [A: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 9.10/5.25        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4817_is__unit__power__iff,axiom,
% 9.10/5.25      ! [A: int,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 9.10/5.25        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4818_dvd__imp__le,axiom,
% 9.10/5.25      ! [K: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ K @ N )
% 9.10/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25         => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4819_nat__mult__dvd__cancel1,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.10/5.25          = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4820_dvd__mult__cancel,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.10/5.25       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.25         => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4821_bezout__add__strong__nat,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( A != zero_zero_nat )
% 9.10/5.25       => ? [D3: nat,X5: nat,Y4: nat] :
% 9.10/5.25            ( ( dvd_dvd_nat @ D3 @ A )
% 9.10/5.25            & ( dvd_dvd_nat @ D3 @ B2 )
% 9.10/5.25            & ( ( times_times_nat @ A @ X5 )
% 9.10/5.25              = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ D3 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4822_zdvd__imp__le,axiom,
% 9.10/5.25      ! [Z: int,N: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ Z @ N )
% 9.10/5.25       => ( ( ord_less_int @ zero_zero_int @ N )
% 9.10/5.25         => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4823_mod__greater__zero__iff__not__dvd,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 9.10/5.25        = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4824_mod__eq__dvd__iff__nat,axiom,
% 9.10/5.25      ! [N: nat,M: nat,Q2: nat] :
% 9.10/5.25        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.25       => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 9.10/5.25            = ( modulo_modulo_nat @ N @ Q2 ) )
% 9.10/5.25          = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4825_prod__decode__aux_Ocases,axiom,
% 9.10/5.25      ! [X3: product_prod_nat_nat] :
% 9.10/5.25        ~ ! [K3: nat,M4: nat] :
% 9.10/5.25            ( X3
% 9.10/5.25           != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4826_finite__divisors__nat,axiom,
% 9.10/5.25      ! [M: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.25       => ( finite_finite_nat
% 9.10/5.25          @ ( collect_nat
% 9.10/5.25            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4827_even__zero,axiom,
% 9.10/5.25      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4828_even__zero,axiom,
% 9.10/5.25      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4829_even__zero,axiom,
% 9.10/5.25      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4830_is__unitE,axiom,
% 9.10/5.25      ! [A: code_integer,C: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.10/5.25       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.25           => ! [B3: code_integer] :
% 9.10/5.25                ( ( B3 != zero_z3403309356797280102nteger )
% 9.10/5.25               => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 9.10/5.25                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 9.10/5.25                      = B3 )
% 9.10/5.25                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 9.10/5.25                        = A )
% 9.10/5.25                     => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 9.10/5.25                          = one_one_Code_integer )
% 9.10/5.25                       => ( ( divide6298287555418463151nteger @ C @ A )
% 9.10/5.25                         != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4831_is__unitE,axiom,
% 9.10/5.25      ! [A: nat,C: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.10/5.25       => ~ ( ( A != zero_zero_nat )
% 9.10/5.25           => ! [B3: nat] :
% 9.10/5.25                ( ( B3 != zero_zero_nat )
% 9.10/5.25               => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 9.10/5.25                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 9.10/5.25                      = B3 )
% 9.10/5.25                   => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 9.10/5.25                        = A )
% 9.10/5.25                     => ( ( ( times_times_nat @ A @ B3 )
% 9.10/5.25                          = one_one_nat )
% 9.10/5.25                       => ( ( divide_divide_nat @ C @ A )
% 9.10/5.25                         != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4832_is__unitE,axiom,
% 9.10/5.25      ! [A: int,C: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.10/5.25       => ~ ( ( A != zero_zero_int )
% 9.10/5.25           => ! [B3: int] :
% 9.10/5.25                ( ( B3 != zero_zero_int )
% 9.10/5.25               => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 9.10/5.25                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 9.10/5.25                      = B3 )
% 9.10/5.25                   => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 9.10/5.25                        = A )
% 9.10/5.25                     => ( ( ( times_times_int @ A @ B3 )
% 9.10/5.25                          = one_one_int )
% 9.10/5.25                       => ( ( divide_divide_int @ C @ A )
% 9.10/5.25                         != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4833_is__unit__div__mult__cancel__left,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 9.10/5.25            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4834_is__unit__div__mult__cancel__left,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( A != zero_zero_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B2 ) )
% 9.10/5.25            = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4835_is__unit__div__mult__cancel__left,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( A != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B2 ) )
% 9.10/5.25            = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4836_is__unit__div__mult__cancel__right,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 9.10/5.25         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B2 @ A ) )
% 9.10/5.25            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4837_is__unit__div__mult__cancel__right,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ( A != zero_zero_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 9.10/5.25         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B2 @ A ) )
% 9.10/5.25            = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4838_is__unit__div__mult__cancel__right,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( A != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 9.10/5.25         => ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ A ) )
% 9.10/5.25            = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4839_evenE,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ~ ! [B3: code_integer] :
% 9.10/5.25              ( A
% 9.10/5.25             != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4840_evenE,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ~ ! [B3: nat] :
% 9.10/5.25              ( A
% 9.10/5.25             != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4841_evenE,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ~ ! [B3: int] :
% 9.10/5.25              ( A
% 9.10/5.25             != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4842_odd__one,axiom,
% 9.10/5.25      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4843_odd__one,axiom,
% 9.10/5.25      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4844_odd__one,axiom,
% 9.10/5.25      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4845_odd__even__add,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.25         => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4846_odd__even__add,axiom,
% 9.10/5.25      ! [A: nat,B2: nat] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.25         => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4847_odd__even__add,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.25         => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4848_bit__eq__rec,axiom,
% 9.10/5.25      ( ( ^ [Y6: code_integer,Z5: code_integer] : ( Y6 = Z5 ) )
% 9.10/5.25      = ( ^ [A5: code_integer,B4: code_integer] :
% 9.10/5.25            ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 )
% 9.10/5.25              = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
% 9.10/5.25            & ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25              = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4849_bit__eq__rec,axiom,
% 9.10/5.25      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.25      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.25            ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
% 9.10/5.25              = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
% 9.10/5.25            & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25              = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4850_bit__eq__rec,axiom,
% 9.10/5.25      ( ( ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.10/5.25      = ( ^ [A5: int,B4: int] :
% 9.10/5.25            ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
% 9.10/5.25              = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
% 9.10/5.25            & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25              = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4851_dvd__power__iff,axiom,
% 9.10/5.25      ! [X3: code_integer,M: nat,N: nat] :
% 9.10/5.25        ( ( X3 != zero_z3403309356797280102nteger )
% 9.10/5.25       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M ) @ ( power_8256067586552552935nteger @ X3 @ N ) )
% 9.10/5.25          = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
% 9.10/5.25            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4852_dvd__power__iff,axiom,
% 9.10/5.25      ! [X3: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( X3 != zero_zero_nat )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M ) @ ( power_power_nat @ X3 @ N ) )
% 9.10/5.25          = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
% 9.10/5.25            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4853_dvd__power__iff,axiom,
% 9.10/5.25      ! [X3: int,M: nat,N: nat] :
% 9.10/5.25        ( ( X3 != zero_zero_int )
% 9.10/5.25       => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ N ) )
% 9.10/5.25          = ( ( dvd_dvd_int @ X3 @ one_one_int )
% 9.10/5.25            | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4854_dvd__power,axiom,
% 9.10/5.25      ! [N: nat,X3: code_integer] :
% 9.10/5.25        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          | ( X3 = one_one_Code_integer ) )
% 9.10/5.25       => ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4855_dvd__power,axiom,
% 9.10/5.25      ! [N: nat,X3: rat] :
% 9.10/5.25        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          | ( X3 = one_one_rat ) )
% 9.10/5.25       => ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4856_dvd__power,axiom,
% 9.10/5.25      ! [N: nat,X3: nat] :
% 9.10/5.25        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          | ( X3 = one_one_nat ) )
% 9.10/5.25       => ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4857_dvd__power,axiom,
% 9.10/5.25      ! [N: nat,X3: real] :
% 9.10/5.25        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          | ( X3 = one_one_real ) )
% 9.10/5.25       => ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4858_dvd__power,axiom,
% 9.10/5.25      ! [N: nat,X3: int] :
% 9.10/5.25        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          | ( X3 = one_one_int ) )
% 9.10/5.25       => ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4859_dvd__power,axiom,
% 9.10/5.25      ! [N: nat,X3: complex] :
% 9.10/5.25        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          | ( X3 = one_one_complex ) )
% 9.10/5.25       => ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4860_even__even__mod__4__iff,axiom,
% 9.10/5.25      ! [N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4861_dvd__mult__cancel2,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 9.10/5.25          = ( N = one_one_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4862_dvd__mult__cancel1,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 9.10/5.25          = ( N = one_one_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4863_dvd__minus__add,axiom,
% 9.10/5.25      ! [Q2: nat,N: nat,R: nat,M: nat] :
% 9.10/5.25        ( ( ord_less_eq_nat @ Q2 @ N )
% 9.10/5.25       => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 9.10/5.25         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 9.10/5.25            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4864_power__dvd__imp__le,axiom,
% 9.10/5.25      ! [I: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.10/5.25       => ( ( ord_less_nat @ one_one_nat @ I )
% 9.10/5.25         => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4865_mod__nat__eqI,axiom,
% 9.10/5.25      ! [R: nat,N: nat,M: nat] :
% 9.10/5.25        ( ( ord_less_nat @ R @ N )
% 9.10/5.25       => ( ( ord_less_eq_nat @ R @ M )
% 9.10/5.25         => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
% 9.10/5.25           => ( ( modulo_modulo_nat @ M @ N )
% 9.10/5.25              = R ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4866_mod__int__pos__iff,axiom,
% 9.10/5.25      ! [K: int,L: int] :
% 9.10/5.25        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 9.10/5.25        = ( ( dvd_dvd_int @ L @ K )
% 9.10/5.25          | ( ( L = zero_zero_int )
% 9.10/5.25            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 9.10/5.25          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4867_even__two__times__div__two,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.10/5.25          = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4868_even__two__times__div__two,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.25          = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4869_even__two__times__div__two,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.10/5.25          = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4870_even__iff__mod__2__eq__zero,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25          = zero_zero_nat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4871_even__iff__mod__2__eq__zero,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25          = zero_zero_int ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4872_even__iff__mod__2__eq__zero,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4873_odd__iff__mod__2__eq__one,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.25        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25          = one_one_nat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4874_odd__iff__mod__2__eq__one,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.25        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25          = one_one_int ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4875_odd__iff__mod__2__eq__one,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.25        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25          = one_one_Code_integer ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4876_power__mono__odd,axiom,
% 9.10/5.25      ! [N: nat,A: real,B2: real] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.25         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4877_power__mono__odd,axiom,
% 9.10/5.25      ! [N: nat,A: rat,B2: rat] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.25         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4878_power__mono__odd,axiom,
% 9.10/5.25      ! [N: nat,A: int,B2: int] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.25         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4879_odd__pos,axiom,
% 9.10/5.25      ! [N: nat] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4880_dvd__power__iff__le,axiom,
% 9.10/5.25      ! [K: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.10/5.25       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 9.10/5.25          = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4881_signed__take__bit__int__less__exp,axiom,
% 9.10/5.25      ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4882_even__unset__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25          | ( M = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4883_even__unset__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25          | ( M = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4884_even__unset__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25          | ( M = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4885_even__set__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25          & ( M != zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4886_even__set__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25          & ( M != zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4887_even__set__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25          & ( M != zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4888_even__flip__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         != ( M = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4889_even__flip__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         != ( M = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4890_even__flip__bit__iff,axiom,
% 9.10/5.25      ! [M: nat,A: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         != ( M = zero_zero_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4891_even__diff__iff,axiom,
% 9.10/5.25      ! [K: int,L: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
% 9.10/5.25        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4892_oddE,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ~ ! [B3: code_integer] :
% 9.10/5.25              ( A
% 9.10/5.25             != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4893_oddE,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ~ ! [B3: nat] :
% 9.10/5.25              ( A
% 9.10/5.25             != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4894_oddE,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25       => ~ ! [B3: int] :
% 9.10/5.25              ( A
% 9.10/5.25             != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4895_parity__cases,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25           != zero_zero_nat ) )
% 9.10/5.25       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25             != one_one_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4896_parity__cases,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25           != zero_zero_int ) )
% 9.10/5.25       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25             != one_one_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4897_parity__cases,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25           != zero_z3403309356797280102nteger ) )
% 9.10/5.25       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25             != one_one_Code_integer ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4898_mod2__eq__if,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25            = zero_zero_nat ) )
% 9.10/5.25        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.25            = one_one_nat ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4899_mod2__eq__if,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25            = zero_zero_int ) )
% 9.10/5.25        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.25            = one_one_int ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4900_mod2__eq__if,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25            = zero_z3403309356797280102nteger ) )
% 9.10/5.25        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.10/5.25         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.25            = one_one_Code_integer ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4901_zero__le__even__power,axiom,
% 9.10/5.25      ! [N: nat,A: real] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4902_zero__le__even__power,axiom,
% 9.10/5.25      ! [N: nat,A: rat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4903_zero__le__even__power,axiom,
% 9.10/5.25      ! [N: nat,A: int] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4904_zero__le__odd__power,axiom,
% 9.10/5.25      ! [N: nat,A: real] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.10/5.25          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4905_zero__le__odd__power,axiom,
% 9.10/5.25      ! [N: nat,A: rat] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.10/5.25          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4906_zero__le__odd__power,axiom,
% 9.10/5.25      ! [N: nat,A: int] :
% 9.10/5.25        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.10/5.25          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4907_zero__le__power__eq,axiom,
% 9.10/5.25      ! [A: real,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4908_zero__le__power__eq,axiom,
% 9.10/5.25      ! [A: rat,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4909_zero__le__power__eq,axiom,
% 9.10/5.25      ! [A: int,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.10/5.25        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4910_signed__take__bit__int__greater__eq__self__iff,axiom,
% 9.10/5.25      ! [K: int,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.10/5.25        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4911_signed__take__bit__int__less__self__iff,axiom,
% 9.10/5.25      ! [N: nat,K: int] :
% 9.10/5.25        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.10/5.25        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4912_zero__less__power__eq,axiom,
% 9.10/5.25      ! [A: real,N: nat] :
% 9.10/5.25        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 9.10/5.25        = ( ( N = zero_zero_nat )
% 9.10/5.25          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( A != zero_zero_real ) )
% 9.10/5.25          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4913_zero__less__power__eq,axiom,
% 9.10/5.25      ! [A: rat,N: nat] :
% 9.10/5.25        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 9.10/5.25        = ( ( N = zero_zero_nat )
% 9.10/5.25          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( A != zero_zero_rat ) )
% 9.10/5.25          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4914_zero__less__power__eq,axiom,
% 9.10/5.25      ! [A: int,N: nat] :
% 9.10/5.25        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 9.10/5.25        = ( ( N = zero_zero_nat )
% 9.10/5.25          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( A != zero_zero_int ) )
% 9.10/5.25          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4915_signed__take__bit__int__less__eq,axiom,
% 9.10/5.25      ! [N: nat,K: int] :
% 9.10/5.25        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.10/5.25       => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4916_even__mask__div__iff_H,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4917_even__mask__div__iff_H,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4918_even__mask__div__iff_H,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4919_power__le__zero__eq,axiom,
% 9.10/5.25      ! [A: real,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 9.10/5.25        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.10/5.25            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4920_power__le__zero__eq,axiom,
% 9.10/5.25      ! [A: rat,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 9.10/5.25        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.10/5.25            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4921_power__le__zero__eq,axiom,
% 9.10/5.25      ! [A: int,N: nat] :
% 9.10/5.25        ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 9.10/5.25        = ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.25          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.10/5.25            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4922_option_Osize__gen_I1_J,axiom,
% 9.10/5.25      ! [X3: nat > nat] :
% 9.10/5.25        ( ( size_option_nat @ X3 @ none_nat )
% 9.10/5.25        = ( suc @ zero_zero_nat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4923_option_Osize__gen_I1_J,axiom,
% 9.10/5.25      ! [X3: product_prod_nat_nat > nat] :
% 9.10/5.25        ( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
% 9.10/5.25        = ( suc @ zero_zero_nat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4924_option_Osize__gen_I1_J,axiom,
% 9.10/5.25      ! [X3: num > nat] :
% 9.10/5.25        ( ( size_option_num @ X3 @ none_num )
% 9.10/5.25        = ( suc @ zero_zero_nat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4925_even__mod__4__div__2,axiom,
% 9.10/5.25      ! [N: nat] :
% 9.10/5.25        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.25          = ( suc @ zero_zero_nat ) )
% 9.10/5.25       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4926_even__mask__div__iff,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            = zero_z3403309356797280102nteger )
% 9.10/5.25          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4927_even__mask__div__iff,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            = zero_zero_nat )
% 9.10/5.25          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4928_even__mask__div__iff,axiom,
% 9.10/5.25      ! [M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            = zero_zero_int )
% 9.10/5.25          | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4929_even__mult__exp__div__exp__iff,axiom,
% 9.10/5.25      ! [A: code_integer,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ( ord_less_nat @ N @ M )
% 9.10/5.25          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            = zero_z3403309356797280102nteger )
% 9.10/5.25          | ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4930_even__mult__exp__div__exp__iff,axiom,
% 9.10/5.25      ! [A: nat,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ( ord_less_nat @ N @ M )
% 9.10/5.25          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            = zero_zero_nat )
% 9.10/5.25          | ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25            & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4931_even__mult__exp__div__exp__iff,axiom,
% 9.10/5.25      ! [A: int,M: nat,N: nat] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.25        = ( ( ord_less_nat @ N @ M )
% 9.10/5.25          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.10/5.25            = zero_zero_int )
% 9.10/5.25          | ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.25            & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4932_ex__min__if__finite,axiom,
% 9.10/5.25      ! [S2: set_real] :
% 9.10/5.25        ( ( finite_finite_real @ S2 )
% 9.10/5.25       => ( ( S2 != bot_bot_set_real )
% 9.10/5.25         => ? [X5: real] :
% 9.10/5.25              ( ( member_real @ X5 @ S2 )
% 9.10/5.25              & ~ ? [Xa: real] :
% 9.10/5.25                    ( ( member_real @ Xa @ S2 )
% 9.10/5.25                    & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4933_ex__min__if__finite,axiom,
% 9.10/5.25      ! [S2: set_rat] :
% 9.10/5.25        ( ( finite_finite_rat @ S2 )
% 9.10/5.25       => ( ( S2 != bot_bot_set_rat )
% 9.10/5.25         => ? [X5: rat] :
% 9.10/5.25              ( ( member_rat @ X5 @ S2 )
% 9.10/5.25              & ~ ? [Xa: rat] :
% 9.10/5.25                    ( ( member_rat @ Xa @ S2 )
% 9.10/5.25                    & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4934_ex__min__if__finite,axiom,
% 9.10/5.25      ! [S2: set_num] :
% 9.10/5.25        ( ( finite_finite_num @ S2 )
% 9.10/5.25       => ( ( S2 != bot_bot_set_num )
% 9.10/5.25         => ? [X5: num] :
% 9.10/5.25              ( ( member_num @ X5 @ S2 )
% 9.10/5.25              & ~ ? [Xa: num] :
% 9.10/5.25                    ( ( member_num @ Xa @ S2 )
% 9.10/5.25                    & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4935_ex__min__if__finite,axiom,
% 9.10/5.25      ! [S2: set_nat] :
% 9.10/5.25        ( ( finite_finite_nat @ S2 )
% 9.10/5.25       => ( ( S2 != bot_bot_set_nat )
% 9.10/5.25         => ? [X5: nat] :
% 9.10/5.25              ( ( member_nat @ X5 @ S2 )
% 9.10/5.25              & ~ ? [Xa: nat] :
% 9.10/5.25                    ( ( member_nat @ Xa @ S2 )
% 9.10/5.25                    & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4936_ex__min__if__finite,axiom,
% 9.10/5.25      ! [S2: set_int] :
% 9.10/5.25        ( ( finite_finite_int @ S2 )
% 9.10/5.25       => ( ( S2 != bot_bot_set_int )
% 9.10/5.25         => ? [X5: int] :
% 9.10/5.25              ( ( member_int @ X5 @ S2 )
% 9.10/5.25              & ~ ? [Xa: int] :
% 9.10/5.25                    ( ( member_int @ Xa @ S2 )
% 9.10/5.25                    & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4937_triangle__def,axiom,
% 9.10/5.25      ( nat_triangle
% 9.10/5.25      = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4938_vebt__buildup_Oelims,axiom,
% 9.10/5.25      ! [X3: nat,Y3: vEBT_VEBT] :
% 9.10/5.25        ( ( ( vEBT_vebt_buildup @ X3 )
% 9.10/5.25          = Y3 )
% 9.10/5.25       => ( ( ( X3 = zero_zero_nat )
% 9.10/5.25           => ( Y3
% 9.10/5.25             != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.10/5.25         => ( ( ( X3
% 9.10/5.25                = ( suc @ zero_zero_nat ) )
% 9.10/5.25             => ( Y3
% 9.10/5.25               != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.10/5.25           => ~ ! [Va2: nat] :
% 9.10/5.25                  ( ( X3
% 9.10/5.25                    = ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.25                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.25                       => ( Y3
% 9.10/5.25                          = ( 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 ) ) ) ) ) ) )
% 9.10/5.25                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.25                       => ( Y3
% 9.10/5.25                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4939_signed__take__bit__rec,axiom,
% 9.10/5.25      ( bit_ri6519982836138164636nteger
% 9.10/5.25      = ( ^ [N2: nat,A5: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4940_signed__take__bit__rec,axiom,
% 9.10/5.25      ( bit_ri631733984087533419it_int
% 9.10/5.25      = ( ^ [N2: nat,A5: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4941_flip__bit__0,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 9.10/5.25        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4942_flip__bit__0,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 9.10/5.25        = ( 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 ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4943_flip__bit__0,axiom,
% 9.10/5.25      ! [A: nat] :
% 9.10/5.25        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 9.10/5.25        = ( 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 ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4944_set__decode__Suc,axiom,
% 9.10/5.25      ! [N: nat,X3: nat] :
% 9.10/5.25        ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X3 ) )
% 9.10/5.25        = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4945_diff__shunt__var,axiom,
% 9.10/5.25      ! [X3: set_nat,Y3: set_nat] :
% 9.10/5.25        ( ( ( minus_minus_set_nat @ X3 @ Y3 )
% 9.10/5.25          = bot_bot_set_nat )
% 9.10/5.25        = ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4946_diff__shunt__var,axiom,
% 9.10/5.25      ! [X3: set_real,Y3: set_real] :
% 9.10/5.25        ( ( ( minus_minus_set_real @ X3 @ Y3 )
% 9.10/5.25          = bot_bot_set_real )
% 9.10/5.25        = ( ord_less_eq_set_real @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4947_diff__shunt__var,axiom,
% 9.10/5.25      ! [X3: set_int,Y3: set_int] :
% 9.10/5.25        ( ( ( minus_minus_set_int @ X3 @ Y3 )
% 9.10/5.25          = bot_bot_set_int )
% 9.10/5.25        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4948_add__scale__eq__noteq,axiom,
% 9.10/5.25      ! [R: complex,A: complex,B2: complex,C: complex,D: complex] :
% 9.10/5.25        ( ( R != zero_zero_complex )
% 9.10/5.25       => ( ( ( A = B2 )
% 9.10/5.25            & ( C != D ) )
% 9.10/5.25         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
% 9.10/5.25           != ( plus_plus_complex @ B2 @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4949_add__scale__eq__noteq,axiom,
% 9.10/5.25      ! [R: real,A: real,B2: real,C: real,D: real] :
% 9.10/5.25        ( ( R != zero_zero_real )
% 9.10/5.25       => ( ( ( A = B2 )
% 9.10/5.25            & ( C != D ) )
% 9.10/5.25         => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 9.10/5.25           != ( plus_plus_real @ B2 @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4950_add__scale__eq__noteq,axiom,
% 9.10/5.25      ! [R: rat,A: rat,B2: rat,C: rat,D: rat] :
% 9.10/5.25        ( ( R != zero_zero_rat )
% 9.10/5.25       => ( ( ( A = B2 )
% 9.10/5.25            & ( C != D ) )
% 9.10/5.25         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 9.10/5.25           != ( plus_plus_rat @ B2 @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4951_add__scale__eq__noteq,axiom,
% 9.10/5.25      ! [R: nat,A: nat,B2: nat,C: nat,D: nat] :
% 9.10/5.25        ( ( R != zero_zero_nat )
% 9.10/5.25       => ( ( ( A = B2 )
% 9.10/5.25            & ( C != D ) )
% 9.10/5.25         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 9.10/5.25           != ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4952_add__scale__eq__noteq,axiom,
% 9.10/5.25      ! [R: int,A: int,B2: int,C: int,D: int] :
% 9.10/5.25        ( ( R != zero_zero_int )
% 9.10/5.25       => ( ( ( A = B2 )
% 9.10/5.25            & ( C != D ) )
% 9.10/5.25         => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 9.10/5.25           != ( plus_plus_int @ B2 @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4953_intind,axiom,
% 9.10/5.25      ! [I: nat,N: nat,P: int > $o,X3: int] :
% 9.10/5.25        ( ( ord_less_nat @ I @ N )
% 9.10/5.25       => ( ( P @ X3 )
% 9.10/5.25         => ( P @ ( nth_int @ ( replicate_int @ N @ X3 ) @ I ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4954_intind,axiom,
% 9.10/5.25      ! [I: nat,N: nat,P: nat > $o,X3: nat] :
% 9.10/5.25        ( ( ord_less_nat @ I @ N )
% 9.10/5.25       => ( ( P @ X3 )
% 9.10/5.25         => ( P @ ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4955_intind,axiom,
% 9.10/5.25      ! [I: nat,N: nat,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 9.10/5.25        ( ( ord_less_nat @ I @ N )
% 9.10/5.25       => ( ( P @ X3 )
% 9.10/5.25         => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I ) ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4956_add_Oinverse__inverse,axiom,
% 9.10/5.25      ! [A: real] :
% 9.10/5.25        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 9.10/5.25        = A ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4957_add_Oinverse__inverse,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 9.10/5.25        = A ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4958_add_Oinverse__inverse,axiom,
% 9.10/5.25      ! [A: complex] :
% 9.10/5.25        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 9.10/5.25        = A ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4959_add_Oinverse__inverse,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.25        = A ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4960_add_Oinverse__inverse,axiom,
% 9.10/5.25      ! [A: rat] :
% 9.10/5.25        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 9.10/5.25        = A ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4961_neg__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( ( uminus_uminus_real @ A )
% 9.10/5.25          = ( uminus_uminus_real @ B2 ) )
% 9.10/5.25        = ( A = B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4962_neg__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( ( uminus_uminus_int @ A )
% 9.10/5.25          = ( uminus_uminus_int @ B2 ) )
% 9.10/5.25        = ( A = B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4963_neg__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( ( uminus1482373934393186551omplex @ A )
% 9.10/5.25          = ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.25        = ( A = B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4964_neg__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.25          = ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.25        = ( A = B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4965_neg__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( ( uminus_uminus_rat @ A )
% 9.10/5.25          = ( uminus_uminus_rat @ B2 ) )
% 9.10/5.25        = ( A = B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4966_verit__minus__simplify_I4_J,axiom,
% 9.10/5.25      ! [B2: real] :
% 9.10/5.25        ( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4967_verit__minus__simplify_I4_J,axiom,
% 9.10/5.25      ! [B2: int] :
% 9.10/5.25        ( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4968_verit__minus__simplify_I4_J,axiom,
% 9.10/5.25      ! [B2: complex] :
% 9.10/5.25        ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4969_verit__minus__simplify_I4_J,axiom,
% 9.10/5.25      ! [B2: code_integer] :
% 9.10/5.25        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4970_verit__minus__simplify_I4_J,axiom,
% 9.10/5.25      ! [B2: rat] :
% 9.10/5.25        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4971_Compl__subset__Compl__iff,axiom,
% 9.10/5.25      ! [A4: set_int,B5: set_int] :
% 9.10/5.25        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( uminus1532241313380277803et_int @ B5 ) )
% 9.10/5.25        = ( ord_less_eq_set_int @ B5 @ A4 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4972_Compl__anti__mono,axiom,
% 9.10/5.25      ! [A4: set_int,B5: set_int] :
% 9.10/5.25        ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.25       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4973_compl__le__compl__iff,axiom,
% 9.10/5.25      ! [X3: set_int,Y3: set_int] :
% 9.10/5.25        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 9.10/5.25        = ( ord_less_eq_set_int @ Y3 @ X3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4974_neg__le__iff__le,axiom,
% 9.10/5.25      ! [B2: real,A: real] :
% 9.10/5.25        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 9.10/5.25        = ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4975_neg__le__iff__le,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.25        = ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4976_neg__le__iff__le,axiom,
% 9.10/5.25      ! [B2: rat,A: rat] :
% 9.10/5.25        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 9.10/5.25        = ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4977_neg__le__iff__le,axiom,
% 9.10/5.25      ! [B2: int,A: int] :
% 9.10/5.25        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 9.10/5.25        = ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4978_add_Oinverse__neutral,axiom,
% 9.10/5.25      ( ( uminus_uminus_real @ zero_zero_real )
% 9.10/5.25      = zero_zero_real ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4979_add_Oinverse__neutral,axiom,
% 9.10/5.25      ( ( uminus_uminus_int @ zero_zero_int )
% 9.10/5.25      = zero_zero_int ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4980_add_Oinverse__neutral,axiom,
% 9.10/5.25      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 9.10/5.25      = zero_zero_complex ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4981_add_Oinverse__neutral,axiom,
% 9.10/5.25      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 9.10/5.25      = zero_z3403309356797280102nteger ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4982_add_Oinverse__neutral,axiom,
% 9.10/5.25      ( ( uminus_uminus_rat @ zero_zero_rat )
% 9.10/5.25      = zero_zero_rat ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4983_neg__0__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: real] :
% 9.10/5.25        ( ( zero_zero_real
% 9.10/5.25          = ( uminus_uminus_real @ A ) )
% 9.10/5.25        = ( zero_zero_real = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4984_neg__0__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( zero_zero_int
% 9.10/5.25          = ( uminus_uminus_int @ A ) )
% 9.10/5.25        = ( zero_zero_int = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4985_neg__0__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: complex] :
% 9.10/5.25        ( ( zero_zero_complex
% 9.10/5.25          = ( uminus1482373934393186551omplex @ A ) )
% 9.10/5.25        = ( zero_zero_complex = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4986_neg__0__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( zero_z3403309356797280102nteger
% 9.10/5.25          = ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.25        = ( zero_z3403309356797280102nteger = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4987_neg__0__equal__iff__equal,axiom,
% 9.10/5.25      ! [A: rat] :
% 9.10/5.25        ( ( zero_zero_rat
% 9.10/5.25          = ( uminus_uminus_rat @ A ) )
% 9.10/5.25        = ( zero_zero_rat = A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4988_neg__equal__0__iff__equal,axiom,
% 9.10/5.25      ! [A: real] :
% 9.10/5.25        ( ( ( uminus_uminus_real @ A )
% 9.10/5.25          = zero_zero_real )
% 9.10/5.25        = ( A = zero_zero_real ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4989_neg__equal__0__iff__equal,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( ( uminus_uminus_int @ A )
% 9.10/5.25          = zero_zero_int )
% 9.10/5.25        = ( A = zero_zero_int ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4990_neg__equal__0__iff__equal,axiom,
% 9.10/5.25      ! [A: complex] :
% 9.10/5.25        ( ( ( uminus1482373934393186551omplex @ A )
% 9.10/5.25          = zero_zero_complex )
% 9.10/5.25        = ( A = zero_zero_complex ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4991_neg__equal__0__iff__equal,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.25          = zero_z3403309356797280102nteger )
% 9.10/5.25        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4992_neg__equal__0__iff__equal,axiom,
% 9.10/5.25      ! [A: rat] :
% 9.10/5.25        ( ( ( uminus_uminus_rat @ A )
% 9.10/5.25          = zero_zero_rat )
% 9.10/5.25        = ( A = zero_zero_rat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4993_equal__neg__zero,axiom,
% 9.10/5.25      ! [A: real] :
% 9.10/5.25        ( ( A
% 9.10/5.25          = ( uminus_uminus_real @ A ) )
% 9.10/5.25        = ( A = zero_zero_real ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4994_equal__neg__zero,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( A
% 9.10/5.25          = ( uminus_uminus_int @ A ) )
% 9.10/5.25        = ( A = zero_zero_int ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4995_equal__neg__zero,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( A
% 9.10/5.25          = ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.25        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4996_equal__neg__zero,axiom,
% 9.10/5.25      ! [A: rat] :
% 9.10/5.25        ( ( A
% 9.10/5.25          = ( uminus_uminus_rat @ A ) )
% 9.10/5.25        = ( A = zero_zero_rat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4997_neg__equal__zero,axiom,
% 9.10/5.25      ! [A: real] :
% 9.10/5.25        ( ( ( uminus_uminus_real @ A )
% 9.10/5.25          = A )
% 9.10/5.25        = ( A = zero_zero_real ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4998_neg__equal__zero,axiom,
% 9.10/5.25      ! [A: int] :
% 9.10/5.25        ( ( ( uminus_uminus_int @ A )
% 9.10/5.25          = A )
% 9.10/5.25        = ( A = zero_zero_int ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_4999_neg__equal__zero,axiom,
% 9.10/5.25      ! [A: code_integer] :
% 9.10/5.25        ( ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.25          = A )
% 9.10/5.25        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5000_neg__equal__zero,axiom,
% 9.10/5.25      ! [A: rat] :
% 9.10/5.25        ( ( ( uminus_uminus_rat @ A )
% 9.10/5.25          = A )
% 9.10/5.25        = ( A = zero_zero_rat ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5001_neg__less__iff__less,axiom,
% 9.10/5.25      ! [B2: real,A: real] :
% 9.10/5.25        ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
% 9.10/5.25        = ( ord_less_real @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5002_neg__less__iff__less,axiom,
% 9.10/5.25      ! [B2: int,A: int] :
% 9.10/5.25        ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
% 9.10/5.25        = ( ord_less_int @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5003_neg__less__iff__less,axiom,
% 9.10/5.25      ! [B2: code_integer,A: code_integer] :
% 9.10/5.25        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.25        = ( ord_le6747313008572928689nteger @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5004_neg__less__iff__less,axiom,
% 9.10/5.25      ! [B2: rat,A: rat] :
% 9.10/5.25        ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) )
% 9.10/5.25        = ( ord_less_rat @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5005_neg__numeral__eq__iff,axiom,
% 9.10/5.25      ! [M: num,N: num] :
% 9.10/5.25        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 9.10/5.25          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.25        = ( M = N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5006_neg__numeral__eq__iff,axiom,
% 9.10/5.25      ! [M: num,N: num] :
% 9.10/5.25        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 9.10/5.25          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.25        = ( M = N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5007_neg__numeral__eq__iff,axiom,
% 9.10/5.25      ! [M: num,N: num] :
% 9.10/5.25        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 9.10/5.25          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.25        = ( M = N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5008_neg__numeral__eq__iff,axiom,
% 9.10/5.25      ! [M: num,N: num] :
% 9.10/5.25        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 9.10/5.25          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.25        = ( M = N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5009_neg__numeral__eq__iff,axiom,
% 9.10/5.25      ! [M: num,N: num] :
% 9.10/5.25        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 9.10/5.25          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.25        = ( M = N ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5010_mult__minus__left,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.25        = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5011_mult__minus__left,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.25        = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5012_mult__minus__left,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 9.10/5.25        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5013_mult__minus__left,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.25        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5014_mult__minus__left,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.25        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5015_minus__mult__minus,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.25        = ( times_times_real @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5016_minus__mult__minus,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.25        = ( times_times_int @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5017_minus__mult__minus,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.25        = ( times_times_complex @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5018_minus__mult__minus,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.25        = ( times_3573771949741848930nteger @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5019_minus__mult__minus,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.25        = ( times_times_rat @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5020_mult__minus__right,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.25        = ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5021_mult__minus__right,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.25        = ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5022_mult__minus__right,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.25        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5023_mult__minus__right,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.25        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5024_mult__minus__right,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.25        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5025_minus__add__distrib,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 9.10/5.25        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5026_minus__add__distrib,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 9.10/5.25        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5027_minus__add__distrib,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 9.10/5.25        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5028_minus__add__distrib,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 9.10/5.25        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5029_minus__add__distrib,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 9.10/5.25        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5030_minus__add__cancel,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5031_minus__add__cancel,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5032_minus__add__cancel,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5033_minus__add__cancel,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5034_minus__add__cancel,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5035_add__minus__cancel,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5036_add__minus__cancel,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5037_add__minus__cancel,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5038_add__minus__cancel,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5039_add__minus__cancel,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) )
% 9.10/5.25        = B2 ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5040_minus__diff__eq,axiom,
% 9.10/5.25      ! [A: real,B2: real] :
% 9.10/5.25        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) )
% 9.10/5.25        = ( minus_minus_real @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5041_minus__diff__eq,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) )
% 9.10/5.25        = ( minus_minus_int @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5042_minus__diff__eq,axiom,
% 9.10/5.25      ! [A: complex,B2: complex] :
% 9.10/5.25        ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) )
% 9.10/5.25        = ( minus_minus_complex @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5043_minus__diff__eq,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 9.10/5.25        = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5044_minus__diff__eq,axiom,
% 9.10/5.25      ! [A: rat,B2: rat] :
% 9.10/5.25        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) )
% 9.10/5.25        = ( minus_minus_rat @ B2 @ A ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5045_div__minus__minus,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.25        = ( divide_divide_int @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5046_div__minus__minus,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.25        = ( divide6298287555418463151nteger @ A @ B2 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5047_dvd__minus__iff,axiom,
% 9.10/5.25      ! [X3: real,Y3: real] :
% 9.10/5.25        ( ( dvd_dvd_real @ X3 @ ( uminus_uminus_real @ Y3 ) )
% 9.10/5.25        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5048_dvd__minus__iff,axiom,
% 9.10/5.25      ! [X3: int,Y3: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
% 9.10/5.25        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5049_dvd__minus__iff,axiom,
% 9.10/5.25      ! [X3: complex,Y3: complex] :
% 9.10/5.25        ( ( dvd_dvd_complex @ X3 @ ( uminus1482373934393186551omplex @ Y3 ) )
% 9.10/5.25        = ( dvd_dvd_complex @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5050_dvd__minus__iff,axiom,
% 9.10/5.25      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ X3 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 9.10/5.25        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5051_dvd__minus__iff,axiom,
% 9.10/5.25      ! [X3: rat,Y3: rat] :
% 9.10/5.25        ( ( dvd_dvd_rat @ X3 @ ( uminus_uminus_rat @ Y3 ) )
% 9.10/5.25        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5052_minus__dvd__iff,axiom,
% 9.10/5.25      ! [X3: real,Y3: real] :
% 9.10/5.25        ( ( dvd_dvd_real @ ( uminus_uminus_real @ X3 ) @ Y3 )
% 9.10/5.25        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5053_minus__dvd__iff,axiom,
% 9.10/5.25      ! [X3: int,Y3: int] :
% 9.10/5.25        ( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
% 9.10/5.25        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5054_minus__dvd__iff,axiom,
% 9.10/5.25      ! [X3: complex,Y3: complex] :
% 9.10/5.25        ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 )
% 9.10/5.25        = ( dvd_dvd_complex @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5055_minus__dvd__iff,axiom,
% 9.10/5.25      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.25        ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X3 ) @ Y3 )
% 9.10/5.25        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5056_minus__dvd__iff,axiom,
% 9.10/5.25      ! [X3: rat,Y3: rat] :
% 9.10/5.25        ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X3 ) @ Y3 )
% 9.10/5.25        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5057_mod__minus__minus,axiom,
% 9.10/5.25      ! [A: int,B2: int] :
% 9.10/5.25        ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.25        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5058_mod__minus__minus,axiom,
% 9.10/5.25      ! [A: code_integer,B2: code_integer] :
% 9.10/5.25        ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.25        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5059_of__bool__less__eq__iff,axiom,
% 9.10/5.25      ! [P: $o,Q: $o] :
% 9.10/5.25        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.10/5.25        = ( P
% 9.10/5.25         => Q ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5060_of__bool__less__eq__iff,axiom,
% 9.10/5.25      ! [P: $o,Q: $o] :
% 9.10/5.25        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.10/5.25        = ( P
% 9.10/5.25         => Q ) ) ).
% 9.10/5.25  
% 9.10/5.25  thf(fact_5061_of__bool__less__eq__iff,axiom,
% 9.10/5.25      ! [P: $o,Q: $o] :
% 9.10/5.25        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.10/5.26        = ( P
% 9.10/5.26         => Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5062_of__bool__less__eq__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 9.10/5.26        = ( P
% 9.10/5.26         => Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5063_of__bool__eq_I1_J,axiom,
% 9.10/5.26      ( ( zero_n1201886186963655149omplex @ $false )
% 9.10/5.26      = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5064_of__bool__eq_I1_J,axiom,
% 9.10/5.26      ( ( zero_n3304061248610475627l_real @ $false )
% 9.10/5.26      = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5065_of__bool__eq_I1_J,axiom,
% 9.10/5.26      ( ( zero_n2052037380579107095ol_rat @ $false )
% 9.10/5.26      = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5066_of__bool__eq_I1_J,axiom,
% 9.10/5.26      ( ( zero_n2687167440665602831ol_nat @ $false )
% 9.10/5.26      = zero_zero_nat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5067_of__bool__eq_I1_J,axiom,
% 9.10/5.26      ( ( zero_n2684676970156552555ol_int @ $false )
% 9.10/5.26      = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5068_of__bool__eq_I1_J,axiom,
% 9.10/5.26      ( ( zero_n356916108424825756nteger @ $false )
% 9.10/5.26      = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5069_of__bool__eq__0__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.10/5.26          = zero_zero_complex )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5070_of__bool__eq__0__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.10/5.26          = zero_zero_real )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5071_of__bool__eq__0__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.10/5.26          = zero_zero_rat )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5072_of__bool__eq__0__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.10/5.26          = zero_zero_nat )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5073_of__bool__eq__0__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.10/5.26          = zero_zero_int )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5074_of__bool__eq__0__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n356916108424825756nteger @ P )
% 9.10/5.26          = zero_z3403309356797280102nteger )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5075_real__add__minus__iff,axiom,
% 9.10/5.26      ! [X3: real,A: real] :
% 9.10/5.26        ( ( ( plus_plus_real @ X3 @ ( uminus_uminus_real @ A ) )
% 9.10/5.26          = zero_zero_real )
% 9.10/5.26        = ( X3 = A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5076_of__bool__less__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 9.10/5.26        = ( ~ P
% 9.10/5.26          & Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5077_of__bool__less__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.10/5.26        = ( ~ P
% 9.10/5.26          & Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5078_of__bool__less__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.10/5.26        = ( ~ P
% 9.10/5.26          & Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5079_of__bool__less__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.10/5.26        = ( ~ P
% 9.10/5.26          & Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5080_of__bool__less__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 9.10/5.26        = ( ~ P
% 9.10/5.26          & Q ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5081_of__bool__eq_I2_J,axiom,
% 9.10/5.26      ( ( zero_n1201886186963655149omplex @ $true )
% 9.10/5.26      = one_one_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5082_of__bool__eq_I2_J,axiom,
% 9.10/5.26      ( ( zero_n3304061248610475627l_real @ $true )
% 9.10/5.26      = one_one_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5083_of__bool__eq_I2_J,axiom,
% 9.10/5.26      ( ( zero_n2052037380579107095ol_rat @ $true )
% 9.10/5.26      = one_one_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5084_of__bool__eq_I2_J,axiom,
% 9.10/5.26      ( ( zero_n2687167440665602831ol_nat @ $true )
% 9.10/5.26      = one_one_nat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5085_of__bool__eq_I2_J,axiom,
% 9.10/5.26      ( ( zero_n2684676970156552555ol_int @ $true )
% 9.10/5.26      = one_one_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5086_of__bool__eq_I2_J,axiom,
% 9.10/5.26      ( ( zero_n356916108424825756nteger @ $true )
% 9.10/5.26      = one_one_Code_integer ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5087_of__bool__eq__1__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.10/5.26          = one_one_complex )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5088_of__bool__eq__1__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.10/5.26          = one_one_real )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5089_of__bool__eq__1__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.10/5.26          = one_one_rat )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5090_of__bool__eq__1__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.10/5.26          = one_one_nat )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5091_of__bool__eq__1__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.10/5.26          = one_one_int )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5092_of__bool__eq__1__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ( zero_n356916108424825756nteger @ P )
% 9.10/5.26          = one_one_Code_integer )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5093_length__replicate,axiom,
% 9.10/5.26      ! [N: nat,X3: vEBT_VEBT] :
% 9.10/5.26        ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) )
% 9.10/5.26        = N ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5094_length__replicate,axiom,
% 9.10/5.26      ! [N: nat,X3: $o] :
% 9.10/5.26        ( ( size_size_list_o @ ( replicate_o @ N @ X3 ) )
% 9.10/5.26        = N ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5095_length__replicate,axiom,
% 9.10/5.26      ! [N: nat,X3: nat] :
% 9.10/5.26        ( ( size_size_list_nat @ ( replicate_nat @ N @ X3 ) )
% 9.10/5.26        = N ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5096_of__bool__or__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n2687167440665602831ol_nat
% 9.10/5.26          @ ( P
% 9.10/5.26            | Q ) )
% 9.10/5.26        = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5097_of__bool__or__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n2684676970156552555ol_int
% 9.10/5.26          @ ( P
% 9.10/5.26            | Q ) )
% 9.10/5.26        = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5098_of__bool__or__iff,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n356916108424825756nteger
% 9.10/5.26          @ ( P
% 9.10/5.26            | Q ) )
% 9.10/5.26        = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5099_neg__less__eq__nonneg,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 9.10/5.26        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5100_neg__less__eq__nonneg,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.10/5.26        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5101_neg__less__eq__nonneg,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.10/5.26        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5102_neg__less__eq__nonneg,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 9.10/5.26        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5103_less__eq__neg__nonpos,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 9.10/5.26        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5104_less__eq__neg__nonpos,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.26        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5105_less__eq__neg__nonpos,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.10/5.26        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5106_less__eq__neg__nonpos,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 9.10/5.26        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5107_neg__le__0__iff__le,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.10/5.26        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5108_neg__le__0__iff__le,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.10/5.26        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5109_neg__le__0__iff__le,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.10/5.26        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5110_neg__le__0__iff__le,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.10/5.26        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5111_neg__0__le__iff__le,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.10/5.26        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5112_neg__0__le__iff__le,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.26        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5113_neg__0__le__iff__le,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.10/5.26        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5114_neg__0__le__iff__le,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.10/5.26        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5115_neg__less__0__iff__less,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.10/5.26        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5116_neg__less__0__iff__less,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.10/5.26        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5117_neg__less__0__iff__less,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.10/5.26        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5118_neg__less__0__iff__less,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.10/5.26        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5119_neg__0__less__iff__less,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.10/5.26        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5120_neg__0__less__iff__less,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.10/5.26        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5121_neg__0__less__iff__less,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.26        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5122_neg__0__less__iff__less,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.10/5.26        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5123_neg__less__pos,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 9.10/5.26        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5124_neg__less__pos,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 9.10/5.26        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5125_neg__less__pos,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.10/5.26        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5126_neg__less__pos,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.10/5.26        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5127_less__neg__neg,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 9.10/5.26        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5128_less__neg__neg,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 9.10/5.26        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5129_less__neg__neg,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.26        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5130_less__neg__neg,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.10/5.26        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5131_ab__left__minus,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.10/5.26        = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5132_ab__left__minus,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.10/5.26        = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5133_ab__left__minus,axiom,
% 9.10/5.26      ! [A: complex] :
% 9.10/5.26        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.10/5.26        = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5134_ab__left__minus,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.10/5.26        = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5135_ab__left__minus,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.10/5.26        = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5136_add_Oright__inverse,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 9.10/5.26        = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5137_add_Oright__inverse,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 9.10/5.26        = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5138_add_Oright__inverse,axiom,
% 9.10/5.26      ! [A: complex] :
% 9.10/5.26        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 9.10/5.26        = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5139_add_Oright__inverse,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.26        = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5140_add_Oright__inverse,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.10/5.26        = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5141_verit__minus__simplify_I3_J,axiom,
% 9.10/5.26      ! [B2: real] :
% 9.10/5.26        ( ( minus_minus_real @ zero_zero_real @ B2 )
% 9.10/5.26        = ( uminus_uminus_real @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5142_verit__minus__simplify_I3_J,axiom,
% 9.10/5.26      ! [B2: int] :
% 9.10/5.26        ( ( minus_minus_int @ zero_zero_int @ B2 )
% 9.10/5.26        = ( uminus_uminus_int @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5143_verit__minus__simplify_I3_J,axiom,
% 9.10/5.26      ! [B2: complex] :
% 9.10/5.26        ( ( minus_minus_complex @ zero_zero_complex @ B2 )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5144_verit__minus__simplify_I3_J,axiom,
% 9.10/5.26      ! [B2: code_integer] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5145_verit__minus__simplify_I3_J,axiom,
% 9.10/5.26      ! [B2: rat] :
% 9.10/5.26        ( ( minus_minus_rat @ zero_zero_rat @ B2 )
% 9.10/5.26        = ( uminus_uminus_rat @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5146_diff__0,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( minus_minus_real @ zero_zero_real @ A )
% 9.10/5.26        = ( uminus_uminus_real @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5147_diff__0,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( minus_minus_int @ zero_zero_int @ A )
% 9.10/5.26        = ( uminus_uminus_int @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5148_diff__0,axiom,
% 9.10/5.26      ! [A: complex] :
% 9.10/5.26        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5149_diff__0,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5150_diff__0,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 9.10/5.26        = ( uminus_uminus_rat @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5151_add__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5152_add__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5153_add__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5154_add__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5155_add__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5156_mult__minus1,axiom,
% 9.10/5.26      ! [Z: real] :
% 9.10/5.26        ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 9.10/5.26        = ( uminus_uminus_real @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5157_mult__minus1,axiom,
% 9.10/5.26      ! [Z: int] :
% 9.10/5.26        ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 9.10/5.26        = ( uminus_uminus_int @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5158_mult__minus1,axiom,
% 9.10/5.26      ! [Z: complex] :
% 9.10/5.26        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5159_mult__minus1,axiom,
% 9.10/5.26      ! [Z: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5160_mult__minus1,axiom,
% 9.10/5.26      ! [Z: rat] :
% 9.10/5.26        ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 9.10/5.26        = ( uminus_uminus_rat @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5161_mult__minus1__right,axiom,
% 9.10/5.26      ! [Z: real] :
% 9.10/5.26        ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26        = ( uminus_uminus_real @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5162_mult__minus1__right,axiom,
% 9.10/5.26      ! [Z: int] :
% 9.10/5.26        ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26        = ( uminus_uminus_int @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5163_mult__minus1__right,axiom,
% 9.10/5.26      ! [Z: complex] :
% 9.10/5.26        ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5164_mult__minus1__right,axiom,
% 9.10/5.26      ! [Z: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5165_mult__minus1__right,axiom,
% 9.10/5.26      ! [Z: rat] :
% 9.10/5.26        ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26        = ( uminus_uminus_rat @ Z ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5166_uminus__add__conv__diff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.26        = ( minus_minus_real @ B2 @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5167_uminus__add__conv__diff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.26        = ( minus_minus_int @ B2 @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5168_uminus__add__conv__diff,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 9.10/5.26        = ( minus_minus_complex @ B2 @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5169_uminus__add__conv__diff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.26        = ( minus_8373710615458151222nteger @ B2 @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5170_uminus__add__conv__diff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.26        = ( minus_minus_rat @ B2 @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5171_diff__minus__eq__add,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( plus_plus_real @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5172_diff__minus__eq__add,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( plus_plus_int @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5173_diff__minus__eq__add,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.26        = ( plus_plus_complex @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5174_diff__minus__eq__add,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( plus_p5714425477246183910nteger @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5175_diff__minus__eq__add,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( plus_plus_rat @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5176_divide__minus1,axiom,
% 9.10/5.26      ! [X3: real] :
% 9.10/5.26        ( ( divide_divide_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26        = ( uminus_uminus_real @ X3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5177_divide__minus1,axiom,
% 9.10/5.26      ! [X3: complex] :
% 9.10/5.26        ( ( divide1717551699836669952omplex @ X3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ X3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5178_divide__minus1,axiom,
% 9.10/5.26      ! [X3: rat] :
% 9.10/5.26        ( ( divide_divide_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26        = ( uminus_uminus_rat @ X3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5179_div__minus1__right,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26        = ( uminus_uminus_int @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5180_div__minus1__right,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5181_minus__mod__self1,axiom,
% 9.10/5.26      ! [B2: int,A: int] :
% 9.10/5.26        ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A ) @ B2 )
% 9.10/5.26        = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5182_minus__mod__self1,axiom,
% 9.10/5.26      ! [B2: code_integer,A: code_integer] :
% 9.10/5.26        ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A ) @ B2 )
% 9.10/5.26        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5183_zero__less__of__bool__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5184_zero__less__of__bool__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5185_zero__less__of__bool__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5186_zero__less__of__bool__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5187_zero__less__of__bool__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 9.10/5.26        = P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5188_of__bool__less__one__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5189_of__bool__less__one__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5190_of__bool__less__one__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5191_of__bool__less__one__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5192_of__bool__less__one__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 9.10/5.26        = ~ P ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5193_of__bool__not__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( zero_n1201886186963655149omplex @ ~ P )
% 9.10/5.26        = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5194_of__bool__not__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( zero_n3304061248610475627l_real @ ~ P )
% 9.10/5.26        = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5195_of__bool__not__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 9.10/5.26        = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5196_of__bool__not__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( zero_n2684676970156552555ol_int @ ~ P )
% 9.10/5.26        = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5197_of__bool__not__iff,axiom,
% 9.10/5.26      ! [P: $o] :
% 9.10/5.26        ( ( zero_n356916108424825756nteger @ ~ P )
% 9.10/5.26        = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5198_Suc__0__mod__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.26        = ( zero_n2687167440665602831ol_nat
% 9.10/5.26          @ ( N
% 9.10/5.26           != ( suc @ zero_zero_nat ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5199_signed__take__bit__of__minus__1,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5200_signed__take__bit__of__minus__1,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5201_Ball__set__replicate,axiom,
% 9.10/5.26      ! [N: nat,A: int,P: int > $o] :
% 9.10/5.26        ( ( ! [X: int] :
% 9.10/5.26              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 9.10/5.26             => ( P @ X ) ) )
% 9.10/5.26        = ( ( P @ A )
% 9.10/5.26          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5202_Ball__set__replicate,axiom,
% 9.10/5.26      ! [N: nat,A: nat,P: nat > $o] :
% 9.10/5.26        ( ( ! [X: nat] :
% 9.10/5.26              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 9.10/5.26             => ( P @ X ) ) )
% 9.10/5.26        = ( ( P @ A )
% 9.10/5.26          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5203_Ball__set__replicate,axiom,
% 9.10/5.26      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.10/5.26        ( ( ! [X: vEBT_VEBT] :
% 9.10/5.26              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 9.10/5.26             => ( P @ X ) ) )
% 9.10/5.26        = ( ( P @ A )
% 9.10/5.26          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5204_Bex__set__replicate,axiom,
% 9.10/5.26      ! [N: nat,A: int,P: int > $o] :
% 9.10/5.26        ( ( ? [X: int] :
% 9.10/5.26              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
% 9.10/5.26              & ( P @ X ) ) )
% 9.10/5.26        = ( ( P @ A )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5205_Bex__set__replicate,axiom,
% 9.10/5.26      ! [N: nat,A: nat,P: nat > $o] :
% 9.10/5.26        ( ( ? [X: nat] :
% 9.10/5.26              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
% 9.10/5.26              & ( P @ X ) ) )
% 9.10/5.26        = ( ( P @ A )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5206_Bex__set__replicate,axiom,
% 9.10/5.26      ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.10/5.26        ( ( ? [X: vEBT_VEBT] :
% 9.10/5.26              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
% 9.10/5.26              & ( P @ X ) ) )
% 9.10/5.26        = ( ( P @ A )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5207_in__set__replicate,axiom,
% 9.10/5.26      ! [X3: real,N: nat,Y3: real] :
% 9.10/5.26        ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ Y3 ) ) )
% 9.10/5.26        = ( ( X3 = Y3 )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5208_in__set__replicate,axiom,
% 9.10/5.26      ! [X3: complex,N: nat,Y3: complex] :
% 9.10/5.26        ( ( member_complex @ X3 @ ( set_complex2 @ ( replicate_complex @ N @ Y3 ) ) )
% 9.10/5.26        = ( ( X3 = Y3 )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5209_in__set__replicate,axiom,
% 9.10/5.26      ! [X3: int,N: nat,Y3: int] :
% 9.10/5.26        ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ Y3 ) ) )
% 9.10/5.26        = ( ( X3 = Y3 )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5210_in__set__replicate,axiom,
% 9.10/5.26      ! [X3: nat,N: nat,Y3: nat] :
% 9.10/5.26        ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ Y3 ) ) )
% 9.10/5.26        = ( ( X3 = Y3 )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5211_in__set__replicate,axiom,
% 9.10/5.26      ! [X3: vEBT_VEBT,N: nat,Y3: vEBT_VEBT] :
% 9.10/5.26        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y3 ) ) )
% 9.10/5.26        = ( ( X3 = Y3 )
% 9.10/5.26          & ( N != zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5212_nth__replicate,axiom,
% 9.10/5.26      ! [I: nat,N: nat,X3: int] :
% 9.10/5.26        ( ( ord_less_nat @ I @ N )
% 9.10/5.26       => ( ( nth_int @ ( replicate_int @ N @ X3 ) @ I )
% 9.10/5.26          = X3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5213_nth__replicate,axiom,
% 9.10/5.26      ! [I: nat,N: nat,X3: nat] :
% 9.10/5.26        ( ( ord_less_nat @ I @ N )
% 9.10/5.26       => ( ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I )
% 9.10/5.26          = X3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5214_nth__replicate,axiom,
% 9.10/5.26      ! [I: nat,N: nat,X3: vEBT_VEBT] :
% 9.10/5.26        ( ( ord_less_nat @ I @ N )
% 9.10/5.26       => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) @ I )
% 9.10/5.26          = X3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5215_dbl__simps_I1_J,axiom,
% 9.10/5.26      ! [K: num] :
% 9.10/5.26        ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 9.10/5.26        = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5216_dbl__simps_I1_J,axiom,
% 9.10/5.26      ! [K: num] :
% 9.10/5.26        ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5217_dbl__simps_I1_J,axiom,
% 9.10/5.26      ! [K: num] :
% 9.10/5.26        ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5218_dbl__simps_I1_J,axiom,
% 9.10/5.26      ! [K: num] :
% 9.10/5.26        ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5219_dbl__simps_I1_J,axiom,
% 9.10/5.26      ! [K: num] :
% 9.10/5.26        ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.10/5.26        = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5220_triangle__Suc,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( nat_triangle @ ( suc @ N ) )
% 9.10/5.26        = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5221_add__neg__numeral__special_I8_J,axiom,
% 9.10/5.26      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.10/5.26      = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5222_add__neg__numeral__special_I8_J,axiom,
% 9.10/5.26      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.10/5.26      = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5223_add__neg__numeral__special_I8_J,axiom,
% 9.10/5.26      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.10/5.26      = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5224_add__neg__numeral__special_I8_J,axiom,
% 9.10/5.26      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.10/5.26      = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5225_add__neg__numeral__special_I8_J,axiom,
% 9.10/5.26      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.10/5.26      = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5226_add__neg__numeral__special_I7_J,axiom,
% 9.10/5.26      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26      = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5227_add__neg__numeral__special_I7_J,axiom,
% 9.10/5.26      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26      = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5228_add__neg__numeral__special_I7_J,axiom,
% 9.10/5.26      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26      = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5229_add__neg__numeral__special_I7_J,axiom,
% 9.10/5.26      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26      = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5230_add__neg__numeral__special_I7_J,axiom,
% 9.10/5.26      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26      = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5231_diff__numeral__special_I12_J,axiom,
% 9.10/5.26      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26      = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5232_diff__numeral__special_I12_J,axiom,
% 9.10/5.26      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26      = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5233_diff__numeral__special_I12_J,axiom,
% 9.10/5.26      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26      = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5234_diff__numeral__special_I12_J,axiom,
% 9.10/5.26      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26      = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5235_diff__numeral__special_I12_J,axiom,
% 9.10/5.26      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26      = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5236_neg__one__eq__numeral__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus_uminus_real @ one_one_real )
% 9.10/5.26          = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5237_neg__one__eq__numeral__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus_uminus_int @ one_one_int )
% 9.10/5.26          = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5238_neg__one__eq__numeral__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 9.10/5.26          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5239_neg__one__eq__numeral__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 9.10/5.26          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5240_neg__one__eq__numeral__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus_uminus_rat @ one_one_rat )
% 9.10/5.26          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5241_numeral__eq__neg__one__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 9.10/5.26          = ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5242_numeral__eq__neg__one__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 9.10/5.26          = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5243_numeral__eq__neg__one__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 9.10/5.26          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5244_numeral__eq__neg__one__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 9.10/5.26          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5245_numeral__eq__neg__one__iff,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 9.10/5.26          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26        = ( N = one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5246_minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 9.10/5.26        = one_one_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5247_minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 9.10/5.26        = one_one_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5248_minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 9.10/5.26        = one_one_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5249_minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 9.10/5.26        = one_one_Code_integer ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5250_minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 9.10/5.26        = one_one_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5251_left__minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat,A: real] :
% 9.10/5.26        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
% 9.10/5.26        = A ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5252_left__minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat,A: int] :
% 9.10/5.26        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
% 9.10/5.26        = A ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5253_left__minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat,A: complex] :
% 9.10/5.26        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 9.10/5.26        = A ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5254_left__minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat,A: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 9.10/5.26        = A ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5255_left__minus__one__mult__self,axiom,
% 9.10/5.26      ! [N: nat,A: rat] :
% 9.10/5.26        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
% 9.10/5.26        = A ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5256_mod__minus1__right,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26        = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5257_mod__minus1__right,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26        = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5258_max__number__of_I2_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26         => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26            = ( numeral_numeral_real @ U ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5259_max__number__of_I2_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26         => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26            = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5260_max__number__of_I2_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26         => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26            = ( numeral_numeral_rat @ U ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5261_max__number__of_I2_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26         => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26            = ( numeral_numeral_int @ U ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5262_max__number__of_I3_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.26         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.26            = ( numeral_numeral_real @ V ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.26         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.10/5.26            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5263_max__number__of_I3_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.10/5.26         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.10/5.26            = ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.10/5.26         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.10/5.26            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5264_max__number__of_I3_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.26         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.26            = ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.26         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.10/5.26            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5265_max__number__of_I3_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.26         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.26            = ( numeral_numeral_int @ V ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.26         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.10/5.26            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5266_max__number__of_I4_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26         => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5267_max__number__of_I4_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26         => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.10/5.26            = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5268_max__number__of_I4_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26         => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5269_max__number__of_I4_J,axiom,
% 9.10/5.26      ! [U: num,V: num] :
% 9.10/5.26        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 9.10/5.26        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26         => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.10/5.26            = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5270_semiring__norm_I168_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: real] :
% 9.10/5.26        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5271_semiring__norm_I168_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: int] :
% 9.10/5.26        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5272_semiring__norm_I168_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: complex] :
% 9.10/5.26        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5273_semiring__norm_I168_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: code_integer] :
% 9.10/5.26        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5274_semiring__norm_I168_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: rat] :
% 9.10/5.26        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5275_diff__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 9.10/5.26        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5276_diff__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5277_diff__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5278_diff__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5279_diff__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 9.10/5.26        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5280_diff__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5281_diff__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5282_diff__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.26        = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5283_diff__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5284_diff__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5285_semiring__norm_I172_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: real] :
% 9.10/5.26        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5286_semiring__norm_I172_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: int] :
% 9.10/5.26        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5287_semiring__norm_I172_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: complex] :
% 9.10/5.26        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5288_semiring__norm_I172_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5289_semiring__norm_I172_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: rat] :
% 9.10/5.26        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5290_semiring__norm_I171_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: real] :
% 9.10/5.26        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5291_semiring__norm_I171_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: int] :
% 9.10/5.26        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5292_semiring__norm_I171_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: complex] :
% 9.10/5.26        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5293_semiring__norm_I171_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5294_semiring__norm_I171_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: rat] :
% 9.10/5.26        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 9.10/5.26        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5295_semiring__norm_I170_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: real] :
% 9.10/5.26        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 9.10/5.26        = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5296_semiring__norm_I170_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: int] :
% 9.10/5.26        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 9.10/5.26        = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5297_semiring__norm_I170_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: complex] :
% 9.10/5.26        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 9.10/5.26        = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5298_semiring__norm_I170_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 9.10/5.26        = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5299_semiring__norm_I170_J,axiom,
% 9.10/5.26      ! [V: num,W: num,Y3: rat] :
% 9.10/5.26        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 9.10/5.26        = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5300_mult__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5301_mult__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5302_mult__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5303_mult__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5304_mult__neg__numeral__simps_I3_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5305_mult__neg__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 9.10/5.26        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5306_mult__neg__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5307_mult__neg__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5308_mult__neg__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5309_mult__neg__numeral__simps_I2_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 9.10/5.26        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5310_mult__neg__numeral__simps_I1_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5311_mult__neg__numeral__simps_I1_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5312_mult__neg__numeral__simps_I1_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.26        = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5313_mult__neg__numeral__simps_I1_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5314_mult__neg__numeral__simps_I1_J,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5315_neg__numeral__le__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5316_neg__numeral__le__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5317_neg__numeral__le__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5318_neg__numeral__le__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( ord_less_eq_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5319_neg__numeral__less__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( ord_less_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5320_neg__numeral__less__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( ord_less_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5321_neg__numeral__less__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( ord_less_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5322_neg__numeral__less__iff,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( ord_less_num @ N @ M ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5323_not__neg__one__le__neg__numeral__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5324_not__neg__one__le__neg__numeral__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5325_not__neg__one__le__neg__numeral__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5326_not__neg__one__le__neg__numeral__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5327_divide__le__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: real,W: num,A: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 9.10/5.26        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5328_divide__le__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: rat,W: num,A: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 9.10/5.26        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5329_le__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: real,B2: real,W: num] :
% 9.10/5.26        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 9.10/5.26        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5330_le__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: rat,B2: rat,W: num] :
% 9.10/5.26        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 9.10/5.26        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5331_divide__eq__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: real,W: num,A: real] :
% 9.10/5.26        ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.26          = A )
% 9.10/5.26        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.26             != zero_zero_real )
% 9.10/5.26           => ( B2
% 9.10/5.26              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 9.10/5.26          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.26              = zero_zero_real )
% 9.10/5.26           => ( A = zero_zero_real ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5332_divide__eq__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: complex,W: num,A: complex] :
% 9.10/5.26        ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.26          = A )
% 9.10/5.26        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.26             != zero_zero_complex )
% 9.10/5.26           => ( B2
% 9.10/5.26              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 9.10/5.26          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.26              = zero_zero_complex )
% 9.10/5.26           => ( A = zero_zero_complex ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5333_divide__eq__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: rat,W: num,A: rat] :
% 9.10/5.26        ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.10/5.26          = A )
% 9.10/5.26        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.26             != zero_zero_rat )
% 9.10/5.26           => ( B2
% 9.10/5.26              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 9.10/5.26          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.26              = zero_zero_rat )
% 9.10/5.26           => ( A = zero_zero_rat ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5334_eq__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: real,B2: real,W: num] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 9.10/5.26        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.26             != zero_zero_real )
% 9.10/5.26           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.26              = B2 ) )
% 9.10/5.26          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.26              = zero_zero_real )
% 9.10/5.26           => ( A = zero_zero_real ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5335_eq__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: complex,B2: complex,W: num] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 9.10/5.26        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.26             != zero_zero_complex )
% 9.10/5.26           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.26              = B2 ) )
% 9.10/5.26          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.26              = zero_zero_complex )
% 9.10/5.26           => ( A = zero_zero_complex ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5336_eq__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: rat,B2: rat,W: num] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 9.10/5.26        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.26             != zero_zero_rat )
% 9.10/5.26           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.10/5.26              = B2 ) )
% 9.10/5.26          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.26              = zero_zero_rat )
% 9.10/5.26           => ( A = zero_zero_rat ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5337_neg__numeral__less__neg__one__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5338_neg__numeral__less__neg__one__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5339_neg__numeral__less__neg__one__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5340_neg__numeral__less__neg__one__iff,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26        = ( M != one ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5341_divide__less__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: real,W: num,A: real] :
% 9.10/5.26        ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 9.10/5.26        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5342_divide__less__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [B2: rat,W: num,A: rat] :
% 9.10/5.26        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 9.10/5.26        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5343_less__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: real,B2: real,W: num] :
% 9.10/5.26        ( ( ord_less_real @ A @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 9.10/5.26        = ( ord_less_real @ B2 @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5344_less__divide__eq__numeral1_I2_J,axiom,
% 9.10/5.26      ! [A: rat,B2: rat,W: num] :
% 9.10/5.26        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 9.10/5.26        = ( ord_less_rat @ B2 @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5345_power2__minus,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.26        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5346_power2__minus,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.26        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5347_power2__minus,axiom,
% 9.10/5.26      ! [A: complex] :
% 9.10/5.26        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.26        = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5348_power2__minus,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.26        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5349_power2__minus,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.26        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5350_odd__of__bool__self,axiom,
% 9.10/5.26      ! [P4: $o] :
% 9.10/5.26        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
% 9.10/5.26        = P4 ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5351_odd__of__bool__self,axiom,
% 9.10/5.26      ! [P4: $o] :
% 9.10/5.26        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
% 9.10/5.26        = P4 ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5352_odd__of__bool__self,axiom,
% 9.10/5.26      ! [P4: $o] :
% 9.10/5.26        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
% 9.10/5.26        = P4 ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5353_add__neg__numeral__special_I9_J,axiom,
% 9.10/5.26      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5354_add__neg__numeral__special_I9_J,axiom,
% 9.10/5.26      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5355_add__neg__numeral__special_I9_J,axiom,
% 9.10/5.26      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5356_add__neg__numeral__special_I9_J,axiom,
% 9.10/5.26      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5357_add__neg__numeral__special_I9_J,axiom,
% 9.10/5.26      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5358_diff__numeral__special_I11_J,axiom,
% 9.10/5.26      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5359_diff__numeral__special_I11_J,axiom,
% 9.10/5.26      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5360_diff__numeral__special_I11_J,axiom,
% 9.10/5.26      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5361_diff__numeral__special_I11_J,axiom,
% 9.10/5.26      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5362_diff__numeral__special_I11_J,axiom,
% 9.10/5.26      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5363_diff__numeral__special_I10_J,axiom,
% 9.10/5.26      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.10/5.26      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5364_diff__numeral__special_I10_J,axiom,
% 9.10/5.26      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.10/5.26      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5365_diff__numeral__special_I10_J,axiom,
% 9.10/5.26      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.10/5.26      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5366_diff__numeral__special_I10_J,axiom,
% 9.10/5.26      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.10/5.26      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5367_diff__numeral__special_I10_J,axiom,
% 9.10/5.26      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.10/5.26      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5368_minus__1__div__2__eq,axiom,
% 9.10/5.26      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.26      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5369_minus__1__div__2__eq,axiom,
% 9.10/5.26      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.26      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5370_minus__1__mod__2__eq,axiom,
% 9.10/5.26      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.26      = one_one_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5371_minus__1__mod__2__eq,axiom,
% 9.10/5.26      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.26      = one_one_Code_integer ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5372_bits__minus__1__mod__2__eq,axiom,
% 9.10/5.26      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.26      = one_one_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5373_bits__minus__1__mod__2__eq,axiom,
% 9.10/5.26      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.26      = one_one_Code_integer ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5374_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [A: real,N: nat] :
% 9.10/5.26        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5375_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [A: int,N: nat] :
% 9.10/5.26        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5376_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [A: complex,N: nat] :
% 9.10/5.26        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5377_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [A: code_integer,N: nat] :
% 9.10/5.26        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5378_Power_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [A: rat,N: nat] :
% 9.10/5.26        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5379_of__bool__half__eq__0,axiom,
% 9.10/5.26      ! [B2: $o] :
% 9.10/5.26        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.26        = zero_zero_nat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5380_of__bool__half__eq__0,axiom,
% 9.10/5.26      ! [B2: $o] :
% 9.10/5.26        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.26        = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5381_of__bool__half__eq__0,axiom,
% 9.10/5.26      ! [B2: $o] :
% 9.10/5.26        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.26        = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5382_power__minus__odd,axiom,
% 9.10/5.26      ! [N: nat,A: real] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.10/5.26          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5383_power__minus__odd,axiom,
% 9.10/5.26      ! [N: nat,A: int] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.10/5.26          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5384_power__minus__odd,axiom,
% 9.10/5.26      ! [N: nat,A: complex] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.10/5.26          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5385_power__minus__odd,axiom,
% 9.10/5.26      ! [N: nat,A: code_integer] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.10/5.26          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5386_power__minus__odd,axiom,
% 9.10/5.26      ! [N: nat,A: rat] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.10/5.26          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5387_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [N: nat,A: real] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.10/5.26          = ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5388_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [N: nat,A: int] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.10/5.26          = ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5389_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [N: nat,A: complex] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.10/5.26          = ( power_power_complex @ A @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5390_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [N: nat,A: code_integer] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.10/5.26          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5391_Parity_Oring__1__class_Opower__minus__even,axiom,
% 9.10/5.26      ! [N: nat,A: rat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.10/5.26          = ( power_power_rat @ A @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5392_diff__numeral__special_I3_J,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5393_diff__numeral__special_I3_J,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5394_diff__numeral__special_I3_J,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 9.10/5.26        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5395_diff__numeral__special_I3_J,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.26        = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5396_diff__numeral__special_I3_J,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.26        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5397_diff__numeral__special_I4_J,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 9.10/5.26        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5398_diff__numeral__special_I4_J,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 9.10/5.26        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5399_diff__numeral__special_I4_J,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5400_diff__numeral__special_I4_J,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5401_diff__numeral__special_I4_J,axiom,
% 9.10/5.26      ! [M: num] :
% 9.10/5.26        ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 9.10/5.26        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5402_signed__take__bit__Suc__minus__bit0,axiom,
% 9.10/5.26      ! [N: nat,K: num] :
% 9.10/5.26        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.10/5.26        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5403_set__decode__0,axiom,
% 9.10/5.26      ! [X3: nat] :
% 9.10/5.26        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
% 9.10/5.26        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5404_dbl__simps_I4_J,axiom,
% 9.10/5.26      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.26      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5405_dbl__simps_I4_J,axiom,
% 9.10/5.26      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.26      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5406_dbl__simps_I4_J,axiom,
% 9.10/5.26      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.26      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5407_dbl__simps_I4_J,axiom,
% 9.10/5.26      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.26      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5408_dbl__simps_I4_J,axiom,
% 9.10/5.26      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.26      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5409_power__minus1__even,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = one_one_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5410_power__minus1__even,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = one_one_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5411_power__minus1__even,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = one_one_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5412_power__minus1__even,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = one_one_Code_integer ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5413_power__minus1__even,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = one_one_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5414_neg__one__odd__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.10/5.26          = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5415_neg__one__odd__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.10/5.26          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5416_neg__one__odd__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.10/5.26          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5417_neg__one__odd__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.10/5.26          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5418_neg__one__odd__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.10/5.26          = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5419_neg__one__even__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.10/5.26          = one_one_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5420_neg__one__even__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.10/5.26          = one_one_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5421_neg__one__even__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.10/5.26          = one_one_complex ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5422_neg__one__even__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.10/5.26          = one_one_Code_integer ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5423_neg__one__even__power,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.26       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.10/5.26          = one_one_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5424_signed__take__bit__0,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5425_signed__take__bit__0,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 9.10/5.26        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5426_one__div__2__pow__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5427_one__div__2__pow__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5428_one__div__2__pow__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5429_bits__1__div__exp,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5430_bits__1__div__exp,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5431_bits__1__div__exp,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5432_one__mod__2__pow__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5433_one__mod__2__pow__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5434_one__mod__2__pow__eq,axiom,
% 9.10/5.26      ! [N: nat] :
% 9.10/5.26        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.26        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5435_signed__take__bit__minus,axiom,
% 9.10/5.26      ! [N: nat,K: int] :
% 9.10/5.26        ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 9.10/5.26        = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5436_dvd__antisym,axiom,
% 9.10/5.26      ! [M: nat,N: nat] :
% 9.10/5.26        ( ( dvd_dvd_nat @ M @ N )
% 9.10/5.26       => ( ( dvd_dvd_nat @ N @ M )
% 9.10/5.26         => ( M = N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5437_equation__minus__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5438_equation__minus__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5439_equation__minus__iff,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5440_equation__minus__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5441_equation__minus__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5442_minus__equation__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ( uminus_uminus_real @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( uminus_uminus_real @ B2 )
% 9.10/5.26          = A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5443_minus__equation__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ( uminus_uminus_int @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( uminus_uminus_int @ B2 )
% 9.10/5.26          = A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5444_minus__equation__iff,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( ( uminus1482373934393186551omplex @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( uminus1482373934393186551omplex @ B2 )
% 9.10/5.26          = A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5445_minus__equation__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( uminus1351360451143612070nteger @ B2 )
% 9.10/5.26          = A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5446_minus__equation__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ( uminus_uminus_rat @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( uminus_uminus_rat @ B2 )
% 9.10/5.26          = A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5447_of__bool__eq__iff,axiom,
% 9.10/5.26      ! [P4: $o,Q2: $o] :
% 9.10/5.26        ( ( ( zero_n2687167440665602831ol_nat @ P4 )
% 9.10/5.26          = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 9.10/5.26        = ( P4 = Q2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5448_of__bool__eq__iff,axiom,
% 9.10/5.26      ! [P4: $o,Q2: $o] :
% 9.10/5.26        ( ( ( zero_n2684676970156552555ol_int @ P4 )
% 9.10/5.26          = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 9.10/5.26        = ( P4 = Q2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5449_of__bool__eq__iff,axiom,
% 9.10/5.26      ! [P4: $o,Q2: $o] :
% 9.10/5.26        ( ( ( zero_n356916108424825756nteger @ P4 )
% 9.10/5.26          = ( zero_n356916108424825756nteger @ Q2 ) )
% 9.10/5.26        = ( P4 = Q2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5450_verit__negate__coefficient_I3_J,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( A = B2 )
% 9.10/5.26       => ( ( uminus_uminus_real @ A )
% 9.10/5.26          = ( uminus_uminus_real @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5451_verit__negate__coefficient_I3_J,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( A = B2 )
% 9.10/5.26       => ( ( uminus_uminus_int @ A )
% 9.10/5.26          = ( uminus_uminus_int @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5452_verit__negate__coefficient_I3_J,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( A = B2 )
% 9.10/5.26       => ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.26          = ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5453_verit__negate__coefficient_I3_J,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( A = B2 )
% 9.10/5.26       => ( ( uminus_uminus_rat @ A )
% 9.10/5.26          = ( uminus_uminus_rat @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5454_compl__mono,axiom,
% 9.10/5.26      ! [X3: set_int,Y3: set_int] :
% 9.10/5.26        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.10/5.26       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5455_compl__le__swap1,axiom,
% 9.10/5.26      ! [Y3: set_int,X3: set_int] :
% 9.10/5.26        ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X3 ) )
% 9.10/5.26       => ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5456_compl__le__swap2,axiom,
% 9.10/5.26      ! [Y3: set_int,X3: set_int] :
% 9.10/5.26        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X3 )
% 9.10/5.26       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y3 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5457_of__bool__conj,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n3304061248610475627l_real
% 9.10/5.26          @ ( P
% 9.10/5.26            & Q ) )
% 9.10/5.26        = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5458_of__bool__conj,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n2052037380579107095ol_rat
% 9.10/5.26          @ ( P
% 9.10/5.26            & Q ) )
% 9.10/5.26        = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5459_of__bool__conj,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n2687167440665602831ol_nat
% 9.10/5.26          @ ( P
% 9.10/5.26            & Q ) )
% 9.10/5.26        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5460_of__bool__conj,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n2684676970156552555ol_int
% 9.10/5.26          @ ( P
% 9.10/5.26            & Q ) )
% 9.10/5.26        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5461_of__bool__conj,axiom,
% 9.10/5.26      ! [P: $o,Q: $o] :
% 9.10/5.26        ( ( zero_n356916108424825756nteger
% 9.10/5.26          @ ( P
% 9.10/5.26            & Q ) )
% 9.10/5.26        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5462_le__imp__neg__le,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.26       => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5463_le__imp__neg__le,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 9.10/5.26       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5464_le__imp__neg__le,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.26       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5465_le__imp__neg__le,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.26       => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5466_minus__le__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.26        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5467_minus__le__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.26        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5468_minus__le__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.26        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5469_minus__le__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.26        = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5470_le__minus__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5471_le__minus__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5472_le__minus__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5473_le__minus__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5474_verit__negate__coefficient_I2_J,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ord_less_real @ A @ B2 )
% 9.10/5.26       => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5475_verit__negate__coefficient_I2_J,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ord_less_int @ A @ B2 )
% 9.10/5.26       => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5476_verit__negate__coefficient_I2_J,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 9.10/5.26       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5477_verit__negate__coefficient_I2_J,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ord_less_rat @ A @ B2 )
% 9.10/5.26       => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5478_minus__less__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.26        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5479_minus__less__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.26        = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5480_minus__less__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.26        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5481_minus__less__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.26        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5482_less__minus__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5483_less__minus__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5484_less__minus__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5485_less__minus__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5486_neg__numeral__neq__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 9.10/5.26       != ( numeral_numeral_real @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5487_neg__numeral__neq__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 9.10/5.26       != ( numeral_numeral_int @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5488_neg__numeral__neq__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 9.10/5.26       != ( numera6690914467698888265omplex @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5489_neg__numeral__neq__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 9.10/5.26       != ( numera6620942414471956472nteger @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5490_neg__numeral__neq__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 9.10/5.26       != ( numeral_numeral_rat @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5491_numeral__neq__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( numeral_numeral_real @ M )
% 9.10/5.26       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5492_numeral__neq__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( numeral_numeral_int @ M )
% 9.10/5.26       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5493_numeral__neq__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( numera6690914467698888265omplex @ M )
% 9.10/5.26       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5494_numeral__neq__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( numera6620942414471956472nteger @ M )
% 9.10/5.26       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5495_numeral__neq__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ( ( numeral_numeral_rat @ M )
% 9.10/5.26       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5496_square__eq__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ( times_times_real @ A @ A )
% 9.10/5.26          = ( times_times_real @ B2 @ B2 ) )
% 9.10/5.26        = ( ( A = B2 )
% 9.10/5.26          | ( A
% 9.10/5.26            = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5497_square__eq__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ( times_times_int @ A @ A )
% 9.10/5.26          = ( times_times_int @ B2 @ B2 ) )
% 9.10/5.26        = ( ( A = B2 )
% 9.10/5.26          | ( A
% 9.10/5.26            = ( uminus_uminus_int @ B2 ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5498_square__eq__iff,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( ( times_times_complex @ A @ A )
% 9.10/5.26          = ( times_times_complex @ B2 @ B2 ) )
% 9.10/5.26        = ( ( A = B2 )
% 9.10/5.26          | ( A
% 9.10/5.26            = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5499_square__eq__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ( times_3573771949741848930nteger @ A @ A )
% 9.10/5.26          = ( times_3573771949741848930nteger @ B2 @ B2 ) )
% 9.10/5.26        = ( ( A = B2 )
% 9.10/5.26          | ( A
% 9.10/5.26            = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5500_square__eq__iff,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ( times_times_rat @ A @ A )
% 9.10/5.26          = ( times_times_rat @ B2 @ B2 ) )
% 9.10/5.26        = ( ( A = B2 )
% 9.10/5.26          | ( A
% 9.10/5.26            = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5501_minus__mult__commute,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.26        = ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5502_minus__mult__commute,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.26        = ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5503_minus__mult__commute,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 9.10/5.26        = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5504_minus__mult__commute,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.26        = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5505_minus__mult__commute,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.26        = ( times_times_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5506_one__neq__neg__one,axiom,
% 9.10/5.26      ( one_one_real
% 9.10/5.26     != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5507_one__neq__neg__one,axiom,
% 9.10/5.26      ( one_one_int
% 9.10/5.26     != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5508_one__neq__neg__one,axiom,
% 9.10/5.26      ( one_one_complex
% 9.10/5.26     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5509_one__neq__neg__one,axiom,
% 9.10/5.26      ( one_one_Code_integer
% 9.10/5.26     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5510_one__neq__neg__one,axiom,
% 9.10/5.26      ( one_one_rat
% 9.10/5.26     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5511_is__num__normalize_I8_J,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5512_is__num__normalize_I8_J,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5513_is__num__normalize_I8_J,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5514_is__num__normalize_I8_J,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 9.10/5.26        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5515_is__num__normalize_I8_J,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5516_add_Oinverse__distrib__swap,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5517_add_Oinverse__distrib__swap,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5518_add_Oinverse__distrib__swap,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5519_add_Oinverse__distrib__swap,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) )
% 9.10/5.26        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5520_add_Oinverse__distrib__swap,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B2 ) )
% 9.10/5.26        = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5521_group__cancel_Oneg1,axiom,
% 9.10/5.26      ! [A4: real,K: real,A: real] :
% 9.10/5.26        ( ( A4
% 9.10/5.26          = ( plus_plus_real @ K @ A ) )
% 9.10/5.26       => ( ( uminus_uminus_real @ A4 )
% 9.10/5.26          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5522_group__cancel_Oneg1,axiom,
% 9.10/5.26      ! [A4: int,K: int,A: int] :
% 9.10/5.26        ( ( A4
% 9.10/5.26          = ( plus_plus_int @ K @ A ) )
% 9.10/5.26       => ( ( uminus_uminus_int @ A4 )
% 9.10/5.26          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5523_group__cancel_Oneg1,axiom,
% 9.10/5.26      ! [A4: complex,K: complex,A: complex] :
% 9.10/5.26        ( ( A4
% 9.10/5.26          = ( plus_plus_complex @ K @ A ) )
% 9.10/5.26       => ( ( uminus1482373934393186551omplex @ A4 )
% 9.10/5.26          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5524_group__cancel_Oneg1,axiom,
% 9.10/5.26      ! [A4: code_integer,K: code_integer,A: code_integer] :
% 9.10/5.26        ( ( A4
% 9.10/5.26          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 9.10/5.26       => ( ( uminus1351360451143612070nteger @ A4 )
% 9.10/5.26          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5525_group__cancel_Oneg1,axiom,
% 9.10/5.26      ! [A4: rat,K: rat,A: rat] :
% 9.10/5.26        ( ( A4
% 9.10/5.26          = ( plus_plus_rat @ K @ A ) )
% 9.10/5.26       => ( ( uminus_uminus_rat @ A4 )
% 9.10/5.26          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5526_minus__diff__minus,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( uminus_uminus_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5527_minus__diff__minus,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5528_minus__diff__minus,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.26        = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5529_minus__diff__minus,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5530_minus__diff__minus,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5531_minus__diff__commute,axiom,
% 9.10/5.26      ! [B2: real,A: real] :
% 9.10/5.26        ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A )
% 9.10/5.26        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5532_minus__diff__commute,axiom,
% 9.10/5.26      ! [B2: int,A: int] :
% 9.10/5.26        ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A )
% 9.10/5.26        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5533_minus__diff__commute,axiom,
% 9.10/5.26      ! [B2: complex,A: complex] :
% 9.10/5.26        ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A )
% 9.10/5.26        = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5534_minus__diff__commute,axiom,
% 9.10/5.26      ! [B2: code_integer,A: code_integer] :
% 9.10/5.26        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A )
% 9.10/5.26        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5535_minus__diff__commute,axiom,
% 9.10/5.26      ! [B2: rat,A: rat] :
% 9.10/5.26        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A )
% 9.10/5.26        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5536_minus__divide__right,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.26        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5537_minus__divide__right,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.26        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5538_minus__divide__right,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.26        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5539_minus__divide__divide,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( divide_divide_real @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5540_minus__divide__divide,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.26        = ( divide1717551699836669952omplex @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5541_minus__divide__divide,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( divide_divide_rat @ A @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5542_minus__divide__left,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.26        = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5543_minus__divide__left,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.26        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5544_minus__divide__left,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.26        = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5545_div__minus__right,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5546_div__minus__right,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5547_mod__minus__eq,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B2 ) ) @ B2 )
% 9.10/5.26        = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5548_mod__minus__eq,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B2 ) ) @ B2 )
% 9.10/5.26        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5549_mod__minus__cong,axiom,
% 9.10/5.26      ! [A: int,B2: int,A2: int] :
% 9.10/5.26        ( ( ( modulo_modulo_int @ A @ B2 )
% 9.10/5.26          = ( modulo_modulo_int @ A2 @ B2 ) )
% 9.10/5.26       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.26          = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5550_mod__minus__cong,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer,A2: code_integer] :
% 9.10/5.26        ( ( ( modulo364778990260209775nteger @ A @ B2 )
% 9.10/5.26          = ( modulo364778990260209775nteger @ A2 @ B2 ) )
% 9.10/5.26       => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.26          = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5551_mod__minus__right,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5552_mod__minus__right,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5553_zero__less__eq__of__bool,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5554_zero__less__eq__of__bool,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5555_zero__less__eq__of__bool,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5556_zero__less__eq__of__bool,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5557_zero__less__eq__of__bool,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5558_of__bool__less__eq__one,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5559_of__bool__less__eq__one,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5560_of__bool__less__eq__one,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5561_of__bool__less__eq__one,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5562_of__bool__less__eq__one,axiom,
% 9.10/5.26      ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5563_of__bool__def,axiom,
% 9.10/5.26      ( zero_n1201886186963655149omplex
% 9.10/5.26      = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5564_of__bool__def,axiom,
% 9.10/5.26      ( zero_n3304061248610475627l_real
% 9.10/5.26      = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5565_of__bool__def,axiom,
% 9.10/5.26      ( zero_n2052037380579107095ol_rat
% 9.10/5.26      = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5566_of__bool__def,axiom,
% 9.10/5.26      ( zero_n2687167440665602831ol_nat
% 9.10/5.26      = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5567_of__bool__def,axiom,
% 9.10/5.26      ( zero_n2684676970156552555ol_int
% 9.10/5.26      = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5568_of__bool__def,axiom,
% 9.10/5.26      ( zero_n356916108424825756nteger
% 9.10/5.26      = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5569_split__of__bool,axiom,
% 9.10/5.26      ! [P: complex > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 9.10/5.26        = ( ( P4
% 9.10/5.26           => ( P @ one_one_complex ) )
% 9.10/5.26          & ( ~ P4
% 9.10/5.26           => ( P @ zero_zero_complex ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5570_split__of__bool,axiom,
% 9.10/5.26      ! [P: real > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 9.10/5.26        = ( ( P4
% 9.10/5.26           => ( P @ one_one_real ) )
% 9.10/5.26          & ( ~ P4
% 9.10/5.26           => ( P @ zero_zero_real ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5571_split__of__bool,axiom,
% 9.10/5.26      ! [P: rat > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 9.10/5.26        = ( ( P4
% 9.10/5.26           => ( P @ one_one_rat ) )
% 9.10/5.26          & ( ~ P4
% 9.10/5.26           => ( P @ zero_zero_rat ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5572_split__of__bool,axiom,
% 9.10/5.26      ! [P: nat > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 9.10/5.26        = ( ( P4
% 9.10/5.26           => ( P @ one_one_nat ) )
% 9.10/5.26          & ( ~ P4
% 9.10/5.26           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5573_split__of__bool,axiom,
% 9.10/5.26      ! [P: int > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 9.10/5.26        = ( ( P4
% 9.10/5.26           => ( P @ one_one_int ) )
% 9.10/5.26          & ( ~ P4
% 9.10/5.26           => ( P @ zero_zero_int ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5574_split__of__bool,axiom,
% 9.10/5.26      ! [P: code_integer > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 9.10/5.26        = ( ( P4
% 9.10/5.26           => ( P @ one_one_Code_integer ) )
% 9.10/5.26          & ( ~ P4
% 9.10/5.26           => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5575_split__of__bool__asm,axiom,
% 9.10/5.26      ! [P: complex > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 9.10/5.26        = ( ~ ( ( P4
% 9.10/5.26                & ~ ( P @ one_one_complex ) )
% 9.10/5.26              | ( ~ P4
% 9.10/5.26                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5576_split__of__bool__asm,axiom,
% 9.10/5.26      ! [P: real > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 9.10/5.26        = ( ~ ( ( P4
% 9.10/5.26                & ~ ( P @ one_one_real ) )
% 9.10/5.26              | ( ~ P4
% 9.10/5.26                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5577_split__of__bool__asm,axiom,
% 9.10/5.26      ! [P: rat > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 9.10/5.26        = ( ~ ( ( P4
% 9.10/5.26                & ~ ( P @ one_one_rat ) )
% 9.10/5.26              | ( ~ P4
% 9.10/5.26                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5578_split__of__bool__asm,axiom,
% 9.10/5.26      ! [P: nat > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 9.10/5.26        = ( ~ ( ( P4
% 9.10/5.26                & ~ ( P @ one_one_nat ) )
% 9.10/5.26              | ( ~ P4
% 9.10/5.26                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5579_split__of__bool__asm,axiom,
% 9.10/5.26      ! [P: int > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 9.10/5.26        = ( ~ ( ( P4
% 9.10/5.26                & ~ ( P @ one_one_int ) )
% 9.10/5.26              | ( ~ P4
% 9.10/5.26                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5580_split__of__bool__asm,axiom,
% 9.10/5.26      ! [P: code_integer > $o,P4: $o] :
% 9.10/5.26        ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 9.10/5.26        = ( ~ ( ( P4
% 9.10/5.26                & ~ ( P @ one_one_Code_integer ) )
% 9.10/5.26              | ( ~ P4
% 9.10/5.26                & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5581_not__numeral__le__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5582_not__numeral__le__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5583_not__numeral__le__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5584_not__numeral__le__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5585_neg__numeral__le__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5586_neg__numeral__le__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5587_neg__numeral__le__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5588_neg__numeral__le__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5589_zero__neq__neg__numeral,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( zero_zero_real
% 9.10/5.26       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5590_zero__neq__neg__numeral,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( zero_zero_int
% 9.10/5.26       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5591_zero__neq__neg__numeral,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( zero_zero_complex
% 9.10/5.26       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5592_zero__neq__neg__numeral,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( zero_z3403309356797280102nteger
% 9.10/5.26       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5593_zero__neq__neg__numeral,axiom,
% 9.10/5.26      ! [N: num] :
% 9.10/5.26        ( zero_zero_rat
% 9.10/5.26       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5594_not__numeral__less__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5595_not__numeral__less__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5596_not__numeral__less__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5597_not__numeral__less__neg__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] :
% 9.10/5.26        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5598_neg__numeral__less__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5599_neg__numeral__less__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5600_neg__numeral__less__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5601_neg__numeral__less__numeral,axiom,
% 9.10/5.26      ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5602_le__minus__one__simps_I2_J,axiom,
% 9.10/5.26      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5603_le__minus__one__simps_I2_J,axiom,
% 9.10/5.26      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5604_le__minus__one__simps_I2_J,axiom,
% 9.10/5.26      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5605_le__minus__one__simps_I2_J,axiom,
% 9.10/5.26      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5606_le__minus__one__simps_I4_J,axiom,
% 9.10/5.26      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5607_le__minus__one__simps_I4_J,axiom,
% 9.10/5.26      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5608_le__minus__one__simps_I4_J,axiom,
% 9.10/5.26      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5609_le__minus__one__simps_I4_J,axiom,
% 9.10/5.26      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5610_zero__neq__neg__one,axiom,
% 9.10/5.26      ( zero_zero_real
% 9.10/5.26     != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5611_zero__neq__neg__one,axiom,
% 9.10/5.26      ( zero_zero_int
% 9.10/5.26     != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5612_zero__neq__neg__one,axiom,
% 9.10/5.26      ( zero_zero_complex
% 9.10/5.26     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5613_zero__neq__neg__one,axiom,
% 9.10/5.26      ( zero_z3403309356797280102nteger
% 9.10/5.26     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5614_zero__neq__neg__one,axiom,
% 9.10/5.26      ( zero_zero_rat
% 9.10/5.26     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5615_neg__eq__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ( uminus_uminus_real @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( plus_plus_real @ A @ B2 )
% 9.10/5.26          = zero_zero_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5616_neg__eq__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ( uminus_uminus_int @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( plus_plus_int @ A @ B2 )
% 9.10/5.26          = zero_zero_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5617_neg__eq__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( ( uminus1482373934393186551omplex @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( plus_plus_complex @ A @ B2 )
% 9.10/5.26          = zero_zero_complex ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5618_neg__eq__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 9.10/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5619_neg__eq__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ( uminus_uminus_rat @ A )
% 9.10/5.26          = B2 )
% 9.10/5.26        = ( ( plus_plus_rat @ A @ B2 )
% 9.10/5.26          = zero_zero_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5620_eq__neg__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus_uminus_real @ B2 ) )
% 9.10/5.26        = ( ( plus_plus_real @ A @ B2 )
% 9.10/5.26          = zero_zero_real ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5621_eq__neg__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus_uminus_int @ B2 ) )
% 9.10/5.26        = ( ( plus_plus_int @ A @ B2 )
% 9.10/5.26          = zero_zero_int ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5622_eq__neg__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.26        = ( ( plus_plus_complex @ A @ B2 )
% 9.10/5.26          = zero_zero_complex ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5623_eq__neg__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.26        = ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 9.10/5.26          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5624_eq__neg__iff__add__eq__0,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( A
% 9.10/5.26          = ( uminus_uminus_rat @ B2 ) )
% 9.10/5.26        = ( ( plus_plus_rat @ A @ B2 )
% 9.10/5.26          = zero_zero_rat ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5625_add_Oinverse__unique,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ( plus_plus_real @ A @ B2 )
% 9.10/5.26          = zero_zero_real )
% 9.10/5.26       => ( ( uminus_uminus_real @ A )
% 9.10/5.26          = B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5626_add_Oinverse__unique,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ( plus_plus_int @ A @ B2 )
% 9.10/5.26          = zero_zero_int )
% 9.10/5.26       => ( ( uminus_uminus_int @ A )
% 9.10/5.26          = B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5627_add_Oinverse__unique,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( ( plus_plus_complex @ A @ B2 )
% 9.10/5.26          = zero_zero_complex )
% 9.10/5.26       => ( ( uminus1482373934393186551omplex @ A )
% 9.10/5.26          = B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5628_add_Oinverse__unique,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 9.10/5.26          = zero_z3403309356797280102nteger )
% 9.10/5.26       => ( ( uminus1351360451143612070nteger @ A )
% 9.10/5.26          = B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5629_add_Oinverse__unique,axiom,
% 9.10/5.26      ! [A: rat,B2: rat] :
% 9.10/5.26        ( ( ( plus_plus_rat @ A @ B2 )
% 9.10/5.26          = zero_zero_rat )
% 9.10/5.26       => ( ( uminus_uminus_rat @ A )
% 9.10/5.26          = B2 ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5630_ab__group__add__class_Oab__left__minus,axiom,
% 9.10/5.26      ! [A: real] :
% 9.10/5.26        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.10/5.26        = zero_zero_real ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5631_ab__group__add__class_Oab__left__minus,axiom,
% 9.10/5.26      ! [A: int] :
% 9.10/5.26        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.10/5.26        = zero_zero_int ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5632_ab__group__add__class_Oab__left__minus,axiom,
% 9.10/5.26      ! [A: complex] :
% 9.10/5.26        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.10/5.26        = zero_zero_complex ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5633_ab__group__add__class_Oab__left__minus,axiom,
% 9.10/5.26      ! [A: code_integer] :
% 9.10/5.26        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.10/5.26        = zero_z3403309356797280102nteger ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5634_ab__group__add__class_Oab__left__minus,axiom,
% 9.10/5.26      ! [A: rat] :
% 9.10/5.26        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.10/5.26        = zero_zero_rat ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5635_add__eq__0__iff,axiom,
% 9.10/5.26      ! [A: real,B2: real] :
% 9.10/5.26        ( ( ( plus_plus_real @ A @ B2 )
% 9.10/5.26          = zero_zero_real )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5636_add__eq__0__iff,axiom,
% 9.10/5.26      ! [A: int,B2: int] :
% 9.10/5.26        ( ( ( plus_plus_int @ A @ B2 )
% 9.10/5.26          = zero_zero_int )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5637_add__eq__0__iff,axiom,
% 9.10/5.26      ! [A: complex,B2: complex] :
% 9.10/5.26        ( ( ( plus_plus_complex @ A @ B2 )
% 9.10/5.26          = zero_zero_complex )
% 9.10/5.26        = ( B2
% 9.10/5.26          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.10/5.26  
% 9.10/5.26  thf(fact_5638_add__eq__0__iff,axiom,
% 9.10/5.26      ! [A: code_integer,B2: code_integer] :
% 9.10/5.26        ( ( ( plus_p5714425477246183910nteger @ A @ B2 )
% 9.10/5.26          = zero_z3403309356797280102nteger )
% 9.10/5.27        = ( B2
% 9.10/5.27          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5639_add__eq__0__iff,axiom,
% 9.10/5.27      ! [A: rat,B2: rat] :
% 9.10/5.27        ( ( ( plus_plus_rat @ A @ B2 )
% 9.10/5.27          = zero_zero_rat )
% 9.10/5.27        = ( B2
% 9.10/5.27          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5640_less__minus__one__simps_I2_J,axiom,
% 9.10/5.27      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5641_less__minus__one__simps_I2_J,axiom,
% 9.10/5.27      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5642_less__minus__one__simps_I2_J,axiom,
% 9.10/5.27      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5643_less__minus__one__simps_I2_J,axiom,
% 9.10/5.27      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5644_less__minus__one__simps_I4_J,axiom,
% 9.10/5.27      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5645_less__minus__one__simps_I4_J,axiom,
% 9.10/5.27      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5646_less__minus__one__simps_I4_J,axiom,
% 9.10/5.27      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5647_less__minus__one__simps_I4_J,axiom,
% 9.10/5.27      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5648_numeral__times__minus__swap,axiom,
% 9.10/5.27      ! [W: num,X3: real] :
% 9.10/5.27        ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.27        = ( times_times_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5649_numeral__times__minus__swap,axiom,
% 9.10/5.27      ! [W: num,X3: int] :
% 9.10/5.27        ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X3 ) )
% 9.10/5.27        = ( times_times_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5650_numeral__times__minus__swap,axiom,
% 9.10/5.27      ! [W: num,X3: complex] :
% 9.10/5.27        ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X3 ) )
% 9.10/5.27        = ( times_times_complex @ X3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5651_numeral__times__minus__swap,axiom,
% 9.10/5.27      ! [W: num,X3: code_integer] :
% 9.10/5.27        ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X3 ) )
% 9.10/5.27        = ( times_3573771949741848930nteger @ X3 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5652_numeral__times__minus__swap,axiom,
% 9.10/5.27      ! [W: num,X3: rat] :
% 9.10/5.27        ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X3 ) )
% 9.10/5.27        = ( times_times_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5653_nonzero__minus__divide__right,axiom,
% 9.10/5.27      ! [B2: real,A: real] :
% 9.10/5.27        ( ( B2 != zero_zero_real )
% 9.10/5.27       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.27          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5654_nonzero__minus__divide__right,axiom,
% 9.10/5.27      ! [B2: complex,A: complex] :
% 9.10/5.27        ( ( B2 != zero_zero_complex )
% 9.10/5.27       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.27          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5655_nonzero__minus__divide__right,axiom,
% 9.10/5.27      ! [B2: rat,A: rat] :
% 9.10/5.27        ( ( B2 != zero_zero_rat )
% 9.10/5.27       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.27          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5656_nonzero__minus__divide__divide,axiom,
% 9.10/5.27      ! [B2: real,A: real] :
% 9.10/5.27        ( ( B2 != zero_zero_real )
% 9.10/5.27       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.27          = ( divide_divide_real @ A @ B2 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5657_nonzero__minus__divide__divide,axiom,
% 9.10/5.27      ! [B2: complex,A: complex] :
% 9.10/5.27        ( ( B2 != zero_zero_complex )
% 9.10/5.27       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.27          = ( divide1717551699836669952omplex @ A @ B2 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5658_nonzero__minus__divide__divide,axiom,
% 9.10/5.27      ! [B2: rat,A: rat] :
% 9.10/5.27        ( ( B2 != zero_zero_rat )
% 9.10/5.27       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.27          = ( divide_divide_rat @ A @ B2 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5659_one__neq__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( one_one_real
% 9.10/5.27       != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5660_one__neq__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( one_one_int
% 9.10/5.27       != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5661_one__neq__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( one_one_complex
% 9.10/5.27       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5662_one__neq__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( one_one_Code_integer
% 9.10/5.27       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5663_one__neq__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( one_one_rat
% 9.10/5.27       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5664_numeral__neq__neg__one,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( numeral_numeral_real @ N )
% 9.10/5.27       != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5665_numeral__neq__neg__one,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( numeral_numeral_int @ N )
% 9.10/5.27       != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5666_numeral__neq__neg__one,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( numera6690914467698888265omplex @ N )
% 9.10/5.27       != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5667_numeral__neq__neg__one,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( numera6620942414471956472nteger @ N )
% 9.10/5.27       != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5668_numeral__neq__neg__one,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( numeral_numeral_rat @ N )
% 9.10/5.27       != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5669_square__eq__1__iff,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ( times_times_real @ X3 @ X3 )
% 9.10/5.27          = one_one_real )
% 9.10/5.27        = ( ( X3 = one_one_real )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5670_square__eq__1__iff,axiom,
% 9.10/5.27      ! [X3: int] :
% 9.10/5.27        ( ( ( times_times_int @ X3 @ X3 )
% 9.10/5.27          = one_one_int )
% 9.10/5.27        = ( ( X3 = one_one_int )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5671_square__eq__1__iff,axiom,
% 9.10/5.27      ! [X3: complex] :
% 9.10/5.27        ( ( ( times_times_complex @ X3 @ X3 )
% 9.10/5.27          = one_one_complex )
% 9.10/5.27        = ( ( X3 = one_one_complex )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5672_square__eq__1__iff,axiom,
% 9.10/5.27      ! [X3: code_integer] :
% 9.10/5.27        ( ( ( times_3573771949741848930nteger @ X3 @ X3 )
% 9.10/5.27          = one_one_Code_integer )
% 9.10/5.27        = ( ( X3 = one_one_Code_integer )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5673_square__eq__1__iff,axiom,
% 9.10/5.27      ! [X3: rat] :
% 9.10/5.27        ( ( ( times_times_rat @ X3 @ X3 )
% 9.10/5.27          = one_one_rat )
% 9.10/5.27        = ( ( X3 = one_one_rat )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5674_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_real
% 9.10/5.27      = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5675_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_int
% 9.10/5.27      = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5676_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_complex
% 9.10/5.27      = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5677_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_8373710615458151222nteger
% 9.10/5.27      = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5678_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_rat
% 9.10/5.27      = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5679_diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_real
% 9.10/5.27      = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5680_diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_int
% 9.10/5.27      = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5681_diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_complex
% 9.10/5.27      = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5682_diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_8373710615458151222nteger
% 9.10/5.27      = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5683_diff__conv__add__uminus,axiom,
% 9.10/5.27      ( minus_minus_rat
% 9.10/5.27      = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5684_group__cancel_Osub2,axiom,
% 9.10/5.27      ! [B5: real,K: real,B2: real,A: real] :
% 9.10/5.27        ( ( B5
% 9.10/5.27          = ( plus_plus_real @ K @ B2 ) )
% 9.10/5.27       => ( ( minus_minus_real @ A @ B5 )
% 9.10/5.27          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5685_group__cancel_Osub2,axiom,
% 9.10/5.27      ! [B5: int,K: int,B2: int,A: int] :
% 9.10/5.27        ( ( B5
% 9.10/5.27          = ( plus_plus_int @ K @ B2 ) )
% 9.10/5.27       => ( ( minus_minus_int @ A @ B5 )
% 9.10/5.27          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5686_group__cancel_Osub2,axiom,
% 9.10/5.27      ! [B5: complex,K: complex,B2: complex,A: complex] :
% 9.10/5.27        ( ( B5
% 9.10/5.27          = ( plus_plus_complex @ K @ B2 ) )
% 9.10/5.27       => ( ( minus_minus_complex @ A @ B5 )
% 9.10/5.27          = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5687_group__cancel_Osub2,axiom,
% 9.10/5.27      ! [B5: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
% 9.10/5.27        ( ( B5
% 9.10/5.27          = ( plus_p5714425477246183910nteger @ K @ B2 ) )
% 9.10/5.27       => ( ( minus_8373710615458151222nteger @ A @ B5 )
% 9.10/5.27          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5688_group__cancel_Osub2,axiom,
% 9.10/5.27      ! [B5: rat,K: rat,B2: rat,A: rat] :
% 9.10/5.27        ( ( B5
% 9.10/5.27          = ( plus_plus_rat @ K @ B2 ) )
% 9.10/5.27       => ( ( minus_minus_rat @ A @ B5 )
% 9.10/5.27          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5689_replicate__length__same,axiom,
% 9.10/5.27      ! [Xs2: list_int,X3: int] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
% 9.10/5.27           => ( X5 = X3 ) )
% 9.10/5.27       => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X3 )
% 9.10/5.27          = Xs2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5690_replicate__length__same,axiom,
% 9.10/5.27      ! [Xs2: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.10/5.27        ( ! [X5: vEBT_VEBT] :
% 9.10/5.27            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.10/5.27           => ( X5 = X3 ) )
% 9.10/5.27       => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X3 )
% 9.10/5.27          = Xs2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5691_replicate__length__same,axiom,
% 9.10/5.27      ! [Xs2: list_o,X3: $o] :
% 9.10/5.27        ( ! [X5: $o] :
% 9.10/5.27            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
% 9.10/5.27           => ( X5 = X3 ) )
% 9.10/5.27       => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X3 )
% 9.10/5.27          = Xs2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5692_replicate__length__same,axiom,
% 9.10/5.27      ! [Xs2: list_nat,X3: nat] :
% 9.10/5.27        ( ! [X5: nat] :
% 9.10/5.27            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
% 9.10/5.27           => ( X5 = X3 ) )
% 9.10/5.27       => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X3 )
% 9.10/5.27          = Xs2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5693_replicate__eqI,axiom,
% 9.10/5.27      ! [Xs2: list_real,N: nat,X3: real] :
% 9.10/5.27        ( ( ( size_size_list_real @ Xs2 )
% 9.10/5.27          = N )
% 9.10/5.27       => ( ! [Y4: real] :
% 9.10/5.27              ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
% 9.10/5.27             => ( Y4 = X3 ) )
% 9.10/5.27         => ( Xs2
% 9.10/5.27            = ( replicate_real @ N @ X3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5694_replicate__eqI,axiom,
% 9.10/5.27      ! [Xs2: list_complex,N: nat,X3: complex] :
% 9.10/5.27        ( ( ( size_s3451745648224563538omplex @ Xs2 )
% 9.10/5.27          = N )
% 9.10/5.27       => ( ! [Y4: complex] :
% 9.10/5.27              ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
% 9.10/5.27             => ( Y4 = X3 ) )
% 9.10/5.27         => ( Xs2
% 9.10/5.27            = ( replicate_complex @ N @ X3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5695_replicate__eqI,axiom,
% 9.10/5.27      ! [Xs2: list_int,N: nat,X3: int] :
% 9.10/5.27        ( ( ( size_size_list_int @ Xs2 )
% 9.10/5.27          = N )
% 9.10/5.27       => ( ! [Y4: int] :
% 9.10/5.27              ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
% 9.10/5.27             => ( Y4 = X3 ) )
% 9.10/5.27         => ( Xs2
% 9.10/5.27            = ( replicate_int @ N @ X3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5696_replicate__eqI,axiom,
% 9.10/5.27      ! [Xs2: list_VEBT_VEBT,N: nat,X3: vEBT_VEBT] :
% 9.10/5.27        ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.10/5.27          = N )
% 9.10/5.27       => ( ! [Y4: vEBT_VEBT] :
% 9.10/5.27              ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 9.10/5.27             => ( Y4 = X3 ) )
% 9.10/5.27         => ( Xs2
% 9.10/5.27            = ( replicate_VEBT_VEBT @ N @ X3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5697_replicate__eqI,axiom,
% 9.10/5.27      ! [Xs2: list_o,N: nat,X3: $o] :
% 9.10/5.27        ( ( ( size_size_list_o @ Xs2 )
% 9.10/5.27          = N )
% 9.10/5.27       => ( ! [Y4: $o] :
% 9.10/5.27              ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
% 9.10/5.27             => ( Y4 = X3 ) )
% 9.10/5.27         => ( Xs2
% 9.10/5.27            = ( replicate_o @ N @ X3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5698_replicate__eqI,axiom,
% 9.10/5.27      ! [Xs2: list_nat,N: nat,X3: nat] :
% 9.10/5.27        ( ( ( size_size_list_nat @ Xs2 )
% 9.10/5.27          = N )
% 9.10/5.27       => ( ! [Y4: nat] :
% 9.10/5.27              ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
% 9.10/5.27             => ( Y4 = X3 ) )
% 9.10/5.27         => ( Xs2
% 9.10/5.27            = ( replicate_nat @ N @ X3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5699_dvd__neg__div,axiom,
% 9.10/5.27      ! [B2: real,A: real] :
% 9.10/5.27        ( ( dvd_dvd_real @ B2 @ A )
% 9.10/5.27       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.27          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5700_dvd__neg__div,axiom,
% 9.10/5.27      ! [B2: int,A: int] :
% 9.10/5.27        ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.27       => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.27          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5701_dvd__neg__div,axiom,
% 9.10/5.27      ! [B2: complex,A: complex] :
% 9.10/5.27        ( ( dvd_dvd_complex @ B2 @ A )
% 9.10/5.27       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B2 )
% 9.10/5.27          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5702_dvd__neg__div,axiom,
% 9.10/5.27      ! [B2: code_integer,A: code_integer] :
% 9.10/5.27        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.27       => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.27          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5703_dvd__neg__div,axiom,
% 9.10/5.27      ! [B2: rat,A: rat] :
% 9.10/5.27        ( ( dvd_dvd_rat @ B2 @ A )
% 9.10/5.27       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.27          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5704_dvd__div__neg,axiom,
% 9.10/5.27      ! [B2: real,A: real] :
% 9.10/5.27        ( ( dvd_dvd_real @ B2 @ A )
% 9.10/5.27       => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B2 ) )
% 9.10/5.27          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5705_dvd__div__neg,axiom,
% 9.10/5.27      ! [B2: int,A: int] :
% 9.10/5.27        ( ( dvd_dvd_int @ B2 @ A )
% 9.10/5.27       => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B2 ) )
% 9.10/5.27          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5706_dvd__div__neg,axiom,
% 9.10/5.27      ! [B2: complex,A: complex] :
% 9.10/5.27        ( ( dvd_dvd_complex @ B2 @ A )
% 9.10/5.27       => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B2 ) )
% 9.10/5.27          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5707_dvd__div__neg,axiom,
% 9.10/5.27      ! [B2: code_integer,A: code_integer] :
% 9.10/5.27        ( ( dvd_dvd_Code_integer @ B2 @ A )
% 9.10/5.27       => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B2 ) )
% 9.10/5.27          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5708_dvd__div__neg,axiom,
% 9.10/5.27      ! [B2: rat,A: rat] :
% 9.10/5.27        ( ( dvd_dvd_rat @ B2 @ A )
% 9.10/5.27       => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B2 ) )
% 9.10/5.27          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5709_subset__Compl__self__eq,axiom,
% 9.10/5.27      ! [A4: set_nat] :
% 9.10/5.27        ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
% 9.10/5.27        = ( A4 = bot_bot_set_nat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5710_subset__Compl__self__eq,axiom,
% 9.10/5.27      ! [A4: set_real] :
% 9.10/5.27        ( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ A4 ) )
% 9.10/5.27        = ( A4 = bot_bot_set_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5711_subset__Compl__self__eq,axiom,
% 9.10/5.27      ! [A4: set_int] :
% 9.10/5.27        ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
% 9.10/5.27        = ( A4 = bot_bot_set_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5712_real__minus__mult__self__le,axiom,
% 9.10/5.27      ! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X3 @ X3 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5713_pos__zmult__eq__1__iff__lemma,axiom,
% 9.10/5.27      ! [M: int,N: int] :
% 9.10/5.27        ( ( ( times_times_int @ M @ N )
% 9.10/5.27          = one_one_int )
% 9.10/5.27       => ( ( M = one_one_int )
% 9.10/5.27          | ( M
% 9.10/5.27            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5714_zmult__eq__1__iff,axiom,
% 9.10/5.27      ! [M: int,N: int] :
% 9.10/5.27        ( ( ( times_times_int @ M @ N )
% 9.10/5.27          = one_one_int )
% 9.10/5.27        = ( ( ( M = one_one_int )
% 9.10/5.27            & ( N = one_one_int ) )
% 9.10/5.27          | ( ( M
% 9.10/5.27              = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.27            & ( N
% 9.10/5.27              = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5715_minus__real__def,axiom,
% 9.10/5.27      ( minus_minus_real
% 9.10/5.27      = ( ^ [X: real,Y: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5716_not__zero__le__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5717_not__zero__le__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5718_not__zero__le__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5719_not__zero__le__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5720_neg__numeral__le__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5721_neg__numeral__le__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5722_neg__numeral__le__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5723_neg__numeral__le__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5724_not__zero__less__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5725_not__zero__less__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5726_not__zero__less__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5727_not__zero__less__neg__numeral,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5728_neg__numeral__less__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5729_neg__numeral__less__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5730_neg__numeral__less__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5731_neg__numeral__less__zero,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5732_le__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5733_le__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5734_le__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5735_le__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5736_le__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5737_le__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5738_le__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5739_le__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5740_less__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5741_less__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5742_less__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5743_less__minus__one__simps_I3_J,axiom,
% 9.10/5.27      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5744_less__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5745_less__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5746_less__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5747_less__minus__one__simps_I1_J,axiom,
% 9.10/5.27      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5748_not__one__le__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5749_not__one__le__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5750_not__one__le__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5751_not__one__le__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5752_not__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5753_not__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5754_not__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5755_not__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5756_neg__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5757_neg__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5758_neg__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5759_neg__numeral__le__neg__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5760_neg__one__le__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5761_neg__one__le__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5762_neg__one__le__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5763_neg__one__le__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5764_neg__numeral__le__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5765_neg__numeral__le__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5766_neg__numeral__le__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5767_neg__numeral__le__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5768_not__neg__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5769_not__neg__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5770_not__neg__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5771_not__neg__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5772_not__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5773_not__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5774_not__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5775_not__one__less__neg__numeral,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5776_not__numeral__less__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5777_not__numeral__less__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5778_not__numeral__less__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5779_not__numeral__less__neg__one,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5780_neg__one__less__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5781_neg__one__less__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5782_neg__one__less__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5783_neg__one__less__numeral,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5784_neg__numeral__less__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5785_neg__numeral__less__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5786_neg__numeral__less__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5787_neg__numeral__less__one,axiom,
% 9.10/5.27      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5788_eq__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: real,B2: real,C: real] :
% 9.10/5.27        ( ( A
% 9.10/5.27          = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( C != zero_zero_real )
% 9.10/5.27           => ( ( times_times_real @ A @ C )
% 9.10/5.27              = ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27          & ( ( C = zero_zero_real )
% 9.10/5.27           => ( A = zero_zero_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5789_eq__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: complex,B2: complex,C: complex] :
% 9.10/5.27        ( ( A
% 9.10/5.27          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( C != zero_zero_complex )
% 9.10/5.27           => ( ( times_times_complex @ A @ C )
% 9.10/5.27              = ( uminus1482373934393186551omplex @ B2 ) ) )
% 9.10/5.27          & ( ( C = zero_zero_complex )
% 9.10/5.27           => ( A = zero_zero_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5790_eq__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.27        ( ( A
% 9.10/5.27          = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( C != zero_zero_rat )
% 9.10/5.27           => ( ( times_times_rat @ A @ C )
% 9.10/5.27              = ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27          & ( ( C = zero_zero_rat )
% 9.10/5.27           => ( A = zero_zero_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5791_minus__divide__eq__eq,axiom,
% 9.10/5.27      ! [B2: real,C: real,A: real] :
% 9.10/5.27        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.27          = A )
% 9.10/5.27        = ( ( ( C != zero_zero_real )
% 9.10/5.27           => ( ( uminus_uminus_real @ B2 )
% 9.10/5.27              = ( times_times_real @ A @ C ) ) )
% 9.10/5.27          & ( ( C = zero_zero_real )
% 9.10/5.27           => ( A = zero_zero_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5792_minus__divide__eq__eq,axiom,
% 9.10/5.27      ! [B2: complex,C: complex,A: complex] :
% 9.10/5.27        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.10/5.27          = A )
% 9.10/5.27        = ( ( ( C != zero_zero_complex )
% 9.10/5.27           => ( ( uminus1482373934393186551omplex @ B2 )
% 9.10/5.27              = ( times_times_complex @ A @ C ) ) )
% 9.10/5.27          & ( ( C = zero_zero_complex )
% 9.10/5.27           => ( A = zero_zero_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5793_minus__divide__eq__eq,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.27        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.27          = A )
% 9.10/5.27        = ( ( ( C != zero_zero_rat )
% 9.10/5.27           => ( ( uminus_uminus_rat @ B2 )
% 9.10/5.27              = ( times_times_rat @ A @ C ) ) )
% 9.10/5.27          & ( ( C = zero_zero_rat )
% 9.10/5.27           => ( A = zero_zero_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5794_nonzero__neg__divide__eq__eq,axiom,
% 9.10/5.27      ! [B2: real,A: real,C: real] :
% 9.10/5.27        ( ( B2 != zero_zero_real )
% 9.10/5.27       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.27            = C )
% 9.10/5.27          = ( ( uminus_uminus_real @ A )
% 9.10/5.27            = ( times_times_real @ C @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5795_nonzero__neg__divide__eq__eq,axiom,
% 9.10/5.27      ! [B2: complex,A: complex,C: complex] :
% 9.10/5.27        ( ( B2 != zero_zero_complex )
% 9.10/5.27       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.27            = C )
% 9.10/5.27          = ( ( uminus1482373934393186551omplex @ A )
% 9.10/5.27            = ( times_times_complex @ C @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5796_nonzero__neg__divide__eq__eq,axiom,
% 9.10/5.27      ! [B2: rat,A: rat,C: rat] :
% 9.10/5.27        ( ( B2 != zero_zero_rat )
% 9.10/5.27       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.27            = C )
% 9.10/5.27          = ( ( uminus_uminus_rat @ A )
% 9.10/5.27            = ( times_times_rat @ C @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5797_nonzero__neg__divide__eq__eq2,axiom,
% 9.10/5.27      ! [B2: real,C: real,A: real] :
% 9.10/5.27        ( ( B2 != zero_zero_real )
% 9.10/5.27       => ( ( C
% 9.10/5.27            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B2 ) ) )
% 9.10/5.27          = ( ( times_times_real @ C @ B2 )
% 9.10/5.27            = ( uminus_uminus_real @ A ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5798_nonzero__neg__divide__eq__eq2,axiom,
% 9.10/5.27      ! [B2: complex,C: complex,A: complex] :
% 9.10/5.27        ( ( B2 != zero_zero_complex )
% 9.10/5.27       => ( ( C
% 9.10/5.27            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.27          = ( ( times_times_complex @ C @ B2 )
% 9.10/5.27            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5799_nonzero__neg__divide__eq__eq2,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.27        ( ( B2 != zero_zero_rat )
% 9.10/5.27       => ( ( C
% 9.10/5.27            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B2 ) ) )
% 9.10/5.27          = ( ( times_times_rat @ C @ B2 )
% 9.10/5.27            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5800_mult__1s__ring__1_I2_J,axiom,
% 9.10/5.27      ! [B2: real] :
% 9.10/5.27        ( ( times_times_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 9.10/5.27        = ( uminus_uminus_real @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5801_mult__1s__ring__1_I2_J,axiom,
% 9.10/5.27      ! [B2: int] :
% 9.10/5.27        ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 9.10/5.27        = ( uminus_uminus_int @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5802_mult__1s__ring__1_I2_J,axiom,
% 9.10/5.27      ! [B2: complex] :
% 9.10/5.27        ( ( times_times_complex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 9.10/5.27        = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5803_mult__1s__ring__1_I2_J,axiom,
% 9.10/5.27      ! [B2: code_integer] :
% 9.10/5.27        ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 9.10/5.27        = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5804_mult__1s__ring__1_I2_J,axiom,
% 9.10/5.27      ! [B2: rat] :
% 9.10/5.27        ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 9.10/5.27        = ( uminus_uminus_rat @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5805_mult__1s__ring__1_I1_J,axiom,
% 9.10/5.27      ! [B2: real] :
% 9.10/5.27        ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B2 )
% 9.10/5.27        = ( uminus_uminus_real @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5806_mult__1s__ring__1_I1_J,axiom,
% 9.10/5.27      ! [B2: int] :
% 9.10/5.27        ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
% 9.10/5.27        = ( uminus_uminus_int @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5807_mult__1s__ring__1_I1_J,axiom,
% 9.10/5.27      ! [B2: complex] :
% 9.10/5.27        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B2 )
% 9.10/5.27        = ( uminus1482373934393186551omplex @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5808_mult__1s__ring__1_I1_J,axiom,
% 9.10/5.27      ! [B2: code_integer] :
% 9.10/5.27        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
% 9.10/5.27        = ( uminus1351360451143612070nteger @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5809_mult__1s__ring__1_I1_J,axiom,
% 9.10/5.27      ! [B2: rat] :
% 9.10/5.27        ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
% 9.10/5.27        = ( uminus_uminus_rat @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5810_divide__eq__minus__1__iff,axiom,
% 9.10/5.27      ! [A: real,B2: real] :
% 9.10/5.27        ( ( ( divide_divide_real @ A @ B2 )
% 9.10/5.27          = ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.27        = ( ( B2 != zero_zero_real )
% 9.10/5.27          & ( A
% 9.10/5.27            = ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5811_divide__eq__minus__1__iff,axiom,
% 9.10/5.27      ! [A: complex,B2: complex] :
% 9.10/5.27        ( ( ( divide1717551699836669952omplex @ A @ B2 )
% 9.10/5.27          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.27        = ( ( B2 != zero_zero_complex )
% 9.10/5.27          & ( A
% 9.10/5.27            = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5812_divide__eq__minus__1__iff,axiom,
% 9.10/5.27      ! [A: rat,B2: rat] :
% 9.10/5.27        ( ( ( divide_divide_rat @ A @ B2 )
% 9.10/5.27          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.27        = ( ( B2 != zero_zero_rat )
% 9.10/5.27          & ( A
% 9.10/5.27            = ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5813_uminus__numeral__One,axiom,
% 9.10/5.27      ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 9.10/5.27      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5814_uminus__numeral__One,axiom,
% 9.10/5.27      ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 9.10/5.27      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5815_uminus__numeral__One,axiom,
% 9.10/5.27      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 9.10/5.27      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5816_uminus__numeral__One,axiom,
% 9.10/5.27      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 9.10/5.27      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5817_uminus__numeral__One,axiom,
% 9.10/5.27      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 9.10/5.27      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5818_power__minus,axiom,
% 9.10/5.27      ! [A: real,N: nat] :
% 9.10/5.27        ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.10/5.27        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5819_power__minus,axiom,
% 9.10/5.27      ! [A: int,N: nat] :
% 9.10/5.27        ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.10/5.27        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5820_power__minus,axiom,
% 9.10/5.27      ! [A: complex,N: nat] :
% 9.10/5.27        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.10/5.27        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5821_power__minus,axiom,
% 9.10/5.27      ! [A: code_integer,N: nat] :
% 9.10/5.27        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.10/5.27        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5822_power__minus,axiom,
% 9.10/5.27      ! [A: rat,N: nat] :
% 9.10/5.27        ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.10/5.27        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5823_power__minus__Bit0,axiom,
% 9.10/5.27      ! [X3: real,K: num] :
% 9.10/5.27        ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.10/5.27        = ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5824_power__minus__Bit0,axiom,
% 9.10/5.27      ! [X3: int,K: num] :
% 9.10/5.27        ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.10/5.27        = ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5825_power__minus__Bit0,axiom,
% 9.10/5.27      ! [X3: complex,K: num] :
% 9.10/5.27        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.10/5.27        = ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5826_power__minus__Bit0,axiom,
% 9.10/5.27      ! [X3: code_integer,K: num] :
% 9.10/5.27        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.10/5.27        = ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5827_power__minus__Bit0,axiom,
% 9.10/5.27      ! [X3: rat,K: num] :
% 9.10/5.27        ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.10/5.27        = ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5828_real__add__less__0__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 9.10/5.27        = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5829_real__0__less__add__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.27        = ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5830_real__add__le__0__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 9.10/5.27        = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5831_real__0__le__add__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.27        = ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5832_pos__minus__divide__less__eq,axiom,
% 9.10/5.27      ! [C: real,B2: real,A: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5833_pos__minus__divide__less__eq,axiom,
% 9.10/5.27      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5834_pos__less__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: real,A: real,B2: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5835_pos__less__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5836_neg__minus__divide__less__eq,axiom,
% 9.10/5.27      ! [C: real,B2: real,A: real] :
% 9.10/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5837_neg__minus__divide__less__eq,axiom,
% 9.10/5.27      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5838_neg__less__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: real,A: real,B2: real] :
% 9.10/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5839_neg__less__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5840_minus__divide__less__eq,axiom,
% 9.10/5.27      ! [B2: real,C: real,A: real] :
% 9.10/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5841_minus__divide__less__eq,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5842_less__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: real,B2: real,C: real] :
% 9.10/5.27        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5843_less__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.27        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5844_divide__eq__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: real,C: real,W: num] :
% 9.10/5.27        ( ( ( divide_divide_real @ B2 @ C )
% 9.10/5.27          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.27        = ( ( ( C != zero_zero_real )
% 9.10/5.27           => ( B2
% 9.10/5.27              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ( C = zero_zero_real )
% 9.10/5.27           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.27              = zero_zero_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5845_divide__eq__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: complex,C: complex,W: num] :
% 9.10/5.27        ( ( ( divide1717551699836669952omplex @ B2 @ C )
% 9.10/5.27          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.27        = ( ( ( C != zero_zero_complex )
% 9.10/5.27           => ( B2
% 9.10/5.27              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ( C = zero_zero_complex )
% 9.10/5.27           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.27              = zero_zero_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5846_divide__eq__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,W: num] :
% 9.10/5.27        ( ( ( divide_divide_rat @ B2 @ C )
% 9.10/5.27          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.10/5.27        = ( ( ( C != zero_zero_rat )
% 9.10/5.27           => ( B2
% 9.10/5.27              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ( C = zero_zero_rat )
% 9.10/5.27           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.27              = zero_zero_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5847_eq__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: real,C: real] :
% 9.10/5.27        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.27          = ( divide_divide_real @ B2 @ C ) )
% 9.10/5.27        = ( ( ( C != zero_zero_real )
% 9.10/5.27           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 9.10/5.27              = B2 ) )
% 9.10/5.27          & ( ( C = zero_zero_real )
% 9.10/5.27           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.27              = zero_zero_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5848_eq__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: complex,C: complex] :
% 9.10/5.27        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.27          = ( divide1717551699836669952omplex @ B2 @ C ) )
% 9.10/5.27        = ( ( ( C != zero_zero_complex )
% 9.10/5.27           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 9.10/5.27              = B2 ) )
% 9.10/5.27          & ( ( C = zero_zero_complex )
% 9.10/5.27           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.27              = zero_zero_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5849_eq__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: rat,C: rat] :
% 9.10/5.27        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.27          = ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.27        = ( ( ( C != zero_zero_rat )
% 9.10/5.27           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 9.10/5.27              = B2 ) )
% 9.10/5.27          & ( ( C = zero_zero_rat )
% 9.10/5.27           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 9.10/5.27              = zero_zero_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5850_subset__decode__imp__le,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 9.10/5.27       => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5851_add__divide__eq__if__simps_I3_J,axiom,
% 9.10/5.27      ! [Z: real,A: real,B2: real] :
% 9.10/5.27        ( ( ( Z = zero_zero_real )
% 9.10/5.27         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 9.10/5.27            = B2 ) )
% 9.10/5.27        & ( ( Z != zero_zero_real )
% 9.10/5.27         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5852_add__divide__eq__if__simps_I3_J,axiom,
% 9.10/5.27      ! [Z: complex,A: complex,B2: complex] :
% 9.10/5.27        ( ( ( Z = zero_zero_complex )
% 9.10/5.27         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 9.10/5.27            = B2 ) )
% 9.10/5.27        & ( ( Z != zero_zero_complex )
% 9.10/5.27         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5853_add__divide__eq__if__simps_I3_J,axiom,
% 9.10/5.27      ! [Z: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ( Z = zero_zero_rat )
% 9.10/5.27         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 9.10/5.27            = B2 ) )
% 9.10/5.27        & ( ( Z != zero_zero_rat )
% 9.10/5.27         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5854_minus__divide__add__eq__iff,axiom,
% 9.10/5.27      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.27        ( ( Z != zero_zero_real )
% 9.10/5.27       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z ) ) @ Y3 )
% 9.10/5.27          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5855_minus__divide__add__eq__iff,axiom,
% 9.10/5.27      ! [Z: complex,X3: complex,Y3: complex] :
% 9.10/5.27        ( ( Z != zero_zero_complex )
% 9.10/5.27       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z ) ) @ Y3 )
% 9.10/5.27          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5856_minus__divide__add__eq__iff,axiom,
% 9.10/5.27      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.27        ( ( Z != zero_zero_rat )
% 9.10/5.27       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z ) ) @ Y3 )
% 9.10/5.27          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5857_add__divide__eq__if__simps_I6_J,axiom,
% 9.10/5.27      ! [Z: real,A: real,B2: real] :
% 9.10/5.27        ( ( ( Z = zero_zero_real )
% 9.10/5.27         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27        & ( ( Z != zero_zero_real )
% 9.10/5.27         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5858_add__divide__eq__if__simps_I6_J,axiom,
% 9.10/5.27      ! [Z: complex,A: complex,B2: complex] :
% 9.10/5.27        ( ( ( Z = zero_zero_complex )
% 9.10/5.27         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( uminus1482373934393186551omplex @ B2 ) ) )
% 9.10/5.27        & ( ( Z != zero_zero_complex )
% 9.10/5.27         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5859_add__divide__eq__if__simps_I6_J,axiom,
% 9.10/5.27      ! [Z: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ( Z = zero_zero_rat )
% 9.10/5.27         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27        & ( ( Z != zero_zero_rat )
% 9.10/5.27         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B2 )
% 9.10/5.27            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5860_add__divide__eq__if__simps_I5_J,axiom,
% 9.10/5.27      ! [Z: real,A: real,B2: real] :
% 9.10/5.27        ( ( ( Z = zero_zero_real )
% 9.10/5.27         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 9.10/5.27            = ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27        & ( ( Z != zero_zero_real )
% 9.10/5.27         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B2 )
% 9.10/5.27            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5861_add__divide__eq__if__simps_I5_J,axiom,
% 9.10/5.27      ! [Z: complex,A: complex,B2: complex] :
% 9.10/5.27        ( ( ( Z = zero_zero_complex )
% 9.10/5.27         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 9.10/5.27            = ( uminus1482373934393186551omplex @ B2 ) ) )
% 9.10/5.27        & ( ( Z != zero_zero_complex )
% 9.10/5.27         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B2 )
% 9.10/5.27            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5862_add__divide__eq__if__simps_I5_J,axiom,
% 9.10/5.27      ! [Z: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ( Z = zero_zero_rat )
% 9.10/5.27         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 9.10/5.27            = ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27        & ( ( Z != zero_zero_rat )
% 9.10/5.27         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B2 )
% 9.10/5.27            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5863_minus__divide__diff__eq__iff,axiom,
% 9.10/5.27      ! [Z: real,X3: real,Y3: real] :
% 9.10/5.27        ( ( Z != zero_zero_real )
% 9.10/5.27       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z ) ) @ Y3 )
% 9.10/5.27          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5864_minus__divide__diff__eq__iff,axiom,
% 9.10/5.27      ! [Z: complex,X3: complex,Y3: complex] :
% 9.10/5.27        ( ( Z != zero_zero_complex )
% 9.10/5.27       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z ) ) @ Y3 )
% 9.10/5.27          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5865_minus__divide__diff__eq__iff,axiom,
% 9.10/5.27      ! [Z: rat,X3: rat,Y3: rat] :
% 9.10/5.27        ( ( Z != zero_zero_rat )
% 9.10/5.27       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z ) ) @ Y3 )
% 9.10/5.27          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5866_even__minus,axiom,
% 9.10/5.27      ! [A: int] :
% 9.10/5.27        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 9.10/5.27        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5867_even__minus,axiom,
% 9.10/5.27      ! [A: code_integer] :
% 9.10/5.27        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.27        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5868_power2__eq__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.27        = ( ( X3 = Y3 )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5869_power2__eq__iff,axiom,
% 9.10/5.27      ! [X3: int,Y3: int] :
% 9.10/5.27        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.27        = ( ( X3 = Y3 )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5870_power2__eq__iff,axiom,
% 9.10/5.27      ! [X3: complex,Y3: complex] :
% 9.10/5.27        ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.27        = ( ( X3 = Y3 )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5871_power2__eq__iff,axiom,
% 9.10/5.27      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.27        ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.27        = ( ( X3 = Y3 )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5872_power2__eq__iff,axiom,
% 9.10/5.27      ! [X3: rat,Y3: rat] :
% 9.10/5.27        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.27        = ( ( X3 = Y3 )
% 9.10/5.27          | ( X3
% 9.10/5.27            = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5873_uminus__power__if,axiom,
% 9.10/5.27      ! [N: nat,A: real] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.10/5.27            = ( power_power_real @ A @ N ) ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 9.10/5.27            = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5874_uminus__power__if,axiom,
% 9.10/5.27      ! [N: nat,A: int] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.10/5.27            = ( power_power_int @ A @ N ) ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 9.10/5.27            = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5875_uminus__power__if,axiom,
% 9.10/5.27      ! [N: nat,A: complex] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.10/5.27            = ( power_power_complex @ A @ N ) ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 9.10/5.27            = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5876_uminus__power__if,axiom,
% 9.10/5.27      ! [N: nat,A: code_integer] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.10/5.27            = ( power_8256067586552552935nteger @ A @ N ) ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 9.10/5.27            = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5877_uminus__power__if,axiom,
% 9.10/5.27      ! [N: nat,A: rat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.10/5.27            = ( power_power_rat @ A @ N ) ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 9.10/5.27            = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5878_verit__less__mono__div__int2,axiom,
% 9.10/5.27      ! [A4: int,B5: int,N: int] :
% 9.10/5.27        ( ( ord_less_eq_int @ A4 @ B5 )
% 9.10/5.27       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 9.10/5.27         => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A4 @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5879_div__eq__minus1,axiom,
% 9.10/5.27      ! [B2: int] :
% 9.10/5.27        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.27       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 9.10/5.27          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5880_of__bool__odd__eq__mod__2,axiom,
% 9.10/5.27      ! [A: nat] :
% 9.10/5.27        ( ( zero_n2687167440665602831ol_nat
% 9.10/5.27          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.27        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5881_of__bool__odd__eq__mod__2,axiom,
% 9.10/5.27      ! [A: int] :
% 9.10/5.27        ( ( zero_n2684676970156552555ol_int
% 9.10/5.27          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.27        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5882_of__bool__odd__eq__mod__2,axiom,
% 9.10/5.27      ! [A: code_integer] :
% 9.10/5.27        ( ( zero_n356916108424825756nteger
% 9.10/5.27          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.10/5.27        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5883_pos__minus__divide__le__eq,axiom,
% 9.10/5.27      ! [C: real,B2: real,A: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5884_pos__minus__divide__le__eq,axiom,
% 9.10/5.27      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5885_pos__le__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: real,A: real,B2: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5886_pos__le__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5887_neg__minus__divide__le__eq,axiom,
% 9.10/5.27      ! [C: real,B2: real,A: real] :
% 9.10/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5888_neg__minus__divide__le__eq,axiom,
% 9.10/5.27      ! [C: rat,B2: rat,A: rat] :
% 9.10/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 9.10/5.27          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5889_neg__le__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: real,A: real,B2: real] :
% 9.10/5.27        ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5890_neg__le__minus__divide__eq,axiom,
% 9.10/5.27      ! [C: rat,A: rat,B2: rat] :
% 9.10/5.27        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5891_minus__divide__le__eq,axiom,
% 9.10/5.27      ! [B2: real,C: real,A: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) @ A )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5892_minus__divide__le__eq,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,A: rat] :
% 9.10/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) @ A )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5893_le__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: real,B2: real,C: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B2 ) ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5894_le__minus__divide__eq,axiom,
% 9.10/5.27      ! [A: rat,B2: rat,C: rat] :
% 9.10/5.27        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C ) ) )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B2 ) ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5895_less__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: real,C: real] :
% 9.10/5.27        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5896_less__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: rat,C: rat] :
% 9.10/5.27        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5897_divide__less__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: real,C: real,W: num] :
% 9.10/5.27        ( ( ord_less_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5898_divide__less__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,W: num] :
% 9.10/5.27        ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5899_power2__eq__1__iff,axiom,
% 9.10/5.27      ! [A: real] :
% 9.10/5.27        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = one_one_real )
% 9.10/5.27        = ( ( A = one_one_real )
% 9.10/5.27          | ( A
% 9.10/5.27            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5900_power2__eq__1__iff,axiom,
% 9.10/5.27      ! [A: int] :
% 9.10/5.27        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = one_one_int )
% 9.10/5.27        = ( ( A = one_one_int )
% 9.10/5.27          | ( A
% 9.10/5.27            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5901_power2__eq__1__iff,axiom,
% 9.10/5.27      ! [A: complex] :
% 9.10/5.27        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = one_one_complex )
% 9.10/5.27        = ( ( A = one_one_complex )
% 9.10/5.27          | ( A
% 9.10/5.27            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5902_power2__eq__1__iff,axiom,
% 9.10/5.27      ! [A: code_integer] :
% 9.10/5.27        ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = one_one_Code_integer )
% 9.10/5.27        = ( ( A = one_one_Code_integer )
% 9.10/5.27          | ( A
% 9.10/5.27            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5903_power2__eq__1__iff,axiom,
% 9.10/5.27      ! [A: rat] :
% 9.10/5.27        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27          = one_one_rat )
% 9.10/5.27        = ( ( A = one_one_rat )
% 9.10/5.27          | ( A
% 9.10/5.27            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5904_minus__one__power__iff,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.10/5.27            = one_one_real ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 9.10/5.27            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5905_minus__one__power__iff,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.10/5.27            = one_one_int ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 9.10/5.27            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5906_minus__one__power__iff,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.10/5.27            = one_one_complex ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 9.10/5.27            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5907_minus__one__power__iff,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.10/5.27            = one_one_Code_integer ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 9.10/5.27            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5908_minus__one__power__iff,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.10/5.27            = one_one_rat ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.27         => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 9.10/5.27            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5909_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.10/5.27      ! [K: nat,N: nat] :
% 9.10/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.27       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 9.10/5.27          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5910_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.10/5.27      ! [K: nat,N: nat] :
% 9.10/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.27       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 9.10/5.27          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5911_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.10/5.27      ! [K: nat,N: nat] :
% 9.10/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.27       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 9.10/5.27          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5912_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.10/5.27      ! [K: nat,N: nat] :
% 9.10/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.27       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 9.10/5.27          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5913_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.10/5.27      ! [K: nat,N: nat] :
% 9.10/5.27        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.27       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 9.10/5.27          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5914_realpow__square__minus__le,axiom,
% 9.10/5.27      ! [U: real,X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5915_signed__take__bit__int__less__eq__self__iff,axiom,
% 9.10/5.27      ! [N: nat,K: int] :
% 9.10/5.27        ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.10/5.27        = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5916_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 9.10/5.27      ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5917_signed__take__bit__int__greater__self__iff,axiom,
% 9.10/5.27      ! [K: int,N: nat] :
% 9.10/5.27        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.10/5.27        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5918_minus__mod__int__eq,axiom,
% 9.10/5.27      ! [L: int,K: int] :
% 9.10/5.27        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 9.10/5.27       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 9.10/5.27          = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5919_zmod__minus1,axiom,
% 9.10/5.27      ! [B2: int] :
% 9.10/5.27        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.27       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
% 9.10/5.27          = ( minus_minus_int @ B2 @ one_one_int ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5920_zminus1__lemma,axiom,
% 9.10/5.27      ! [A: int,B2: int,Q2: int,R: int] :
% 9.10/5.27        ( ( eucl_rel_int @ A @ B2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.27       => ( ( B2 != zero_zero_int )
% 9.10/5.27         => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5921_bits__induct,axiom,
% 9.10/5.27      ! [P: nat > $o,A: nat] :
% 9.10/5.27        ( ! [A3: nat] :
% 9.10/5.27            ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27              = A3 )
% 9.10/5.27           => ( P @ A3 ) )
% 9.10/5.27       => ( ! [A3: nat,B3: $o] :
% 9.10/5.27              ( ( P @ A3 )
% 9.10/5.27             => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.27                  = A3 )
% 9.10/5.27               => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 9.10/5.27         => ( P @ A ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5922_bits__induct,axiom,
% 9.10/5.27      ! [P: int > $o,A: int] :
% 9.10/5.27        ( ! [A3: int] :
% 9.10/5.27            ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.27              = A3 )
% 9.10/5.27           => ( P @ A3 ) )
% 9.10/5.27       => ( ! [A3: int,B3: $o] :
% 9.10/5.27              ( ( P @ A3 )
% 9.10/5.27             => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.27                  = A3 )
% 9.10/5.27               => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 9.10/5.27         => ( P @ A ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5923_bits__induct,axiom,
% 9.10/5.27      ! [P: code_integer > $o,A: code_integer] :
% 9.10/5.27        ( ! [A3: code_integer] :
% 9.10/5.27            ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.27              = A3 )
% 9.10/5.27           => ( P @ A3 ) )
% 9.10/5.27       => ( ! [A3: code_integer,B3: $o] :
% 9.10/5.27              ( ( P @ A3 )
% 9.10/5.27             => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.27                  = A3 )
% 9.10/5.27               => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 9.10/5.27         => ( P @ A ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5924_le__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: real,C: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C ) )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5925_le__divide__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [W: num,B2: rat,C: rat] :
% 9.10/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C ) )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5926_divide__le__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: real,C: real,W: num] :
% 9.10/5.27        ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.27        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.27           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.10/5.27               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5927_divide__le__eq__numeral_I2_J,axiom,
% 9.10/5.27      ! [B2: rat,C: rat,W: num] :
% 9.10/5.27        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 9.10/5.27        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 9.10/5.27          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.10/5.27           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B2 ) )
% 9.10/5.27              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.10/5.27               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5928_square__le__1,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.27       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.27         => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5929_square__le__1,axiom,
% 9.10/5.27      ! [X3: code_integer] :
% 9.10/5.27        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
% 9.10/5.27       => ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
% 9.10/5.27         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5930_square__le__1,axiom,
% 9.10/5.27      ! [X3: rat] :
% 9.10/5.27        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
% 9.10/5.27       => ( ( ord_less_eq_rat @ X3 @ one_one_rat )
% 9.10/5.27         => ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5931_square__le__1,axiom,
% 9.10/5.27      ! [X3: int] :
% 9.10/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
% 9.10/5.27       => ( ( ord_less_eq_int @ X3 @ one_one_int )
% 9.10/5.27         => ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5932_minus__power__mult__self,axiom,
% 9.10/5.27      ! [A: real,N: nat] :
% 9.10/5.27        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 9.10/5.27        = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5933_minus__power__mult__self,axiom,
% 9.10/5.27      ! [A: int,N: nat] :
% 9.10/5.27        ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 9.10/5.27        = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5934_minus__power__mult__self,axiom,
% 9.10/5.27      ! [A: complex,N: nat] :
% 9.10/5.27        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 9.10/5.27        = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5935_minus__power__mult__self,axiom,
% 9.10/5.27      ! [A: code_integer,N: nat] :
% 9.10/5.27        ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 9.10/5.27        = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5936_minus__power__mult__self,axiom,
% 9.10/5.27      ! [A: rat,N: nat] :
% 9.10/5.27        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 9.10/5.27        = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5937_signed__take__bit__int__eq__self,axiom,
% 9.10/5.27      ! [N: nat,K: int] :
% 9.10/5.27        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.10/5.27       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27         => ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.10/5.27            = K ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5938_signed__take__bit__int__eq__self__iff,axiom,
% 9.10/5.27      ! [N: nat,K: int] :
% 9.10/5.27        ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.10/5.27          = K )
% 9.10/5.27        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.10/5.27          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5939_minus__1__div__exp__eq__int,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5940_div__pos__neg__trivial,axiom,
% 9.10/5.27      ! [K: int,L: int] :
% 9.10/5.27        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.27       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.10/5.27         => ( ( divide_divide_int @ K @ L )
% 9.10/5.27            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5941_add__0__iff,axiom,
% 9.10/5.27      ! [B2: complex,A: complex] :
% 9.10/5.27        ( ( B2
% 9.10/5.27          = ( plus_plus_complex @ B2 @ A ) )
% 9.10/5.27        = ( A = zero_zero_complex ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5942_add__0__iff,axiom,
% 9.10/5.27      ! [B2: real,A: real] :
% 9.10/5.27        ( ( B2
% 9.10/5.27          = ( plus_plus_real @ B2 @ A ) )
% 9.10/5.27        = ( A = zero_zero_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5943_add__0__iff,axiom,
% 9.10/5.27      ! [B2: rat,A: rat] :
% 9.10/5.27        ( ( B2
% 9.10/5.27          = ( plus_plus_rat @ B2 @ A ) )
% 9.10/5.27        = ( A = zero_zero_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5944_add__0__iff,axiom,
% 9.10/5.27      ! [B2: nat,A: nat] :
% 9.10/5.27        ( ( B2
% 9.10/5.27          = ( plus_plus_nat @ B2 @ A ) )
% 9.10/5.27        = ( A = zero_zero_nat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5945_add__0__iff,axiom,
% 9.10/5.27      ! [B2: int,A: int] :
% 9.10/5.27        ( ( B2
% 9.10/5.27          = ( plus_plus_int @ B2 @ A ) )
% 9.10/5.27        = ( A = zero_zero_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5946_exp__mod__exp,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5947_exp__mod__exp,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5948_exp__mod__exp,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5949_crossproduct__noteq,axiom,
% 9.10/5.27      ! [A: real,B2: real,C: real,D: real] :
% 9.10/5.27        ( ( ( A != B2 )
% 9.10/5.27          & ( C != D ) )
% 9.10/5.27        = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) )
% 9.10/5.27         != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5950_crossproduct__noteq,axiom,
% 9.10/5.27      ! [A: rat,B2: rat,C: rat,D: rat] :
% 9.10/5.27        ( ( ( A != B2 )
% 9.10/5.27          & ( C != D ) )
% 9.10/5.27        = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B2 @ D ) )
% 9.10/5.27         != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B2 @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5951_crossproduct__noteq,axiom,
% 9.10/5.27      ! [A: nat,B2: nat,C: nat,D: nat] :
% 9.10/5.27        ( ( ( A != B2 )
% 9.10/5.27          & ( C != D ) )
% 9.10/5.27        = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D ) )
% 9.10/5.27         != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5952_crossproduct__noteq,axiom,
% 9.10/5.27      ! [A: int,B2: int,C: int,D: int] :
% 9.10/5.27        ( ( ( A != B2 )
% 9.10/5.27          & ( C != D ) )
% 9.10/5.27        = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) )
% 9.10/5.27         != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5953_crossproduct__eq,axiom,
% 9.10/5.27      ! [W: real,Y3: real,X3: real,Z: real] :
% 9.10/5.27        ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X3 @ Z ) )
% 9.10/5.27          = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X3 @ Y3 ) ) )
% 9.10/5.27        = ( ( W = X3 )
% 9.10/5.27          | ( Y3 = Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5954_crossproduct__eq,axiom,
% 9.10/5.27      ! [W: rat,Y3: rat,X3: rat,Z: rat] :
% 9.10/5.27        ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X3 @ Z ) )
% 9.10/5.27          = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X3 @ Y3 ) ) )
% 9.10/5.27        = ( ( W = X3 )
% 9.10/5.27          | ( Y3 = Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5955_crossproduct__eq,axiom,
% 9.10/5.27      ! [W: nat,Y3: nat,X3: nat,Z: nat] :
% 9.10/5.27        ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X3 @ Z ) )
% 9.10/5.27          = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X3 @ Y3 ) ) )
% 9.10/5.27        = ( ( W = X3 )
% 9.10/5.27          | ( Y3 = Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5956_crossproduct__eq,axiom,
% 9.10/5.27      ! [W: int,Y3: int,X3: int,Z: int] :
% 9.10/5.27        ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X3 @ Z ) )
% 9.10/5.27          = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X3 @ Y3 ) ) )
% 9.10/5.27        = ( ( W = X3 )
% 9.10/5.27          | ( Y3 = Z ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5957_power__minus1__odd,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.27        = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5958_power__minus1__odd,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.27        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5959_power__minus1__odd,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.27        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5960_power__minus1__odd,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.27        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5961_power__minus1__odd,axiom,
% 9.10/5.27      ! [N: nat] :
% 9.10/5.27        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.27        = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5962_int__bit__induct,axiom,
% 9.10/5.27      ! [P: int > $o,K: int] :
% 9.10/5.27        ( ( P @ zero_zero_int )
% 9.10/5.27       => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.27         => ( ! [K3: int] :
% 9.10/5.27                ( ( P @ K3 )
% 9.10/5.27               => ( ( K3 != zero_zero_int )
% 9.10/5.27                 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.27           => ( ! [K3: int] :
% 9.10/5.27                  ( ( P @ K3 )
% 9.10/5.27                 => ( ( K3
% 9.10/5.27                     != ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.27                   => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 9.10/5.27             => ( P @ K ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5963_signed__take__bit__int__greater__eq,axiom,
% 9.10/5.27      ! [K: int,N: nat] :
% 9.10/5.27        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.27       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5964_set__decode__def,axiom,
% 9.10/5.27      ( nat_set_decode
% 9.10/5.27      = ( ^ [X: nat] :
% 9.10/5.27            ( collect_nat
% 9.10/5.27            @ ^ [N2: nat] :
% 9.10/5.27                ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5965_exp__div__exp__eq,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( times_times_nat
% 9.10/5.27          @ ( zero_n2687167440665602831ol_nat
% 9.10/5.27            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.10/5.27               != zero_zero_nat )
% 9.10/5.27              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.10/5.27          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5966_exp__div__exp__eq,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( times_times_int
% 9.10/5.27          @ ( zero_n2684676970156552555ol_int
% 9.10/5.27            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.10/5.27               != zero_zero_int )
% 9.10/5.27              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.10/5.27          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5967_exp__div__exp__eq,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.27        = ( times_3573771949741848930nteger
% 9.10/5.27          @ ( zero_n356916108424825756nteger
% 9.10/5.27            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 9.10/5.27               != zero_z3403309356797280102nteger )
% 9.10/5.27              & ( ord_less_eq_nat @ N @ M ) ) )
% 9.10/5.27          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5968_vebt__buildup_Osimps_I3_J,axiom,
% 9.10/5.27      ! [Va: nat] :
% 9.10/5.27        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.10/5.27         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.10/5.27            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.10/5.27        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.10/5.27         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.10/5.27            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5969_Divides_Oadjust__div__eq,axiom,
% 9.10/5.27      ! [Q2: int,R: int] :
% 9.10/5.27        ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
% 9.10/5.27        = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5970_signed__take__bit__Suc__minus__bit1,axiom,
% 9.10/5.27      ! [N: nat,K: num] :
% 9.10/5.27        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.10/5.27        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5971_vebt__buildup_Opelims,axiom,
% 9.10/5.27      ! [X3: nat,Y3: vEBT_VEBT] :
% 9.10/5.27        ( ( ( vEBT_vebt_buildup @ X3 )
% 9.10/5.27          = Y3 )
% 9.10/5.27       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
% 9.10/5.27         => ( ( ( X3 = zero_zero_nat )
% 9.10/5.27             => ( ( Y3
% 9.10/5.27                  = ( vEBT_Leaf @ $false @ $false ) )
% 9.10/5.27               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 9.10/5.27           => ( ( ( X3
% 9.10/5.27                  = ( suc @ zero_zero_nat ) )
% 9.10/5.27               => ( ( Y3
% 9.10/5.27                    = ( vEBT_Leaf @ $false @ $false ) )
% 9.10/5.27                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 9.10/5.27             => ~ ! [Va2: nat] :
% 9.10/5.27                    ( ( X3
% 9.10/5.27                      = ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.27                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.27                         => ( Y3
% 9.10/5.27                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.10/5.27                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.10/5.27                         => ( Y3
% 9.10/5.27                            = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.10/5.27                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5972_ln__one__minus__pos__lower__bound,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.27         => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5973_artanh__def,axiom,
% 9.10/5.27      ( artanh_real
% 9.10/5.27      = ( ^ [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 ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5974_Sum__Icc__int,axiom,
% 9.10/5.27      ! [M: int,N: int] :
% 9.10/5.27        ( ( ord_less_eq_int @ M @ N )
% 9.10/5.27       => ( ( groups4538972089207619220nt_int
% 9.10/5.27            @ ^ [X: int] : X
% 9.10/5.27            @ ( set_or1266510415728281911st_int @ M @ N ) )
% 9.10/5.27          = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5975_divmod__step__def,axiom,
% 9.10/5.27      ( unique5026877609467782581ep_nat
% 9.10/5.27      = ( ^ [L2: num] :
% 9.10/5.27            ( produc2626176000494625587at_nat
% 9.10/5.27            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5976_divmod__step__def,axiom,
% 9.10/5.27      ( unique5024387138958732305ep_int
% 9.10/5.27      = ( ^ [L2: num] :
% 9.10/5.27            ( produc4245557441103728435nt_int
% 9.10/5.27            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5977_divmod__step__def,axiom,
% 9.10/5.27      ( unique4921790084139445826nteger
% 9.10/5.27      = ( ^ [L2: num] :
% 9.10/5.27            ( produc6916734918728496179nteger
% 9.10/5.27            @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5978_semiring__norm_I90_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ( bit1 @ M )
% 9.10/5.27          = ( bit1 @ N ) )
% 9.10/5.27        = ( M = N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5979_verit__eq__simplify_I9_J,axiom,
% 9.10/5.27      ! [X32: num,Y32: num] :
% 9.10/5.27        ( ( ( bit1 @ X32 )
% 9.10/5.27          = ( bit1 @ Y32 ) )
% 9.10/5.27        = ( X32 = Y32 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5980_semiring__norm_I88_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( bit0 @ M )
% 9.10/5.27       != ( bit1 @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5981_semiring__norm_I89_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( bit1 @ M )
% 9.10/5.27       != ( bit0 @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5982_semiring__norm_I84_J,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( one
% 9.10/5.27       != ( bit1 @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5983_semiring__norm_I86_J,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ( ( bit1 @ M )
% 9.10/5.27       != one ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5984_semiring__norm_I73_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5985_semiring__norm_I80_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( ord_less_num @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5986_case__prod__conv,axiom,
% 9.10/5.27      ! [F: nat > nat > nat,A: nat,B2: nat] :
% 9.10/5.27        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
% 9.10/5.27        = ( F @ A @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5987_case__prod__conv,axiom,
% 9.10/5.27      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B2: nat] :
% 9.10/5.27        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
% 9.10/5.27        = ( F @ A @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5988_case__prod__conv,axiom,
% 9.10/5.27      ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B2: nat] :
% 9.10/5.27        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B2 ) )
% 9.10/5.27        = ( F @ A @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5989_case__prod__conv,axiom,
% 9.10/5.27      ! [F: int > int > product_prod_int_int,A: int,B2: int] :
% 9.10/5.27        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B2 ) )
% 9.10/5.27        = ( F @ A @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5990_case__prod__conv,axiom,
% 9.10/5.27      ! [F: int > int > $o,A: int,B2: int] :
% 9.10/5.27        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) )
% 9.10/5.27        = ( F @ A @ B2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5991_ln__inj__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.27         => ( ( ( ln_ln_real @ X3 )
% 9.10/5.27              = ( ln_ln_real @ Y3 ) )
% 9.10/5.27            = ( X3 = Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5992_ln__less__cancel__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.27         => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 9.10/5.27            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5993_semiring__norm_I9_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.27        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5994_semiring__norm_I7_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5995_semiring__norm_I15_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.27        = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5996_semiring__norm_I14_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5997_semiring__norm_I72_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5998_semiring__norm_I81_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.27        = ( ord_less_num @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_5999_semiring__norm_I70_J,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6000_semiring__norm_I77_J,axiom,
% 9.10/5.27      ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6001_zdiv__numeral__Bit1,axiom,
% 9.10/5.27      ! [V: num,W: num] :
% 9.10/5.27        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.10/5.27        = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6002_semiring__norm_I3_J,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 9.10/5.27        = ( bit1 @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6003_semiring__norm_I4_J,axiom,
% 9.10/5.27      ! [N: num] :
% 9.10/5.27        ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 9.10/5.27        = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6004_semiring__norm_I5_J,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 9.10/5.27        = ( bit1 @ M ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6005_semiring__norm_I8_J,axiom,
% 9.10/5.27      ! [M: num] :
% 9.10/5.27        ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 9.10/5.27        = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6006_semiring__norm_I10_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6007_ln__le__cancel__iff,axiom,
% 9.10/5.27      ! [X3: real,Y3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.27         => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 9.10/5.27            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6008_ln__less__zero__iff,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 9.10/5.27          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6009_ln__gt__zero__iff,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.27          = ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6010_ln__eq__zero__iff,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ( ln_ln_real @ X3 )
% 9.10/5.27            = zero_zero_real )
% 9.10/5.27          = ( X3 = one_one_real ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6011_semiring__norm_I16_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6012_semiring__norm_I74_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.27        = ( ord_less_num @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6013_semiring__norm_I79_J,axiom,
% 9.10/5.27      ! [M: num,N: num] :
% 9.10/5.27        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.27        = ( ord_less_eq_num @ M @ N ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6014_ln__ge__zero__iff,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.27          = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6015_ln__le__zero__iff,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 9.10/5.27          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6016_Suc__div__eq__add3__div__numeral,axiom,
% 9.10/5.27      ! [M: nat,V: num] :
% 9.10/5.27        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 9.10/5.27        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6017_div__Suc__eq__div__add3,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 9.10/5.27        = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6018_mod__Suc__eq__mod__add3,axiom,
% 9.10/5.27      ! [M: nat,N: nat] :
% 9.10/5.27        ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 9.10/5.27        = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6019_Suc__mod__eq__add3__mod__numeral,axiom,
% 9.10/5.27      ! [M: nat,V: num] :
% 9.10/5.27        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 9.10/5.27        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6020_zmod__numeral__Bit1,axiom,
% 9.10/5.27      ! [V: num,W: num] :
% 9.10/5.27        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 9.10/5.27        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6021_signed__take__bit__Suc__bit1,axiom,
% 9.10/5.27      ! [N: nat,K: num] :
% 9.10/5.27        ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.10/5.27        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6022_verit__eq__simplify_I14_J,axiom,
% 9.10/5.27      ! [X2: num,X32: num] :
% 9.10/5.27        ( ( bit0 @ X2 )
% 9.10/5.27       != ( bit1 @ X32 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6023_verit__eq__simplify_I12_J,axiom,
% 9.10/5.27      ! [X32: num] :
% 9.10/5.27        ( one
% 9.10/5.27       != ( bit1 @ X32 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6024_old_Oprod_Ocase,axiom,
% 9.10/5.27      ! [F: nat > nat > nat,X1: nat,X2: nat] :
% 9.10/5.27        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
% 9.10/5.27        = ( F @ X1 @ X2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6025_old_Oprod_Ocase,axiom,
% 9.10/5.27      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X2: nat] :
% 9.10/5.27        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
% 9.10/5.27        = ( F @ X1 @ X2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6026_old_Oprod_Ocase,axiom,
% 9.10/5.27      ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X2: nat] :
% 9.10/5.27        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
% 9.10/5.27        = ( F @ X1 @ X2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6027_old_Oprod_Ocase,axiom,
% 9.10/5.27      ! [F: int > int > product_prod_int_int,X1: int,X2: int] :
% 9.10/5.27        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
% 9.10/5.27        = ( F @ X1 @ X2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6028_old_Oprod_Ocase,axiom,
% 9.10/5.27      ! [F: int > int > $o,X1: int,X2: int] :
% 9.10/5.27        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X2 ) )
% 9.10/5.27        = ( F @ X1 @ X2 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6029_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 9.10/5.27        ( ! [I3: nat] :
% 9.10/5.27            ( ( member_nat @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6030_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_real,F: real > rat,G: real > rat] :
% 9.10/5.27        ( ! [I3: real] :
% 9.10/5.27            ( ( member_real @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6031_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_int,F: int > rat,G: int > rat] :
% 9.10/5.27        ( ! [I3: int] :
% 9.10/5.27            ( ( member_int @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6032_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 9.10/5.27        ( ! [I3: complex] :
% 9.10/5.27            ( ( member_complex @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6033_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_real,F: real > nat,G: real > nat] :
% 9.10/5.27        ( ! [I3: real] :
% 9.10/5.27            ( ( member_real @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6034_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_int,F: int > nat,G: int > nat] :
% 9.10/5.27        ( ! [I3: int] :
% 9.10/5.27            ( ( member_int @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6035_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 9.10/5.27        ( ! [I3: complex] :
% 9.10/5.27            ( ( member_complex @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6036_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_nat,F: nat > int,G: nat > int] :
% 9.10/5.27        ( ! [I3: nat] :
% 9.10/5.27            ( ( member_nat @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6037_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_real,F: real > int,G: real > int] :
% 9.10/5.27        ( ! [I3: real] :
% 9.10/5.27            ( ( member_real @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6038_sum__mono,axiom,
% 9.10/5.27      ! [K5: set_complex,F: complex > int,G: complex > int] :
% 9.10/5.27        ( ! [I3: complex] :
% 9.10/5.27            ( ( member_complex @ I3 @ K5 )
% 9.10/5.27           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27       => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6039_sum__product,axiom,
% 9.10/5.27      ! [F: int > int,A4: set_int,G: int > int,B5: set_int] :
% 9.10/5.27        ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
% 9.10/5.27        = ( groups4538972089207619220nt_int
% 9.10/5.27          @ ^ [I4: int] :
% 9.10/5.27              ( groups4538972089207619220nt_int
% 9.10/5.27              @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
% 9.10/5.27              @ B5 )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6040_sum__product,axiom,
% 9.10/5.27      ! [F: complex > complex,A4: set_complex,G: complex > complex,B5: set_complex] :
% 9.10/5.27        ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
% 9.10/5.27        = ( groups7754918857620584856omplex
% 9.10/5.27          @ ^ [I4: complex] :
% 9.10/5.27              ( groups7754918857620584856omplex
% 9.10/5.27              @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
% 9.10/5.27              @ B5 )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6041_sum__product,axiom,
% 9.10/5.27      ! [F: nat > nat,A4: set_nat,G: nat > nat,B5: set_nat] :
% 9.10/5.27        ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
% 9.10/5.27        = ( groups3542108847815614940at_nat
% 9.10/5.27          @ ^ [I4: nat] :
% 9.10/5.27              ( groups3542108847815614940at_nat
% 9.10/5.27              @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
% 9.10/5.27              @ B5 )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6042_sum__product,axiom,
% 9.10/5.27      ! [F: nat > real,A4: set_nat,G: nat > real,B5: set_nat] :
% 9.10/5.27        ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
% 9.10/5.27        = ( groups6591440286371151544t_real
% 9.10/5.27          @ ^ [I4: nat] :
% 9.10/5.27              ( groups6591440286371151544t_real
% 9.10/5.27              @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
% 9.10/5.27              @ B5 )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6043_sum__distrib__right,axiom,
% 9.10/5.27      ! [F: int > int,A4: set_int,R: int] :
% 9.10/5.27        ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ R )
% 9.10/5.27        = ( groups4538972089207619220nt_int
% 9.10/5.27          @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6044_sum__distrib__right,axiom,
% 9.10/5.27      ! [F: complex > complex,A4: set_complex,R: complex] :
% 9.10/5.27        ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
% 9.10/5.27        = ( groups7754918857620584856omplex
% 9.10/5.27          @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6045_sum__distrib__right,axiom,
% 9.10/5.27      ! [F: nat > nat,A4: set_nat,R: nat] :
% 9.10/5.27        ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ R )
% 9.10/5.27        = ( groups3542108847815614940at_nat
% 9.10/5.27          @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6046_sum__distrib__right,axiom,
% 9.10/5.27      ! [F: nat > real,A4: set_nat,R: real] :
% 9.10/5.27        ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
% 9.10/5.27        = ( groups6591440286371151544t_real
% 9.10/5.27          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6047_sum__distrib__left,axiom,
% 9.10/5.27      ! [R: int,F: int > int,A4: set_int] :
% 9.10/5.27        ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A4 ) )
% 9.10/5.27        = ( groups4538972089207619220nt_int
% 9.10/5.27          @ ^ [N2: int] : ( times_times_int @ R @ ( F @ N2 ) )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6048_sum__distrib__left,axiom,
% 9.10/5.27      ! [R: complex,F: complex > complex,A4: set_complex] :
% 9.10/5.27        ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A4 ) )
% 9.10/5.27        = ( groups7754918857620584856omplex
% 9.10/5.27          @ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6049_sum__distrib__left,axiom,
% 9.10/5.27      ! [R: nat,F: nat > nat,A4: set_nat] :
% 9.10/5.27        ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 9.10/5.27        = ( groups3542108847815614940at_nat
% 9.10/5.27          @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6050_sum__distrib__left,axiom,
% 9.10/5.27      ! [R: real,F: nat > real,A4: set_nat] :
% 9.10/5.27        ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A4 ) )
% 9.10/5.27        = ( groups6591440286371151544t_real
% 9.10/5.27          @ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6051_sum_Odistrib,axiom,
% 9.10/5.27      ! [G: int > int,H2: int > int,A4: set_int] :
% 9.10/5.27        ( ( groups4538972089207619220nt_int
% 9.10/5.27          @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6052_sum_Odistrib,axiom,
% 9.10/5.27      ! [G: complex > complex,H2: complex > complex,A4: set_complex] :
% 9.10/5.27        ( ( groups7754918857620584856omplex
% 9.10/5.27          @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6053_sum_Odistrib,axiom,
% 9.10/5.27      ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
% 9.10/5.27        ( ( groups3542108847815614940at_nat
% 9.10/5.27          @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6054_sum_Odistrib,axiom,
% 9.10/5.27      ! [G: nat > real,H2: nat > real,A4: set_nat] :
% 9.10/5.27        ( ( groups6591440286371151544t_real
% 9.10/5.27          @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6055_sum__subtractf,axiom,
% 9.10/5.27      ! [F: int > int,G: int > int,A4: set_int] :
% 9.10/5.27        ( ( groups4538972089207619220nt_int
% 9.10/5.27          @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6056_sum__subtractf,axiom,
% 9.10/5.27      ! [F: complex > complex,G: complex > complex,A4: set_complex] :
% 9.10/5.27        ( ( groups7754918857620584856omplex
% 9.10/5.27          @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6057_sum__subtractf,axiom,
% 9.10/5.27      ! [F: nat > real,G: nat > real,A4: set_nat] :
% 9.10/5.27        ( ( groups6591440286371151544t_real
% 9.10/5.27          @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.27          @ A4 )
% 9.10/5.27        = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6058_sum__divide__distrib,axiom,
% 9.10/5.27      ! [F: complex > complex,A4: set_complex,R: complex] :
% 9.10/5.27        ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
% 9.10/5.27        = ( groups7754918857620584856omplex
% 9.10/5.27          @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6059_sum__divide__distrib,axiom,
% 9.10/5.27      ! [F: nat > real,A4: set_nat,R: real] :
% 9.10/5.27        ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
% 9.10/5.27        = ( groups6591440286371151544t_real
% 9.10/5.27          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R )
% 9.10/5.27          @ A4 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6060_mod__sum__eq,axiom,
% 9.10/5.27      ! [F: int > int,A: int,A4: set_int] :
% 9.10/5.27        ( ( modulo_modulo_int
% 9.10/5.27          @ ( groups4538972089207619220nt_int
% 9.10/5.27            @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 9.10/5.27            @ A4 )
% 9.10/5.27          @ A )
% 9.10/5.27        = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6061_mod__sum__eq,axiom,
% 9.10/5.27      ! [F: nat > nat,A: nat,A4: set_nat] :
% 9.10/5.27        ( ( modulo_modulo_nat
% 9.10/5.27          @ ( groups3542108847815614940at_nat
% 9.10/5.27            @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 9.10/5.27            @ A4 )
% 9.10/5.27          @ A )
% 9.10/5.27        = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6062_cond__case__prod__eta,axiom,
% 9.10/5.27      ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
% 9.10/5.27        ( ! [X5: nat,Y4: nat] :
% 9.10/5.27            ( ( F @ X5 @ Y4 )
% 9.10/5.27            = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 9.10/5.27       => ( ( produc6842872674320459806at_nat @ F )
% 9.10/5.27          = G ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6063_cond__case__prod__eta,axiom,
% 9.10/5.27      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 9.10/5.27        ( ! [X5: nat,Y4: nat] :
% 9.10/5.27            ( ( F @ X5 @ Y4 )
% 9.10/5.27            = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 9.10/5.27       => ( ( produc27273713700761075at_nat @ F )
% 9.10/5.27          = G ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6064_cond__case__prod__eta,axiom,
% 9.10/5.27      ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.10/5.27        ( ! [X5: nat,Y4: nat] :
% 9.10/5.27            ( ( F @ X5 @ Y4 )
% 9.10/5.27            = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
% 9.10/5.27       => ( ( produc8739625826339149834_nat_o @ F )
% 9.10/5.27          = G ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6065_cond__case__prod__eta,axiom,
% 9.10/5.27      ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 9.10/5.27        ( ! [X5: int,Y4: int] :
% 9.10/5.27            ( ( F @ X5 @ Y4 )
% 9.10/5.27            = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 9.10/5.27       => ( ( produc4245557441103728435nt_int @ F )
% 9.10/5.27          = G ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6066_cond__case__prod__eta,axiom,
% 9.10/5.27      ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 9.10/5.27        ( ! [X5: int,Y4: int] :
% 9.10/5.27            ( ( F @ X5 @ Y4 )
% 9.10/5.27            = ( G @ ( product_Pair_int_int @ X5 @ Y4 ) ) )
% 9.10/5.27       => ( ( produc4947309494688390418_int_o @ F )
% 9.10/5.27          = G ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6067_case__prod__eta,axiom,
% 9.10/5.27      ! [F: product_prod_nat_nat > nat] :
% 9.10/5.27        ( ( produc6842872674320459806at_nat
% 9.10/5.27          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 9.10/5.27        = F ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6068_case__prod__eta,axiom,
% 9.10/5.27      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 9.10/5.27        ( ( produc27273713700761075at_nat
% 9.10/5.27          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 9.10/5.27        = F ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6069_case__prod__eta,axiom,
% 9.10/5.27      ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.10/5.27        ( ( produc8739625826339149834_nat_o
% 9.10/5.27          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 9.10/5.27        = F ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6070_case__prod__eta,axiom,
% 9.10/5.27      ! [F: product_prod_int_int > product_prod_int_int] :
% 9.10/5.27        ( ( produc4245557441103728435nt_int
% 9.10/5.27          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 9.10/5.27        = F ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6071_case__prod__eta,axiom,
% 9.10/5.27      ! [F: product_prod_int_int > $o] :
% 9.10/5.27        ( ( produc4947309494688390418_int_o
% 9.10/5.27          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 9.10/5.27        = F ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6072_case__prodE2,axiom,
% 9.10/5.27      ! [Q: nat > $o,P: nat > nat > nat,Z: product_prod_nat_nat] :
% 9.10/5.27        ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z ) )
% 9.10/5.27       => ~ ! [X5: nat,Y4: nat] :
% 9.10/5.27              ( ( Z
% 9.10/5.27                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 9.10/5.27             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6073_case__prodE2,axiom,
% 9.10/5.27      ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
% 9.10/5.27        ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
% 9.10/5.27       => ~ ! [X5: nat,Y4: nat] :
% 9.10/5.27              ( ( Z
% 9.10/5.27                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 9.10/5.27             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6074_case__prodE2,axiom,
% 9.10/5.27      ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
% 9.10/5.27        ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
% 9.10/5.27       => ~ ! [X5: nat,Y4: nat] :
% 9.10/5.27              ( ( Z
% 9.10/5.27                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 9.10/5.27             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6075_case__prodE2,axiom,
% 9.10/5.27      ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
% 9.10/5.27        ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
% 9.10/5.27       => ~ ! [X5: int,Y4: int] :
% 9.10/5.27              ( ( Z
% 9.10/5.27                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.27             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6076_case__prodE2,axiom,
% 9.10/5.27      ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
% 9.10/5.27        ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
% 9.10/5.27       => ~ ! [X5: int,Y4: int] :
% 9.10/5.27              ( ( Z
% 9.10/5.27                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.27             => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6077_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_real,F: real > real] :
% 9.10/5.27        ( ! [X5: real] :
% 9.10/5.27            ( ( member_real @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6078_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_int,F: int > real] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6079_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_complex,F: complex > real] :
% 9.10/5.27        ( ! [X5: complex] :
% 9.10/5.27            ( ( member_complex @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6080_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_nat,F: nat > rat] :
% 9.10/5.27        ( ! [X5: nat] :
% 9.10/5.27            ( ( member_nat @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6081_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_real,F: real > rat] :
% 9.10/5.27        ( ! [X5: real] :
% 9.10/5.27            ( ( member_real @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6082_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_int,F: int > rat] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6083_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_complex,F: complex > rat] :
% 9.10/5.27        ( ! [X5: complex] :
% 9.10/5.27            ( ( member_complex @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6084_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_real,F: real > nat] :
% 9.10/5.27        ( ! [X5: real] :
% 9.10/5.27            ( ( member_real @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6085_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_int,F: int > nat] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6086_sum__nonneg,axiom,
% 9.10/5.27      ! [A4: set_complex,F: complex > nat] :
% 9.10/5.27        ( ! [X5: complex] :
% 9.10/5.27            ( ( member_complex @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.27       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6087_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_real,F: real > real] :
% 9.10/5.27        ( ! [X5: real] :
% 9.10/5.27            ( ( member_real @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 9.10/5.27       => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6088_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_int,F: int > real] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 9.10/5.27       => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6089_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_complex,F: complex > real] :
% 9.10/5.27        ( ! [X5: complex] :
% 9.10/5.27            ( ( member_complex @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 9.10/5.27       => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ zero_zero_real ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6090_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_nat,F: nat > rat] :
% 9.10/5.27        ( ! [X5: nat] :
% 9.10/5.27            ( ( member_nat @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6091_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_real,F: real > rat] :
% 9.10/5.27        ( ! [X5: real] :
% 9.10/5.27            ( ( member_real @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6092_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_int,F: int > rat] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6093_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_complex,F: complex > rat] :
% 9.10/5.27        ( ! [X5: complex] :
% 9.10/5.27            ( ( member_complex @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.10/5.27       => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ zero_zero_rat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6094_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_real,F: real > nat] :
% 9.10/5.27        ( ! [X5: real] :
% 9.10/5.27            ( ( member_real @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 9.10/5.27       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6095_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_int,F: int > nat] :
% 9.10/5.27        ( ! [X5: int] :
% 9.10/5.27            ( ( member_int @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 9.10/5.27       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6096_sum__nonpos,axiom,
% 9.10/5.27      ! [A4: set_complex,F: complex > nat] :
% 9.10/5.27        ( ! [X5: complex] :
% 9.10/5.27            ( ( member_complex @ X5 @ A4 )
% 9.10/5.27           => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 9.10/5.27       => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ zero_zero_nat ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6097_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
% 9.10/5.27        ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 9.10/5.27          = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: real] :
% 9.10/5.27              ( ( member_real @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_real @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_real @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6098_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
% 9.10/5.27        ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 9.10/5.27          = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: nat] :
% 9.10/5.27              ( ( member_nat @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_nat @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_nat @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6099_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
% 9.10/5.27        ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 9.10/5.27          = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: int] :
% 9.10/5.27              ( ( member_int @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_int @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_int @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6100_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
% 9.10/5.27        ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 9.10/5.27          = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: complex] :
% 9.10/5.27              ( ( member_complex @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_complex @ I @ I5 )
% 9.10/5.27           => ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6101_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
% 9.10/5.27        ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 9.10/5.27          = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: real] :
% 9.10/5.27              ( ( member_real @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_real @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_real @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6102_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
% 9.10/5.27        ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 9.10/5.27          = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: int] :
% 9.10/5.27              ( ( member_int @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_int @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_int @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6103_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
% 9.10/5.27        ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 9.10/5.27          = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: complex] :
% 9.10/5.27              ( ( member_complex @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_complex @ I @ I5 )
% 9.10/5.27           => ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6104_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: real > int,I5: set_real,G: real > int,I: real] :
% 9.10/5.27        ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 9.10/5.27          = ( groups1932886352136224148al_int @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: real] :
% 9.10/5.27              ( ( member_real @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_real @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_real @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6105_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
% 9.10/5.27        ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 9.10/5.27          = ( groups3539618377306564664at_int @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: nat] :
% 9.10/5.27              ( ( member_nat @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_nat @ I @ I5 )
% 9.10/5.27           => ( ( finite_finite_nat @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6106_sum__mono__inv,axiom,
% 9.10/5.27      ! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
% 9.10/5.27        ( ( ( groups5690904116761175830ex_int @ F @ I5 )
% 9.10/5.27          = ( groups5690904116761175830ex_int @ G @ I5 ) )
% 9.10/5.27       => ( ! [I3: complex] :
% 9.10/5.27              ( ( member_complex @ I3 @ I5 )
% 9.10/5.27             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.10/5.27         => ( ( member_complex @ I @ I5 )
% 9.10/5.27           => ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.27             => ( ( F @ I )
% 9.10/5.27                = ( G @ I ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6107_ln__less__self,axiom,
% 9.10/5.27      ! [X3: real] :
% 9.10/5.27        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.27       => ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6108_xor__num_Ocases,axiom,
% 9.10/5.27      ! [X3: product_prod_num_num] :
% 9.10/5.27        ( ( X3
% 9.10/5.27         != ( product_Pair_num_num @ one @ one ) )
% 9.10/5.27       => ( ! [N3: num] :
% 9.10/5.27              ( X3
% 9.10/5.27             != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
% 9.10/5.27         => ( ! [N3: num] :
% 9.10/5.27                ( X3
% 9.10/5.27               != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
% 9.10/5.27           => ( ! [M4: num] :
% 9.10/5.27                  ( X3
% 9.10/5.27                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 9.10/5.27             => ( ! [M4: num,N3: num] :
% 9.10/5.27                    ( X3
% 9.10/5.27                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
% 9.10/5.27               => ( ! [M4: num,N3: num] :
% 9.10/5.27                      ( X3
% 9.10/5.27                     != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
% 9.10/5.27                 => ( ! [M4: num] :
% 9.10/5.27                        ( X3
% 9.10/5.27                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 9.10/5.27                   => ( ! [M4: num,N3: num] :
% 9.10/5.27                          ( X3
% 9.10/5.27                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
% 9.10/5.27                     => ~ ! [M4: num,N3: num] :
% 9.10/5.27                            ( X3
% 9.10/5.27                           != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6109_num_Oexhaust,axiom,
% 9.10/5.27      ! [Y3: num] :
% 9.10/5.27        ( ( Y3 != one )
% 9.10/5.27       => ( ! [X22: num] :
% 9.10/5.27              ( Y3
% 9.10/5.27             != ( bit0 @ X22 ) )
% 9.10/5.27         => ~ ! [X33: num] :
% 9.10/5.27                ( Y3
% 9.10/5.27               != ( bit1 @ X33 ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6110_sum__le__included,axiom,
% 9.10/5.27      ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 9.10/5.27        ( ( finite_finite_int @ S )
% 9.10/5.27       => ( ( finite_finite_int @ T )
% 9.10/5.27         => ( ! [X5: int] :
% 9.10/5.27                ( ( member_int @ X5 @ T )
% 9.10/5.27               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 9.10/5.27           => ( ! [X5: int] :
% 9.10/5.27                  ( ( member_int @ X5 @ S )
% 9.10/5.27                 => ? [Xa: int] :
% 9.10/5.27                      ( ( member_int @ Xa @ T )
% 9.10/5.27                      & ( ( I @ Xa )
% 9.10/5.27                        = X5 )
% 9.10/5.27                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.27             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6111_sum__le__included,axiom,
% 9.10/5.27      ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 9.10/5.27        ( ( finite_finite_int @ S )
% 9.10/5.27       => ( ( finite3207457112153483333omplex @ T )
% 9.10/5.27         => ( ! [X5: complex] :
% 9.10/5.27                ( ( member_complex @ X5 @ T )
% 9.10/5.27               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 9.10/5.27           => ( ! [X5: int] :
% 9.10/5.27                  ( ( member_int @ X5 @ S )
% 9.10/5.27                 => ? [Xa: complex] :
% 9.10/5.27                      ( ( member_complex @ Xa @ T )
% 9.10/5.27                      & ( ( I @ Xa )
% 9.10/5.27                        = X5 )
% 9.10/5.27                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.27             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6112_sum__le__included,axiom,
% 9.10/5.27      ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 9.10/5.27        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.27       => ( ( finite_finite_int @ T )
% 9.10/5.27         => ( ! [X5: int] :
% 9.10/5.27                ( ( member_int @ X5 @ T )
% 9.10/5.27               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 9.10/5.27           => ( ! [X5: complex] :
% 9.10/5.27                  ( ( member_complex @ X5 @ S )
% 9.10/5.27                 => ? [Xa: int] :
% 9.10/5.27                      ( ( member_int @ Xa @ T )
% 9.10/5.27                      & ( ( I @ Xa )
% 9.10/5.27                        = X5 )
% 9.10/5.27                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.27             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6113_sum__le__included,axiom,
% 9.10/5.27      ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 9.10/5.27        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.27       => ( ( finite3207457112153483333omplex @ T )
% 9.10/5.27         => ( ! [X5: complex] :
% 9.10/5.27                ( ( member_complex @ X5 @ T )
% 9.10/5.27               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
% 9.10/5.27           => ( ! [X5: complex] :
% 9.10/5.27                  ( ( member_complex @ X5 @ S )
% 9.10/5.27                 => ? [Xa: complex] :
% 9.10/5.27                      ( ( member_complex @ Xa @ T )
% 9.10/5.27                      & ( ( I @ Xa )
% 9.10/5.27                        = X5 )
% 9.10/5.27                      & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.27             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 9.10/5.27  
% 9.10/5.27  thf(fact_6114_sum__le__included,axiom,
% 9.10/5.27      ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 9.10/5.27        ( ( finite_finite_nat @ S )
% 9.10/5.27       => ( ( finite_finite_nat @ T )
% 9.10/5.27         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ T )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S )
% 9.10/5.28                 => ? [Xa: nat] :
% 9.10/5.28                      ( ( member_nat @ Xa @ T )
% 9.10/5.28                      & ( ( I @ Xa )
% 9.10/5.28                        = X5 )
% 9.10/5.28                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.28             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6115_sum__le__included,axiom,
% 9.10/5.28      ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ S )
% 9.10/5.28       => ( ( finite_finite_int @ T )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ T )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S )
% 9.10/5.28                 => ? [Xa: int] :
% 9.10/5.28                      ( ( member_int @ Xa @ T )
% 9.10/5.28                      & ( ( I @ Xa )
% 9.10/5.28                        = X5 )
% 9.10/5.28                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.28             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6116_sum__le__included,axiom,
% 9.10/5.28      ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ S )
% 9.10/5.28       => ( ( finite3207457112153483333omplex @ T )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ T )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S )
% 9.10/5.28                 => ? [Xa: complex] :
% 9.10/5.28                      ( ( member_complex @ Xa @ T )
% 9.10/5.28                      & ( ( I @ Xa )
% 9.10/5.28                        = X5 )
% 9.10/5.28                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.28             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6117_sum__le__included,axiom,
% 9.10/5.28      ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ( finite_finite_nat @ T )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ T )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S )
% 9.10/5.28                 => ? [Xa: nat] :
% 9.10/5.28                      ( ( member_nat @ Xa @ T )
% 9.10/5.28                      & ( ( I @ Xa )
% 9.10/5.28                        = X5 )
% 9.10/5.28                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.28             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6118_sum__le__included,axiom,
% 9.10/5.28      ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ( finite_finite_int @ T )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ T )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S )
% 9.10/5.28                 => ? [Xa: int] :
% 9.10/5.28                      ( ( member_int @ Xa @ T )
% 9.10/5.28                      & ( ( I @ Xa )
% 9.10/5.28                        = X5 )
% 9.10/5.28                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.28             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6119_sum__le__included,axiom,
% 9.10/5.28      ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ( finite3207457112153483333omplex @ T )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ T )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S )
% 9.10/5.28                 => ? [Xa: complex] :
% 9.10/5.28                      ( ( member_complex @ Xa @ T )
% 9.10/5.28                      & ( ( I @ Xa )
% 9.10/5.28                        = X5 )
% 9.10/5.28                      & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
% 9.10/5.28             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6120_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_real,F: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ A4 )
% 9.10/5.28       => ( ! [X5: real] :
% 9.10/5.28              ( ( member_real @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups8097168146408367636l_real @ F @ A4 )
% 9.10/5.28              = zero_zero_real )
% 9.10/5.28            = ( ! [X: real] :
% 9.10/5.28                  ( ( member_real @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6121_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups8778361861064173332t_real @ F @ A4 )
% 9.10/5.28              = zero_zero_real )
% 9.10/5.28            = ( ! [X: int] :
% 9.10/5.28                  ( ( member_int @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6122_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups5808333547571424918x_real @ F @ A4 )
% 9.10/5.28              = zero_zero_real )
% 9.10/5.28            = ( ! [X: complex] :
% 9.10/5.28                  ( ( member_complex @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_real ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6123_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_real,F: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ A4 )
% 9.10/5.28       => ( ! [X5: real] :
% 9.10/5.28              ( ( member_real @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups1300246762558778688al_rat @ F @ A4 )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28            = ( ! [X: real] :
% 9.10/5.28                  ( ( member_real @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6124_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ! [X5: nat] :
% 9.10/5.28              ( ( member_nat @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28            = ( ! [X: nat] :
% 9.10/5.28                  ( ( member_nat @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6125_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28            = ( ! [X: int] :
% 9.10/5.28                  ( ( member_int @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6126_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28            = ( ! [X: complex] :
% 9.10/5.28                  ( ( member_complex @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6127_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_real,F: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ A4 )
% 9.10/5.28       => ( ! [X5: real] :
% 9.10/5.28              ( ( member_real @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups1935376822645274424al_nat @ F @ A4 )
% 9.10/5.28              = zero_zero_nat )
% 9.10/5.28            = ( ! [X: real] :
% 9.10/5.28                  ( ( member_real @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6128_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > nat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups4541462559716669496nt_nat @ F @ A4 )
% 9.10/5.28              = zero_zero_nat )
% 9.10/5.28            = ( ! [X: int] :
% 9.10/5.28                  ( ( member_int @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6129_sum__nonneg__eq__0__iff,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.28         => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
% 9.10/5.28              = zero_zero_nat )
% 9.10/5.28            = ( ! [X: complex] :
% 9.10/5.28                  ( ( member_complex @ X @ A4 )
% 9.10/5.28                 => ( ( F @ X )
% 9.10/5.28                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6130_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > real,G: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: int] :
% 9.10/5.28                ( ( member_int @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6131_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: complex] :
% 9.10/5.28                ( ( member_complex @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6132_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ! [X5: nat] :
% 9.10/5.28              ( ( member_nat @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: nat] :
% 9.10/5.28                ( ( member_nat @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6133_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > rat,G: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: int] :
% 9.10/5.28                ( ( member_int @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6134_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: complex] :
% 9.10/5.28                ( ( member_complex @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6135_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > nat,G: int > nat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: int] :
% 9.10/5.28                ( ( member_int @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6136_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > nat,G: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: complex] :
% 9.10/5.28                ( ( member_complex @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6137_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_nat,F: nat > int,G: nat > int] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ! [X5: nat] :
% 9.10/5.28              ( ( member_nat @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: nat] :
% 9.10/5.28                ( ( member_nat @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6138_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > int,G: complex > int] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ! [X5: complex] :
% 9.10/5.28              ( ( member_complex @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: complex] :
% 9.10/5.28                ( ( member_complex @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6139_sum__strict__mono__ex1,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > int,G: int > int] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ! [X5: int] :
% 9.10/5.28              ( ( member_int @ X5 @ A4 )
% 9.10/5.28             => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28         => ( ? [X6: int] :
% 9.10/5.28                ( ( member_int @ X6 @ A4 )
% 9.10/5.28                & ( ord_less_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
% 9.10/5.28           => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6140_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.10/5.28        ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 9.10/5.28       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_nat @ S2 )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6141_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 9.10/5.28        ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 9.10/5.28       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_int @ S2 )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6142_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 9.10/5.28        ( ( R2 @ zero_zero_real @ zero_zero_real )
% 9.10/5.28       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_int @ S2 )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6143_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.10/5.28        ( ( R2 @ zero_zero_real @ zero_zero_real )
% 9.10/5.28       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite3207457112153483333omplex @ S2 )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6144_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 9.10/5.28        ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 9.10/5.28       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_nat @ S2 )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6145_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 9.10/5.28        ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 9.10/5.28       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_int @ S2 )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6146_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.10/5.28        ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 9.10/5.28       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite3207457112153483333omplex @ S2 )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6147_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 9.10/5.28        ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 9.10/5.28       => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_int @ S2 )
% 9.10/5.28           => ( ! [X5: int] :
% 9.10/5.28                  ( ( member_int @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6148_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 9.10/5.28        ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 9.10/5.28       => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite3207457112153483333omplex @ S2 )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6149_sum_Orelated,axiom,
% 9.10/5.28      ! [R2: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
% 9.10/5.28        ( ( R2 @ zero_zero_int @ zero_zero_int )
% 9.10/5.28       => ( ! [X15: int,Y15: int,X22: int,Y22: int] :
% 9.10/5.28              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.28                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.28             => ( R2 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X22 @ Y22 ) ) )
% 9.10/5.28         => ( ( finite_finite_nat @ S2 )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28             => ( R2 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6150_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_complex )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6151_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > real,G: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_int )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6152_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_real,F: real > real,G: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_real )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6153_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_complex )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6154_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_nat )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6155_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > rat,G: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_int )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6156_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_real,F: real > rat,G: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_real )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6157_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_complex,F: complex > nat,G: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_complex )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6158_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_int,F: int > nat,G: int > nat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_int )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6159_sum__strict__mono,axiom,
% 9.10/5.28      ! [A4: set_real,F: real > nat,G: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ A4 )
% 9.10/5.28       => ( ( A4 != bot_bot_set_real )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ A4 )
% 9.10/5.28               => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.28           => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6160_ln__bound,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6161_ln__gt__zero__imp__gt__one,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.28       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28         => ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6162_ln__less__zero,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.28         => ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6163_ln__gt__zero,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.28       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6164_ln__ge__zero,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.28       => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6165_numeral__Bit1,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6166_numeral__Bit1,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6167_numeral__Bit1,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6168_numeral__Bit1,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6169_numeral__Bit1,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6170_eval__nat__numeral_I3_J,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 9.10/5.28        = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6171_cong__exp__iff__simps_I10_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6172_cong__exp__iff__simps_I10_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6173_cong__exp__iff__simps_I10_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6174_cong__exp__iff__simps_I12_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6175_cong__exp__iff__simps_I12_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6176_cong__exp__iff__simps_I12_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6177_cong__exp__iff__simps_I13_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 9.10/5.28          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6178_cong__exp__iff__simps_I13_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 9.10/5.28          = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6179_cong__exp__iff__simps_I13_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num,N: num] :
% 9.10/5.28        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 9.10/5.28          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6180_power__minus__Bit1,axiom,
% 9.10/5.28      ! [X3: real,K: num] :
% 9.10/5.28        ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.10/5.28        = ( uminus_uminus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6181_power__minus__Bit1,axiom,
% 9.10/5.28      ! [X3: int,K: num] :
% 9.10/5.28        ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.10/5.28        = ( uminus_uminus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6182_power__minus__Bit1,axiom,
% 9.10/5.28      ! [X3: complex,K: num] :
% 9.10/5.28        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.10/5.28        = ( uminus1482373934393186551omplex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6183_power__minus__Bit1,axiom,
% 9.10/5.28      ! [X3: code_integer,K: num] :
% 9.10/5.28        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.10/5.28        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6184_power__minus__Bit1,axiom,
% 9.10/5.28      ! [X3: rat,K: num] :
% 9.10/5.28        ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.10/5.28        = ( uminus_uminus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6185_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_real,F: real > real,B5: real,I: real] :
% 9.10/5.28        ( ( finite_finite_real @ S )
% 9.10/5.28       => ( ! [I3: real] :
% 9.10/5.28              ( ( member_real @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_real @ I @ S )
% 9.10/5.28             => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6186_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_int,F: int > real,B5: real,I: int] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ! [I3: int] :
% 9.10/5.28              ( ( member_int @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_int @ I @ S )
% 9.10/5.28             => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6187_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_complex,F: complex > real,B5: real,I: complex] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.28       => ( ! [I3: complex] :
% 9.10/5.28              ( ( member_complex @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_complex @ I @ S )
% 9.10/5.28             => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6188_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_real,F: real > rat,B5: rat,I: real] :
% 9.10/5.28        ( ( finite_finite_real @ S )
% 9.10/5.28       => ( ! [I3: real] :
% 9.10/5.28              ( ( member_real @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_real @ I @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6189_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_nat,F: nat > rat,B5: rat,I: nat] :
% 9.10/5.28        ( ( finite_finite_nat @ S )
% 9.10/5.28       => ( ! [I3: nat] :
% 9.10/5.28              ( ( member_nat @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_nat @ I @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6190_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_int,F: int > rat,B5: rat,I: int] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ! [I3: int] :
% 9.10/5.28              ( ( member_int @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_int @ I @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6191_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_complex,F: complex > rat,B5: rat,I: complex] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.28       => ( ! [I3: complex] :
% 9.10/5.28              ( ( member_complex @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_complex @ I @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6192_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_real,F: real > nat,B5: nat,I: real] :
% 9.10/5.28        ( ( finite_finite_real @ S )
% 9.10/5.28       => ( ! [I3: real] :
% 9.10/5.28              ( ( member_real @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_real @ I @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6193_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_int,F: int > nat,B5: nat,I: int] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ! [I3: int] :
% 9.10/5.28              ( ( member_int @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_int @ I @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6194_sum__nonneg__leq__bound,axiom,
% 9.10/5.28      ! [S: set_complex,F: complex > nat,B5: nat,I: complex] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.28       => ( ! [I3: complex] :
% 9.10/5.28              ( ( member_complex @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 9.10/5.28              = B5 )
% 9.10/5.28           => ( ( member_complex @ I @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6195_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_real,F: real > real,I: real] :
% 9.10/5.28        ( ( finite_finite_real @ S )
% 9.10/5.28       => ( ! [I3: real] :
% 9.10/5.28              ( ( member_real @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 9.10/5.28              = zero_zero_real )
% 9.10/5.28           => ( ( member_real @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_real ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6196_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_int,F: int > real,I: int] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ! [I3: int] :
% 9.10/5.28              ( ( member_int @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 9.10/5.28              = zero_zero_real )
% 9.10/5.28           => ( ( member_int @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_real ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6197_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_complex,F: complex > real,I: complex] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.28       => ( ! [I3: complex] :
% 9.10/5.28              ( ( member_complex @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 9.10/5.28              = zero_zero_real )
% 9.10/5.28           => ( ( member_complex @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_real ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6198_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_real,F: real > rat,I: real] :
% 9.10/5.28        ( ( finite_finite_real @ S )
% 9.10/5.28       => ( ! [I3: real] :
% 9.10/5.28              ( ( member_real @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28           => ( ( member_real @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6199_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_nat,F: nat > rat,I: nat] :
% 9.10/5.28        ( ( finite_finite_nat @ S )
% 9.10/5.28       => ( ! [I3: nat] :
% 9.10/5.28              ( ( member_nat @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28           => ( ( member_nat @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6200_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_int,F: int > rat,I: int] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ! [I3: int] :
% 9.10/5.28              ( ( member_int @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28           => ( ( member_int @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6201_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_complex,F: complex > rat,I: complex] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.28       => ( ! [I3: complex] :
% 9.10/5.28              ( ( member_complex @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 9.10/5.28              = zero_zero_rat )
% 9.10/5.28           => ( ( member_complex @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_rat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6202_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_real,F: real > nat,I: real] :
% 9.10/5.28        ( ( finite_finite_real @ S )
% 9.10/5.28       => ( ! [I3: real] :
% 9.10/5.28              ( ( member_real @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 9.10/5.28              = zero_zero_nat )
% 9.10/5.28           => ( ( member_real @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_nat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6203_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_int,F: int > nat,I: int] :
% 9.10/5.28        ( ( finite_finite_int @ S )
% 9.10/5.28       => ( ! [I3: int] :
% 9.10/5.28              ( ( member_int @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 9.10/5.28              = zero_zero_nat )
% 9.10/5.28           => ( ( member_int @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_nat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6204_sum__nonneg__0,axiom,
% 9.10/5.28      ! [S: set_complex,F: complex > nat,I: complex] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ S )
% 9.10/5.28       => ( ! [I3: complex] :
% 9.10/5.28              ( ( member_complex @ I3 @ S )
% 9.10/5.28             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 9.10/5.28              = zero_zero_nat )
% 9.10/5.28           => ( ( member_complex @ I @ S )
% 9.10/5.28             => ( ( F @ I )
% 9.10/5.28                = zero_zero_nat ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6205_numeral__code_I3_J,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6206_numeral__code_I3_J,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6207_numeral__code_I3_J,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6208_numeral__code_I3_J,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6209_numeral__code_I3_J,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 9.10/5.28        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6210_power__numeral__odd,axiom,
% 9.10/5.28      ! [Z: complex,W: num] :
% 9.10/5.28        ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 9.10/5.28        = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6211_power__numeral__odd,axiom,
% 9.10/5.28      ! [Z: real,W: num] :
% 9.10/5.28        ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 9.10/5.28        = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6212_power__numeral__odd,axiom,
% 9.10/5.28      ! [Z: rat,W: num] :
% 9.10/5.28        ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 9.10/5.28        = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6213_power__numeral__odd,axiom,
% 9.10/5.28      ! [Z: nat,W: num] :
% 9.10/5.28        ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 9.10/5.28        = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6214_power__numeral__odd,axiom,
% 9.10/5.28      ! [Z: int,W: num] :
% 9.10/5.28        ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 9.10/5.28        = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6215_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_real,I: real,F: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ I5 )
% 9.10/5.28       => ( ( member_real @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: real] :
% 9.10/5.28                  ( ( member_real @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6216_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_int,I: int,F: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ I5 )
% 9.10/5.28       => ( ( member_int @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: int] :
% 9.10/5.28                  ( ( member_int @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6217_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_complex,I: complex,F: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.28       => ( ( member_complex @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: complex] :
% 9.10/5.28                  ( ( member_complex @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6218_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_real,I: real,F: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ I5 )
% 9.10/5.28       => ( ( member_real @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: real] :
% 9.10/5.28                  ( ( member_real @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6219_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_nat,I: nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ I5 )
% 9.10/5.28       => ( ( member_nat @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: nat] :
% 9.10/5.28                  ( ( member_nat @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6220_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_int,I: int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ I5 )
% 9.10/5.28       => ( ( member_int @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: int] :
% 9.10/5.28                  ( ( member_int @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6221_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_complex,I: complex,F: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.28       => ( ( member_complex @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: complex] :
% 9.10/5.28                  ( ( member_complex @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6222_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_real,I: real,F: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ I5 )
% 9.10/5.28       => ( ( member_real @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: real] :
% 9.10/5.28                  ( ( member_real @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6223_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_int,I: int,F: int > nat] :
% 9.10/5.28        ( ( finite_finite_int @ I5 )
% 9.10/5.28       => ( ( member_int @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: int] :
% 9.10/5.28                  ( ( member_int @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6224_sum__pos2,axiom,
% 9.10/5.28      ! [I5: set_complex,I: complex,F: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.28       => ( ( member_complex @ I @ I5 )
% 9.10/5.28         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.10/5.28           => ( ! [I3: complex] :
% 9.10/5.28                  ( ( member_complex @ I3 @ I5 )
% 9.10/5.28                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28             => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6225_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_complex,F: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_complex )
% 9.10/5.28         => ( ! [I3: complex] :
% 9.10/5.28                ( ( member_complex @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6226_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_int,F: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_int )
% 9.10/5.28         => ( ! [I3: int] :
% 9.10/5.28                ( ( member_int @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6227_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_real,F: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_real )
% 9.10/5.28         => ( ! [I3: real] :
% 9.10/5.28                ( ( member_real @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6228_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_complex,F: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_complex )
% 9.10/5.28         => ( ! [I3: complex] :
% 9.10/5.28                ( ( member_complex @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6229_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_nat )
% 9.10/5.28         => ( ! [I3: nat] :
% 9.10/5.28                ( ( member_nat @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6230_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_int )
% 9.10/5.28         => ( ! [I3: int] :
% 9.10/5.28                ( ( member_int @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6231_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_real,F: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_real )
% 9.10/5.28         => ( ! [I3: real] :
% 9.10/5.28                ( ( member_real @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6232_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_complex,F: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_complex )
% 9.10/5.28         => ( ! [I3: complex] :
% 9.10/5.28                ( ( member_complex @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6233_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_int,F: int > nat] :
% 9.10/5.28        ( ( finite_finite_int @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_int )
% 9.10/5.28         => ( ! [I3: int] :
% 9.10/5.28                ( ( member_int @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6234_sum__pos,axiom,
% 9.10/5.28      ! [I5: set_real,F: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ I5 )
% 9.10/5.28       => ( ( I5 != bot_bot_set_real )
% 9.10/5.28         => ( ! [I3: real] :
% 9.10/5.28                ( ( member_real @ I3 @ I5 )
% 9.10/5.28               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.10/5.28           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6235_ln__ge__zero__imp__ge__one,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.28       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28         => ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6236_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_complex ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_complex ) )
% 9.10/5.28               => ( ( ( groups5754745047067104278omplex @ G @ A4 )
% 9.10/5.28                    = ( groups5754745047067104278omplex @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups5754745047067104278omplex @ G @ C4 )
% 9.10/5.28                    = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6237_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: nat] :
% 9.10/5.28                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_complex ) )
% 9.10/5.28             => ( ! [B3: nat] :
% 9.10/5.28                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_complex ) )
% 9.10/5.28               => ( ( ( groups2073611262835488442omplex @ G @ A4 )
% 9.10/5.28                    = ( groups2073611262835488442omplex @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups2073611262835488442omplex @ G @ C4 )
% 9.10/5.28                    = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6238_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_real ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_real ) )
% 9.10/5.28               => ( ( ( groups8097168146408367636l_real @ G @ A4 )
% 9.10/5.28                    = ( groups8097168146408367636l_real @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups8097168146408367636l_real @ G @ C4 )
% 9.10/5.28                    = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6239_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: complex] :
% 9.10/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_real ) )
% 9.10/5.28             => ( ! [B3: complex] :
% 9.10/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_real ) )
% 9.10/5.28               => ( ( ( groups5808333547571424918x_real @ G @ A4 )
% 9.10/5.28                    = ( groups5808333547571424918x_real @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups5808333547571424918x_real @ G @ C4 )
% 9.10/5.28                    = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6240_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_rat ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_rat ) )
% 9.10/5.28               => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
% 9.10/5.28                    = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 9.10/5.28                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6241_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: nat] :
% 9.10/5.28                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_rat ) )
% 9.10/5.28             => ( ! [B3: nat] :
% 9.10/5.28                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_rat ) )
% 9.10/5.28               => ( ( ( groups2906978787729119204at_rat @ G @ A4 )
% 9.10/5.28                    = ( groups2906978787729119204at_rat @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups2906978787729119204at_rat @ G @ C4 )
% 9.10/5.28                    = ( groups2906978787729119204at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6242_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: complex] :
% 9.10/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_rat ) )
% 9.10/5.28             => ( ! [B3: complex] :
% 9.10/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_rat ) )
% 9.10/5.28               => ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
% 9.10/5.28                    = ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 9.10/5.28                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6243_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_nat ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_nat ) )
% 9.10/5.28               => ( ( ( groups1935376822645274424al_nat @ G @ A4 )
% 9.10/5.28                    = ( groups1935376822645274424al_nat @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 9.10/5.28                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6244_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: complex] :
% 9.10/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_nat ) )
% 9.10/5.28             => ( ! [B3: complex] :
% 9.10/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_nat ) )
% 9.10/5.28               => ( ( ( groups5693394587270226106ex_nat @ G @ A4 )
% 9.10/5.28                    = ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 9.10/5.28                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6245_sum_Osame__carrier,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_int ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_int ) )
% 9.10/5.28               => ( ( ( groups1932886352136224148al_int @ G @ A4 )
% 9.10/5.28                    = ( groups1932886352136224148al_int @ H2 @ B5 ) )
% 9.10/5.28                  = ( ( groups1932886352136224148al_int @ G @ C4 )
% 9.10/5.28                    = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6246_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_complex ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_complex ) )
% 9.10/5.28               => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 9.10/5.28                    = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups5754745047067104278omplex @ G @ A4 )
% 9.10/5.28                    = ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6247_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: nat] :
% 9.10/5.28                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_complex ) )
% 9.10/5.28             => ( ! [B3: nat] :
% 9.10/5.28                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_complex ) )
% 9.10/5.28               => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 9.10/5.28                    = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups2073611262835488442omplex @ G @ A4 )
% 9.10/5.28                    = ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6248_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_real ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_real ) )
% 9.10/5.28               => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 9.10/5.28                    = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups8097168146408367636l_real @ G @ A4 )
% 9.10/5.28                    = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6249_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: complex] :
% 9.10/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_real ) )
% 9.10/5.28             => ( ! [B3: complex] :
% 9.10/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_real ) )
% 9.10/5.28               => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 9.10/5.28                    = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups5808333547571424918x_real @ G @ A4 )
% 9.10/5.28                    = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6250_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_rat ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_rat ) )
% 9.10/5.28               => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 9.10/5.28                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups1300246762558778688al_rat @ G @ A4 )
% 9.10/5.28                    = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6251_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: nat] :
% 9.10/5.28                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_rat ) )
% 9.10/5.28             => ( ! [B3: nat] :
% 9.10/5.28                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_rat ) )
% 9.10/5.28               => ( ( ( groups2906978787729119204at_rat @ G @ C4 )
% 9.10/5.28                    = ( groups2906978787729119204at_rat @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups2906978787729119204at_rat @ G @ A4 )
% 9.10/5.28                    = ( groups2906978787729119204at_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6252_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: complex] :
% 9.10/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_rat ) )
% 9.10/5.28             => ( ! [B3: complex] :
% 9.10/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_rat ) )
% 9.10/5.28               => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 9.10/5.28                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups5058264527183730370ex_rat @ G @ A4 )
% 9.10/5.28                    = ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6253_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_nat ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_nat ) )
% 9.10/5.28               => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 9.10/5.28                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups1935376822645274424al_nat @ G @ A4 )
% 9.10/5.28                    = ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6254_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: complex] :
% 9.10/5.28                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_nat ) )
% 9.10/5.28             => ( ! [B3: complex] :
% 9.10/5.28                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_nat ) )
% 9.10/5.28               => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 9.10/5.28                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups5693394587270226106ex_nat @ G @ A4 )
% 9.10/5.28                    = ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6255_sum_Osame__carrierI,axiom,
% 9.10/5.28      ! [C4: set_real,A4: set_real,B5: set_real,G: real > int,H2: real > int] :
% 9.10/5.28        ( ( finite_finite_real @ C4 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.28         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.28           => ( ! [A3: real] :
% 9.10/5.28                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.28                 => ( ( G @ A3 )
% 9.10/5.28                    = zero_zero_int ) )
% 9.10/5.28             => ( ! [B3: real] :
% 9.10/5.28                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.28                   => ( ( H2 @ B3 )
% 9.10/5.28                      = zero_zero_int ) )
% 9.10/5.28               => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 9.10/5.28                    = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 9.10/5.28                 => ( ( groups1932886352136224148al_int @ G @ A4 )
% 9.10/5.28                    = ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6256_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ( groups2073611262835488442omplex @ G @ S2 )
% 9.10/5.28              = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6257_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ( groups5808333547571424918x_real @ G @ S2 )
% 9.10/5.28              = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6258_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 9.10/5.28              = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6259_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 9.10/5.28              = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6260_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_nat ) )
% 9.10/5.28           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 9.10/5.28              = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6261_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_int ) )
% 9.10/5.28           => ( ( groups3539618377306564664at_int @ G @ S2 )
% 9.10/5.28              = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6262_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_int ) )
% 9.10/5.28           => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 9.10/5.28              = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6263_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_int,S2: set_int,G: int > complex] :
% 9.10/5.28        ( ( finite_finite_int @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ( groups3049146728041665814omplex @ G @ S2 )
% 9.10/5.28              = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6264_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_int,S2: set_int,G: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ( groups8778361861064173332t_real @ G @ S2 )
% 9.10/5.28              = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6265_sum_Omono__neutral__left,axiom,
% 9.10/5.28      ! [T3: set_int,S2: set_int,G: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ( groups3906332499630173760nt_rat @ G @ S2 )
% 9.10/5.28              = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6266_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ( groups2073611262835488442omplex @ G @ T3 )
% 9.10/5.28              = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6267_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ( groups5808333547571424918x_real @ G @ T3 )
% 9.10/5.28              = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6268_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 9.10/5.28              = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6269_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 9.10/5.28              = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6270_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_nat ) )
% 9.10/5.28           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 9.10/5.28              = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6271_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_int ) )
% 9.10/5.28           => ( ( groups3539618377306564664at_int @ G @ T3 )
% 9.10/5.28              = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6272_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_int ) )
% 9.10/5.28           => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 9.10/5.28              = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6273_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_int,S2: set_int,G: int > complex] :
% 9.10/5.28        ( ( finite_finite_int @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ( groups3049146728041665814omplex @ G @ T3 )
% 9.10/5.28              = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6274_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_int,S2: set_int,G: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ( groups8778361861064173332t_real @ G @ T3 )
% 9.10/5.28              = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6275_sum_Omono__neutral__right,axiom,
% 9.10/5.28      ! [T3: set_int,S2: set_int,G: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: int] :
% 9.10/5.28                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 9.10/5.28              = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6276_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5754745047067104278omplex @ G @ S2 )
% 9.10/5.28                = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6277_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups2073611262835488442omplex @ G @ S2 )
% 9.10/5.28                = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6278_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups8097168146408367636l_real @ G @ S2 )
% 9.10/5.28                = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6279_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5808333547571424918x_real @ G @ S2 )
% 9.10/5.28                = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6280_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups1300246762558778688al_rat @ G @ S2 )
% 9.10/5.28                = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6281_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 9.10/5.28                = ( groups2906978787729119204at_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6282_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 9.10/5.28                = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6283_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_nat ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups1935376822645274424al_nat @ G @ S2 )
% 9.10/5.28                = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6284_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_nat ) )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 9.10/5.28                = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6285_sum_Omono__neutral__cong__left,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( H2 @ X5 )
% 9.10/5.28                  = zero_zero_int ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups1932886352136224148al_int @ G @ S2 )
% 9.10/5.28                = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6286_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5754745047067104278omplex @ G @ T3 )
% 9.10/5.28                = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6287_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_complex ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups2073611262835488442omplex @ G @ T3 )
% 9.10/5.28                = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6288_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups8097168146408367636l_real @ G @ T3 )
% 9.10/5.28                = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6289_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_real ) )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5808333547571424918x_real @ G @ T3 )
% 9.10/5.28                = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6290_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 9.10/5.28                = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6291_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_nat,S2: set_nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: nat] :
% 9.10/5.28                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ! [X5: nat] :
% 9.10/5.28                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 9.10/5.28                = ( groups2906978787729119204at_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6292_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_rat ) )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 9.10/5.28                = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6293_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_nat ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 9.10/5.28                = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6294_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: complex] :
% 9.10/5.28                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_nat ) )
% 9.10/5.28           => ( ! [X5: complex] :
% 9.10/5.28                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 9.10/5.28                = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6295_sum_Omono__neutral__cong__right,axiom,
% 9.10/5.28      ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
% 9.10/5.28        ( ( finite_finite_real @ T3 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.28         => ( ! [X5: real] :
% 9.10/5.28                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.28               => ( ( G @ X5 )
% 9.10/5.28                  = zero_zero_int ) )
% 9.10/5.28           => ( ! [X5: real] :
% 9.10/5.28                  ( ( member_real @ X5 @ S2 )
% 9.10/5.28                 => ( ( G @ X5 )
% 9.10/5.28                    = ( H2 @ X5 ) ) )
% 9.10/5.28             => ( ( groups1932886352136224148al_int @ G @ T3 )
% 9.10/5.28                = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6296_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,G: complex > real] :
% 9.10/5.28        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28         => ( ( groups5808333547571424918x_real @ G @ A4 )
% 9.10/5.28            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6297_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_nat,A4: set_nat,G: nat > rat] :
% 9.10/5.28        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.28       => ( ( finite_finite_nat @ A4 )
% 9.10/5.28         => ( ( groups2906978787729119204at_rat @ G @ A4 )
% 9.10/5.28            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6298_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,G: complex > rat] :
% 9.10/5.28        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28         => ( ( groups5058264527183730370ex_rat @ G @ A4 )
% 9.10/5.28            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6299_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,G: complex > nat] :
% 9.10/5.28        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28         => ( ( groups5693394587270226106ex_nat @ G @ A4 )
% 9.10/5.28            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6300_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_nat,A4: set_nat,G: nat > int] :
% 9.10/5.28        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.28       => ( ( finite_finite_nat @ A4 )
% 9.10/5.28         => ( ( groups3539618377306564664at_int @ G @ A4 )
% 9.10/5.28            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6301_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,G: complex > int] :
% 9.10/5.28        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28         => ( ( groups5690904116761175830ex_int @ G @ A4 )
% 9.10/5.28            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6302_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_int,A4: set_int,G: int > real] :
% 9.10/5.28        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28       => ( ( finite_finite_int @ A4 )
% 9.10/5.28         => ( ( groups8778361861064173332t_real @ G @ A4 )
% 9.10/5.28            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6303_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_int,A4: set_int,G: int > rat] :
% 9.10/5.28        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28       => ( ( finite_finite_int @ A4 )
% 9.10/5.28         => ( ( groups3906332499630173760nt_rat @ G @ A4 )
% 9.10/5.28            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6304_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_int,A4: set_int,G: int > nat] :
% 9.10/5.28        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28       => ( ( finite_finite_int @ A4 )
% 9.10/5.28         => ( ( groups4541462559716669496nt_nat @ G @ A4 )
% 9.10/5.28            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6305_sum_Osubset__diff,axiom,
% 9.10/5.28      ! [B5: set_int,A4: set_int,G: int > int] :
% 9.10/5.28        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28       => ( ( finite_finite_int @ A4 )
% 9.10/5.28         => ( ( groups4538972089207619220nt_int @ G @ A4 )
% 9.10/5.28            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6306_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_nat,B5: set_nat,F: nat > complex] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.28         => ( ( groups2073611262835488442omplex @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ A4 ) @ ( groups2073611262835488442omplex @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6307_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_complex,B5: set_complex,F: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6308_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_nat,B5: set_nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.28         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6309_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_complex,B5: set_complex,F: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6310_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_nat,B5: set_nat,F: nat > int] :
% 9.10/5.28        ( ( finite_finite_nat @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.28         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6311_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_complex,B5: set_complex,F: complex > int] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.28         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6312_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_int,B5: set_int,F: int > complex] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28         => ( ( groups3049146728041665814omplex @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_complex @ ( groups3049146728041665814omplex @ F @ A4 ) @ ( groups3049146728041665814omplex @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6313_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_int,B5: set_int,F: int > real] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6314_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_int,B5: set_int,F: int > rat] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6315_sum__diff,axiom,
% 9.10/5.28      ! [A4: set_int,B5: set_int,F: int > int] :
% 9.10/5.28        ( ( finite_finite_int @ A4 )
% 9.10/5.28       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.28         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 9.10/5.28            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6316_ln__add__one__self__le__self,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6317_ln__add__one__self__le__self2,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.28       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6318_ln__mult,axiom,
% 9.10/5.28      ! [X3: real,Y3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.28         => ( ( ln_ln_real @ ( times_times_real @ X3 @ Y3 ) )
% 9.10/5.28            = ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6319_ln__eq__minus__one,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ( ln_ln_real @ X3 )
% 9.10/5.28            = ( minus_minus_real @ X3 @ one_one_real ) )
% 9.10/5.28         => ( X3 = one_one_real ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6320_ln__div,axiom,
% 9.10/5.28      ! [X3: real,Y3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.28         => ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.10/5.28            = ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6321_numeral__Bit1__div__2,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.28        = ( numeral_numeral_nat @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6322_numeral__Bit1__div__2,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.28        = ( numeral_numeral_int @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6323_odd__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6324_odd__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6325_odd__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6326_cong__exp__iff__simps_I3_J,axiom,
% 9.10/5.28      ! [N: num,Q2: num] :
% 9.10/5.28        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != zero_zero_nat ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6327_cong__exp__iff__simps_I3_J,axiom,
% 9.10/5.28      ! [N: num,Q2: num] :
% 9.10/5.28        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != zero_zero_int ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6328_cong__exp__iff__simps_I3_J,axiom,
% 9.10/5.28      ! [N: num,Q2: num] :
% 9.10/5.28        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.10/5.28       != zero_z3403309356797280102nteger ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6329_power3__eq__cube,axiom,
% 9.10/5.28      ! [A: complex] :
% 9.10/5.28        ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.10/5.28        = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6330_power3__eq__cube,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.10/5.28        = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6331_power3__eq__cube,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.10/5.28        = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6332_power3__eq__cube,axiom,
% 9.10/5.28      ! [A: nat] :
% 9.10/5.28        ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.10/5.28        = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6333_power3__eq__cube,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.10/5.28        = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6334_numeral__3__eq__3,axiom,
% 9.10/5.28      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 9.10/5.28      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6335_Suc3__eq__add__3,axiom,
% 9.10/5.28      ! [N: nat] :
% 9.10/5.28        ( ( suc @ ( suc @ ( suc @ N ) ) )
% 9.10/5.28        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6336_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,F: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: real] :
% 9.10/5.28                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6337_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,F: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: complex] :
% 9.10/5.28                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6338_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,F: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: real] :
% 9.10/5.28                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6339_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_nat,A4: set_nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: nat] :
% 9.10/5.28                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6340_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,F: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: complex] :
% 9.10/5.28                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6341_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,F: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: real] :
% 9.10/5.28                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6342_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,F: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: complex] :
% 9.10/5.28                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6343_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,F: real > int] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: real] :
% 9.10/5.28                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6344_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_nat,A4: set_nat,F: nat > int] :
% 9.10/5.28        ( ( finite_finite_nat @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: nat] :
% 9.10/5.28                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6345_sum__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,F: complex > int] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ! [B3: complex] :
% 9.10/5.28                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 9.10/5.28           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6346_num_Osize_I6_J,axiom,
% 9.10/5.28      ! [X32: num] :
% 9.10/5.28        ( ( size_size_num @ ( bit1 @ X32 ) )
% 9.10/5.28        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6347_ln__le__minus__one,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6348_ln__one__minus__pos__upper__bound,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.28         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6349_ln__diff__le,axiom,
% 9.10/5.28      ! [X3: real,Y3: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.28         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6350_cong__exp__iff__simps_I7_J,axiom,
% 9.10/5.28      ! [Q2: num,N: num] :
% 9.10/5.28        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 9.10/5.28          = zero_zero_nat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6351_cong__exp__iff__simps_I7_J,axiom,
% 9.10/5.28      ! [Q2: num,N: num] :
% 9.10/5.28        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 9.10/5.28          = zero_zero_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6352_cong__exp__iff__simps_I7_J,axiom,
% 9.10/5.28      ! [Q2: num,N: num] :
% 9.10/5.28        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 9.10/5.28          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6353_cong__exp__iff__simps_I11_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num] :
% 9.10/5.28        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 9.10/5.28          = zero_zero_nat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6354_cong__exp__iff__simps_I11_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num] :
% 9.10/5.28        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 9.10/5.28          = zero_zero_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6355_cong__exp__iff__simps_I11_J,axiom,
% 9.10/5.28      ! [M: num,Q2: num] :
% 9.10/5.28        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 9.10/5.28          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 9.10/5.28        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 9.10/5.28          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6356_Suc__div__eq__add3__div,axiom,
% 9.10/5.28      ! [M: nat,N: nat] :
% 9.10/5.28        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 9.10/5.28        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6357_Suc__mod__eq__add3__mod,axiom,
% 9.10/5.28      ! [M: nat,N: nat] :
% 9.10/5.28        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 9.10/5.28        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6358_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,B2: real,F: real > real] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: real] :
% 9.10/5.28                    ( ( member_real @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6359_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > real] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_real @ zero_zero_real @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: complex] :
% 9.10/5.28                    ( ( member_complex @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6360_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,B2: real,F: real > rat] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: real] :
% 9.10/5.28                    ( ( member_real @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6361_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_nat,A4: set_nat,B2: nat,F: nat > rat] :
% 9.10/5.28        ( ( finite_finite_nat @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.28         => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: nat] :
% 9.10/5.28                    ( ( member_nat @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6362_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > rat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: complex] :
% 9.10/5.28                    ( ( member_complex @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6363_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,B2: real,F: real > nat] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: real] :
% 9.10/5.28                    ( ( member_real @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6364_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > nat] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: complex] :
% 9.10/5.28                    ( ( member_complex @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6365_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_real,A4: set_real,B2: real,F: real > int] :
% 9.10/5.28        ( ( finite_finite_real @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.28         => ( ( member_real @ B2 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: real] :
% 9.10/5.28                    ( ( member_real @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6366_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_nat,A4: set_nat,B2: nat,F: nat > int] :
% 9.10/5.28        ( ( finite_finite_nat @ B5 )
% 9.10/5.28       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.28         => ( ( member_nat @ B2 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: nat] :
% 9.10/5.28                    ( ( member_nat @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6367_sum__strict__mono2,axiom,
% 9.10/5.28      ! [B5: set_complex,A4: set_complex,B2: complex,F: complex > int] :
% 9.10/5.28        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.28       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.28         => ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.28           => ( ( ord_less_int @ zero_zero_int @ ( F @ B2 ) )
% 9.10/5.28             => ( ! [X5: complex] :
% 9.10/5.28                    ( ( member_complex @ X5 @ B5 )
% 9.10/5.28                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.28               => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6368_divmod__step__nat__def,axiom,
% 9.10/5.28      ( unique5026877609467782581ep_nat
% 9.10/5.28      = ( ^ [L2: num] :
% 9.10/5.28            ( produc2626176000494625587at_nat
% 9.10/5.28            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6369_ln__one__plus__pos__lower__bound,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.28       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.28         => ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6370_divmod__step__int__def,axiom,
% 9.10/5.28      ( unique5024387138958732305ep_int
% 9.10/5.28      = ( ^ [L2: num] :
% 9.10/5.28            ( produc4245557441103728435nt_int
% 9.10/5.28            @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6371_odd__mod__4__div__2,axiom,
% 9.10/5.28      ! [N: nat] :
% 9.10/5.28        ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.28          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.10/5.28       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6372_mod__exhaust__less__4,axiom,
% 9.10/5.28      ! [M: nat] :
% 9.10/5.28        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.28          = zero_zero_nat )
% 9.10/5.28        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.28          = one_one_nat )
% 9.10/5.28        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.28          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.28        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.28          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6373_ln__2__less__1,axiom,
% 9.10/5.28      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6374_divmod__algorithm__code_I6_J,axiom,
% 9.10/5.28      ! [M: num,N: num] :
% 9.10/5.28        ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.28        = ( produc4245557441103728435nt_int
% 9.10/5.28          @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 9.10/5.28          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6375_divmod__algorithm__code_I6_J,axiom,
% 9.10/5.28      ! [M: num,N: num] :
% 9.10/5.28        ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.28        = ( produc2626176000494625587at_nat
% 9.10/5.28          @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 9.10/5.28          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6376_divmod__algorithm__code_I6_J,axiom,
% 9.10/5.28      ! [M: num,N: num] :
% 9.10/5.28        ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.28        = ( produc6916734918728496179nteger
% 9.10/5.28          @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 9.10/5.28          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6377_signed__take__bit__numeral__minus__bit1,axiom,
% 9.10/5.28      ! [L: num,K: num] :
% 9.10/5.28        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.10/5.28        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6378_dbl__dec__simps_I4_J,axiom,
% 9.10/5.28      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.28      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6379_dbl__dec__simps_I4_J,axiom,
% 9.10/5.28      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.28      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6380_dbl__dec__simps_I4_J,axiom,
% 9.10/5.28      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.28      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6381_dbl__dec__simps_I4_J,axiom,
% 9.10/5.28      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.28      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6382_dbl__dec__simps_I4_J,axiom,
% 9.10/5.28      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.28      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6383_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 9.10/5.28      ! [X3: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.28       => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.28         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6384_abs__idempotent,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 9.10/5.28        = ( abs_abs_real @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6385_abs__idempotent,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 9.10/5.28        = ( abs_abs_int @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6386_abs__idempotent,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.10/5.28        = ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6387_abs__idempotent,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 9.10/5.28        = ( abs_abs_rat @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6388_abs__abs,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 9.10/5.28        = ( abs_abs_real @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6389_abs__abs,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 9.10/5.28        = ( abs_abs_int @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6390_abs__abs,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.10/5.28        = ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6391_abs__abs,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 9.10/5.28        = ( abs_abs_rat @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6392_case__prodI,axiom,
% 9.10/5.28      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.28        ( ( F @ A @ B2 )
% 9.10/5.28       => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6393_case__prodI,axiom,
% 9.10/5.28      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.28        ( ( F @ A @ B2 )
% 9.10/5.28       => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6394_case__prodI,axiom,
% 9.10/5.28      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
% 9.10/5.28        ( ( F @ A @ B2 )
% 9.10/5.28       => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6395_case__prodI,axiom,
% 9.10/5.28      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 9.10/5.28        ( ( F @ A @ B2 )
% 9.10/5.28       => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6396_case__prodI,axiom,
% 9.10/5.28      ! [F: int > int > $o,A: int,B2: int] :
% 9.10/5.28        ( ( F @ A @ B2 )
% 9.10/5.28       => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6397_case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 9.10/5.28        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.10/5.28           => ( C @ A3 @ B3 ) )
% 9.10/5.28       => ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6398_case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 9.10/5.28        ( ! [A3: produc6241069584506657477e_term > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc8603105652947943368nteger @ A3 @ B3 ) )
% 9.10/5.28           => ( C @ A3 @ B3 ) )
% 9.10/5.28       => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6399_case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 9.10/5.28        ( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc5700946648718959541nt_int @ A3 @ B3 ) )
% 9.10/5.28           => ( C @ A3 @ B3 ) )
% 9.10/5.28       => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6400_case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 9.10/5.28        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.10/5.28           => ( C @ A3 @ B3 ) )
% 9.10/5.28       => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6401_case__prodI2,axiom,
% 9.10/5.28      ! [P4: product_prod_int_int,C: int > int > $o] :
% 9.10/5.28        ( ! [A3: int,B3: int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.10/5.28           => ( C @ A3 @ B3 ) )
% 9.10/5.28       => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6402_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: nat,C: int > int > set_nat,A: int,B2: int] :
% 9.10/5.28        ( ( member_nat @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6403_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: real,C: int > int > set_real,A: int,B2: int] :
% 9.10/5.28        ( ( member_real @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6404_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: int,C: int > int > set_int,A: int,B2: int] :
% 9.10/5.28        ( ( member_int @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6405_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: complex,C: int > int > set_complex,A: int,B2: int] :
% 9.10/5.28        ( ( member_complex @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6406_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.28        ( ( member_nat @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6407_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.28        ( ( member_real @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6408_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.28        ( ( member_int @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6409_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.28        ( ( member_complex @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ ( produc6137756002093451184nteger @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6410_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 9.10/5.28        ( ( member_nat @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6411_mem__case__prodI,axiom,
% 9.10/5.28      ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 9.10/5.28        ( ( member_real @ Z @ ( C @ A @ B2 ) )
% 9.10/5.28       => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B2 ) ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6412_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: product_prod_int_int,Z: nat,C: int > int > set_nat] :
% 9.10/5.28        ( ! [A3: int,B3: int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.10/5.28           => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6413_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: product_prod_int_int,Z: real,C: int > int > set_real] :
% 9.10/5.28        ( ! [A3: int,B3: int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.10/5.28           => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6414_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: product_prod_int_int,Z: int,C: int > int > set_int] :
% 9.10/5.28        ( ! [A3: int,B3: int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.10/5.28           => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6415_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: product_prod_int_int,Z: complex,C: int > int > set_complex] :
% 9.10/5.28        ( ! [A3: int,B3: int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( product_Pair_int_int @ A3 @ B3 ) )
% 9.10/5.28           => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6416_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc8763457246119570046nteger,Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
% 9.10/5.28        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.10/5.28           => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6417_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc8763457246119570046nteger,Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
% 9.10/5.28        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.10/5.28           => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6418_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc8763457246119570046nteger,Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
% 9.10/5.28        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.10/5.28           => ( member_int @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6419_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc8763457246119570046nteger,Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
% 9.10/5.28        ( ! [A3: code_integer > option6357759511663192854e_term,B3: produc8923325533196201883nteger] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc6137756002093451184nteger @ A3 @ B3 ) )
% 9.10/5.28           => ( member_complex @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6420_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc7773217078559923341nt_int,Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
% 9.10/5.28        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.10/5.28           => ( member_nat @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6421_mem__case__prodI2,axiom,
% 9.10/5.28      ! [P4: produc7773217078559923341nt_int,Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
% 9.10/5.28        ( ! [A3: int > option6357759511663192854e_term,B3: product_prod_int_int] :
% 9.10/5.28            ( ( P4
% 9.10/5.28              = ( produc4305682042979456191nt_int @ A3 @ B3 ) )
% 9.10/5.28           => ( member_real @ Z @ ( C @ A3 @ B3 ) ) )
% 9.10/5.28       => ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6422_case__prodI2_H,axiom,
% 9.10/5.28      ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
% 9.10/5.28        ( ! [A3: nat,B3: nat] :
% 9.10/5.28            ( ( ( product_Pair_nat_nat @ A3 @ B3 )
% 9.10/5.28              = P4 )
% 9.10/5.28           => ( C @ A3 @ B3 @ X3 ) )
% 9.10/5.28       => ( produc8739625826339149834_nat_o @ C @ P4 @ X3 ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6423_abs__0__eq,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( zero_z3403309356797280102nteger
% 9.10/5.28          = ( abs_abs_Code_integer @ A ) )
% 9.10/5.28        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6424_abs__0__eq,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( zero_zero_real
% 9.10/5.28          = ( abs_abs_real @ A ) )
% 9.10/5.28        = ( A = zero_zero_real ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6425_abs__0__eq,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( zero_zero_rat
% 9.10/5.28          = ( abs_abs_rat @ A ) )
% 9.10/5.28        = ( A = zero_zero_rat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6426_abs__0__eq,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( zero_zero_int
% 9.10/5.28          = ( abs_abs_int @ A ) )
% 9.10/5.28        = ( A = zero_zero_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6427_abs__eq__0,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( ( abs_abs_Code_integer @ A )
% 9.10/5.28          = zero_z3403309356797280102nteger )
% 9.10/5.28        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6428_abs__eq__0,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( ( abs_abs_real @ A )
% 9.10/5.28          = zero_zero_real )
% 9.10/5.28        = ( A = zero_zero_real ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6429_abs__eq__0,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( ( abs_abs_rat @ A )
% 9.10/5.28          = zero_zero_rat )
% 9.10/5.28        = ( A = zero_zero_rat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6430_abs__eq__0,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( ( abs_abs_int @ A )
% 9.10/5.28          = zero_zero_int )
% 9.10/5.28        = ( A = zero_zero_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6431_abs__zero,axiom,
% 9.10/5.28      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.10/5.28      = zero_z3403309356797280102nteger ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6432_abs__zero,axiom,
% 9.10/5.28      ( ( abs_abs_real @ zero_zero_real )
% 9.10/5.28      = zero_zero_real ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6433_abs__zero,axiom,
% 9.10/5.28      ( ( abs_abs_rat @ zero_zero_rat )
% 9.10/5.28      = zero_zero_rat ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6434_abs__zero,axiom,
% 9.10/5.28      ( ( abs_abs_int @ zero_zero_int )
% 9.10/5.28      = zero_zero_int ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6435_abs__0,axiom,
% 9.10/5.28      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.10/5.28      = zero_z3403309356797280102nteger ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6436_abs__0,axiom,
% 9.10/5.28      ( ( abs_abs_complex @ zero_zero_complex )
% 9.10/5.28      = zero_zero_complex ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6437_abs__0,axiom,
% 9.10/5.28      ( ( abs_abs_real @ zero_zero_real )
% 9.10/5.28      = zero_zero_real ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6438_abs__0,axiom,
% 9.10/5.28      ( ( abs_abs_rat @ zero_zero_rat )
% 9.10/5.28      = zero_zero_rat ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6439_abs__0,axiom,
% 9.10/5.28      ( ( abs_abs_int @ zero_zero_int )
% 9.10/5.28      = zero_zero_int ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6440_abs__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 9.10/5.28        = ( numera6620942414471956472nteger @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6441_abs__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 9.10/5.28        = ( numeral_numeral_real @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6442_abs__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 9.10/5.28        = ( numeral_numeral_rat @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6443_abs__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 9.10/5.28        = ( numeral_numeral_int @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6444_abs__mult__self__eq,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 9.10/5.28        = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6445_abs__mult__self__eq,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 9.10/5.28        = ( times_times_real @ A @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6446_abs__mult__self__eq,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 9.10/5.28        = ( times_times_rat @ A @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6447_abs__mult__self__eq,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 9.10/5.28        = ( times_times_int @ A @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6448_abs__1,axiom,
% 9.10/5.28      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 9.10/5.28      = one_one_Code_integer ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6449_abs__1,axiom,
% 9.10/5.28      ( ( abs_abs_complex @ one_one_complex )
% 9.10/5.28      = one_one_complex ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6450_abs__1,axiom,
% 9.10/5.28      ( ( abs_abs_real @ one_one_real )
% 9.10/5.28      = one_one_real ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6451_abs__1,axiom,
% 9.10/5.28      ( ( abs_abs_rat @ one_one_rat )
% 9.10/5.28      = one_one_rat ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6452_abs__1,axiom,
% 9.10/5.28      ( ( abs_abs_int @ one_one_int )
% 9.10/5.28      = one_one_int ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6453_abs__add__abs,axiom,
% 9.10/5.28      ! [A: code_integer,B2: code_integer] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) )
% 9.10/5.28        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6454_abs__add__abs,axiom,
% 9.10/5.28      ! [A: real,B2: real] :
% 9.10/5.28        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) )
% 9.10/5.28        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6455_abs__add__abs,axiom,
% 9.10/5.28      ! [A: rat,B2: rat] :
% 9.10/5.28        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) )
% 9.10/5.28        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6456_abs__add__abs,axiom,
% 9.10/5.28      ! [A: int,B2: int] :
% 9.10/5.28        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) )
% 9.10/5.28        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6457_abs__divide,axiom,
% 9.10/5.28      ! [A: complex,B2: complex] :
% 9.10/5.28        ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.28        = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6458_abs__divide,axiom,
% 9.10/5.28      ! [A: real,B2: real] :
% 9.10/5.28        ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.28        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6459_abs__divide,axiom,
% 9.10/5.28      ! [A: rat,B2: rat] :
% 9.10/5.28        ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.28        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6460_abs__minus__cancel,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 9.10/5.28        = ( abs_abs_real @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6461_abs__minus__cancel,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 9.10/5.28        = ( abs_abs_int @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6462_abs__minus__cancel,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.28        = ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6463_abs__minus__cancel,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 9.10/5.28        = ( abs_abs_rat @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6464_abs__minus,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 9.10/5.28        = ( abs_abs_real @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6465_abs__minus,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 9.10/5.28        = ( abs_abs_int @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6466_abs__minus,axiom,
% 9.10/5.28      ! [A: complex] :
% 9.10/5.28        ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 9.10/5.28        = ( abs_abs_complex @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6467_abs__minus,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 9.10/5.28        = ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6468_abs__minus,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 9.10/5.28        = ( abs_abs_rat @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6469_abs__dvd__iff,axiom,
% 9.10/5.28      ! [M: real,K: real] :
% 9.10/5.28        ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 9.10/5.28        = ( dvd_dvd_real @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6470_abs__dvd__iff,axiom,
% 9.10/5.28      ! [M: int,K: int] :
% 9.10/5.28        ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 9.10/5.28        = ( dvd_dvd_int @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6471_abs__dvd__iff,axiom,
% 9.10/5.28      ! [M: code_integer,K: code_integer] :
% 9.10/5.28        ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 9.10/5.28        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6472_abs__dvd__iff,axiom,
% 9.10/5.28      ! [M: rat,K: rat] :
% 9.10/5.28        ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 9.10/5.28        = ( dvd_dvd_rat @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6473_dvd__abs__iff,axiom,
% 9.10/5.28      ! [M: real,K: real] :
% 9.10/5.28        ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 9.10/5.28        = ( dvd_dvd_real @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6474_dvd__abs__iff,axiom,
% 9.10/5.28      ! [M: int,K: int] :
% 9.10/5.28        ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 9.10/5.28        = ( dvd_dvd_int @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6475_dvd__abs__iff,axiom,
% 9.10/5.28      ! [M: code_integer,K: code_integer] :
% 9.10/5.28        ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 9.10/5.28        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6476_dvd__abs__iff,axiom,
% 9.10/5.28      ! [M: rat,K: rat] :
% 9.10/5.28        ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 9.10/5.28        = ( dvd_dvd_rat @ M @ K ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6477_abs__bool__eq,axiom,
% 9.10/5.28      ! [P: $o] :
% 9.10/5.28        ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 9.10/5.28        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6478_abs__bool__eq,axiom,
% 9.10/5.28      ! [P: $o] :
% 9.10/5.28        ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 9.10/5.28        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6479_abs__bool__eq,axiom,
% 9.10/5.28      ! [P: $o] :
% 9.10/5.28        ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.10/5.28        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6480_abs__bool__eq,axiom,
% 9.10/5.28      ! [P: $o] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 9.10/5.28        = ( zero_n356916108424825756nteger @ P ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6481_dbl__dec__simps_I3_J,axiom,
% 9.10/5.28      ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 9.10/5.28      = one_one_complex ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6482_dbl__dec__simps_I3_J,axiom,
% 9.10/5.28      ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 9.10/5.28      = one_one_real ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6483_dbl__dec__simps_I3_J,axiom,
% 9.10/5.28      ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 9.10/5.28      = one_one_rat ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6484_dbl__dec__simps_I3_J,axiom,
% 9.10/5.28      ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 9.10/5.28      = one_one_int ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6485_abs__le__zero__iff,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 9.10/5.28        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6486_abs__le__zero__iff,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 9.10/5.28        = ( A = zero_zero_real ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6487_abs__le__zero__iff,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 9.10/5.28        = ( A = zero_zero_rat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6488_abs__le__zero__iff,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 9.10/5.28        = ( A = zero_zero_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6489_abs__le__self__iff,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 9.10/5.28        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6490_abs__le__self__iff,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 9.10/5.28        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6491_abs__le__self__iff,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 9.10/5.28        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6492_abs__le__self__iff,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 9.10/5.28        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6493_abs__of__nonneg,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.28       => ( ( abs_abs_Code_integer @ A )
% 9.10/5.28          = A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6494_abs__of__nonneg,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.28       => ( ( abs_abs_real @ A )
% 9.10/5.28          = A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6495_abs__of__nonneg,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.28       => ( ( abs_abs_rat @ A )
% 9.10/5.28          = A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6496_abs__of__nonneg,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.28       => ( ( abs_abs_int @ A )
% 9.10/5.28          = A ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6497_zero__less__abs__iff,axiom,
% 9.10/5.28      ! [A: code_integer] :
% 9.10/5.28        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 9.10/5.28        = ( A != zero_z3403309356797280102nteger ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6498_zero__less__abs__iff,axiom,
% 9.10/5.28      ! [A: real] :
% 9.10/5.28        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 9.10/5.28        = ( A != zero_zero_real ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6499_zero__less__abs__iff,axiom,
% 9.10/5.28      ! [A: rat] :
% 9.10/5.28        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 9.10/5.28        = ( A != zero_zero_rat ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6500_zero__less__abs__iff,axiom,
% 9.10/5.28      ! [A: int] :
% 9.10/5.28        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 9.10/5.28        = ( A != zero_zero_int ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6501_abs__neg__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.28        = ( numeral_numeral_real @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6502_abs__neg__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.28        = ( numeral_numeral_int @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6503_abs__neg__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.10/5.28        = ( numera6620942414471956472nteger @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6504_abs__neg__numeral,axiom,
% 9.10/5.28      ! [N: num] :
% 9.10/5.28        ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.10/5.28        = ( numeral_numeral_rat @ N ) ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6505_abs__neg__one,axiom,
% 9.10/5.28      ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.28      = one_one_real ) ).
% 9.10/5.28  
% 9.10/5.28  thf(fact_6506_abs__neg__one,axiom,
% 9.10/5.28      ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.28      = one_one_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6507_abs__neg__one,axiom,
% 9.10/5.29      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.29      = one_one_Code_integer ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6508_abs__neg__one,axiom,
% 9.10/5.29      ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.29      = one_one_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6509_abs__power__minus,axiom,
% 9.10/5.29      ! [A: real,N: nat] :
% 9.10/5.29        ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 9.10/5.29        = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6510_abs__power__minus,axiom,
% 9.10/5.29      ! [A: int,N: nat] :
% 9.10/5.29        ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 9.10/5.29        = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6511_abs__power__minus,axiom,
% 9.10/5.29      ! [A: code_integer,N: nat] :
% 9.10/5.29        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 9.10/5.29        = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6512_abs__power__minus,axiom,
% 9.10/5.29      ! [A: rat,N: nat] :
% 9.10/5.29        ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 9.10/5.29        = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6513_pred__numeral__simps_I1_J,axiom,
% 9.10/5.29      ( ( pred_numeral @ one )
% 9.10/5.29      = zero_zero_nat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6514_eq__numeral__Suc,axiom,
% 9.10/5.29      ! [K: num,N: nat] :
% 9.10/5.29        ( ( ( numeral_numeral_nat @ K )
% 9.10/5.29          = ( suc @ N ) )
% 9.10/5.29        = ( ( pred_numeral @ K )
% 9.10/5.29          = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6515_Suc__eq__numeral,axiom,
% 9.10/5.29      ! [N: nat,K: num] :
% 9.10/5.29        ( ( ( suc @ N )
% 9.10/5.29          = ( numeral_numeral_nat @ K ) )
% 9.10/5.29        = ( N
% 9.10/5.29          = ( pred_numeral @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6516_sum__abs,axiom,
% 9.10/5.29      ! [F: int > int,A4: set_int] :
% 9.10/5.29        ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
% 9.10/5.29        @ ( groups4538972089207619220nt_int
% 9.10/5.29          @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6517_sum__abs,axiom,
% 9.10/5.29      ! [F: nat > real,A4: set_nat] :
% 9.10/5.29        ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
% 9.10/5.29        @ ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6518_zero__le__divide__abs__iff,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) )
% 9.10/5.29        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.29          | ( B2 = zero_zero_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6519_zero__le__divide__abs__iff,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) )
% 9.10/5.29        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.29          | ( B2 = zero_zero_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6520_divide__le__0__abs__iff,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
% 9.10/5.29        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.10/5.29          | ( B2 = zero_zero_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6521_divide__le__0__abs__iff,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
% 9.10/5.29        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.10/5.29          | ( B2 = zero_zero_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6522_abs__of__nonpos,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.10/5.29       => ( ( abs_abs_real @ A )
% 9.10/5.29          = ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6523_abs__of__nonpos,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 9.10/5.29       => ( ( abs_abs_Code_integer @ A )
% 9.10/5.29          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6524_abs__of__nonpos,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.10/5.29       => ( ( abs_abs_rat @ A )
% 9.10/5.29          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6525_abs__of__nonpos,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.10/5.29       => ( ( abs_abs_int @ A )
% 9.10/5.29          = ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6526_less__numeral__Suc,axiom,
% 9.10/5.29      ! [K: num,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.10/5.29        = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6527_less__Suc__numeral,axiom,
% 9.10/5.29      ! [N: nat,K: num] :
% 9.10/5.29        ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.29        = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6528_artanh__minus__real,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.29       => ( ( artanh_real @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.29          = ( uminus_uminus_real @ ( artanh_real @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6529_pred__numeral__simps_I3_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( pred_numeral @ ( bit1 @ K ) )
% 9.10/5.29        = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6530_le__numeral__Suc,axiom,
% 9.10/5.29      ! [K: num,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6531_le__Suc__numeral,axiom,
% 9.10/5.29      ! [N: nat,K: num] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.29        = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6532_diff__Suc__numeral,axiom,
% 9.10/5.29      ! [N: nat,K: num] :
% 9.10/5.29        ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.29        = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6533_diff__numeral__Suc,axiom,
% 9.10/5.29      ! [K: num,N: nat] :
% 9.10/5.29        ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.10/5.29        = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6534_max__Suc__numeral,axiom,
% 9.10/5.29      ! [N: nat,K: num] :
% 9.10/5.29        ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.29        = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6535_max__numeral__Suc,axiom,
% 9.10/5.29      ! [K: num,N: nat] :
% 9.10/5.29        ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.10/5.29        = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6536_sum__abs__ge__zero,axiom,
% 9.10/5.29      ! [F: int > int,A4: set_int] :
% 9.10/5.29        ( ord_less_eq_int @ zero_zero_int
% 9.10/5.29        @ ( groups4538972089207619220nt_int
% 9.10/5.29          @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6537_sum__abs__ge__zero,axiom,
% 9.10/5.29      ! [F: nat > real,A4: set_nat] :
% 9.10/5.29        ( ord_less_eq_real @ zero_zero_real
% 9.10/5.29        @ ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6538_dbl__dec__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 9.10/5.29      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6539_dbl__dec__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 9.10/5.29      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6540_dbl__dec__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 9.10/5.29      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6541_dbl__dec__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 9.10/5.29      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6542_dbl__dec__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 9.10/5.29      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6543_numeral__div__minus__numeral,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.29        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6544_minus__numeral__div__numeral,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.29        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6545_zero__less__power__abs__iff,axiom,
% 9.10/5.29      ! [A: code_integer,N: nat] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 9.10/5.29        = ( ( A != zero_z3403309356797280102nteger )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6546_zero__less__power__abs__iff,axiom,
% 9.10/5.29      ! [A: real,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 9.10/5.29        = ( ( A != zero_zero_real )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6547_zero__less__power__abs__iff,axiom,
% 9.10/5.29      ! [A: rat,N: nat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 9.10/5.29        = ( ( A != zero_zero_rat )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6548_zero__less__power__abs__iff,axiom,
% 9.10/5.29      ! [A: int,N: nat] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 9.10/5.29        = ( ( A != zero_zero_int )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6549_abs__power2,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6550_abs__power2,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6551_abs__power2,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6552_abs__power2,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6553_power2__abs,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6554_power2__abs,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29        = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6555_power2__abs,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29        = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6556_power2__abs,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29        = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6557_sum_Ocl__ivl__Suc,axiom,
% 9.10/5.29      ! [N: nat,M: nat,G: nat > complex] :
% 9.10/5.29        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = zero_zero_complex ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6558_sum_Ocl__ivl__Suc,axiom,
% 9.10/5.29      ! [N: nat,M: nat,G: nat > rat] :
% 9.10/5.29        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = zero_zero_rat ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6559_sum_Ocl__ivl__Suc,axiom,
% 9.10/5.29      ! [N: nat,M: nat,G: nat > int] :
% 9.10/5.29        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = zero_zero_int ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6560_sum_Ocl__ivl__Suc,axiom,
% 9.10/5.29      ! [N: nat,M: nat,G: nat > nat] :
% 9.10/5.29        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = zero_zero_nat ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6561_sum_Ocl__ivl__Suc,axiom,
% 9.10/5.29      ! [N: nat,M: nat,G: nat > real] :
% 9.10/5.29        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = zero_zero_real ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.29         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6562_dvd__numeral__simp,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.29        = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6563_dvd__numeral__simp,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6564_dvd__numeral__simp,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 9.10/5.29        = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6565_divmod__algorithm__code_I2_J,axiom,
% 9.10/5.29      ! [M: num] :
% 9.10/5.29        ( ( unique5052692396658037445od_int @ M @ one )
% 9.10/5.29        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6566_divmod__algorithm__code_I2_J,axiom,
% 9.10/5.29      ! [M: num] :
% 9.10/5.29        ( ( unique5055182867167087721od_nat @ M @ one )
% 9.10/5.29        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6567_divmod__algorithm__code_I2_J,axiom,
% 9.10/5.29      ! [M: num] :
% 9.10/5.29        ( ( unique3479559517661332726nteger @ M @ one )
% 9.10/5.29        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6568_sum__zero__power,axiom,
% 9.10/5.29      ! [A4: set_nat,C: nat > complex] :
% 9.10/5.29        ( ( ( ( finite_finite_nat @ A4 )
% 9.10/5.29            & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = ( C @ zero_zero_nat ) ) )
% 9.10/5.29        & ( ~ ( ( finite_finite_nat @ A4 )
% 9.10/5.29              & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = zero_zero_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6569_sum__zero__power,axiom,
% 9.10/5.29      ! [A4: set_nat,C: nat > rat] :
% 9.10/5.29        ( ( ( ( finite_finite_nat @ A4 )
% 9.10/5.29            & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = ( C @ zero_zero_nat ) ) )
% 9.10/5.29        & ( ~ ( ( finite_finite_nat @ A4 )
% 9.10/5.29              & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = zero_zero_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6570_sum__zero__power,axiom,
% 9.10/5.29      ! [A4: set_nat,C: nat > real] :
% 9.10/5.29        ( ( ( ( finite_finite_nat @ A4 )
% 9.10/5.29            & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = ( C @ zero_zero_nat ) ) )
% 9.10/5.29        & ( ~ ( ( finite_finite_nat @ A4 )
% 9.10/5.29              & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = zero_zero_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6571_power__even__abs__numeral,axiom,
% 9.10/5.29      ! [W: num,A: code_integer] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29          = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6572_power__even__abs__numeral,axiom,
% 9.10/5.29      ! [W: num,A: rat] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29          = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6573_power__even__abs__numeral,axiom,
% 9.10/5.29      ! [W: num,A: real] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29       => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29          = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6574_power__even__abs__numeral,axiom,
% 9.10/5.29      ! [W: num,A: int] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29       => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 9.10/5.29          = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6575_divmod__algorithm__code_I3_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 9.10/5.29        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6576_divmod__algorithm__code_I3_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 9.10/5.29        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6577_divmod__algorithm__code_I3_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 9.10/5.29        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6578_divmod__algorithm__code_I4_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 9.10/5.29        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6579_divmod__algorithm__code_I4_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 9.10/5.29        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6580_divmod__algorithm__code_I4_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 9.10/5.29        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6581_sum__zero__power_H,axiom,
% 9.10/5.29      ! [A4: set_nat,C: nat > complex,D: nat > complex] :
% 9.10/5.29        ( ( ( ( finite_finite_nat @ A4 )
% 9.10/5.29            & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.10/5.29        & ( ~ ( ( finite_finite_nat @ A4 )
% 9.10/5.29              & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = zero_zero_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6582_sum__zero__power_H,axiom,
% 9.10/5.29      ! [A4: set_nat,C: nat > rat,D: nat > rat] :
% 9.10/5.29        ( ( ( ( finite_finite_nat @ A4 )
% 9.10/5.29            & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.10/5.29        & ( ~ ( ( finite_finite_nat @ A4 )
% 9.10/5.29              & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = zero_zero_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6583_sum__zero__power_H,axiom,
% 9.10/5.29      ! [A4: set_nat,C: nat > real,D: nat > real] :
% 9.10/5.29        ( ( ( ( finite_finite_nat @ A4 )
% 9.10/5.29            & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.10/5.29        & ( ~ ( ( finite_finite_nat @ A4 )
% 9.10/5.29              & ( member_nat @ zero_zero_nat @ A4 ) )
% 9.10/5.29         => ( ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 9.10/5.29              @ A4 )
% 9.10/5.29            = zero_zero_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6584_one__div__minus__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.29        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6585_minus__one__div__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.29        = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6586_signed__take__bit__numeral__bit0,axiom,
% 9.10/5.29      ! [L: num,K: num] :
% 9.10/5.29        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.10/5.29        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6587_divmod__algorithm__code_I5_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.10/5.29        = ( produc4245557441103728435nt_int
% 9.10/5.29          @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 9.10/5.29          @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6588_divmod__algorithm__code_I5_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.10/5.29        = ( produc2626176000494625587at_nat
% 9.10/5.29          @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 9.10/5.29          @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6589_divmod__algorithm__code_I5_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.10/5.29        = ( produc6916734918728496179nteger
% 9.10/5.29          @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 9.10/5.29          @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6590_signed__take__bit__numeral__minus__bit0,axiom,
% 9.10/5.29      ! [L: num,K: num] :
% 9.10/5.29        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.10/5.29        = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6591_divmod__algorithm__code_I8_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( ( ord_less_num @ M @ N )
% 9.10/5.29         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_num @ M @ N )
% 9.10/5.29         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6592_divmod__algorithm__code_I8_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( ( ord_less_num @ M @ N )
% 9.10/5.29         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_num @ M @ N )
% 9.10/5.29         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6593_divmod__algorithm__code_I8_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( ( ord_less_num @ M @ N )
% 9.10/5.29         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_num @ M @ N )
% 9.10/5.29         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6594_divmod__algorithm__code_I7_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( ( ord_less_eq_num @ M @ N )
% 9.10/5.29         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.10/5.29         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6595_divmod__algorithm__code_I7_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( ( ord_less_eq_num @ M @ N )
% 9.10/5.29         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.10/5.29         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6596_divmod__algorithm__code_I7_J,axiom,
% 9.10/5.29      ! [M: num,N: num] :
% 9.10/5.29        ( ( ( ord_less_eq_num @ M @ N )
% 9.10/5.29         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_num @ M @ N )
% 9.10/5.29         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.29            = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6597_signed__take__bit__numeral__bit1,axiom,
% 9.10/5.29      ! [L: num,K: num] :
% 9.10/5.29        ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.10/5.29        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6598_abs__ge__self,axiom,
% 9.10/5.29      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6599_abs__ge__self,axiom,
% 9.10/5.29      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6600_abs__ge__self,axiom,
% 9.10/5.29      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6601_abs__ge__self,axiom,
% 9.10/5.29      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6602_abs__le__D1,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 9.10/5.29       => ( ord_less_eq_real @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6603_abs__le__D1,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 9.10/5.29       => ( ord_le3102999989581377725nteger @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6604_abs__le__D1,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 9.10/5.29       => ( ord_less_eq_rat @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6605_abs__le__D1,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 9.10/5.29       => ( ord_less_eq_int @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6606_abs__eq__0__iff,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ( ( ( abs_abs_Code_integer @ A )
% 9.10/5.29          = zero_z3403309356797280102nteger )
% 9.10/5.29        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6607_abs__eq__0__iff,axiom,
% 9.10/5.29      ! [A: complex] :
% 9.10/5.29        ( ( ( abs_abs_complex @ A )
% 9.10/5.29          = zero_zero_complex )
% 9.10/5.29        = ( A = zero_zero_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6608_abs__eq__0__iff,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ( ( ( abs_abs_real @ A )
% 9.10/5.29          = zero_zero_real )
% 9.10/5.29        = ( A = zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6609_abs__eq__0__iff,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ( ( ( abs_abs_rat @ A )
% 9.10/5.29          = zero_zero_rat )
% 9.10/5.29        = ( A = zero_zero_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6610_abs__eq__0__iff,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ( ( ( abs_abs_int @ A )
% 9.10/5.29          = zero_zero_int )
% 9.10/5.29        = ( A = zero_zero_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6611_abs__mult,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 9.10/5.29        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6612_abs__mult,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.29        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6613_abs__mult,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.29        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6614_abs__mult,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 9.10/5.29        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6615_abs__one,axiom,
% 9.10/5.29      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 9.10/5.29      = one_one_Code_integer ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6616_abs__one,axiom,
% 9.10/5.29      ( ( abs_abs_real @ one_one_real )
% 9.10/5.29      = one_one_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6617_abs__one,axiom,
% 9.10/5.29      ( ( abs_abs_rat @ one_one_rat )
% 9.10/5.29      = one_one_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6618_abs__one,axiom,
% 9.10/5.29      ( ( abs_abs_int @ one_one_int )
% 9.10/5.29      = one_one_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6619_abs__minus__commute,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) )
% 9.10/5.29        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6620_abs__minus__commute,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) )
% 9.10/5.29        = ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6621_abs__minus__commute,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) )
% 9.10/5.29        = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6622_abs__minus__commute,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) )
% 9.10/5.29        = ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6623_abs__eq__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ( abs_abs_real @ X3 )
% 9.10/5.29          = ( abs_abs_real @ Y3 ) )
% 9.10/5.29        = ( ( X3 = Y3 )
% 9.10/5.29          | ( X3
% 9.10/5.29            = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6624_abs__eq__iff,axiom,
% 9.10/5.29      ! [X3: int,Y3: int] :
% 9.10/5.29        ( ( ( abs_abs_int @ X3 )
% 9.10/5.29          = ( abs_abs_int @ Y3 ) )
% 9.10/5.29        = ( ( X3 = Y3 )
% 9.10/5.29          | ( X3
% 9.10/5.29            = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6625_abs__eq__iff,axiom,
% 9.10/5.29      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.29        ( ( ( abs_abs_Code_integer @ X3 )
% 9.10/5.29          = ( abs_abs_Code_integer @ Y3 ) )
% 9.10/5.29        = ( ( X3 = Y3 )
% 9.10/5.29          | ( X3
% 9.10/5.29            = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6626_abs__eq__iff,axiom,
% 9.10/5.29      ! [X3: rat,Y3: rat] :
% 9.10/5.29        ( ( ( abs_abs_rat @ X3 )
% 9.10/5.29          = ( abs_abs_rat @ Y3 ) )
% 9.10/5.29        = ( ( X3 = Y3 )
% 9.10/5.29          | ( X3
% 9.10/5.29            = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6627_power__abs,axiom,
% 9.10/5.29      ! [A: code_integer,N: nat] :
% 9.10/5.29        ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 9.10/5.29        = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6628_power__abs,axiom,
% 9.10/5.29      ! [A: rat,N: nat] :
% 9.10/5.29        ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 9.10/5.29        = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6629_power__abs,axiom,
% 9.10/5.29      ! [A: real,N: nat] :
% 9.10/5.29        ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 9.10/5.29        = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6630_power__abs,axiom,
% 9.10/5.29      ! [A: int,N: nat] :
% 9.10/5.29        ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 9.10/5.29        = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6631_dvd__if__abs__eq,axiom,
% 9.10/5.29      ! [L: real,K: real] :
% 9.10/5.29        ( ( ( abs_abs_real @ L )
% 9.10/5.29          = ( abs_abs_real @ K ) )
% 9.10/5.29       => ( dvd_dvd_real @ L @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6632_dvd__if__abs__eq,axiom,
% 9.10/5.29      ! [L: int,K: int] :
% 9.10/5.29        ( ( ( abs_abs_int @ L )
% 9.10/5.29          = ( abs_abs_int @ K ) )
% 9.10/5.29       => ( dvd_dvd_int @ L @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6633_dvd__if__abs__eq,axiom,
% 9.10/5.29      ! [L: code_integer,K: code_integer] :
% 9.10/5.29        ( ( ( abs_abs_Code_integer @ L )
% 9.10/5.29          = ( abs_abs_Code_integer @ K ) )
% 9.10/5.29       => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6634_dvd__if__abs__eq,axiom,
% 9.10/5.29      ! [L: rat,K: rat] :
% 9.10/5.29        ( ( ( abs_abs_rat @ L )
% 9.10/5.29          = ( abs_abs_rat @ K ) )
% 9.10/5.29       => ( dvd_dvd_rat @ L @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6635_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: nat,C: int > int > set_nat,P4: product_prod_int_int] :
% 9.10/5.29        ( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: int,Y4: int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6636_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: real,C: int > int > set_real,P4: product_prod_int_int] :
% 9.10/5.29        ( ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: int,Y4: int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6637_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: int,C: int > int > set_int,P4: product_prod_int_int] :
% 9.10/5.29        ( ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: int,Y4: int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6638_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: complex,C: int > int > set_complex,P4: product_prod_int_int] :
% 9.10/5.29        ( ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: int,Y4: int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6639_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
% 9.10/5.29        ( ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6640_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
% 9.10/5.29        ( ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6641_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
% 9.10/5.29        ( ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_int @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6642_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P4: produc8763457246119570046nteger] :
% 9.10/5.29        ( ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_complex @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6643_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
% 9.10/5.29        ( ( member_nat @ Z @ ( produc8289552606927098482et_nat @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_nat @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6644_mem__case__prodE,axiom,
% 9.10/5.29      ! [Z: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
% 9.10/5.29        ( ( member_real @ Z @ ( produc8709739885379107790t_real @ C @ P4 ) )
% 9.10/5.29       => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( member_real @ Z @ ( C @ X5 @ Y4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6645_case__prodD,axiom,
% 9.10/5.29      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.29        ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B2 ) )
% 9.10/5.29       => ( F @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6646_case__prodD,axiom,
% 9.10/5.29      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B2: produc8923325533196201883nteger] :
% 9.10/5.29        ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B2 ) )
% 9.10/5.29       => ( F @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6647_case__prodD,axiom,
% 9.10/5.29      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
% 9.10/5.29        ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B2 ) )
% 9.10/5.29       => ( F @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6648_case__prodD,axiom,
% 9.10/5.29      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B2: product_prod_int_int] :
% 9.10/5.29        ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B2 ) )
% 9.10/5.29       => ( F @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6649_case__prodD,axiom,
% 9.10/5.29      ! [F: int > int > $o,A: int,B2: int] :
% 9.10/5.29        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B2 ) )
% 9.10/5.29       => ( F @ A @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6650_case__prodE,axiom,
% 9.10/5.29      ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
% 9.10/5.29        ( ( produc127349428274296955eger_o @ C @ P4 )
% 9.10/5.29       => ~ ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc6137756002093451184nteger @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6651_case__prodE,axiom,
% 9.10/5.29      ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
% 9.10/5.29        ( ( produc6253627499356882019eger_o @ C @ P4 )
% 9.10/5.29       => ~ ! [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc8603105652947943368nteger @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6652_case__prodE,axiom,
% 9.10/5.29      ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
% 9.10/5.29        ( ( produc1573362020775583542_int_o @ C @ P4 )
% 9.10/5.29       => ~ ! [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc5700946648718959541nt_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6653_case__prodE,axiom,
% 9.10/5.29      ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
% 9.10/5.29        ( ( produc2558449545302689196_int_o @ C @ P4 )
% 9.10/5.29       => ~ ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( produc4305682042979456191nt_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6654_case__prodE,axiom,
% 9.10/5.29      ! [C: int > int > $o,P4: product_prod_int_int] :
% 9.10/5.29        ( ( produc4947309494688390418_int_o @ C @ P4 )
% 9.10/5.29       => ~ ! [X5: int,Y4: int] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( product_Pair_int_int @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( C @ X5 @ Y4 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6655_case__prodD_H,axiom,
% 9.10/5.29      ! [R2: nat > nat > product_prod_nat_nat > $o,A: nat,B2: nat,C: product_prod_nat_nat] :
% 9.10/5.29        ( ( produc8739625826339149834_nat_o @ R2 @ ( product_Pair_nat_nat @ A @ B2 ) @ C )
% 9.10/5.29       => ( R2 @ A @ B2 @ C ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6656_case__prodE_H,axiom,
% 9.10/5.29      ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
% 9.10/5.29        ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
% 9.10/5.29       => ~ ! [X5: nat,Y4: nat] :
% 9.10/5.29              ( ( P4
% 9.10/5.29                = ( product_Pair_nat_nat @ X5 @ Y4 ) )
% 9.10/5.29             => ~ ( C @ X5 @ Y4 @ Z ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6657_abs__ge__zero,axiom,
% 9.10/5.29      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6658_abs__ge__zero,axiom,
% 9.10/5.29      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6659_abs__ge__zero,axiom,
% 9.10/5.29      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6660_abs__ge__zero,axiom,
% 9.10/5.29      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6661_abs__not__less__zero,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6662_abs__not__less__zero,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6663_abs__not__less__zero,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6664_abs__not__less__zero,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6665_abs__of__pos,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.29       => ( ( abs_abs_Code_integer @ A )
% 9.10/5.29          = A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6666_abs__of__pos,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.29       => ( ( abs_abs_real @ A )
% 9.10/5.29          = A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6667_abs__of__pos,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.29       => ( ( abs_abs_rat @ A )
% 9.10/5.29          = A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6668_abs__of__pos,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ A )
% 9.10/5.29       => ( ( abs_abs_int @ A )
% 9.10/5.29          = A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6669_abs__triangle__ineq,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6670_abs__triangle__ineq,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6671_abs__triangle__ineq,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6672_abs__triangle__ineq,axiom,
% 9.10/5.29      ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6673_abs__mult__less,axiom,
% 9.10/5.29      ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 9.10/5.29       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D )
% 9.10/5.29         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6674_abs__mult__less,axiom,
% 9.10/5.29      ! [A: real,C: real,B2: real,D: real] :
% 9.10/5.29        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 9.10/5.29       => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D )
% 9.10/5.29         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6675_abs__mult__less,axiom,
% 9.10/5.29      ! [A: rat,C: rat,B2: rat,D: rat] :
% 9.10/5.29        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 9.10/5.29       => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D )
% 9.10/5.29         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6676_abs__mult__less,axiom,
% 9.10/5.29      ! [A: int,C: int,B2: int,D: int] :
% 9.10/5.29        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 9.10/5.29       => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
% 9.10/5.29         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6677_abs__triangle__ineq2,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6678_abs__triangle__ineq2,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6679_abs__triangle__ineq2,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6680_abs__triangle__ineq2,axiom,
% 9.10/5.29      ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6681_abs__triangle__ineq3,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6682_abs__triangle__ineq3,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6683_abs__triangle__ineq3,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6684_abs__triangle__ineq3,axiom,
% 9.10/5.29      ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6685_abs__triangle__ineq2__sym,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6686_abs__triangle__ineq2__sym,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6687_abs__triangle__ineq2__sym,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6688_abs__triangle__ineq2__sym,axiom,
% 9.10/5.29      ! [A: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6689_nonzero__abs__divide,axiom,
% 9.10/5.29      ! [B2: real,A: real] :
% 9.10/5.29        ( ( B2 != zero_zero_real )
% 9.10/5.29       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.29          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6690_nonzero__abs__divide,axiom,
% 9.10/5.29      ! [B2: rat,A: rat] :
% 9.10/5.29        ( ( B2 != zero_zero_rat )
% 9.10/5.29       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.29          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6691_abs__leI,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
% 9.10/5.29         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6692_abs__leI,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 9.10/5.29       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 )
% 9.10/5.29         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6693_abs__leI,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.29       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 )
% 9.10/5.29         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6694_abs__leI,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.29       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
% 9.10/5.29         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6695_abs__le__D2,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 9.10/5.29       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6696_abs__le__D2,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 9.10/5.29       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6697_abs__le__D2,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 9.10/5.29       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6698_abs__le__D2,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 9.10/5.29       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6699_abs__le__iff,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.29          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6700_abs__le__iff,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_le3102999989581377725nteger @ A @ B2 )
% 9.10/5.29          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6701_abs__le__iff,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_less_eq_rat @ A @ B2 )
% 9.10/5.29          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6702_abs__le__iff,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_less_eq_int @ A @ B2 )
% 9.10/5.29          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6703_abs__ge__minus__self,axiom,
% 9.10/5.29      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6704_abs__ge__minus__self,axiom,
% 9.10/5.29      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6705_abs__ge__minus__self,axiom,
% 9.10/5.29      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6706_abs__ge__minus__self,axiom,
% 9.10/5.29      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6707_abs__less__iff,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_less_real @ A @ B2 )
% 9.10/5.29          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6708_abs__less__iff,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_less_int @ A @ B2 )
% 9.10/5.29          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6709_abs__less__iff,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_le6747313008572928689nteger @ A @ B2 )
% 9.10/5.29          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6710_abs__less__iff,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B2 )
% 9.10/5.29        = ( ( ord_less_rat @ A @ B2 )
% 9.10/5.29          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6711_sin__bound__lemma,axiom,
% 9.10/5.29      ! [X3: real,Y3: real,U: real,V: real] :
% 9.10/5.29        ( ( X3 = Y3 )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 9.10/5.29         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X3 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6712_numeral__eq__Suc,axiom,
% 9.10/5.29      ( numeral_numeral_nat
% 9.10/5.29      = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6713_sum__subtractf__nat,axiom,
% 9.10/5.29      ! [A4: set_real,G: real > nat,F: real > nat] :
% 9.10/5.29        ( ! [X5: real] :
% 9.10/5.29            ( ( member_real @ X5 @ A4 )
% 9.10/5.29           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 9.10/5.29       => ( ( groups1935376822645274424al_nat
% 9.10/5.29            @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.29            @ A4 )
% 9.10/5.29          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6714_sum__subtractf__nat,axiom,
% 9.10/5.29      ! [A4: set_int,G: int > nat,F: int > nat] :
% 9.10/5.29        ( ! [X5: int] :
% 9.10/5.29            ( ( member_int @ X5 @ A4 )
% 9.10/5.29           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 9.10/5.29       => ( ( groups4541462559716669496nt_nat
% 9.10/5.29            @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.29            @ A4 )
% 9.10/5.29          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6715_sum__subtractf__nat,axiom,
% 9.10/5.29      ! [A4: set_complex,G: complex > nat,F: complex > nat] :
% 9.10/5.29        ( ! [X5: complex] :
% 9.10/5.29            ( ( member_complex @ X5 @ A4 )
% 9.10/5.29           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 9.10/5.29       => ( ( groups5693394587270226106ex_nat
% 9.10/5.29            @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.29            @ A4 )
% 9.10/5.29          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6716_sum__subtractf__nat,axiom,
% 9.10/5.29      ! [A4: set_nat,G: nat > nat,F: nat > nat] :
% 9.10/5.29        ( ! [X5: nat] :
% 9.10/5.29            ( ( member_nat @ X5 @ A4 )
% 9.10/5.29           => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
% 9.10/5.29       => ( ( groups3542108847815614940at_nat
% 9.10/5.29            @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.10/5.29            @ A4 )
% 9.10/5.29          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6717_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 9.10/5.29      ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 9.10/5.29        = ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6718_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 9.10/5.29      ! [G: nat > real,M: nat,K: nat,N: nat] :
% 9.10/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 9.10/5.29        = ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6719_dense__eq0__I,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ! [E2: real] :
% 9.10/5.29            ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.10/5.29           => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E2 ) )
% 9.10/5.29       => ( X3 = zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6720_dense__eq0__I,axiom,
% 9.10/5.29      ! [X3: rat] :
% 9.10/5.29        ( ! [E2: rat] :
% 9.10/5.29            ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.10/5.29           => ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E2 ) )
% 9.10/5.29       => ( X3 = zero_zero_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6721_abs__mult__pos,axiom,
% 9.10/5.29      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 9.10/5.29       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X3 )
% 9.10/5.29          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6722_abs__mult__pos,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X3 )
% 9.10/5.29          = ( abs_abs_real @ ( times_times_real @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6723_abs__mult__pos,axiom,
% 9.10/5.29      ! [X3: rat,Y3: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.10/5.29       => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X3 )
% 9.10/5.29          = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6724_abs__mult__pos,axiom,
% 9.10/5.29      ! [X3: int,Y3: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.29       => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X3 )
% 9.10/5.29          = ( abs_abs_int @ ( times_times_int @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6725_abs__eq__mult,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.29            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 9.10/5.29          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.10/5.29            | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
% 9.10/5.29       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B2 ) )
% 9.10/5.29          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6726_abs__eq__mult,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.29            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.10/5.29          & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.10/5.29            | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
% 9.10/5.29       => ( ( abs_abs_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.29          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6727_abs__eq__mult,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.29            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.10/5.29          & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.10/5.29            | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
% 9.10/5.29       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.29          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6728_abs__eq__mult,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.29            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.10/5.29          & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.29            | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
% 9.10/5.29       => ( ( abs_abs_int @ ( times_times_int @ A @ B2 ) )
% 9.10/5.29          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6729_abs__eq__iff_H,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( ( abs_abs_real @ A )
% 9.10/5.29          = B2 )
% 9.10/5.29        = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
% 9.10/5.29          & ( ( A = B2 )
% 9.10/5.29            | ( A
% 9.10/5.29              = ( uminus_uminus_real @ B2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6730_abs__eq__iff_H,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( ( abs_abs_Code_integer @ A )
% 9.10/5.29          = B2 )
% 9.10/5.29        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
% 9.10/5.29          & ( ( A = B2 )
% 9.10/5.29            | ( A
% 9.10/5.29              = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6731_abs__eq__iff_H,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( ( abs_abs_rat @ A )
% 9.10/5.29          = B2 )
% 9.10/5.29        = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
% 9.10/5.29          & ( ( A = B2 )
% 9.10/5.29            | ( A
% 9.10/5.29              = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6732_abs__eq__iff_H,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( ( abs_abs_int @ A )
% 9.10/5.29          = B2 )
% 9.10/5.29        = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.29          & ( ( A = B2 )
% 9.10/5.29            | ( A
% 9.10/5.29              = ( uminus_uminus_int @ B2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6733_eq__abs__iff_H,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( A
% 9.10/5.29          = ( abs_abs_real @ B2 ) )
% 9.10/5.29        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.29          & ( ( B2 = A )
% 9.10/5.29            | ( B2
% 9.10/5.29              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6734_eq__abs__iff_H,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( A
% 9.10/5.29          = ( abs_abs_Code_integer @ B2 ) )
% 9.10/5.29        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.10/5.29          & ( ( B2 = A )
% 9.10/5.29            | ( B2
% 9.10/5.29              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6735_eq__abs__iff_H,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] :
% 9.10/5.29        ( ( A
% 9.10/5.29          = ( abs_abs_rat @ B2 ) )
% 9.10/5.29        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.10/5.29          & ( ( B2 = A )
% 9.10/5.29            | ( B2
% 9.10/5.29              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6736_eq__abs__iff_H,axiom,
% 9.10/5.29      ! [A: int,B2: int] :
% 9.10/5.29        ( ( A
% 9.10/5.29          = ( abs_abs_int @ B2 ) )
% 9.10/5.29        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.10/5.29          & ( ( B2 = A )
% 9.10/5.29            | ( B2
% 9.10/5.29              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6737_abs__minus__le__zero,axiom,
% 9.10/5.29      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6738_abs__minus__le__zero,axiom,
% 9.10/5.29      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6739_abs__minus__le__zero,axiom,
% 9.10/5.29      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6740_abs__minus__le__zero,axiom,
% 9.10/5.29      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6741_abs__div__pos,axiom,
% 9.10/5.29      ! [Y3: real,X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.29       => ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y3 )
% 9.10/5.29          = ( abs_abs_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6742_abs__div__pos,axiom,
% 9.10/5.29      ! [Y3: rat,X3: rat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.10/5.29       => ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y3 )
% 9.10/5.29          = ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6743_zero__le__power__abs,axiom,
% 9.10/5.29      ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6744_zero__le__power__abs,axiom,
% 9.10/5.29      ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6745_zero__le__power__abs,axiom,
% 9.10/5.29      ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6746_zero__le__power__abs,axiom,
% 9.10/5.29      ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6747_abs__if,axiom,
% 9.10/5.29      ( abs_abs_real
% 9.10/5.29      = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6748_abs__if,axiom,
% 9.10/5.29      ( abs_abs_int
% 9.10/5.29      = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6749_abs__if,axiom,
% 9.10/5.29      ( abs_abs_Code_integer
% 9.10/5.29      = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6750_abs__if,axiom,
% 9.10/5.29      ( abs_abs_rat
% 9.10/5.29      = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6751_abs__if__raw,axiom,
% 9.10/5.29      ( abs_abs_real
% 9.10/5.29      = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6752_abs__if__raw,axiom,
% 9.10/5.29      ( abs_abs_int
% 9.10/5.29      = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6753_abs__if__raw,axiom,
% 9.10/5.29      ( abs_abs_Code_integer
% 9.10/5.29      = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6754_abs__if__raw,axiom,
% 9.10/5.29      ( abs_abs_rat
% 9.10/5.29      = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6755_abs__of__neg,axiom,
% 9.10/5.29      ! [A: real] :
% 9.10/5.29        ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.29       => ( ( abs_abs_real @ A )
% 9.10/5.29          = ( uminus_uminus_real @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6756_abs__of__neg,axiom,
% 9.10/5.29      ! [A: int] :
% 9.10/5.29        ( ( ord_less_int @ A @ zero_zero_int )
% 9.10/5.29       => ( ( abs_abs_int @ A )
% 9.10/5.29          = ( uminus_uminus_int @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6757_abs__of__neg,axiom,
% 9.10/5.29      ! [A: code_integer] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 9.10/5.29       => ( ( abs_abs_Code_integer @ A )
% 9.10/5.29          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6758_abs__of__neg,axiom,
% 9.10/5.29      ! [A: rat] :
% 9.10/5.29        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.29       => ( ( abs_abs_rat @ A )
% 9.10/5.29          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6759_abs__diff__le__iff,axiom,
% 9.10/5.29      ! [X3: code_integer,A: code_integer,R: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6760_abs__diff__le__iff,axiom,
% 9.10/5.29      ! [X3: real,A: real,R: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6761_abs__diff__le__iff,axiom,
% 9.10/5.29      ! [X3: rat,A: rat,R: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6762_abs__diff__le__iff,axiom,
% 9.10/5.29      ! [X3: int,A: int,R: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6763_abs__triangle__ineq4,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6764_abs__triangle__ineq4,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6765_abs__triangle__ineq4,axiom,
% 9.10/5.29      ! [A: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6766_abs__triangle__ineq4,axiom,
% 9.10/5.29      ! [A: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6767_abs__diff__triangle__ineq,axiom,
% 9.10/5.29      ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B2 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6768_abs__diff__triangle__ineq,axiom,
% 9.10/5.29      ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6769_abs__diff__triangle__ineq,axiom,
% 9.10/5.29      ! [A: rat,B2: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B2 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6770_abs__diff__triangle__ineq,axiom,
% 9.10/5.29      ! [A: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6771_abs__diff__less__iff,axiom,
% 9.10/5.29      ! [X3: code_integer,A: code_integer,R: code_integer] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6772_abs__diff__less__iff,axiom,
% 9.10/5.29      ! [X3: real,A: real,R: real] :
% 9.10/5.29        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6773_abs__diff__less__iff,axiom,
% 9.10/5.29      ! [X3: rat,A: rat,R: rat] :
% 9.10/5.29        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6774_abs__diff__less__iff,axiom,
% 9.10/5.29      ! [X3: int,A: int,R: int] :
% 9.10/5.29        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R )
% 9.10/5.29        = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X3 )
% 9.10/5.29          & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6775_sum__SucD,axiom,
% 9.10/5.29      ! [F: nat > nat,A4: set_nat,N: nat] :
% 9.10/5.29        ( ( ( groups3542108847815614940at_nat @ F @ A4 )
% 9.10/5.29          = ( suc @ N ) )
% 9.10/5.29       => ? [X5: nat] :
% 9.10/5.29            ( ( member_nat @ X5 @ A4 )
% 9.10/5.29            & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6776_abs__real__def,axiom,
% 9.10/5.29      ( abs_abs_real
% 9.10/5.29      = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6777_lemma__interval__lt,axiom,
% 9.10/5.29      ! [A: real,X3: real,B2: real] :
% 9.10/5.29        ( ( ord_less_real @ A @ X3 )
% 9.10/5.29       => ( ( ord_less_real @ X3 @ B2 )
% 9.10/5.29         => ? [D3: real] :
% 9.10/5.29              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.29              & ! [Y5: real] :
% 9.10/5.29                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D3 )
% 9.10/5.29                 => ( ( ord_less_real @ A @ Y5 )
% 9.10/5.29                    & ( ord_less_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6778_sum__power__add,axiom,
% 9.10/5.29      ! [X3: complex,M: nat,I5: set_nat] :
% 9.10/5.29        ( ( groups2073611262835488442omplex
% 9.10/5.29          @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 9.10/5.29          @ I5 )
% 9.10/5.29        = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6779_sum__power__add,axiom,
% 9.10/5.29      ! [X3: rat,M: nat,I5: set_nat] :
% 9.10/5.29        ( ( groups2906978787729119204at_rat
% 9.10/5.29          @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 9.10/5.29          @ I5 )
% 9.10/5.29        = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6780_sum__power__add,axiom,
% 9.10/5.29      ! [X3: int,M: nat,I5: set_nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int
% 9.10/5.29          @ ^ [I4: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 9.10/5.29          @ I5 )
% 9.10/5.29        = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6781_sum__power__add,axiom,
% 9.10/5.29      ! [X3: real,M: nat,I5: set_nat] :
% 9.10/5.29        ( ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I4 ) )
% 9.10/5.29          @ I5 )
% 9.10/5.29        = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6782_sum_OatLeastAtMost__rev,axiom,
% 9.10/5.29      ! [G: nat > nat,N: nat,M: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 9.10/5.29        = ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6783_sum_OatLeastAtMost__rev,axiom,
% 9.10/5.29      ! [G: nat > real,N: nat,M: nat] :
% 9.10/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 9.10/5.29        = ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6784_pred__numeral__def,axiom,
% 9.10/5.29      ( pred_numeral
% 9.10/5.29      = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6785_sum__nth__roots,axiom,
% 9.10/5.29      ! [N: nat,C: complex] :
% 9.10/5.29        ( ( ord_less_nat @ one_one_nat @ N )
% 9.10/5.29       => ( ( groups7754918857620584856omplex
% 9.10/5.29            @ ^ [X: complex] : X
% 9.10/5.29            @ ( collect_complex
% 9.10/5.29              @ ^ [Z2: complex] :
% 9.10/5.29                  ( ( power_power_complex @ Z2 @ N )
% 9.10/5.29                  = C ) ) )
% 9.10/5.29          = zero_zero_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6786_sum__roots__unity,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ one_one_nat @ N )
% 9.10/5.29       => ( ( groups7754918857620584856omplex
% 9.10/5.29            @ ^ [X: complex] : X
% 9.10/5.29            @ ( collect_complex
% 9.10/5.29              @ ^ [Z2: complex] :
% 9.10/5.29                  ( ( power_power_complex @ Z2 @ N )
% 9.10/5.29                  = one_one_complex ) ) )
% 9.10/5.29          = zero_zero_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6787_abs__add__one__gt__zero,axiom,
% 9.10/5.29      ! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6788_abs__add__one__gt__zero,axiom,
% 9.10/5.29      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6789_abs__add__one__gt__zero,axiom,
% 9.10/5.29      ! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6790_abs__add__one__gt__zero,axiom,
% 9.10/5.29      ! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6791_sum__diff__nat,axiom,
% 9.10/5.29      ! [B5: set_complex,A4: set_complex,F: complex > nat] :
% 9.10/5.29        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.29       => ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.29         => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B5 ) )
% 9.10/5.29            = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6792_sum__diff__nat,axiom,
% 9.10/5.29      ! [B5: set_int,A4: set_int,F: int > nat] :
% 9.10/5.29        ( ( finite_finite_int @ B5 )
% 9.10/5.29       => ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.29         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ B5 ) )
% 9.10/5.29            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6793_sum__diff__nat,axiom,
% 9.10/5.29      ! [B5: set_nat,A4: set_nat,F: nat > nat] :
% 9.10/5.29        ( ( finite_finite_nat @ B5 )
% 9.10/5.29       => ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.29         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B5 ) )
% 9.10/5.29            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6794_sum_OatLeast0__atMost__Suc,axiom,
% 9.10/5.29      ! [G: nat > rat,N: nat] :
% 9.10/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6795_sum_OatLeast0__atMost__Suc,axiom,
% 9.10/5.29      ! [G: nat > int,N: nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6796_sum_OatLeast0__atMost__Suc,axiom,
% 9.10/5.29      ! [G: nat > nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6797_sum_OatLeast0__atMost__Suc,axiom,
% 9.10/5.29      ! [G: nat > real,N: nat] :
% 9.10/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6798_sum_OatLeast__Suc__atMost,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > rat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6799_sum_OatLeast__Suc__atMost,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > int] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6800_sum_OatLeast__Suc__atMost,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6801_sum_OatLeast__Suc__atMost,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > real] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6802_sum_Onat__ivl__Suc_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > rat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6803_sum_Onat__ivl__Suc_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > int] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6804_sum_Onat__ivl__Suc_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6805_sum_Onat__ivl__Suc_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > real] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6806_lemma__interval,axiom,
% 9.10/5.29      ! [A: real,X3: real,B2: real] :
% 9.10/5.29        ( ( ord_less_real @ A @ X3 )
% 9.10/5.29       => ( ( ord_less_real @ X3 @ B2 )
% 9.10/5.29         => ? [D3: real] :
% 9.10/5.29              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.29              & ! [Y5: real] :
% 9.10/5.29                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D3 )
% 9.10/5.29                 => ( ( ord_less_eq_real @ A @ Y5 )
% 9.10/5.29                    & ( ord_less_eq_real @ Y5 @ B2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6807_sum_OSuc__reindex__ivl,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > rat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_rat @ ( G @ M )
% 9.10/5.29            @ ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6808_sum_OSuc__reindex__ivl,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > int] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_int @ ( G @ M )
% 9.10/5.29            @ ( groups3539618377306564664at_int
% 9.10/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6809_sum_OSuc__reindex__ivl,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_nat @ ( G @ M )
% 9.10/5.29            @ ( groups3542108847815614940at_nat
% 9.10/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6810_sum_OSuc__reindex__ivl,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > real] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.29          = ( plus_plus_real @ ( G @ M )
% 9.10/5.29            @ ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6811_sum__Suc__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > complex] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups2073611262835488442omplex
% 9.10/5.29            @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_complex @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6812_sum__Suc__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > rat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups2906978787729119204at_rat
% 9.10/5.29            @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6813_sum__Suc__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > int] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups3539618377306564664at_int
% 9.10/5.29            @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6814_sum__Suc__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > real] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.29       => ( ( groups6591440286371151544t_real
% 9.10/5.29            @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6815_abs__le__square__iff,axiom,
% 9.10/5.29      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 9.10/5.29        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6816_abs__le__square__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6817_abs__le__square__iff,axiom,
% 9.10/5.29      ! [X3: rat,Y3: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6818_abs__le__square__iff,axiom,
% 9.10/5.29      ! [X3: int,Y3: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6819_abs__square__eq__1,axiom,
% 9.10/5.29      ! [X3: code_integer] :
% 9.10/5.29        ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          = one_one_Code_integer )
% 9.10/5.29        = ( ( abs_abs_Code_integer @ X3 )
% 9.10/5.29          = one_one_Code_integer ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6820_abs__square__eq__1,axiom,
% 9.10/5.29      ! [X3: rat] :
% 9.10/5.29        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          = one_one_rat )
% 9.10/5.29        = ( ( abs_abs_rat @ X3 )
% 9.10/5.29          = one_one_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6821_abs__square__eq__1,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          = one_one_real )
% 9.10/5.29        = ( ( abs_abs_real @ X3 )
% 9.10/5.29          = one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6822_abs__square__eq__1,axiom,
% 9.10/5.29      ! [X3: int] :
% 9.10/5.29        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          = one_one_int )
% 9.10/5.29        = ( ( abs_abs_int @ X3 )
% 9.10/5.29          = one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6823_power__even__abs,axiom,
% 9.10/5.29      ! [N: nat,A: code_integer] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 9.10/5.29          = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6824_power__even__abs,axiom,
% 9.10/5.29      ! [N: nat,A: rat] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 9.10/5.29          = ( power_power_rat @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6825_power__even__abs,axiom,
% 9.10/5.29      ! [N: nat,A: real] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 9.10/5.29          = ( power_power_real @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6826_power__even__abs,axiom,
% 9.10/5.29      ! [N: nat,A: int] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 9.10/5.29          = ( power_power_int @ A @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6827_sum_Oub__add__nat,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > rat,P4: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.29       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.29          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6828_sum_Oub__add__nat,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > int,P4: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.29       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.29          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6829_sum_Oub__add__nat,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > nat,P4: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.29       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.29          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6830_sum_Oub__add__nat,axiom,
% 9.10/5.29      ! [M: nat,N: nat,G: nat > real,P4: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.29       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.29          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6831_divmod__int__def,axiom,
% 9.10/5.29      ( unique5052692396658037445od_int
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6832_power2__le__iff__abs__le,axiom,
% 9.10/5.29      ! [Y3: code_integer,X3: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 9.10/5.29       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6833_power2__le__iff__abs__le,axiom,
% 9.10/5.29      ! [Y3: real,X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29          = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6834_power2__le__iff__abs__le,axiom,
% 9.10/5.29      ! [Y3: rat,X3: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.10/5.29       => ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29          = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6835_power2__le__iff__abs__le,axiom,
% 9.10/5.29      ! [Y3: int,X3: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.29       => ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.29          = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6836_abs__sqrt__wlog,axiom,
% 9.10/5.29      ! [P: code_integer > code_integer > $o,X3: code_integer] :
% 9.10/5.29        ( ! [X5: code_integer] :
% 9.10/5.29            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
% 9.10/5.29           => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.29       => ( P @ ( abs_abs_Code_integer @ X3 ) @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6837_abs__sqrt__wlog,axiom,
% 9.10/5.29      ! [P: real > real > $o,X3: real] :
% 9.10/5.29        ( ! [X5: real] :
% 9.10/5.29            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.10/5.29           => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.29       => ( P @ ( abs_abs_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6838_abs__sqrt__wlog,axiom,
% 9.10/5.29      ! [P: rat > rat > $o,X3: rat] :
% 9.10/5.29        ( ! [X5: rat] :
% 9.10/5.29            ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
% 9.10/5.29           => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.29       => ( P @ ( abs_abs_rat @ X3 ) @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6839_abs__sqrt__wlog,axiom,
% 9.10/5.29      ! [P: int > int > $o,X3: int] :
% 9.10/5.29        ( ! [X5: int] :
% 9.10/5.29            ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 9.10/5.29           => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.29       => ( P @ ( abs_abs_int @ X3 ) @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6840_abs__square__le__1,axiom,
% 9.10/5.29      ! [X3: code_integer] :
% 9.10/5.29        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.10/5.29        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6841_abs__square__le__1,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.10/5.29        = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6842_abs__square__le__1,axiom,
% 9.10/5.29      ! [X3: rat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.10/5.29        = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6843_abs__square__le__1,axiom,
% 9.10/5.29      ! [X3: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.10/5.29        = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6844_divmod__def,axiom,
% 9.10/5.29      ( unique5052692396658037445od_int
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6845_divmod__def,axiom,
% 9.10/5.29      ( unique5055182867167087721od_nat
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6846_divmod__def,axiom,
% 9.10/5.29      ( unique3479559517661332726nteger
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6847_abs__square__less__1,axiom,
% 9.10/5.29      ! [X3: code_integer] :
% 9.10/5.29        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.10/5.29        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6848_abs__square__less__1,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.10/5.29        = ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6849_abs__square__less__1,axiom,
% 9.10/5.29      ! [X3: rat] :
% 9.10/5.29        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.10/5.29        = ( ord_less_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6850_abs__square__less__1,axiom,
% 9.10/5.29      ! [X3: int] :
% 9.10/5.29        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.10/5.29        = ( ord_less_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6851_power__mono__even,axiom,
% 9.10/5.29      ! [N: nat,A: code_integer,B2: code_integer] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B2 ) )
% 9.10/5.29         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B2 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6852_power__mono__even,axiom,
% 9.10/5.29      ! [N: nat,A: real,B2: real] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B2 ) )
% 9.10/5.29         => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B2 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6853_power__mono__even,axiom,
% 9.10/5.29      ! [N: nat,A: rat,B2: rat] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B2 ) )
% 9.10/5.29         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B2 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6854_power__mono__even,axiom,
% 9.10/5.29      ! [N: nat,A: int,B2: int] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B2 ) )
% 9.10/5.29         => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B2 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6855_divmod_H__nat__def,axiom,
% 9.10/5.29      ( unique5055182867167087721od_nat
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6856_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B2: code_integer,Delta: code_integer] :
% 9.10/5.29        ( ! [I3: nat] :
% 9.10/5.29            ( ( member_nat @ I3 @ I5 )
% 9.10/5.29           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups7501900531339628137nteger @ X3 @ I5 )
% 9.10/5.29            = one_one_Code_integer )
% 9.10/5.29         => ( ! [I3: nat] :
% 9.10/5.29                ( ( member_nat @ I3 @ I5 )
% 9.10/5.29               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_le3102999989581377725nteger
% 9.10/5.29              @ ( abs_abs_Code_integer
% 9.10/5.29                @ ( minus_8373710615458151222nteger
% 9.10/5.29                  @ ( groups7501900531339628137nteger
% 9.10/5.29                    @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6857_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B2: code_integer,Delta: code_integer] :
% 9.10/5.29        ( ! [I3: real] :
% 9.10/5.29            ( ( member_real @ I3 @ I5 )
% 9.10/5.29           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups7713935264441627589nteger @ X3 @ I5 )
% 9.10/5.29            = one_one_Code_integer )
% 9.10/5.29         => ( ! [I3: real] :
% 9.10/5.29                ( ( member_real @ I3 @ I5 )
% 9.10/5.29               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_le3102999989581377725nteger
% 9.10/5.29              @ ( abs_abs_Code_integer
% 9.10/5.29                @ ( minus_8373710615458151222nteger
% 9.10/5.29                  @ ( groups7713935264441627589nteger
% 9.10/5.29                    @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6858_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B2: code_integer,Delta: code_integer] :
% 9.10/5.29        ( ! [I3: int] :
% 9.10/5.29            ( ( member_int @ I3 @ I5 )
% 9.10/5.29           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups7873554091576472773nteger @ X3 @ I5 )
% 9.10/5.29            = one_one_Code_integer )
% 9.10/5.29         => ( ! [I3: int] :
% 9.10/5.29                ( ( member_int @ I3 @ I5 )
% 9.10/5.29               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_le3102999989581377725nteger
% 9.10/5.29              @ ( abs_abs_Code_integer
% 9.10/5.29                @ ( minus_8373710615458151222nteger
% 9.10/5.29                  @ ( groups7873554091576472773nteger
% 9.10/5.29                    @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6859_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_complex,X3: complex > code_integer,A: complex > code_integer,B2: code_integer,Delta: code_integer] :
% 9.10/5.29        ( ! [I3: complex] :
% 9.10/5.29            ( ( member_complex @ I3 @ I5 )
% 9.10/5.29           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups6621422865394947399nteger @ X3 @ I5 )
% 9.10/5.29            = one_one_Code_integer )
% 9.10/5.29         => ( ! [I3: complex] :
% 9.10/5.29                ( ( member_complex @ I3 @ I5 )
% 9.10/5.29               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_le3102999989581377725nteger
% 9.10/5.29              @ ( abs_abs_Code_integer
% 9.10/5.29                @ ( minus_8373710615458151222nteger
% 9.10/5.29                  @ ( groups6621422865394947399nteger
% 9.10/5.29                    @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6860_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_real,X3: real > real,A: real > real,B2: real,Delta: real] :
% 9.10/5.29        ( ! [I3: real] :
% 9.10/5.29            ( ( member_real @ I3 @ I5 )
% 9.10/5.29           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups8097168146408367636l_real @ X3 @ I5 )
% 9.10/5.29            = one_one_real )
% 9.10/5.29         => ( ! [I3: real] :
% 9.10/5.29                ( ( member_real @ I3 @ I5 )
% 9.10/5.29               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_less_eq_real
% 9.10/5.29              @ ( abs_abs_real
% 9.10/5.29                @ ( minus_minus_real
% 9.10/5.29                  @ ( groups8097168146408367636l_real
% 9.10/5.29                    @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6861_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_int,X3: int > real,A: int > real,B2: real,Delta: real] :
% 9.10/5.29        ( ! [I3: int] :
% 9.10/5.29            ( ( member_int @ I3 @ I5 )
% 9.10/5.29           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups8778361861064173332t_real @ X3 @ I5 )
% 9.10/5.29            = one_one_real )
% 9.10/5.29         => ( ! [I3: int] :
% 9.10/5.29                ( ( member_int @ I3 @ I5 )
% 9.10/5.29               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_less_eq_real
% 9.10/5.29              @ ( abs_abs_real
% 9.10/5.29                @ ( minus_minus_real
% 9.10/5.29                  @ ( groups8778361861064173332t_real
% 9.10/5.29                    @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6862_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_complex,X3: complex > real,A: complex > real,B2: real,Delta: real] :
% 9.10/5.29        ( ! [I3: complex] :
% 9.10/5.29            ( ( member_complex @ I3 @ I5 )
% 9.10/5.29           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups5808333547571424918x_real @ X3 @ I5 )
% 9.10/5.29            = one_one_real )
% 9.10/5.29         => ( ! [I3: complex] :
% 9.10/5.29                ( ( member_complex @ I3 @ I5 )
% 9.10/5.29               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_less_eq_real
% 9.10/5.29              @ ( abs_abs_real
% 9.10/5.29                @ ( minus_minus_real
% 9.10/5.29                  @ ( groups5808333547571424918x_real
% 9.10/5.29                    @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6863_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_nat,X3: nat > rat,A: nat > rat,B2: rat,Delta: rat] :
% 9.10/5.29        ( ! [I3: nat] :
% 9.10/5.29            ( ( member_nat @ I3 @ I5 )
% 9.10/5.29           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups2906978787729119204at_rat @ X3 @ I5 )
% 9.10/5.29            = one_one_rat )
% 9.10/5.29         => ( ! [I3: nat] :
% 9.10/5.29                ( ( member_nat @ I3 @ I5 )
% 9.10/5.29               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_less_eq_rat
% 9.10/5.29              @ ( abs_abs_rat
% 9.10/5.29                @ ( minus_minus_rat
% 9.10/5.29                  @ ( groups2906978787729119204at_rat
% 9.10/5.29                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6864_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_real,X3: real > rat,A: real > rat,B2: rat,Delta: rat] :
% 9.10/5.29        ( ! [I3: real] :
% 9.10/5.29            ( ( member_real @ I3 @ I5 )
% 9.10/5.29           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups1300246762558778688al_rat @ X3 @ I5 )
% 9.10/5.29            = one_one_rat )
% 9.10/5.29         => ( ! [I3: real] :
% 9.10/5.29                ( ( member_real @ I3 @ I5 )
% 9.10/5.29               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_less_eq_rat
% 9.10/5.29              @ ( abs_abs_rat
% 9.10/5.29                @ ( minus_minus_rat
% 9.10/5.29                  @ ( groups1300246762558778688al_rat
% 9.10/5.29                    @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6865_convex__sum__bound__le,axiom,
% 9.10/5.29      ! [I5: set_int,X3: int > rat,A: int > rat,B2: rat,Delta: rat] :
% 9.10/5.29        ( ! [I3: int] :
% 9.10/5.29            ( ( member_int @ I3 @ I5 )
% 9.10/5.29           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 9.10/5.29       => ( ( ( groups3906332499630173760nt_rat @ X3 @ I5 )
% 9.10/5.29            = one_one_rat )
% 9.10/5.29         => ( ! [I3: int] :
% 9.10/5.29                ( ( member_int @ I3 @ I5 )
% 9.10/5.29               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B2 ) ) @ Delta ) )
% 9.10/5.29           => ( ord_less_eq_rat
% 9.10/5.29              @ ( abs_abs_rat
% 9.10/5.29                @ ( minus_minus_rat
% 9.10/5.29                  @ ( groups3906332499630173760nt_rat
% 9.10/5.29                    @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
% 9.10/5.29                    @ I5 )
% 9.10/5.29                  @ B2 ) )
% 9.10/5.29              @ Delta ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6866_dbl__dec__def,axiom,
% 9.10/5.29      ( neg_nu6511756317524482435omplex
% 9.10/5.29      = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6867_dbl__dec__def,axiom,
% 9.10/5.29      ( neg_nu6075765906172075777c_real
% 9.10/5.29      = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6868_dbl__dec__def,axiom,
% 9.10/5.29      ( neg_nu3179335615603231917ec_rat
% 9.10/5.29      = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6869_dbl__dec__def,axiom,
% 9.10/5.29      ( neg_nu3811975205180677377ec_int
% 9.10/5.29      = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6870_sum__natinterval__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > complex] :
% 9.10/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6871_sum__natinterval__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > rat] :
% 9.10/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6872_sum__natinterval__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > int] :
% 9.10/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups3539618377306564664at_int
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups3539618377306564664at_int
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_int ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6873_sum__natinterval__diff,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > real] :
% 9.10/5.29        ( ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.10/5.29        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29              @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6874_sum__telescope_H_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > complex] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups2073611262835488442omplex
% 9.10/5.29            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.10/5.29          = ( minus_minus_complex @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6875_sum__telescope_H_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > rat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups2906978787729119204at_rat
% 9.10/5.29            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.10/5.29          = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6876_sum__telescope_H_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > int] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups3539618377306564664at_int
% 9.10/5.29            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.10/5.29          = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6877_sum__telescope_H_H,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > real] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( groups6591440286371151544t_real
% 9.10/5.29            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.10/5.29          = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6878_mask__eq__sum__exp,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 9.10/5.29        = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.29          @ ( collect_nat
% 9.10/5.29            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6879_mask__eq__sum__exp,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 9.10/5.29        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          @ ( collect_nat
% 9.10/5.29            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6880_sum__gp__multiplied,axiom,
% 9.10/5.29      ! [M: nat,N: nat,X3: complex] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.10/5.29          = ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6881_sum__gp__multiplied,axiom,
% 9.10/5.29      ! [M: nat,N: nat,X3: rat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.10/5.29          = ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6882_sum__gp__multiplied,axiom,
% 9.10/5.29      ! [M: nat,N: nat,X3: int] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.10/5.29          = ( minus_minus_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6883_sum__gp__multiplied,axiom,
% 9.10/5.29      ! [M: nat,N: nat,X3: real] :
% 9.10/5.29        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29       => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.10/5.29          = ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6884_sum_Oin__pairs,axiom,
% 9.10/5.29      ! [G: nat > rat,M: nat,N: nat] :
% 9.10/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.29        = ( groups2906978787729119204at_rat
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6885_sum_Oin__pairs,axiom,
% 9.10/5.29      ! [G: nat > int,M: nat,N: nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.29        = ( groups3539618377306564664at_int
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6886_sum_Oin__pairs,axiom,
% 9.10/5.29      ! [G: nat > nat,M: nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.29        = ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6887_sum_Oin__pairs,axiom,
% 9.10/5.29      ! [G: nat > real,M: nat,N: nat] :
% 9.10/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.29        = ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6888_mask__eq__sum__exp__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.29        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          @ ( collect_nat
% 9.10/5.29            @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6889_gauss__sum__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [X: nat] : X
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6890_arith__series__nat,axiom,
% 9.10/5.29      ! [A: nat,D: nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6891_Sum__Icc__nat,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [X: nat] : X
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6892_divmod__divmod__step,axiom,
% 9.10/5.29      ( unique5055182867167087721od_nat
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6893_divmod__divmod__step,axiom,
% 9.10/5.29      ( unique5052692396658037445od_int
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6894_divmod__divmod__step,axiom,
% 9.10/5.29      ( unique3479559517661332726nteger
% 9.10/5.29      = ( ^ [M3: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6895_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.29         => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6896_abs__ln__one__plus__x__minus__x__bound,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.29       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6897_divmod__nat__if,axiom,
% 9.10/5.29      ( divmod_nat
% 9.10/5.29      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.29            ( if_Pro6206227464963214023at_nat
% 9.10/5.29            @ ( ( N2 = zero_zero_nat )
% 9.10/5.29              | ( ord_less_nat @ M3 @ N2 ) )
% 9.10/5.29            @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 9.10/5.29            @ ( produc2626176000494625587at_nat
% 9.10/5.29              @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 9.10/5.29              @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6898_tanh__ln__real,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( tanh_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.29          = ( 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 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6899_arctan__double,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.29       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X3 ) )
% 9.10/5.29          = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6900_dbl__inc__simps_I3_J,axiom,
% 9.10/5.29      ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 9.10/5.29      = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6901_dbl__inc__simps_I3_J,axiom,
% 9.10/5.29      ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 9.10/5.29      = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6902_dbl__inc__simps_I3_J,axiom,
% 9.10/5.29      ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 9.10/5.29      = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6903_dbl__inc__simps_I3_J,axiom,
% 9.10/5.29      ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 9.10/5.29      = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6904_sum__gp,axiom,
% 9.10/5.29      ! [N: nat,M: nat,X3: complex] :
% 9.10/5.29        ( ( ( ord_less_nat @ N @ M )
% 9.10/5.29         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_complex ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ N @ M )
% 9.10/5.29         => ( ( ( X3 = one_one_complex )
% 9.10/5.29             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29                = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.10/5.29            & ( ( X3 != one_one_complex )
% 9.10/5.29             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29                = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6905_sum__gp,axiom,
% 9.10/5.29      ! [N: nat,M: nat,X3: rat] :
% 9.10/5.29        ( ( ( ord_less_nat @ N @ M )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_rat ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ N @ M )
% 9.10/5.29         => ( ( ( X3 = one_one_rat )
% 9.10/5.29             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29                = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.10/5.29            & ( ( X3 != one_one_rat )
% 9.10/5.29             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29                = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6906_sum__gp,axiom,
% 9.10/5.29      ! [N: nat,M: nat,X3: real] :
% 9.10/5.29        ( ( ( ord_less_nat @ N @ M )
% 9.10/5.29         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29            = zero_zero_real ) )
% 9.10/5.29        & ( ~ ( ord_less_nat @ N @ M )
% 9.10/5.29         => ( ( ( X3 = one_one_real )
% 9.10/5.29             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29                = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.10/5.29            & ( ( X3 != one_one_real )
% 9.10/5.29             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.29                = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6907_of__nat__eq__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ( semiri1314217659103216013at_int @ M )
% 9.10/5.29          = ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( M = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6908_of__nat__eq__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ( semiri5074537144036343181t_real @ M )
% 9.10/5.29          = ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( M = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6909_of__nat__eq__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.10/5.29          = ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( M = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6910_int__eq__iff__numeral,axiom,
% 9.10/5.29      ! [M: nat,V: num] :
% 9.10/5.29        ( ( ( semiri1314217659103216013at_int @ M )
% 9.10/5.29          = ( numeral_numeral_int @ V ) )
% 9.10/5.29        = ( M
% 9.10/5.29          = ( numeral_numeral_nat @ V ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6911_abs__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.10/5.29        = ( semiri4939895301339042750nteger @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6912_abs__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29        = ( semiri681578069525770553at_rat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6913_abs__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6914_abs__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6915_negative__zle,axiom,
% 9.10/5.29      ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6916_tanh__real__less__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
% 9.10/5.29        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6917_tanh__real__le__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6918_of__nat__eq__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ( semiri8010041392384452111omplex @ M )
% 9.10/5.29          = zero_zero_complex )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6919_of__nat__eq__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ( semiri681578069525770553at_rat @ M )
% 9.10/5.29          = zero_zero_rat )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6920_of__nat__eq__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ( semiri1314217659103216013at_int @ M )
% 9.10/5.29          = zero_zero_int )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6921_of__nat__eq__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ( semiri5074537144036343181t_real @ M )
% 9.10/5.29          = zero_zero_real )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6922_of__nat__eq__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.10/5.29          = zero_zero_nat )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6923_of__nat__0__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( zero_zero_complex
% 9.10/5.29          = ( semiri8010041392384452111omplex @ N ) )
% 9.10/5.29        = ( zero_zero_nat = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6924_of__nat__0__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( zero_zero_rat
% 9.10/5.29          = ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29        = ( zero_zero_nat = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6925_of__nat__0__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( zero_zero_int
% 9.10/5.29          = ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( zero_zero_nat = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6926_of__nat__0__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( zero_zero_real
% 9.10/5.29          = ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( zero_zero_nat = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6927_of__nat__0__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( zero_zero_nat
% 9.10/5.29          = ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( zero_zero_nat = N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6928_of__nat__0,axiom,
% 9.10/5.29      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 9.10/5.29      = zero_zero_complex ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6929_of__nat__0,axiom,
% 9.10/5.29      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 9.10/5.29      = zero_zero_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6930_of__nat__0,axiom,
% 9.10/5.29      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.10/5.29      = zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6931_of__nat__0,axiom,
% 9.10/5.29      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 9.10/5.29      = zero_zero_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6932_of__nat__0,axiom,
% 9.10/5.29      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 9.10/5.29      = zero_zero_nat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6933_of__nat__less__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6934_of__nat__less__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6935_of__nat__less__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6936_of__nat__less__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6937_of__nat__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( numera6690914467698888265omplex @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6938_of__nat__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( numeral_numeral_rat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6939_of__nat__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( numeral_numeral_int @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6940_of__nat__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( numeral_numeral_real @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6941_of__nat__numeral,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( numeral_numeral_nat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6942_of__nat__le__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6943_of__nat__le__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6944_of__nat__le__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6945_of__nat__le__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6946_of__nat__add,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.29        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6947_of__nat__add,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6948_of__nat__add,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.29        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6949_of__nat__add,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.29        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6950_of__nat__mult,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 9.10/5.29        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6951_of__nat__mult,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 9.10/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6952_of__nat__mult,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 9.10/5.29        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6953_of__nat__mult,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 9.10/5.29        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6954_of__nat__eq__1__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ( semiri8010041392384452111omplex @ N )
% 9.10/5.29          = one_one_complex )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6955_of__nat__eq__1__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ( semiri681578069525770553at_rat @ N )
% 9.10/5.29          = one_one_rat )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6956_of__nat__eq__1__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ( semiri1314217659103216013at_int @ N )
% 9.10/5.29          = one_one_int )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6957_of__nat__eq__1__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ( semiri5074537144036343181t_real @ N )
% 9.10/5.29          = one_one_real )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6958_of__nat__eq__1__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ( semiri1316708129612266289at_nat @ N )
% 9.10/5.29          = one_one_nat )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6959_of__nat__1__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( one_one_complex
% 9.10/5.29          = ( semiri8010041392384452111omplex @ N ) )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6960_of__nat__1__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( one_one_rat
% 9.10/5.29          = ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6961_of__nat__1__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( one_one_int
% 9.10/5.29          = ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6962_of__nat__1__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( one_one_real
% 9.10/5.29          = ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6963_of__nat__1__eq__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( one_one_nat
% 9.10/5.29          = ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( N = one_one_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6964_of__nat__1,axiom,
% 9.10/5.29      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 9.10/5.29      = one_one_complex ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6965_of__nat__1,axiom,
% 9.10/5.29      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 9.10/5.29      = one_one_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6966_of__nat__1,axiom,
% 9.10/5.29      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.10/5.29      = one_one_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6967_of__nat__1,axiom,
% 9.10/5.29      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 9.10/5.29      = one_one_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6968_of__nat__1,axiom,
% 9.10/5.29      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 9.10/5.29      = one_one_nat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6969_of__nat__power,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 9.10/5.29        = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6970_of__nat__power,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 9.10/5.29        = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6971_of__nat__power,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 9.10/5.29        = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6972_of__nat__power,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 9.10/5.29        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6973_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
% 9.10/5.29          = ( semiri8010041392384452111omplex @ X3 ) )
% 9.10/5.29        = ( ( power_power_nat @ B2 @ W )
% 9.10/5.29          = X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6974_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
% 9.10/5.29          = ( semiri1314217659103216013at_int @ X3 ) )
% 9.10/5.29        = ( ( power_power_nat @ B2 @ W )
% 9.10/5.29          = X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6975_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
% 9.10/5.29          = ( semiri5074537144036343181t_real @ X3 ) )
% 9.10/5.29        = ( ( power_power_nat @ B2 @ W )
% 9.10/5.29          = X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6976_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
% 9.10/5.29          = ( semiri1316708129612266289at_nat @ X3 ) )
% 9.10/5.29        = ( ( power_power_nat @ B2 @ W )
% 9.10/5.29          = X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6977_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ( semiri8010041392384452111omplex @ X3 )
% 9.10/5.29          = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
% 9.10/5.29        = ( X3
% 9.10/5.29          = ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6978_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ( semiri1314217659103216013at_int @ X3 )
% 9.10/5.29          = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 9.10/5.29        = ( X3
% 9.10/5.29          = ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6979_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ( semiri5074537144036343181t_real @ X3 )
% 9.10/5.29          = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 9.10/5.29        = ( X3
% 9.10/5.29          = ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6980_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ( semiri1316708129612266289at_nat @ X3 )
% 9.10/5.29          = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 9.10/5.29        = ( X3
% 9.10/5.29          = ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6981_negative__zless,axiom,
% 9.10/5.29      ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6982_zabs__less__one__iff,axiom,
% 9.10/5.29      ! [Z: int] :
% 9.10/5.29        ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 9.10/5.29        = ( Z = zero_zero_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6983_zero__less__arctan__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X3 ) )
% 9.10/5.29        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6984_arctan__less__zero__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( arctan @ X3 ) @ zero_zero_real )
% 9.10/5.29        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6985_zero__le__arctan__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6986_arctan__le__zero__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( arctan @ X3 ) @ zero_zero_real )
% 9.10/5.29        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6987_tanh__real__pos__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
% 9.10/5.29        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6988_tanh__real__neg__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
% 9.10/5.29        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6989_tanh__real__nonpos__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( tanh_real @ X3 ) @ zero_zero_real )
% 9.10/5.29        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6990_tanh__real__nonneg__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6991_dbl__inc__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 9.10/5.29      = one_one_complex ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6992_dbl__inc__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 9.10/5.29      = one_one_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6993_dbl__inc__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 9.10/5.29      = one_one_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6994_dbl__inc__simps_I2_J,axiom,
% 9.10/5.29      ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 9.10/5.29      = one_one_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6995_of__nat__of__bool,axiom,
% 9.10/5.29      ! [P: $o] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.10/5.29        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6996_of__nat__of__bool,axiom,
% 9.10/5.29      ! [P: $o] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.10/5.29        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6997_of__nat__of__bool,axiom,
% 9.10/5.29      ! [P: $o] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.10/5.29        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6998_of__nat__of__bool,axiom,
% 9.10/5.29      ! [P: $o] :
% 9.10/5.29        ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.10/5.29        = ( zero_n356916108424825756nteger @ P ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_6999_dbl__inc__simps_I4_J,axiom,
% 9.10/5.29      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.29      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7000_dbl__inc__simps_I4_J,axiom,
% 9.10/5.29      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.29      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7001_dbl__inc__simps_I4_J,axiom,
% 9.10/5.29      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.10/5.29      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7002_dbl__inc__simps_I4_J,axiom,
% 9.10/5.29      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.10/5.29      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7003_dbl__inc__simps_I4_J,axiom,
% 9.10/5.29      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.29      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7004_dbl__inc__simps_I5_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 9.10/5.29        = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7005_dbl__inc__simps_I5_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 9.10/5.29        = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7006_dbl__inc__simps_I5_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 9.10/5.29        = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7007_dbl__inc__simps_I5_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 9.10/5.29        = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7008_of__nat__le__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7009_of__nat__le__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7010_of__nat__le__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7011_of__nat__le__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 9.10/5.29        = ( M = zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7012_of__nat__Suc,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 9.10/5.29        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7013_of__nat__Suc,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 9.10/5.29        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7014_of__nat__Suc,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 9.10/5.29        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7015_of__nat__Suc,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 9.10/5.29        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7016_of__nat__Suc,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 9.10/5.29        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7017_numeral__less__real__of__nat__iff,axiom,
% 9.10/5.29      ! [W: num,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7018_real__of__nat__less__numeral__iff,axiom,
% 9.10/5.29      ! [N: nat,W: num] :
% 9.10/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 9.10/5.29        = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7019_numeral__le__real__of__nat__iff,axiom,
% 9.10/5.29      ! [N: num,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7020_of__nat__0__less__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7021_of__nat__0__less__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7022_of__nat__0__less__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7023_of__nat__0__less__iff,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7024_dbl__inc__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 9.10/5.29        = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7025_dbl__inc__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.10/5.29        = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7026_dbl__inc__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 9.10/5.29        = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7027_dbl__inc__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.10/5.29        = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7028_dbl__inc__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.10/5.29        = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7029_dbl__dec__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 9.10/5.29        = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7030_dbl__dec__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.10/5.29        = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7031_dbl__dec__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 9.10/5.29        = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7032_dbl__dec__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 9.10/5.29        = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7033_dbl__dec__simps_I1_J,axiom,
% 9.10/5.29      ! [K: num] :
% 9.10/5.29        ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.10/5.29        = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7034_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7035_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7036_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7037_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7038_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7039_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7040_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7041_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7042_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [Y3: nat,X3: num,N: nat] :
% 9.10/5.29        ( ( ( semiri8010041392384452111omplex @ Y3 )
% 9.10/5.29          = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N ) )
% 9.10/5.29        = ( Y3
% 9.10/5.29          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7043_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [Y3: nat,X3: num,N: nat] :
% 9.10/5.29        ( ( ( semiri681578069525770553at_rat @ Y3 )
% 9.10/5.29          = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
% 9.10/5.29        = ( Y3
% 9.10/5.29          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7044_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [Y3: nat,X3: num,N: nat] :
% 9.10/5.29        ( ( ( semiri1314217659103216013at_int @ Y3 )
% 9.10/5.29          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
% 9.10/5.29        = ( Y3
% 9.10/5.29          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7045_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [Y3: nat,X3: num,N: nat] :
% 9.10/5.29        ( ( ( semiri5074537144036343181t_real @ Y3 )
% 9.10/5.29          = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
% 9.10/5.29        = ( Y3
% 9.10/5.29          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7046_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [Y3: nat,X3: num,N: nat] :
% 9.10/5.29        ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 9.10/5.29          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 9.10/5.29        = ( Y3
% 9.10/5.29          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7047_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: num,N: nat,Y3: nat] :
% 9.10/5.29        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N )
% 9.10/5.29          = ( semiri8010041392384452111omplex @ Y3 ) )
% 9.10/5.29        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.29          = Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7048_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: num,N: nat,Y3: nat] :
% 9.10/5.29        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N )
% 9.10/5.29          = ( semiri681578069525770553at_rat @ Y3 ) )
% 9.10/5.29        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.29          = Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7049_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: num,N: nat,Y3: nat] :
% 9.10/5.29        ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
% 9.10/5.29          = ( semiri1314217659103216013at_int @ Y3 ) )
% 9.10/5.29        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.29          = Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7050_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: num,N: nat,Y3: nat] :
% 9.10/5.29        ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N )
% 9.10/5.29          = ( semiri5074537144036343181t_real @ Y3 ) )
% 9.10/5.29        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.29          = Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7051_numeral__power__eq__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: num,N: nat,Y3: nat] :
% 9.10/5.29        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.29          = ( semiri1316708129612266289at_nat @ Y3 ) )
% 9.10/5.29        = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.29          = Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7052_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7053_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7054_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7055_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.10/5.29      ! [B2: nat,W: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7056_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7057_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7058_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7059_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,B2: nat,W: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B2 @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7060_of__nat__zero__less__power__iff,axiom,
% 9.10/5.29      ! [X3: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N ) )
% 9.10/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7061_of__nat__zero__less__power__iff,axiom,
% 9.10/5.29      ! [X3: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N ) )
% 9.10/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7062_of__nat__zero__less__power__iff,axiom,
% 9.10/5.29      ! [X3: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N ) )
% 9.10/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7063_of__nat__zero__less__power__iff,axiom,
% 9.10/5.29      ! [X3: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N ) )
% 9.10/5.29        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7064_even__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 9.10/5.29        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7065_even__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7066_even__of__nat,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7067_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7068_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7069_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7070_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.10/5.29        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7071_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7072_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7073_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7074_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7075_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7076_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7077_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7078_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.10/5.29      ! [I: num,N: nat,X3: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7079_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7080_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7081_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7082_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.10/5.29      ! [X3: nat,I: num,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7083_Collect__case__prod__mono,axiom,
% 9.10/5.29      ! [A4: int > int > $o,B5: int > int > $o] :
% 9.10/5.29        ( ( ord_le6741204236512500942_int_o @ A4 @ B5 )
% 9.10/5.29       => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7084_mult__of__nat__commute,axiom,
% 9.10/5.29      ! [X3: nat,Y3: rat] :
% 9.10/5.29        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y3 )
% 9.10/5.29        = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7085_mult__of__nat__commute,axiom,
% 9.10/5.29      ! [X3: nat,Y3: int] :
% 9.10/5.29        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y3 )
% 9.10/5.29        = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7086_mult__of__nat__commute,axiom,
% 9.10/5.29      ! [X3: nat,Y3: real] :
% 9.10/5.29        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y3 )
% 9.10/5.29        = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7087_mult__of__nat__commute,axiom,
% 9.10/5.29      ! [X3: nat,Y3: nat] :
% 9.10/5.29        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y3 )
% 9.10/5.29        = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7088_arctan__less__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 9.10/5.29        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7089_arctan__monotone,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.29       => ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7090_arctan__monotone_H,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.29       => ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7091_arctan__le__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7092_of__nat__0__le__iff,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7093_of__nat__0__le__iff,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7094_of__nat__0__le__iff,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7095_of__nat__0__le__iff,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7096_of__nat__less__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7097_of__nat__less__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7098_of__nat__less__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7099_of__nat__less__0__iff,axiom,
% 9.10/5.29      ! [M: nat] :
% 9.10/5.29        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7100_of__nat__neq__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 9.10/5.29       != zero_zero_complex ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7101_of__nat__neq__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 9.10/5.29       != zero_zero_rat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7102_of__nat__neq__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 9.10/5.29       != zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7103_of__nat__neq__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 9.10/5.29       != zero_zero_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7104_of__nat__neq__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 9.10/5.29       != zero_zero_nat ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7105_div__mult2__eq_H,axiom,
% 9.10/5.29      ! [A: int,M: nat,N: nat] :
% 9.10/5.29        ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.29        = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7106_div__mult2__eq_H,axiom,
% 9.10/5.29      ! [A: nat,M: nat,N: nat] :
% 9.10/5.29        ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 9.10/5.29        = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7107_less__imp__of__nat__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ M @ N )
% 9.10/5.29       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7108_less__imp__of__nat__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ M @ N )
% 9.10/5.29       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7109_less__imp__of__nat__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ M @ N )
% 9.10/5.29       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7110_less__imp__of__nat__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ M @ N )
% 9.10/5.29       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7111_of__nat__less__imp__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.10/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7112_of__nat__less__imp__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7113_of__nat__less__imp__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7114_of__nat__less__imp__less,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29       => ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7115_of__nat__mono,axiom,
% 9.10/5.29      ! [I: nat,J: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.29       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7116_of__nat__mono,axiom,
% 9.10/5.29      ! [I: nat,J: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.29       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7117_of__nat__mono,axiom,
% 9.10/5.29      ! [I: nat,J: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.29       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7118_of__nat__mono,axiom,
% 9.10/5.29      ! [I: nat,J: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.29       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7119_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.29        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7120_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7121_of__nat__dvd__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 9.10/5.29        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7122_of__nat__dvd__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7123_of__nat__dvd__iff,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.10/5.29        = ( dvd_dvd_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7124_int__ops_I1_J,axiom,
% 9.10/5.29      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.10/5.29      = zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7125_int__ops_I3_J,axiom,
% 9.10/5.29      ! [N: num] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 9.10/5.29        = ( numeral_numeral_int @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7126_abs__zmult__eq__1,axiom,
% 9.10/5.29      ! [M: int,N: int] :
% 9.10/5.29        ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 9.10/5.29          = one_one_int )
% 9.10/5.29       => ( ( abs_abs_int @ M )
% 9.10/5.29          = one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7127_nat__int__comparison_I2_J,axiom,
% 9.10/5.29      ( ord_less_nat
% 9.10/5.29      = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7128_zle__int,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.29        = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7129_nat__int__comparison_I3_J,axiom,
% 9.10/5.29      ( ord_less_eq_nat
% 9.10/5.29      = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7130_nonneg__int__cases,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ( K
% 9.10/5.29             != ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7131_zero__le__imp__eq__int,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.29       => ? [N3: nat] :
% 9.10/5.29            ( K
% 9.10/5.29            = ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7132_of__nat__mod,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 9.10/5.29        = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7133_of__nat__mod,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 9.10/5.29        = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7134_of__nat__mod,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 9.10/5.29        = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7135_zadd__int__left,axiom,
% 9.10/5.29      ! [M: nat,N: nat,Z: int] :
% 9.10/5.29        ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 9.10/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7136_int__plus,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 9.10/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7137_int__ops_I5_J,axiom,
% 9.10/5.29      ! [A: nat,B2: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B2 ) )
% 9.10/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7138_int__ops_I2_J,axiom,
% 9.10/5.29      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.10/5.29      = one_one_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7139_int__ops_I7_J,axiom,
% 9.10/5.29      ! [A: nat,B2: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
% 9.10/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7140_zle__iff__zadd,axiom,
% 9.10/5.29      ( ord_less_eq_int
% 9.10/5.29      = ( ^ [W3: int,Z2: int] :
% 9.10/5.29          ? [N2: nat] :
% 9.10/5.29            ( Z2
% 9.10/5.29            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7141_not__int__zless__negative,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7142_zdiv__int,axiom,
% 9.10/5.29      ! [A: nat,B2: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
% 9.10/5.29        = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7143_of__nat__max,axiom,
% 9.10/5.29      ! [X3: nat,Y3: nat] :
% 9.10/5.29        ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X3 @ Y3 ) )
% 9.10/5.29        = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X3 ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7144_of__nat__max,axiom,
% 9.10/5.29      ! [X3: nat,Y3: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X3 @ Y3 ) )
% 9.10/5.29        = ( ord_max_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7145_of__nat__max,axiom,
% 9.10/5.29      ! [X3: nat,Y3: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X3 @ Y3 ) )
% 9.10/5.29        = ( ord_max_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7146_of__nat__max,axiom,
% 9.10/5.29      ! [X3: nat,Y3: nat] :
% 9.10/5.29        ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X3 @ Y3 ) )
% 9.10/5.29        = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7147_tanh__real__lt__1,axiom,
% 9.10/5.29      ! [X3: real] : ( ord_less_real @ ( tanh_real @ X3 ) @ one_one_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7148_nat__less__as__int,axiom,
% 9.10/5.29      ( ord_less_nat
% 9.10/5.29      = ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7149_nat__leq__as__int,axiom,
% 9.10/5.29      ( ord_less_eq_nat
% 9.10/5.29      = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7150_of__nat__diff,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7151_of__nat__diff,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7152_of__nat__diff,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7153_of__nat__diff,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7154_of__nat__diff,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 9.10/5.29          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7155_zabs__def,axiom,
% 9.10/5.29      ( abs_abs_int
% 9.10/5.29      = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7156_reals__Archimedean3,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ! [Y5: real] :
% 9.10/5.29          ? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7157_int__cases4,axiom,
% 9.10/5.29      ! [M: int] :
% 9.10/5.29        ( ! [N3: nat] :
% 9.10/5.29            ( M
% 9.10/5.29           != ( semiri1314217659103216013at_int @ N3 ) )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.10/5.29             => ( M
% 9.10/5.29               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7158_dvd__imp__le__int,axiom,
% 9.10/5.29      ! [I: int,D: int] :
% 9.10/5.29        ( ( I != zero_zero_int )
% 9.10/5.29       => ( ( dvd_dvd_int @ D @ I )
% 9.10/5.29         => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7159_real__of__nat__div4,axiom,
% 9.10/5.29      ! [N: nat,X3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7160_int__zle__neg,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.10/5.29        = ( ( N = zero_zero_nat )
% 9.10/5.29          & ( M = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7161_int__ops_I4_J,axiom,
% 9.10/5.29      ! [A: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 9.10/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7162_int__Suc,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 9.10/5.29        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7163_zless__iff__Suc__zadd,axiom,
% 9.10/5.29      ( ord_less_int
% 9.10/5.29      = ( ^ [W3: int,Z2: int] :
% 9.10/5.29          ? [N2: nat] :
% 9.10/5.29            ( Z2
% 9.10/5.29            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7164_abs__mod__less,axiom,
% 9.10/5.29      ! [L: int,K: int] :
% 9.10/5.29        ( ( L != zero_zero_int )
% 9.10/5.29       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7165_negative__zle__0,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7166_nonpos__int__cases,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ( K
% 9.10/5.29             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7167_real__of__nat__div,axiom,
% 9.10/5.29      ! [D: nat,N: nat] :
% 9.10/5.29        ( ( dvd_dvd_nat @ D @ N )
% 9.10/5.29       => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 9.10/5.29          = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7168_tanh__real__gt__neg1,axiom,
% 9.10/5.29      ! [X3: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7169_mod__mult2__eq_H,axiom,
% 9.10/5.29      ! [A: code_integer,M: nat,N: nat] :
% 9.10/5.29        ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 9.10/5.29        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7170_mod__mult2__eq_H,axiom,
% 9.10/5.29      ! [A: int,M: nat,N: nat] :
% 9.10/5.29        ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.29        = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7171_mod__mult2__eq_H,axiom,
% 9.10/5.29      ! [A: nat,M: nat,N: nat] :
% 9.10/5.29        ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 9.10/5.29        = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7172_field__char__0__class_Oof__nat__div,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.29        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7173_field__char__0__class_Oof__nat__div,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.29        = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7174_field__char__0__class_Oof__nat__div,axiom,
% 9.10/5.29      ! [M: nat,N: nat] :
% 9.10/5.29        ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 9.10/5.29        = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7175_pos__int__cases,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ( ( K
% 9.10/5.29                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.10/5.29             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7176_zero__less__imp__eq__int,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.29       => ? [N3: nat] :
% 9.10/5.29            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.10/5.29            & ( K
% 9.10/5.29              = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7177_int__cases3,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( K != zero_zero_int )
% 9.10/5.29       => ( ! [N3: nat] :
% 9.10/5.29              ( ( K
% 9.10/5.29                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.10/5.29             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 9.10/5.29         => ~ ! [N3: nat] :
% 9.10/5.29                ( ( K
% 9.10/5.29                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.10/5.29               => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7178_nat__less__real__le,axiom,
% 9.10/5.29      ( ord_less_nat
% 9.10/5.29      = ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7179_nat__le__real__less,axiom,
% 9.10/5.29      ( ord_less_eq_nat
% 9.10/5.29      = ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7180_zmult__zless__mono2__lemma,axiom,
% 9.10/5.29      ! [I: int,J: int,K: nat] :
% 9.10/5.29        ( ( ord_less_int @ I @ J )
% 9.10/5.29       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.29         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7181_zdvd__mult__cancel1,axiom,
% 9.10/5.29      ! [M: int,N: int] :
% 9.10/5.29        ( ( M != zero_zero_int )
% 9.10/5.29       => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 9.10/5.29          = ( ( abs_abs_int @ N )
% 9.10/5.29            = one_one_int ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7182_not__zle__0__negative,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7183_negD,axiom,
% 9.10/5.29      ! [X3: int] :
% 9.10/5.29        ( ( ord_less_int @ X3 @ zero_zero_int )
% 9.10/5.29       => ? [N3: nat] :
% 9.10/5.29            ( X3
% 9.10/5.29            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7184_negative__zless__0,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7185_dbl__inc__def,axiom,
% 9.10/5.29      ( neg_nu8557863876264182079omplex
% 9.10/5.29      = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7186_dbl__inc__def,axiom,
% 9.10/5.29      ( neg_nu8295874005876285629c_real
% 9.10/5.29      = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7187_dbl__inc__def,axiom,
% 9.10/5.29      ( neg_nu5219082963157363817nc_rat
% 9.10/5.29      = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7188_dbl__inc__def,axiom,
% 9.10/5.29      ( neg_nu5851722552734809277nc_int
% 9.10/5.29      = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7189_int__ops_I6_J,axiom,
% 9.10/5.29      ! [A: nat,B2: nat] :
% 9.10/5.29        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 9.10/5.29         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 9.10/5.29            = zero_zero_int ) )
% 9.10/5.29        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
% 9.10/5.29         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
% 9.10/5.29            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7190_real__of__nat__div__aux,axiom,
% 9.10/5.29      ! [X3: nat,D: nat] :
% 9.10/5.29        ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ D ) )
% 9.10/5.29        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X3 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X3 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7191_of__nat__less__two__power,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7192_of__nat__less__two__power,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7193_of__nat__less__two__power,axiom,
% 9.10/5.29      ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7194_inverse__of__nat__le,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( N != zero_zero_nat )
% 9.10/5.29         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7195_inverse__of__nat__le,axiom,
% 9.10/5.29      ! [N: nat,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.29       => ( ( N != zero_zero_nat )
% 9.10/5.29         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7196_even__abs__add__iff,axiom,
% 9.10/5.29      ! [K: int,L: int] :
% 9.10/5.29        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
% 9.10/5.29        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7197_even__add__abs__iff,axiom,
% 9.10/5.29      ! [K: int,L: int] :
% 9.10/5.29        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
% 9.10/5.29        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7198_real__archimedian__rdiv__eq__0,axiom,
% 9.10/5.29      ! [X3: real,C: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.10/5.29         => ( ! [M4: nat] :
% 9.10/5.29                ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 9.10/5.29               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X3 ) @ C ) )
% 9.10/5.29           => ( X3 = zero_zero_real ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7199_neg__int__cases,axiom,
% 9.10/5.29      ! [K: int] :
% 9.10/5.29        ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ( ( K
% 9.10/5.29                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.10/5.29             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7200_zdiff__int__split,axiom,
% 9.10/5.29      ! [P: int > $o,X3: nat,Y3: nat] :
% 9.10/5.29        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y3 ) ) )
% 9.10/5.29        = ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.10/5.29           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 9.10/5.29          & ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.29           => ( P @ zero_zero_int ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7201_real__of__nat__div2,axiom,
% 9.10/5.29      ! [N: nat,X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7202_real__of__nat__div3,axiom,
% 9.10/5.29      ! [N: nat,X3: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X3 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X3 ) ) ) @ one_one_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7203_ln__realpow,axiom,
% 9.10/5.29      ! [X3: real,N: nat] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( ln_ln_real @ ( power_power_real @ X3 @ N ) )
% 9.10/5.29          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7204_nat__intermed__int__val,axiom,
% 9.10/5.29      ! [M: nat,N: nat,F: nat > int,K: int] :
% 9.10/5.29        ( ! [I3: nat] :
% 9.10/5.29            ( ( ( ord_less_eq_nat @ M @ I3 )
% 9.10/5.29              & ( ord_less_nat @ I3 @ N ) )
% 9.10/5.29           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 9.10/5.29       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.29         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 9.10/5.29           => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.10/5.29             => ? [I3: nat] :
% 9.10/5.29                  ( ( ord_less_eq_nat @ M @ I3 )
% 9.10/5.29                  & ( ord_less_eq_nat @ I3 @ N )
% 9.10/5.29                  & ( ( F @ I3 )
% 9.10/5.29                    = K ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7205_incr__lemma,axiom,
% 9.10/5.29      ! [D: int,Z: int,X3: int] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.29       => ( ord_less_int @ Z @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7206_decr__lemma,axiom,
% 9.10/5.29      ! [D: int,X3: int,Z: int] :
% 9.10/5.29        ( ( ord_less_int @ zero_zero_int @ D )
% 9.10/5.29       => ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7207_linear__plus__1__le__power,axiom,
% 9.10/5.29      ! [X3: real,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X3 @ one_one_real ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7208_Bernoulli__inequality,axiom,
% 9.10/5.29      ! [X3: real,N: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.29       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7209_divmod__nat__def,axiom,
% 9.10/5.29      ( divmod_nat
% 9.10/5.29      = ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7210_nat__ivt__aux,axiom,
% 9.10/5.29      ! [N: nat,F: nat > int,K: int] :
% 9.10/5.29        ( ! [I3: nat] :
% 9.10/5.29            ( ( ord_less_nat @ I3 @ N )
% 9.10/5.29           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 9.10/5.29       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.10/5.29         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.10/5.29           => ? [I3: nat] :
% 9.10/5.29                ( ( ord_less_eq_nat @ I3 @ N )
% 9.10/5.29                & ( ( F @ I3 )
% 9.10/5.29                  = K ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7211_nat0__intermed__int__val,axiom,
% 9.10/5.29      ! [N: nat,F: nat > int,K: int] :
% 9.10/5.29        ( ! [I3: nat] :
% 9.10/5.29            ( ( ord_less_nat @ I3 @ N )
% 9.10/5.29           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 9.10/5.29       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.10/5.29         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.10/5.29           => ? [I3: nat] :
% 9.10/5.29                ( ( ord_less_eq_nat @ I3 @ N )
% 9.10/5.29                & ( ( F @ I3 )
% 9.10/5.29                  = K ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7212_double__arith__series,axiom,
% 9.10/5.29      ! [A: complex,D: complex,N: nat] :
% 9.10/5.29        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.10/5.29          @ ( groups2073611262835488442omplex
% 9.10/5.29            @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7213_double__arith__series,axiom,
% 9.10/5.29      ! [A: rat,D: rat,N: nat] :
% 9.10/5.29        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.10/5.29          @ ( groups2906978787729119204at_rat
% 9.10/5.29            @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7214_double__arith__series,axiom,
% 9.10/5.29      ! [A: int,D: int,N: nat] :
% 9.10/5.29        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.10/5.29          @ ( groups3539618377306564664at_int
% 9.10/5.29            @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7215_double__arith__series,axiom,
% 9.10/5.29      ! [A: nat,D: nat,N: nat] :
% 9.10/5.29        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.10/5.29          @ ( groups3542108847815614940at_nat
% 9.10/5.29            @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7216_double__arith__series,axiom,
% 9.10/5.29      ! [A: real,D: real,N: nat] :
% 9.10/5.29        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.10/5.29          @ ( groups6591440286371151544t_real
% 9.10/5.29            @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 9.10/5.29            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7217_double__gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7218_double__gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7219_double__gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7220_double__gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7221_double__gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.29        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7222_arctan__add,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.29       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.29         => ( ( plus_plus_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 9.10/5.29            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7223_arith__series,axiom,
% 9.10/5.29      ! [A: int,D: int,N: nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.29        = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7224_arith__series,axiom,
% 9.10/5.29      ! [A: nat,D: nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat
% 9.10/5.29          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 9.10/5.29          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7225_gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.29        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7226_gauss__sum,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7227_double__gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.10/5.29        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7228_double__gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.10/5.29        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7229_double__gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.10/5.29        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7230_double__gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.10/5.29        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7231_double__gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.10/5.29        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7232_Bernoulli__inequality__even,axiom,
% 9.10/5.29      ! [N: nat,X3: real] :
% 9.10/5.29        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.29       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X3 ) @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7233_sum__gp__offset,axiom,
% 9.10/5.29      ! [X3: complex,M: nat,N: nat] :
% 9.10/5.29        ( ( ( X3 = one_one_complex )
% 9.10/5.29         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.29            = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 9.10/5.29        & ( ( X3 != one_one_complex )
% 9.10/5.29         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.29            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7234_sum__gp__offset,axiom,
% 9.10/5.29      ! [X3: rat,M: nat,N: nat] :
% 9.10/5.29        ( ( ( X3 = one_one_rat )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.29            = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 9.10/5.29        & ( ( X3 != one_one_rat )
% 9.10/5.29         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.29            = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7235_sum__gp__offset,axiom,
% 9.10/5.29      ! [X3: real,M: nat,N: nat] :
% 9.10/5.29        ( ( ( X3 = one_one_real )
% 9.10/5.29         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.29            = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 9.10/5.29        & ( ( X3 != one_one_real )
% 9.10/5.29         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.29            = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7236_gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.10/5.29        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7237_gauss__sum__from__Suc__0,axiom,
% 9.10/5.29      ! [N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.10/5.29        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7238_of__nat__code__if,axiom,
% 9.10/5.29      ( semiri8010041392384452111omplex
% 9.10/5.29      = ( ^ [N2: nat] :
% 9.10/5.29            ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
% 9.10/5.29            @ ( produc1917071388513777916omplex
% 9.10/5.29              @ ^ [M3: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
% 9.10/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7239_of__nat__code__if,axiom,
% 9.10/5.29      ( semiri681578069525770553at_rat
% 9.10/5.29      = ( ^ [N2: nat] :
% 9.10/5.29            ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
% 9.10/5.29            @ ( produc6207742614233964070at_rat
% 9.10/5.29              @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
% 9.10/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7240_of__nat__code__if,axiom,
% 9.10/5.29      ( semiri1314217659103216013at_int
% 9.10/5.29      = ( ^ [N2: nat] :
% 9.10/5.29            ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
% 9.10/5.29            @ ( produc6840382203811409530at_int
% 9.10/5.29              @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
% 9.10/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7241_of__nat__code__if,axiom,
% 9.10/5.29      ( semiri5074537144036343181t_real
% 9.10/5.29      = ( ^ [N2: nat] :
% 9.10/5.29            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 9.10/5.29            @ ( produc1703576794950452218t_real
% 9.10/5.29              @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
% 9.10/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7242_of__nat__code__if,axiom,
% 9.10/5.29      ( semiri1316708129612266289at_nat
% 9.10/5.29      = ( ^ [N2: nat] :
% 9.10/5.29            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 9.10/5.29            @ ( produc6842872674320459806at_nat
% 9.10/5.29              @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
% 9.10/5.29              @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7243_nat__approx__posE,axiom,
% 9.10/5.29      ! [E: rat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7244_nat__approx__posE,axiom,
% 9.10/5.29      ! [E: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.29       => ~ ! [N3: nat] :
% 9.10/5.29              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7245_monoseq__arctan__series,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.29       => ( topolo6980174941875973593q_real
% 9.10/5.29          @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7246_lemma__termdiff3,axiom,
% 9.10/5.29      ! [H2: real,Z: real,K5: real,N: nat] :
% 9.10/5.29        ( ( H2 != zero_zero_real )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 9.10/5.29         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7247_lemma__termdiff3,axiom,
% 9.10/5.29      ! [H2: complex,Z: complex,K5: real,N: nat] :
% 9.10/5.29        ( ( H2 != zero_zero_complex )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 9.10/5.29         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7248_ex__less__of__nat__mult,axiom,
% 9.10/5.29      ! [X3: rat,Y3: rat] :
% 9.10/5.29        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.10/5.29       => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7249_ex__less__of__nat__mult,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7250_ln__series,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.29         => ( ( ln_ln_real @ X3 )
% 9.10/5.29            = ( suminf_real
% 9.10/5.29              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7251_powser__zero,axiom,
% 9.10/5.29      ! [F: nat > complex] :
% 9.10/5.29        ( ( suminf_complex
% 9.10/5.29          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 9.10/5.29        = ( F @ zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7252_powser__zero,axiom,
% 9.10/5.29      ! [F: nat > real] :
% 9.10/5.29        ( ( suminf_real
% 9.10/5.29          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 9.10/5.29        = ( F @ zero_zero_nat ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7253_int__if,axiom,
% 9.10/5.29      ! [P: $o,A: nat,B2: nat] :
% 9.10/5.29        ( ( P
% 9.10/5.29         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 9.10/5.29            = ( semiri1314217659103216013at_int @ A ) ) )
% 9.10/5.29        & ( ~ P
% 9.10/5.29         => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
% 9.10/5.29            = ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7254_nat__int__comparison_I1_J,axiom,
% 9.10/5.29      ( ( ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.29      = ( ^ [A5: nat,B4: nat] :
% 9.10/5.29            ( ( semiri1314217659103216013at_int @ A5 )
% 9.10/5.29            = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7255_complex__mod__minus__le__complex__mod,axiom,
% 9.10/5.29      ! [X3: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7256_complex__mod__triangle__ineq2,axiom,
% 9.10/5.29      ! [B2: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7257_monoseq__realpow,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.29       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.29         => ( topolo6980174941875973593q_real @ ( power_power_real @ X3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7258_real__arch__simple,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29      ? [N3: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7259_real__arch__simple,axiom,
% 9.10/5.29      ! [X3: rat] :
% 9.10/5.29      ? [N3: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7260_reals__Archimedean2,axiom,
% 9.10/5.29      ! [X3: rat] :
% 9.10/5.29      ? [N3: nat] : ( ord_less_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7261_reals__Archimedean2,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29      ? [N3: nat] : ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7262_arctan__series,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.29       => ( ( arctan @ X3 )
% 9.10/5.29          = ( suminf_real
% 9.10/5.29            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7263_norm__divide__numeral,axiom,
% 9.10/5.29      ! [A: real,W: num] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.29        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7264_norm__divide__numeral,axiom,
% 9.10/5.29      ! [A: complex,W: num] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.29        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7265_norm__mult__numeral1,axiom,
% 9.10/5.29      ! [W: num,A: real] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 9.10/5.29        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7266_norm__mult__numeral1,axiom,
% 9.10/5.29      ! [W: num,A: complex] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 9.10/5.29        = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7267_norm__mult__numeral2,axiom,
% 9.10/5.29      ! [A: real,W: num] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.29        = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7268_norm__mult__numeral2,axiom,
% 9.10/5.29      ! [A: complex,W: num] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.29        = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7269_norm__neg__numeral,axiom,
% 9.10/5.29      ! [W: num] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.29        = ( numeral_numeral_real @ W ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7270_norm__neg__numeral,axiom,
% 9.10/5.29      ! [W: num] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.29        = ( numeral_numeral_real @ W ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7271_norm__le__zero__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ zero_zero_real )
% 9.10/5.29        = ( X3 = zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7272_norm__le__zero__iff,axiom,
% 9.10/5.29      ! [X3: complex] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real )
% 9.10/5.29        = ( X3 = zero_zero_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7273_zero__less__norm__iff,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X3 ) )
% 9.10/5.29        = ( X3 != zero_zero_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7274_zero__less__norm__iff,axiom,
% 9.10/5.29      ! [X3: complex] :
% 9.10/5.29        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) )
% 9.10/5.29        = ( X3 != zero_zero_complex ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7275_norm__numeral,axiom,
% 9.10/5.29      ! [W: num] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 9.10/5.29        = ( numeral_numeral_real @ W ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7276_norm__numeral,axiom,
% 9.10/5.29      ! [W: num] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.29        = ( numeral_numeral_real @ W ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7277_norm__minus__commute,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) )
% 9.10/5.29        = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7278_norm__minus__commute,axiom,
% 9.10/5.29      ! [A: complex,B2: complex] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) )
% 9.10/5.29        = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7279_norm__not__less__zero,axiom,
% 9.10/5.29      ! [X3: complex] :
% 9.10/5.29        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7280_norm__ge__zero,axiom,
% 9.10/5.29      ! [X3: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7281_norm__mult,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) )
% 9.10/5.29        = ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7282_norm__mult,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) )
% 9.10/5.29        = ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7283_sum__norm__le,axiom,
% 9.10/5.29      ! [S2: set_real,F: real > complex,G: real > real] :
% 9.10/5.29        ( ! [X5: real] :
% 9.10/5.29            ( ( member_real @ X5 @ S2 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7284_sum__norm__le,axiom,
% 9.10/5.29      ! [S2: set_int,F: int > complex,G: int > real] :
% 9.10/5.29        ( ! [X5: int] :
% 9.10/5.29            ( ( member_int @ X5 @ S2 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7285_sum__norm__le,axiom,
% 9.10/5.29      ! [S2: set_nat,F: nat > complex,G: nat > real] :
% 9.10/5.29        ( ! [X5: nat] :
% 9.10/5.29            ( ( member_nat @ X5 @ S2 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7286_sum__norm__le,axiom,
% 9.10/5.29      ! [S2: set_complex,F: complex > complex,G: complex > real] :
% 9.10/5.29        ( ! [X5: complex] :
% 9.10/5.29            ( ( member_complex @ X5 @ S2 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7287_sum__norm__le,axiom,
% 9.10/5.29      ! [S2: set_nat,F: nat > real,G: nat > real] :
% 9.10/5.29        ( ! [X5: nat] :
% 9.10/5.29            ( ( member_nat @ X5 @ S2 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7288_norm__divide,axiom,
% 9.10/5.29      ! [A: real,B2: real] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.29        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7289_norm__divide,axiom,
% 9.10/5.29      ! [A: complex,B2: complex] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.29        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7290_norm__power,axiom,
% 9.10/5.29      ! [X3: real,N: nat] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) )
% 9.10/5.29        = ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7291_norm__power,axiom,
% 9.10/5.29      ! [X3: complex,N: nat] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) )
% 9.10/5.29        = ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7292_norm__sum,axiom,
% 9.10/5.29      ! [F: nat > complex,A4: set_nat] :
% 9.10/5.29        ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A4 ) )
% 9.10/5.29        @ ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7293_norm__sum,axiom,
% 9.10/5.29      ! [F: complex > complex,A4: set_complex] :
% 9.10/5.29        ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A4 ) )
% 9.10/5.29        @ ( groups5808333547571424918x_real
% 9.10/5.29          @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7294_norm__sum,axiom,
% 9.10/5.29      ! [F: nat > real,A4: set_nat] :
% 9.10/5.29        ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
% 9.10/5.29        @ ( groups6591440286371151544t_real
% 9.10/5.29          @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 9.10/5.29          @ A4 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7295_norm__uminus__minus,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) )
% 9.10/5.29        = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7296_norm__uminus__minus,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex] :
% 9.10/5.29        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 ) )
% 9.10/5.29        = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7297_nonzero__norm__divide,axiom,
% 9.10/5.29      ! [B2: real,A: real] :
% 9.10/5.29        ( ( B2 != zero_zero_real )
% 9.10/5.29       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.29          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7298_nonzero__norm__divide,axiom,
% 9.10/5.29      ! [B2: complex,A: complex] :
% 9.10/5.29        ( ( B2 != zero_zero_complex )
% 9.10/5.29       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.29          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7299_power__eq__imp__eq__norm,axiom,
% 9.10/5.29      ! [W: real,N: nat,Z: real] :
% 9.10/5.29        ( ( ( power_power_real @ W @ N )
% 9.10/5.29          = ( power_power_real @ Z @ N ) )
% 9.10/5.29       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.29         => ( ( real_V7735802525324610683m_real @ W )
% 9.10/5.29            = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7300_power__eq__imp__eq__norm,axiom,
% 9.10/5.29      ! [W: complex,N: nat,Z: complex] :
% 9.10/5.29        ( ( ( power_power_complex @ W @ N )
% 9.10/5.29          = ( power_power_complex @ Z @ N ) )
% 9.10/5.29       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.29         => ( ( real_V1022390504157884413omplex @ W )
% 9.10/5.29            = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7301_norm__mult__less,axiom,
% 9.10/5.29      ! [X3: real,R: real,Y3: real,S: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R )
% 9.10/5.29       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 9.10/5.29         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7302_norm__mult__less,axiom,
% 9.10/5.29      ! [X3: complex,R: real,Y3: complex,S: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R )
% 9.10/5.29       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 9.10/5.29         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7303_norm__mult__ineq,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7304_norm__mult__ineq,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7305_norm__triangle__lt,axiom,
% 9.10/5.29      ! [X3: real,Y3: real,E: real] :
% 9.10/5.29        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 9.10/5.29       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7306_norm__triangle__lt,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex,E: real] :
% 9.10/5.29        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 9.10/5.29       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7307_norm__add__less,axiom,
% 9.10/5.29      ! [X3: real,R: real,Y3: real,S: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R )
% 9.10/5.29       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 9.10/5.29         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7308_norm__add__less,axiom,
% 9.10/5.29      ! [X3: complex,R: real,Y3: complex,S: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R )
% 9.10/5.29       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 9.10/5.29         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7309_norm__power__ineq,axiom,
% 9.10/5.29      ! [X3: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X3 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X3 ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7310_norm__power__ineq,axiom,
% 9.10/5.29      ! [X3: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X3 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X3 ) @ N ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7311_norm__triangle__mono,axiom,
% 9.10/5.29      ! [A: real,R: real,B2: real,S: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S )
% 9.10/5.29         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7312_norm__triangle__mono,axiom,
% 9.10/5.29      ! [A: complex,R: real,B2: complex,S: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S )
% 9.10/5.29         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7313_norm__triangle__ineq,axiom,
% 9.10/5.29      ! [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 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7314_norm__triangle__ineq,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7315_norm__triangle__le,axiom,
% 9.10/5.29      ! [X3: real,Y3: real,E: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7316_norm__triangle__le,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex,E: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7317_norm__add__leD,axiom,
% 9.10/5.29      ! [A: real,B2: real,C: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) @ C )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7318_norm__add__leD,axiom,
% 9.10/5.29      ! [A: complex,B2: complex,C: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) @ C )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7319_norm__diff__triangle__less,axiom,
% 9.10/5.29      ! [X3: real,Y3: real,E1: real,Z: real,E22: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
% 9.10/5.29       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 9.10/5.29         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7320_norm__diff__triangle__less,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
% 9.10/5.29       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 9.10/5.29         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7321_norm__triangle__sub,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7322_norm__triangle__sub,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7323_norm__triangle__ineq4,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7324_norm__triangle__ineq4,axiom,
% 9.10/5.29      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7325_norm__diff__triangle__le,axiom,
% 9.10/5.29      ! [X3: real,Y3: real,E1: real,Z: real,E22: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 9.10/5.29         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7326_norm__diff__triangle__le,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 9.10/5.29         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7327_norm__triangle__le__diff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real,E: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7328_norm__triangle__le__diff,axiom,
% 9.10/5.29      ! [X3: complex,Y3: complex,E: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 9.10/5.29       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7329_norm__diff__ineq,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7330_norm__diff__ineq,axiom,
% 9.10/5.29      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7331_norm__triangle__ineq2,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7332_norm__triangle__ineq2,axiom,
% 9.10/5.29      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7333_power__eq__1__iff,axiom,
% 9.10/5.29      ! [W: real,N: nat] :
% 9.10/5.29        ( ( ( power_power_real @ W @ N )
% 9.10/5.29          = one_one_real )
% 9.10/5.29       => ( ( ( real_V7735802525324610683m_real @ W )
% 9.10/5.29            = one_one_real )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7334_power__eq__1__iff,axiom,
% 9.10/5.29      ! [W: complex,N: nat] :
% 9.10/5.29        ( ( ( power_power_complex @ W @ N )
% 9.10/5.29          = one_one_complex )
% 9.10/5.29       => ( ( ( real_V1022390504157884413omplex @ W )
% 9.10/5.29            = one_one_real )
% 9.10/5.29          | ( N = zero_zero_nat ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7335_norm__diff__triangle__ineq,axiom,
% 9.10/5.29      ! [A: real,B2: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7336_norm__diff__triangle__ineq,axiom,
% 9.10/5.29      ! [A: complex,B2: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B2 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7337_norm__triangle__ineq3,axiom,
% 9.10/5.29      ! [A: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7338_norm__triangle__ineq3,axiom,
% 9.10/5.29      ! [A: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B2 ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7339_square__norm__one,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          = one_one_real )
% 9.10/5.29       => ( ( real_V7735802525324610683m_real @ X3 )
% 9.10/5.29          = one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7340_square__norm__one,axiom,
% 9.10/5.29      ! [X3: complex] :
% 9.10/5.29        ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.29          = one_one_complex )
% 9.10/5.29       => ( ( real_V1022390504157884413omplex @ X3 )
% 9.10/5.29          = one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7341_norm__power__diff,axiom,
% 9.10/5.29      ! [Z: real,W: real,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 9.10/5.29         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7342_norm__power__diff,axiom,
% 9.10/5.29      ! [Z: complex,W: complex,M: nat] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 9.10/5.29       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 9.10/5.29         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7343_suminf__geometric,axiom,
% 9.10/5.29      ! [C: real] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.10/5.29       => ( ( suminf_real @ ( power_power_real @ C ) )
% 9.10/5.29          = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7344_suminf__geometric,axiom,
% 9.10/5.29      ! [C: complex] :
% 9.10/5.29        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.10/5.29       => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 9.10/5.29          = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7345_pi__series,axiom,
% 9.10/5.29      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.29      = ( suminf_real
% 9.10/5.29        @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7346_lemma__termdiff2,axiom,
% 9.10/5.29      ! [H2: complex,Z: complex,N: nat] :
% 9.10/5.29        ( ( H2 != zero_zero_complex )
% 9.10/5.29       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.29          = ( times_times_complex @ H2
% 9.10/5.29            @ ( groups2073611262835488442omplex
% 9.10/5.29              @ ^ [P5: nat] :
% 9.10/5.29                  ( groups2073611262835488442omplex
% 9.10/5.29                  @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 9.10/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 9.10/5.29              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7347_lemma__termdiff2,axiom,
% 9.10/5.29      ! [H2: rat,Z: rat,N: nat] :
% 9.10/5.29        ( ( H2 != zero_zero_rat )
% 9.10/5.29       => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.29          = ( times_times_rat @ H2
% 9.10/5.29            @ ( groups2906978787729119204at_rat
% 9.10/5.29              @ ^ [P5: nat] :
% 9.10/5.29                  ( groups2906978787729119204at_rat
% 9.10/5.29                  @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 9.10/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 9.10/5.29              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7348_lemma__termdiff2,axiom,
% 9.10/5.29      ! [H2: real,Z: real,N: nat] :
% 9.10/5.29        ( ( H2 != zero_zero_real )
% 9.10/5.29       => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.29          = ( times_times_real @ H2
% 9.10/5.29            @ ( groups6591440286371151544t_real
% 9.10/5.29              @ ^ [P5: nat] :
% 9.10/5.29                  ( groups6591440286371151544t_real
% 9.10/5.29                  @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 9.10/5.29                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 9.10/5.29              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7349_summable__arctan__series,axiom,
% 9.10/5.29      ! [X3: real] :
% 9.10/5.29        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.29       => ( summable_real
% 9.10/5.29          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7350_pred__subset__eq2,axiom,
% 9.10/5.29      ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 9.10/5.29        ( ( ord_le6741204236512500942_int_o
% 9.10/5.29          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
% 9.10/5.29          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) )
% 9.10/5.29        = ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7351_pred__subset__eq2,axiom,
% 9.10/5.29      ! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 9.10/5.29        ( ( ord_le3636971675376928563eger_o
% 9.10/5.29          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R2 )
% 9.10/5.29          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S2 ) )
% 9.10/5.29        = ( ord_le3216752416896350996nteger @ R2 @ S2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7352_pred__subset__eq2,axiom,
% 9.10/5.29      ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 9.10/5.29        ( ( ord_le4340812435750786203eger_o
% 9.10/5.29          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 )
% 9.10/5.29          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) )
% 9.10/5.29        = ( ord_le653643898420964396nteger @ R2 @ S2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7353_pred__subset__eq2,axiom,
% 9.10/5.29      ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 9.10/5.29        ( ( ord_le5643404153117327598_int_o
% 9.10/5.29          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 )
% 9.10/5.29          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) )
% 9.10/5.29        = ( ord_le8725513860283290265nt_int @ R2 @ S2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7354_pred__subset__eq2,axiom,
% 9.10/5.29      ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 9.10/5.29        ( ( ord_le2124322318746777828_int_o
% 9.10/5.29          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 )
% 9.10/5.29          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) )
% 9.10/5.29        = ( ord_le135402666524580259nt_int @ R2 @ S2 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7355_infinite__int__iff__unbounded,axiom,
% 9.10/5.29      ! [S2: set_int] :
% 9.10/5.29        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.10/5.29        = ( ! [M3: int] :
% 9.10/5.29            ? [N2: int] :
% 9.10/5.29              ( ( ord_less_int @ M3 @ ( abs_abs_int @ N2 ) )
% 9.10/5.29              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7356_lessThan__iff,axiom,
% 9.10/5.29      ! [I: rat,K: rat] :
% 9.10/5.29        ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 9.10/5.29        = ( ord_less_rat @ I @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7357_lessThan__iff,axiom,
% 9.10/5.29      ! [I: num,K: num] :
% 9.10/5.29        ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 9.10/5.29        = ( ord_less_num @ I @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7358_lessThan__iff,axiom,
% 9.10/5.29      ! [I: nat,K: nat] :
% 9.10/5.29        ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 9.10/5.29        = ( ord_less_nat @ I @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7359_lessThan__iff,axiom,
% 9.10/5.29      ! [I: int,K: int] :
% 9.10/5.29        ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 9.10/5.29        = ( ord_less_int @ I @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7360_lessThan__iff,axiom,
% 9.10/5.29      ! [I: real,K: real] :
% 9.10/5.29        ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 9.10/5.29        = ( ord_less_real @ I @ K ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7361_summable__iff__shift,axiom,
% 9.10/5.29      ! [F: nat > real,K: nat] :
% 9.10/5.29        ( ( summable_real
% 9.10/5.29          @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 9.10/5.29        = ( summable_real @ F ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7362_lessThan__subset__iff,axiom,
% 9.10/5.29      ! [X3: rat,Y3: rat] :
% 9.10/5.29        ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7363_lessThan__subset__iff,axiom,
% 9.10/5.29      ! [X3: num,Y3: num] :
% 9.10/5.29        ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7364_lessThan__subset__iff,axiom,
% 9.10/5.29      ! [X3: nat,Y3: nat] :
% 9.10/5.29        ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7365_lessThan__subset__iff,axiom,
% 9.10/5.29      ! [X3: int,Y3: int] :
% 9.10/5.29        ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7366_lessThan__subset__iff,axiom,
% 9.10/5.29      ! [X3: real,Y3: real] :
% 9.10/5.29        ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 9.10/5.29        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7367_summable__cmult__iff,axiom,
% 9.10/5.29      ! [C: complex,F: nat > complex] :
% 9.10/5.29        ( ( summable_complex
% 9.10/5.29          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 9.10/5.29        = ( ( C = zero_zero_complex )
% 9.10/5.29          | ( summable_complex @ F ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7368_summable__cmult__iff,axiom,
% 9.10/5.29      ! [C: real,F: nat > real] :
% 9.10/5.29        ( ( summable_real
% 9.10/5.29          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.10/5.29        = ( ( C = zero_zero_real )
% 9.10/5.29          | ( summable_real @ F ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7369_summable__divide__iff,axiom,
% 9.10/5.29      ! [F: nat > complex,C: complex] :
% 9.10/5.29        ( ( summable_complex
% 9.10/5.29          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 9.10/5.29        = ( ( C = zero_zero_complex )
% 9.10/5.29          | ( summable_complex @ F ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7370_summable__divide__iff,axiom,
% 9.10/5.29      ! [F: nat > real,C: real] :
% 9.10/5.29        ( ( summable_real
% 9.10/5.29          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 9.10/5.29        = ( ( C = zero_zero_real )
% 9.10/5.29          | ( summable_real @ F ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7371_sum_OlessThan__Suc,axiom,
% 9.10/5.29      ! [G: nat > rat,N: nat] :
% 9.10/5.29        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7372_sum_OlessThan__Suc,axiom,
% 9.10/5.29      ! [G: nat > int,N: nat] :
% 9.10/5.29        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7373_sum_OlessThan__Suc,axiom,
% 9.10/5.29      ! [G: nat > nat,N: nat] :
% 9.10/5.29        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7374_sum_OlessThan__Suc,axiom,
% 9.10/5.29      ! [G: nat > real,N: nat] :
% 9.10/5.29        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.29        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7375_summable__geometric__iff,axiom,
% 9.10/5.29      ! [C: real] :
% 9.10/5.29        ( ( summable_real @ ( power_power_real @ C ) )
% 9.10/5.29        = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7376_summable__geometric__iff,axiom,
% 9.10/5.29      ! [C: complex] :
% 9.10/5.29        ( ( summable_complex @ ( power_power_complex @ C ) )
% 9.10/5.29        = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7377_summable__comparison__test_H,axiom,
% 9.10/5.29      ! [G: nat > real,N4: nat,F: nat > real] :
% 9.10/5.29        ( ( summable_real @ G )
% 9.10/5.29       => ( ! [N3: nat] :
% 9.10/5.29              ( ( ord_less_eq_nat @ N4 @ N3 )
% 9.10/5.29             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.10/5.29         => ( summable_real @ F ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7378_summable__comparison__test_H,axiom,
% 9.10/5.29      ! [G: nat > real,N4: nat,F: nat > complex] :
% 9.10/5.29        ( ( summable_real @ G )
% 9.10/5.29       => ( ! [N3: nat] :
% 9.10/5.29              ( ( ord_less_eq_nat @ N4 @ N3 )
% 9.10/5.29             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.10/5.29         => ( summable_complex @ F ) ) ) ).
% 9.10/5.29  
% 9.10/5.29  thf(fact_7379_summable__comparison__test,axiom,
% 9.10/5.29      ! [F: nat > real,G: nat > real] :
% 9.10/5.29        ( ? [N7: nat] :
% 9.10/5.29          ! [N3: nat] :
% 9.10/5.29            ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.10/5.29           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( summable_real @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7380_summable__comparison__test,axiom,
% 9.10/5.30      ! [F: nat > complex,G: nat > real] :
% 9.10/5.30        ( ? [N7: nat] :
% 9.10/5.30          ! [N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.10/5.30           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( summable_complex @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7381_summable__ignore__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > real,K: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7382_summable__add,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7383_summable__add,axiom,
% 9.10/5.30      ! [F: nat > nat,G: nat > nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ( summable_nat @ G )
% 9.10/5.30         => ( summable_nat
% 9.10/5.30            @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7384_summable__add,axiom,
% 9.10/5.30      ! [F: nat > int,G: nat > int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ( summable_int @ G )
% 9.10/5.30         => ( summable_int
% 9.10/5.30            @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7385_summable__diff,axiom,
% 9.10/5.30      ! [F: nat > complex,G: nat > complex] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( ( summable_complex @ G )
% 9.10/5.30         => ( summable_complex
% 9.10/5.30            @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7386_summable__diff,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7387_summable__divide,axiom,
% 9.10/5.30      ! [F: nat > complex,C: complex] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7388_summable__divide,axiom,
% 9.10/5.30      ! [F: nat > real,C: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7389_summable__mult2,axiom,
% 9.10/5.30      ! [F: nat > real,C: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7390_summable__mult,axiom,
% 9.10/5.30      ! [F: nat > real,C: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7391_suminf__le__const,axiom,
% 9.10/5.30      ! [F: nat > int,X3: int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.10/5.30         => ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7392_suminf__le__const,axiom,
% 9.10/5.30      ! [F: nat > nat,X3: nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.10/5.30         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7393_suminf__le__const,axiom,
% 9.10/5.30      ! [F: nat > real,X3: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.10/5.30         => ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7394_summableI__nonneg__bounded,axiom,
% 9.10/5.30      ! [F: nat > int,X3: int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.10/5.30         => ( summable_int @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7395_summableI__nonneg__bounded,axiom,
% 9.10/5.30      ! [F: nat > nat,X3: nat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.10/5.30         => ( summable_nat @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7396_summableI__nonneg__bounded,axiom,
% 9.10/5.30      ! [F: nat > real,X3: real] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.10/5.30         => ( summable_real @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7397_suminf__le,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.30       => ( ( summable_real @ F )
% 9.10/5.30         => ( ( summable_real @ G )
% 9.10/5.30           => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7398_suminf__le,axiom,
% 9.10/5.30      ! [F: nat > nat,G: nat > nat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.30       => ( ( summable_nat @ F )
% 9.10/5.30         => ( ( summable_nat @ G )
% 9.10/5.30           => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7399_suminf__le,axiom,
% 9.10/5.30      ! [F: nat > int,G: nat > int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.30       => ( ( summable_int @ F )
% 9.10/5.30         => ( ( summable_int @ G )
% 9.10/5.30           => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7400_summable__mult__D,axiom,
% 9.10/5.30      ! [C: complex,F: nat > complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 9.10/5.30       => ( ( C != zero_zero_complex )
% 9.10/5.30         => ( summable_complex @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7401_summable__mult__D,axiom,
% 9.10/5.30      ! [C: real,F: nat > real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.10/5.30       => ( ( C != zero_zero_real )
% 9.10/5.30         => ( summable_real @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7402_summable__zero__power,axiom,
% 9.10/5.30      summable_real @ ( power_power_real @ zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7403_summable__zero__power,axiom,
% 9.10/5.30      summable_int @ ( power_power_int @ zero_zero_int ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7404_summable__zero__power,axiom,
% 9.10/5.30      summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7405_lessThan__def,axiom,
% 9.10/5.30      ( set_ord_lessThan_rat
% 9.10/5.30      = ( ^ [U2: rat] :
% 9.10/5.30            ( collect_rat
% 9.10/5.30            @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7406_lessThan__def,axiom,
% 9.10/5.30      ( set_ord_lessThan_num
% 9.10/5.30      = ( ^ [U2: num] :
% 9.10/5.30            ( collect_num
% 9.10/5.30            @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7407_lessThan__def,axiom,
% 9.10/5.30      ( set_ord_lessThan_nat
% 9.10/5.30      = ( ^ [U2: nat] :
% 9.10/5.30            ( collect_nat
% 9.10/5.30            @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7408_lessThan__def,axiom,
% 9.10/5.30      ( set_ord_lessThan_int
% 9.10/5.30      = ( ^ [U2: int] :
% 9.10/5.30            ( collect_int
% 9.10/5.30            @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7409_lessThan__def,axiom,
% 9.10/5.30      ( set_or5984915006950818249n_real
% 9.10/5.30      = ( ^ [U2: real] :
% 9.10/5.30            ( collect_real
% 9.10/5.30            @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7410_suminf__mult,axiom,
% 9.10/5.30      ! [F: nat > real,C: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.10/5.30          = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7411_suminf__mult2,axiom,
% 9.10/5.30      ! [F: nat > real,C: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 9.10/5.30          = ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7412_suminf__add,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 9.10/5.30            = ( suminf_real
% 9.10/5.30              @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7413_suminf__add,axiom,
% 9.10/5.30      ! [F: nat > nat,G: nat > nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ( summable_nat @ G )
% 9.10/5.30         => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 9.10/5.30            = ( suminf_nat
% 9.10/5.30              @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7414_suminf__add,axiom,
% 9.10/5.30      ! [F: nat > int,G: nat > int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ( summable_int @ G )
% 9.10/5.30         => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 9.10/5.30            = ( suminf_int
% 9.10/5.30              @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7415_suminf__diff,axiom,
% 9.10/5.30      ! [F: nat > complex,G: nat > complex] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( ( summable_complex @ G )
% 9.10/5.30         => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
% 9.10/5.30            = ( suminf_complex
% 9.10/5.30              @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7416_suminf__diff,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 9.10/5.30            = ( suminf_real
% 9.10/5.30              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7417_suminf__divide,axiom,
% 9.10/5.30      ! [F: nat > complex,C: complex] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( ( suminf_complex
% 9.10/5.30            @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 9.10/5.30          = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7418_suminf__divide,axiom,
% 9.10/5.30      ! [F: nat > real,C: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 9.10/5.30          = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7419_suminf__split__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > real,K: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( suminf_real @ F )
% 9.10/5.30          = ( plus_plus_real
% 9.10/5.30            @ ( suminf_real
% 9.10/5.30              @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 9.10/5.30            @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7420_suminf__minus__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > complex,K: nat] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( ( suminf_complex
% 9.10/5.30            @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 9.10/5.30          = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7421_suminf__minus__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > real,K: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 9.10/5.30          = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7422_finite__nat__bounded,axiom,
% 9.10/5.30      ! [S2: set_nat] :
% 9.10/5.30        ( ( finite_finite_nat @ S2 )
% 9.10/5.30       => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7423_finite__nat__iff__bounded,axiom,
% 9.10/5.30      ( finite_finite_nat
% 9.10/5.30      = ( ^ [S4: set_nat] :
% 9.10/5.30          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7424_sum__less__suminf,axiom,
% 9.10/5.30      ! [F: nat > int,N: nat] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [M4: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N @ M4 )
% 9.10/5.30             => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 9.10/5.30         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7425_sum__less__suminf,axiom,
% 9.10/5.30      ! [F: nat > nat,N: nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [M4: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N @ M4 )
% 9.10/5.30             => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 9.10/5.30         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7426_sum__less__suminf,axiom,
% 9.10/5.30      ! [F: nat > real,N: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [M4: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N @ M4 )
% 9.10/5.30             => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 9.10/5.30         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7427_powser__insidea,axiom,
% 9.10/5.30      ! [F: nat > real,X3: real,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 9.10/5.30       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7428_powser__insidea,axiom,
% 9.10/5.30      ! [F: nat > complex,X3: complex,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 9.10/5.30       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7429_suminf__eq__zero__iff,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ( suminf_real @ F )
% 9.10/5.30              = zero_zero_real )
% 9.10/5.30            = ( ! [N2: nat] :
% 9.10/5.30                  ( ( F @ N2 )
% 9.10/5.30                  = zero_zero_real ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7430_suminf__eq__zero__iff,axiom,
% 9.10/5.30      ! [F: nat > nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ( suminf_nat @ F )
% 9.10/5.30              = zero_zero_nat )
% 9.10/5.30            = ( ! [N2: nat] :
% 9.10/5.30                  ( ( F @ N2 )
% 9.10/5.30                  = zero_zero_nat ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7431_suminf__eq__zero__iff,axiom,
% 9.10/5.30      ! [F: nat > int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ( suminf_int @ F )
% 9.10/5.30              = zero_zero_int )
% 9.10/5.30            = ( ! [N2: nat] :
% 9.10/5.30                  ( ( F @ N2 )
% 9.10/5.30                  = zero_zero_int ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7432_suminf__nonneg,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.10/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7433_suminf__nonneg,axiom,
% 9.10/5.30      ! [F: nat > nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.10/5.30         => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7434_suminf__nonneg,axiom,
% 9.10/5.30      ! [F: nat > int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.10/5.30         => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7435_suminf__pos,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7436_suminf__pos,axiom,
% 9.10/5.30      ! [F: nat > nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.10/5.30         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7437_suminf__pos,axiom,
% 9.10/5.30      ! [F: nat > int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 9.10/5.30         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7438_lessThan__strict__subset__iff,axiom,
% 9.10/5.30      ! [M: rat,N: rat] :
% 9.10/5.30        ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 9.10/5.30        = ( ord_less_rat @ M @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7439_lessThan__strict__subset__iff,axiom,
% 9.10/5.30      ! [M: num,N: num] :
% 9.10/5.30        ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 9.10/5.30        = ( ord_less_num @ M @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7440_lessThan__strict__subset__iff,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30        = ( ord_less_nat @ M @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7441_lessThan__strict__subset__iff,axiom,
% 9.10/5.30      ! [M: int,N: int] :
% 9.10/5.30        ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 9.10/5.30        = ( ord_less_int @ M @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7442_lessThan__strict__subset__iff,axiom,
% 9.10/5.30      ! [M: real,N: real] :
% 9.10/5.30        ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 9.10/5.30        = ( ord_less_real @ M @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7443_summable__0__powser,axiom,
% 9.10/5.30      ! [F: nat > complex] :
% 9.10/5.30        ( summable_complex
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7444_summable__0__powser,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( summable_real
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7445_summable__zero__power_H,axiom,
% 9.10/5.30      ! [F: nat > complex] :
% 9.10/5.30        ( summable_complex
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7446_summable__zero__power_H,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( summable_real
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7447_summable__zero__power_H,axiom,
% 9.10/5.30      ! [F: nat > int] :
% 9.10/5.30        ( summable_int
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7448_powser__split__head_I3_J,axiom,
% 9.10/5.30      ! [F: nat > complex,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30       => ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7449_powser__split__head_I3_J,axiom,
% 9.10/5.30      ! [F: nat > real,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30       => ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7450_summable__powser__split__head,axiom,
% 9.10/5.30      ! [F: nat > complex,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30        = ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7451_summable__powser__split__head,axiom,
% 9.10/5.30      ! [F: nat > real,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30        = ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7452_summable__powser__ignore__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > complex,M: nat,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30        = ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7453_summable__powser__ignore__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > real,M: nat,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30        = ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7454_pi__not__less__zero,axiom,
% 9.10/5.30      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7455_pi__gt__zero,axiom,
% 9.10/5.30      ord_less_real @ zero_zero_real @ pi ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7456_pi__ge__zero,axiom,
% 9.10/5.30      ord_less_eq_real @ zero_zero_real @ pi ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7457_sum__less__suminf2,axiom,
% 9.10/5.30      ! [F: nat > int,N: nat,I: nat] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [M4: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N @ M4 )
% 9.10/5.30             => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 9.10/5.30         => ( ( ord_less_eq_nat @ N @ I )
% 9.10/5.30           => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.10/5.30             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7458_sum__less__suminf2,axiom,
% 9.10/5.30      ! [F: nat > nat,N: nat,I: nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [M4: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N @ M4 )
% 9.10/5.30             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 9.10/5.30         => ( ( ord_less_eq_nat @ N @ I )
% 9.10/5.30           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.10/5.30             => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7459_sum__less__suminf2,axiom,
% 9.10/5.30      ! [F: nat > real,N: nat,I: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [M4: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N @ M4 )
% 9.10/5.30             => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 9.10/5.30         => ( ( ord_less_eq_nat @ N @ I )
% 9.10/5.30           => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.10/5.30             => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7460_summable__norm__comparison__test,axiom,
% 9.10/5.30      ! [F: nat > complex,G: nat > real] :
% 9.10/5.30        ( ? [N7: nat] :
% 9.10/5.30          ! [N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.10/5.30           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7461_summable__rabs__comparison__test,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real] :
% 9.10/5.30        ( ? [N7: nat] :
% 9.10/5.30          ! [N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.10/5.30           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.10/5.30       => ( ( summable_real @ G )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7462_summable__rabs,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 9.10/5.30          @ ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7463_suminf__pos2,axiom,
% 9.10/5.30      ! [F: nat > real,I: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.10/5.30           => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7464_suminf__pos2,axiom,
% 9.10/5.30      ! [F: nat > nat,I: nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.10/5.30           => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7465_suminf__pos2,axiom,
% 9.10/5.30      ! [F: nat > int,I: nat] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.10/5.30           => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7466_suminf__pos__iff,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 9.10/5.30            = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7467_suminf__pos__iff,axiom,
% 9.10/5.30      ! [F: nat > nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 9.10/5.30            = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7468_suminf__pos__iff,axiom,
% 9.10/5.30      ! [F: nat > int] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.10/5.30         => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 9.10/5.30            = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7469_summable__geometric,axiom,
% 9.10/5.30      ! [C: real] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.10/5.30       => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7470_summable__geometric,axiom,
% 9.10/5.30      ! [C: complex] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.10/5.30       => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7471_complete__algebra__summable__geometric,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ one_one_real )
% 9.10/5.30       => ( summable_real @ ( power_power_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7472_complete__algebra__summable__geometric,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ one_one_real )
% 9.10/5.30       => ( summable_complex @ ( power_power_complex @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7473_suminf__split__head,axiom,
% 9.10/5.30      ! [F: nat > complex] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( ( suminf_complex
% 9.10/5.30            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 9.10/5.30          = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7474_suminf__split__head,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 9.10/5.30          = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7475_summable__norm,axiom,
% 9.10/5.30      ! [F: nat > real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 9.10/5.30          @ ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7476_summable__norm,axiom,
% 9.10/5.30      ! [F: nat > complex] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 9.10/5.30          @ ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7477_sum_Onat__diff__reindex,axiom,
% 9.10/5.30      ! [G: nat > nat,N: nat] :
% 9.10/5.30        ( ( groups3542108847815614940at_nat
% 9.10/5.30          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30        = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7478_sum_Onat__diff__reindex,axiom,
% 9.10/5.30      ! [G: nat > real,N: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30        = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7479_sum__le__suminf,axiom,
% 9.10/5.30      ! [F: nat > int,I5: set_nat] :
% 9.10/5.30        ( ( summable_int @ F )
% 9.10/5.30       => ( ( finite_finite_nat @ I5 )
% 9.10/5.30         => ( ! [N3: nat] :
% 9.10/5.30                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.10/5.30               => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 9.10/5.30           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7480_sum__le__suminf,axiom,
% 9.10/5.30      ! [F: nat > nat,I5: set_nat] :
% 9.10/5.30        ( ( summable_nat @ F )
% 9.10/5.30       => ( ( finite_finite_nat @ I5 )
% 9.10/5.30         => ( ! [N3: nat] :
% 9.10/5.30                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.10/5.30               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 9.10/5.30           => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7481_sum__le__suminf,axiom,
% 9.10/5.30      ! [F: nat > real,I5: set_nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( finite_finite_nat @ I5 )
% 9.10/5.30         => ( ! [N3: nat] :
% 9.10/5.30                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.10/5.30               => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 9.10/5.30           => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7482_sum__diff__distrib,axiom,
% 9.10/5.30      ! [Q: int > nat,P: int > nat,N: int] :
% 9.10/5.30        ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 9.10/5.30       => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 9.10/5.30          = ( groups4541462559716669496nt_nat
% 9.10/5.30            @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.10/5.30            @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7483_sum__diff__distrib,axiom,
% 9.10/5.30      ! [Q: real > nat,P: real > nat,N: real] :
% 9.10/5.30        ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 9.10/5.30       => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 9.10/5.30          = ( groups1935376822645274424al_nat
% 9.10/5.30            @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.10/5.30            @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7484_sum__diff__distrib,axiom,
% 9.10/5.30      ! [Q: nat > nat,P: nat > nat,N: nat] :
% 9.10/5.30        ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 9.10/5.30       => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 9.10/5.30          = ( groups3542108847815614940at_nat
% 9.10/5.30            @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7485_powser__inside,axiom,
% 9.10/5.30      ! [F: nat > real,X3: real,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 9.10/5.30       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7486_powser__inside,axiom,
% 9.10/5.30      ! [F: nat > complex,X3: complex,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 9.10/5.30       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 9.10/5.30         => ( summable_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7487_pi__less__4,axiom,
% 9.10/5.30      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7488_pi__ge__two,axiom,
% 9.10/5.30      ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7489_pi__half__neq__two,axiom,
% 9.10/5.30      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30     != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7490_sum__pos__lt__pair,axiom,
% 9.10/5.30      ! [F: nat > real,K: nat] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 9.10/5.30         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7491_powser__split__head_I1_J,axiom,
% 9.10/5.30      ! [F: nat > complex,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30       => ( ( suminf_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30          = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 9.10/5.30            @ ( times_times_complex
% 9.10/5.30              @ ( suminf_complex
% 9.10/5.30                @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30              @ Z ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7492_powser__split__head_I1_J,axiom,
% 9.10/5.30      ! [F: nat > real,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30       => ( ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30          = ( plus_plus_real @ ( F @ zero_zero_nat )
% 9.10/5.30            @ ( times_times_real
% 9.10/5.30              @ ( suminf_real
% 9.10/5.30                @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30              @ Z ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7493_sum_OlessThan__Suc__shift,axiom,
% 9.10/5.30      ! [G: nat > rat,N: nat] :
% 9.10/5.30        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.30        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.10/5.30          @ ( groups2906978787729119204at_rat
% 9.10/5.30            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7494_sum_OlessThan__Suc__shift,axiom,
% 9.10/5.30      ! [G: nat > int,N: nat] :
% 9.10/5.30        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.30        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.10/5.30          @ ( groups3539618377306564664at_int
% 9.10/5.30            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7495_sum_OlessThan__Suc__shift,axiom,
% 9.10/5.30      ! [G: nat > nat,N: nat] :
% 9.10/5.30        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.30        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.10/5.30          @ ( groups3542108847815614940at_nat
% 9.10/5.30            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7496_sum_OlessThan__Suc__shift,axiom,
% 9.10/5.30      ! [G: nat > real,N: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.30        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7497_powser__split__head_I2_J,axiom,
% 9.10/5.30      ! [F: nat > complex,Z: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30       => ( ( times_times_complex
% 9.10/5.30            @ ( suminf_complex
% 9.10/5.30              @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30            @ Z )
% 9.10/5.30          = ( minus_minus_complex
% 9.10/5.30            @ ( suminf_complex
% 9.10/5.30              @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.10/5.30            @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7498_powser__split__head_I2_J,axiom,
% 9.10/5.30      ! [F: nat > real,Z: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30       => ( ( times_times_real
% 9.10/5.30            @ ( suminf_real
% 9.10/5.30              @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30            @ Z )
% 9.10/5.30          = ( minus_minus_real
% 9.10/5.30            @ ( suminf_real
% 9.10/5.30              @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.10/5.30            @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7499_sumr__diff__mult__const2,axiom,
% 9.10/5.30      ! [F: nat > complex,N: nat,R: complex] :
% 9.10/5.30        ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R ) )
% 9.10/5.30        = ( groups2073611262835488442omplex
% 9.10/5.30          @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ R )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7500_sumr__diff__mult__const2,axiom,
% 9.10/5.30      ! [F: nat > rat,N: nat,R: rat] :
% 9.10/5.30        ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R ) )
% 9.10/5.30        = ( groups2906978787729119204at_rat
% 9.10/5.30          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7501_sumr__diff__mult__const2,axiom,
% 9.10/5.30      ! [F: nat > int,N: nat,R: int] :
% 9.10/5.30        ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
% 9.10/5.30        = ( groups3539618377306564664at_int
% 9.10/5.30          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7502_sumr__diff__mult__const2,axiom,
% 9.10/5.30      ! [F: nat > real,N: nat,R: real] :
% 9.10/5.30        ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
% 9.10/5.30        = ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7503_sum__lessThan__telescope,axiom,
% 9.10/5.30      ! [F: nat > complex,M: nat] :
% 9.10/5.30        ( ( groups2073611262835488442omplex
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_complex @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7504_sum__lessThan__telescope,axiom,
% 9.10/5.30      ! [F: nat > rat,M: nat] :
% 9.10/5.30        ( ( groups2906978787729119204at_rat
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7505_sum__lessThan__telescope,axiom,
% 9.10/5.30      ! [F: nat > int,M: nat] :
% 9.10/5.30        ( ( groups3539618377306564664at_int
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7506_sum__lessThan__telescope,axiom,
% 9.10/5.30      ! [F: nat > real,M: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7507_sum__lessThan__telescope_H,axiom,
% 9.10/5.30      ! [F: nat > complex,M: nat] :
% 9.10/5.30        ( ( groups2073611262835488442omplex
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7508_sum__lessThan__telescope_H,axiom,
% 9.10/5.30      ! [F: nat > rat,M: nat] :
% 9.10/5.30        ( ( groups2906978787729119204at_rat
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7509_sum__lessThan__telescope_H,axiom,
% 9.10/5.30      ! [F: nat > int,M: nat] :
% 9.10/5.30        ( ( groups3539618377306564664at_int
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7510_sum__lessThan__telescope_H,axiom,
% 9.10/5.30      ! [F: nat > real,M: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7511_summable__partial__sum__bound,axiom,
% 9.10/5.30      ! [F: nat > complex,E: real] :
% 9.10/5.30        ( ( summable_complex @ F )
% 9.10/5.30       => ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.30         => ~ ! [N8: nat] :
% 9.10/5.30                ~ ! [M2: nat] :
% 9.10/5.30                    ( ( ord_less_eq_nat @ N8 @ M2 )
% 9.10/5.30                   => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7512_summable__partial__sum__bound,axiom,
% 9.10/5.30      ! [F: nat > real,E: real] :
% 9.10/5.30        ( ( summable_real @ F )
% 9.10/5.30       => ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.30         => ~ ! [N8: nat] :
% 9.10/5.30                ~ ! [M2: nat] :
% 9.10/5.30                    ( ( ord_less_eq_nat @ N8 @ M2 )
% 9.10/5.30                   => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N9 ) ) ) @ E ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7513_suminf__exist__split,axiom,
% 9.10/5.30      ! [R: real,F: nat > real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ R )
% 9.10/5.30       => ( ( summable_real @ F )
% 9.10/5.30         => ? [N8: nat] :
% 9.10/5.30            ! [N9: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N8 @ N9 )
% 9.10/5.30             => ( ord_less_real
% 9.10/5.30                @ ( real_V7735802525324610683m_real
% 9.10/5.30                  @ ( suminf_real
% 9.10/5.30                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 9.10/5.30                @ R ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7514_suminf__exist__split,axiom,
% 9.10/5.30      ! [R: real,F: nat > complex] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ R )
% 9.10/5.30       => ( ( summable_complex @ F )
% 9.10/5.30         => ? [N8: nat] :
% 9.10/5.30            ! [N9: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N8 @ N9 )
% 9.10/5.30             => ( ord_less_real
% 9.10/5.30                @ ( real_V1022390504157884413omplex
% 9.10/5.30                  @ ( suminf_complex
% 9.10/5.30                    @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N9 ) ) ) )
% 9.10/5.30                @ R ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7515_summable__power__series,axiom,
% 9.10/5.30      ! [F: nat > real,Z: real] :
% 9.10/5.30        ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 9.10/5.30       => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 9.10/5.30           => ( ( ord_less_real @ Z @ one_one_real )
% 9.10/5.30             => ( summable_real
% 9.10/5.30                @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7516_Abel__lemma,axiom,
% 9.10/5.30      ! [R: real,R0: real,A: nat > complex,M7: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ R )
% 9.10/5.30       => ( ( ord_less_real @ R @ R0 )
% 9.10/5.30         => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
% 9.10/5.30           => ( summable_real
% 9.10/5.30              @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R @ N2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7517_bot__empty__eq2,axiom,
% 9.10/5.30      ( bot_bot_int_int_o
% 9.10/5.30      = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7518_bot__empty__eq2,axiom,
% 9.10/5.30      ( bot_bo5358457235160185703eger_o
% 9.10/5.30      = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7519_bot__empty__eq2,axiom,
% 9.10/5.30      ( bot_bo3000040243691356879eger_o
% 9.10/5.30      = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7520_bot__empty__eq2,axiom,
% 9.10/5.30      ( bot_bo8662317086119403298_int_o
% 9.10/5.30      = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7521_bot__empty__eq2,axiom,
% 9.10/5.30      ( bot_bo1403522918969695512_int_o
% 9.10/5.30      = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7522_pred__equals__eq2,axiom,
% 9.10/5.30      ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 9.10/5.30        ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) )
% 9.10/5.30          = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S2 ) ) )
% 9.10/5.30        = ( R2 = S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7523_pred__equals__eq2,axiom,
% 9.10/5.30      ! [R2: set_Pr8056137968301705908nteger,S2: set_Pr8056137968301705908nteger] :
% 9.10/5.30        ( ( ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R2 ) )
% 9.10/5.30          = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S2 ) ) )
% 9.10/5.30        = ( R2 = S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7524_pred__equals__eq2,axiom,
% 9.10/5.30      ! [R2: set_Pr1281608226676607948nteger,S2: set_Pr1281608226676607948nteger] :
% 9.10/5.30        ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R2 ) )
% 9.10/5.30          = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S2 ) ) )
% 9.10/5.30        = ( R2 = S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7525_pred__equals__eq2,axiom,
% 9.10/5.30      ! [R2: set_Pr9222295170931077689nt_int,S2: set_Pr9222295170931077689nt_int] :
% 9.10/5.30        ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R2 ) )
% 9.10/5.30          = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S2 ) ) )
% 9.10/5.30        = ( R2 = S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7526_pred__equals__eq2,axiom,
% 9.10/5.30      ! [R2: set_Pr1872883991513573699nt_int,S2: set_Pr1872883991513573699nt_int] :
% 9.10/5.30        ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R2 ) )
% 9.10/5.30          = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S2 ) ) )
% 9.10/5.30        = ( R2 = S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7527_summable__ratio__test,axiom,
% 9.10/5.30      ! [C: real,N4: nat,F: nat > real] :
% 9.10/5.30        ( ( ord_less_real @ C @ one_one_real )
% 9.10/5.30       => ( ! [N3: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N4 @ N3 )
% 9.10/5.30             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 9.10/5.30         => ( summable_real @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7528_summable__ratio__test,axiom,
% 9.10/5.30      ! [C: real,N4: nat,F: nat > complex] :
% 9.10/5.30        ( ( ord_less_real @ C @ one_one_real )
% 9.10/5.30       => ( ! [N3: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ N4 @ N3 )
% 9.10/5.30             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 9.10/5.30         => ( summable_complex @ F ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7529_pi__half__neq__zero,axiom,
% 9.10/5.30      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30     != zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7530_pi__half__less__two,axiom,
% 9.10/5.30      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7531_pi__half__le__two,axiom,
% 9.10/5.30      ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7532_power__diff__1__eq,axiom,
% 9.10/5.30      ! [X3: complex,N: nat] :
% 9.10/5.30        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex )
% 9.10/5.30        = ( times_times_complex @ ( minus_minus_complex @ X3 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7533_power__diff__1__eq,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat )
% 9.10/5.30        = ( times_times_rat @ ( minus_minus_rat @ X3 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7534_power__diff__1__eq,axiom,
% 9.10/5.30      ! [X3: int,N: nat] :
% 9.10/5.30        ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ one_one_int )
% 9.10/5.30        = ( times_times_int @ ( minus_minus_int @ X3 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7535_power__diff__1__eq,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real )
% 9.10/5.30        = ( times_times_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7536_one__diff__power__eq,axiom,
% 9.10/5.30      ! [X3: complex,N: nat] :
% 9.10/5.30        ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
% 9.10/5.30        = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7537_one__diff__power__eq,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7538_one__diff__power__eq,axiom,
% 9.10/5.30      ! [X3: int,N: nat] :
% 9.10/5.30        ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
% 9.10/5.30        = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7539_one__diff__power__eq,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
% 9.10/5.30        = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7540_geometric__sum,axiom,
% 9.10/5.30      ! [X3: complex,N: nat] :
% 9.10/5.30        ( ( X3 != one_one_complex )
% 9.10/5.30       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7541_geometric__sum,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( X3 != one_one_rat )
% 9.10/5.30       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30          = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7542_geometric__sum,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( X3 != one_one_real )
% 9.10/5.30       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7543_sum__gp__strict,axiom,
% 9.10/5.30      ! [X3: complex,N: nat] :
% 9.10/5.30        ( ( ( X3 = one_one_complex )
% 9.10/5.30         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( semiri8010041392384452111omplex @ N ) ) )
% 9.10/5.30        & ( ( X3 != one_one_complex )
% 9.10/5.30         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7544_sum__gp__strict,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( ( X3 = one_one_rat )
% 9.10/5.30         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( semiri681578069525770553at_rat @ N ) ) )
% 9.10/5.30        & ( ( X3 != one_one_rat )
% 9.10/5.30         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7545_sum__gp__strict,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( ( X3 = one_one_real )
% 9.10/5.30         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.30        & ( ( X3 != one_one_real )
% 9.10/5.30         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7546_lemma__termdiff1,axiom,
% 9.10/5.30      ! [Z: complex,H2: complex,M: nat] :
% 9.10/5.30        ( ( groups2073611262835488442omplex
% 9.10/5.30          @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( groups2073611262835488442omplex
% 9.10/5.30          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7547_lemma__termdiff1,axiom,
% 9.10/5.30      ! [Z: rat,H2: rat,M: nat] :
% 9.10/5.30        ( ( groups2906978787729119204at_rat
% 9.10/5.30          @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( groups2906978787729119204at_rat
% 9.10/5.30          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7548_lemma__termdiff1,axiom,
% 9.10/5.30      ! [Z: int,H2: int,M: nat] :
% 9.10/5.30        ( ( groups3539618377306564664at_int
% 9.10/5.30          @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( groups3539618377306564664at_int
% 9.10/5.30          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7549_lemma__termdiff1,axiom,
% 9.10/5.30      ! [Z: real,H2: real,M: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) )
% 9.10/5.30        = ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7550_pi__half__gt__zero,axiom,
% 9.10/5.30      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7551_power__diff__sumr2,axiom,
% 9.10/5.30      ! [X3: complex,N: nat,Y3: complex] :
% 9.10/5.30        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 9.10/5.30        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 9.10/5.30          @ ( groups2073611262835488442omplex
% 9.10/5.30            @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7552_power__diff__sumr2,axiom,
% 9.10/5.30      ! [X3: rat,N: nat,Y3: rat] :
% 9.10/5.30        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 9.10/5.30          @ ( groups2906978787729119204at_rat
% 9.10/5.30            @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X3 @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7553_power__diff__sumr2,axiom,
% 9.10/5.30      ! [X3: int,N: nat,Y3: int] :
% 9.10/5.30        ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 9.10/5.30        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 9.10/5.30          @ ( groups3539618377306564664at_int
% 9.10/5.30            @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X3 @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7554_power__diff__sumr2,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Y3: real] :
% 9.10/5.30        ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 9.10/5.30        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X3 @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7555_diff__power__eq__sum,axiom,
% 9.10/5.30      ! [X3: complex,N: nat,Y3: complex] :
% 9.10/5.30        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 9.10/5.30        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 9.10/5.30          @ ( groups2073611262835488442omplex
% 9.10/5.30            @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P5 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7556_diff__power__eq__sum,axiom,
% 9.10/5.30      ! [X3: rat,N: nat,Y3: rat] :
% 9.10/5.30        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 9.10/5.30        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 9.10/5.30          @ ( groups2906978787729119204at_rat
% 9.10/5.30            @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P5 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7557_diff__power__eq__sum,axiom,
% 9.10/5.30      ! [X3: int,N: nat,Y3: int] :
% 9.10/5.30        ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 9.10/5.30        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 9.10/5.30          @ ( groups3539618377306564664at_int
% 9.10/5.30            @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X3 @ P5 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7558_diff__power__eq__sum,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Y3: real] :
% 9.10/5.30        ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 9.10/5.30        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X3 @ P5 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7559_pi__half__ge__zero,axiom,
% 9.10/5.30      ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7560_m2pi__less__pi,axiom,
% 9.10/5.30      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7561_arctan__ubound,axiom,
% 9.10/5.30      ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7562_arctan__one,axiom,
% 9.10/5.30      ( ( arctan @ one_one_real )
% 9.10/5.30      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7563_real__sum__nat__ivl__bounded2,axiom,
% 9.10/5.30      ! [N: nat,F: nat > rat,K5: rat,K: nat] :
% 9.10/5.30        ( ! [P7: nat] :
% 9.10/5.30            ( ( ord_less_nat @ P7 @ N )
% 9.10/5.30           => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 9.10/5.30       => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 9.10/5.30         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7564_real__sum__nat__ivl__bounded2,axiom,
% 9.10/5.30      ! [N: nat,F: nat > int,K5: int,K: nat] :
% 9.10/5.30        ( ! [P7: nat] :
% 9.10/5.30            ( ( ord_less_nat @ P7 @ N )
% 9.10/5.30           => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 9.10/5.30       => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 9.10/5.30         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7565_real__sum__nat__ivl__bounded2,axiom,
% 9.10/5.30      ! [N: nat,F: nat > nat,K5: nat,K: nat] :
% 9.10/5.30        ( ! [P7: nat] :
% 9.10/5.30            ( ( ord_less_nat @ P7 @ N )
% 9.10/5.30           => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 9.10/5.30       => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 9.10/5.30         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7566_real__sum__nat__ivl__bounded2,axiom,
% 9.10/5.30      ! [N: nat,F: nat > real,K5: real,K: nat] :
% 9.10/5.30        ( ! [P7: nat] :
% 9.10/5.30            ( ( ord_less_nat @ P7 @ N )
% 9.10/5.30           => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 9.10/5.30         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7567_one__diff__power__eq_H,axiom,
% 9.10/5.30      ! [X3: complex,N: nat] :
% 9.10/5.30        ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N ) )
% 9.10/5.30        = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
% 9.10/5.30          @ ( groups2073611262835488442omplex
% 9.10/5.30            @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7568_one__diff__power__eq_H,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
% 9.10/5.30          @ ( groups2906978787729119204at_rat
% 9.10/5.30            @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7569_one__diff__power__eq_H,axiom,
% 9.10/5.30      ! [X3: int,N: nat] :
% 9.10/5.30        ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N ) )
% 9.10/5.30        = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
% 9.10/5.30          @ ( groups3539618377306564664at_int
% 9.10/5.30            @ ^ [I4: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7570_one__diff__power__eq_H,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N ) )
% 9.10/5.30        = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [I4: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7571_subrelI,axiom,
% 9.10/5.30      ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 9.10/5.30        ( ! [X5: int,Y4: int] :
% 9.10/5.30            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R )
% 9.10/5.30           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S ) )
% 9.10/5.30       => ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7572_subrelI,axiom,
% 9.10/5.30      ! [R: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
% 9.10/5.30        ( ! [X5: code_integer > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.30            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y4 ) @ R )
% 9.10/5.30           => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X5 @ Y4 ) @ S ) )
% 9.10/5.30       => ( ord_le3216752416896350996nteger @ R @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7573_subrelI,axiom,
% 9.10/5.30      ! [R: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
% 9.10/5.30        ( ! [X5: produc6241069584506657477e_term > option6357759511663192854e_term,Y4: produc8923325533196201883nteger] :
% 9.10/5.30            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y4 ) @ R )
% 9.10/5.30           => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X5 @ Y4 ) @ S ) )
% 9.10/5.30       => ( ord_le653643898420964396nteger @ R @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7574_subrelI,axiom,
% 9.10/5.30      ! [R: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
% 9.10/5.30        ( ! [X5: produc8551481072490612790e_term > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.30            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y4 ) @ R )
% 9.10/5.30           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X5 @ Y4 ) @ S ) )
% 9.10/5.30       => ( ord_le8725513860283290265nt_int @ R @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7575_subrelI,axiom,
% 9.10/5.30      ! [R: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
% 9.10/5.30        ( ! [X5: int > option6357759511663192854e_term,Y4: product_prod_int_int] :
% 9.10/5.30            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y4 ) @ R )
% 9.10/5.30           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X5 @ Y4 ) @ S ) )
% 9.10/5.30       => ( ord_le135402666524580259nt_int @ R @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7576_minus__pi__half__less__zero,axiom,
% 9.10/5.30      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7577_arctan__lbound,axiom,
% 9.10/5.30      ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7578_arctan__bounded,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 9.10/5.30        & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7579_unbounded__k__infinite,axiom,
% 9.10/5.30      ! [K: nat,S2: set_nat] :
% 9.10/5.30        ( ! [M4: nat] :
% 9.10/5.30            ( ( ord_less_nat @ K @ M4 )
% 9.10/5.30           => ? [N9: nat] :
% 9.10/5.30                ( ( ord_less_nat @ M4 @ N9 )
% 9.10/5.30                & ( member_nat @ N9 @ S2 ) ) )
% 9.10/5.30       => ~ ( finite_finite_nat @ S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7580_infinite__nat__iff__unbounded,axiom,
% 9.10/5.30      ! [S2: set_nat] :
% 9.10/5.30        ( ( ~ ( finite_finite_nat @ S2 ) )
% 9.10/5.30        = ( ! [M3: nat] :
% 9.10/5.30            ? [N2: nat] :
% 9.10/5.30              ( ( ord_less_nat @ M3 @ N2 )
% 9.10/5.30              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7581_infinite__nat__iff__unbounded__le,axiom,
% 9.10/5.30      ! [S2: set_nat] :
% 9.10/5.30        ( ( ~ ( finite_finite_nat @ S2 ) )
% 9.10/5.30        = ( ! [M3: nat] :
% 9.10/5.30            ? [N2: nat] :
% 9.10/5.30              ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30              & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7582_sum__split__even__odd,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real,N: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.30        = ( plus_plus_real
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7583_pred__subset__eq,axiom,
% 9.10/5.30      ! [R2: set_nat,S2: set_nat] :
% 9.10/5.30        ( ( ord_less_eq_nat_o
% 9.10/5.30          @ ^ [X: nat] : ( member_nat @ X @ R2 )
% 9.10/5.30          @ ^ [X: nat] : ( member_nat @ X @ S2 ) )
% 9.10/5.30        = ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7584_pred__subset__eq,axiom,
% 9.10/5.30      ! [R2: set_real,S2: set_real] :
% 9.10/5.30        ( ( ord_less_eq_real_o
% 9.10/5.30          @ ^ [X: real] : ( member_real @ X @ R2 )
% 9.10/5.30          @ ^ [X: real] : ( member_real @ X @ S2 ) )
% 9.10/5.30        = ( ord_less_eq_set_real @ R2 @ S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7585_pred__subset__eq,axiom,
% 9.10/5.30      ! [R2: set_complex,S2: set_complex] :
% 9.10/5.30        ( ( ord_le4573692005234683329plex_o
% 9.10/5.30          @ ^ [X: complex] : ( member_complex @ X @ R2 )
% 9.10/5.30          @ ^ [X: complex] : ( member_complex @ X @ S2 ) )
% 9.10/5.30        = ( ord_le211207098394363844omplex @ R2 @ S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7586_pred__subset__eq,axiom,
% 9.10/5.30      ! [R2: set_int,S2: set_int] :
% 9.10/5.30        ( ( ord_less_eq_int_o
% 9.10/5.30          @ ^ [X: int] : ( member_int @ X @ R2 )
% 9.10/5.30          @ ^ [X: int] : ( member_int @ X @ S2 ) )
% 9.10/5.30        = ( ord_less_eq_set_int @ R2 @ S2 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7587_machin__Euler,axiom,
% 9.10/5.30      ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7588_machin,axiom,
% 9.10/5.30      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.30      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7589_infinite__int__iff__unbounded__le,axiom,
% 9.10/5.30      ! [S2: set_int] :
% 9.10/5.30        ( ( ~ ( finite_finite_int @ S2 ) )
% 9.10/5.30        = ( ! [M3: int] :
% 9.10/5.30            ? [N2: int] :
% 9.10/5.30              ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
% 9.10/5.30              & ( member_int @ N2 @ S2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7590_sin__cos__npi,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7591_sumr__cos__zero__one,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real
% 9.10/5.30          @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 9.10/5.30          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.30        = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7592_cos__pi__eq__zero,axiom,
% 9.10/5.30      ! [M: nat] :
% 9.10/5.30        ( ( 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 ) ) ) )
% 9.10/5.30        = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7593_accp__subset,axiom,
% 9.10/5.30      ! [R1: product_prod_num_num > product_prod_num_num > $o,R22: product_prod_num_num > product_prod_num_num > $o] :
% 9.10/5.30        ( ( ord_le2556027599737686990_num_o @ R1 @ R22 )
% 9.10/5.30       => ( ord_le2239182809043710856_num_o @ ( accp_P3113834385874906142um_num @ R22 ) @ ( accp_P3113834385874906142um_num @ R1 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7594_accp__subset,axiom,
% 9.10/5.30      ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.10/5.30        ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
% 9.10/5.30       => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7595_accp__subset,axiom,
% 9.10/5.30      ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
% 9.10/5.30        ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
% 9.10/5.30       => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7596_accp__subset,axiom,
% 9.10/5.30      ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
% 9.10/5.30        ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
% 9.10/5.30       => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7597_accp__subset,axiom,
% 9.10/5.30      ! [R1: nat > nat > $o,R22: nat > nat > $o] :
% 9.10/5.30        ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
% 9.10/5.30       => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7598_geometric__deriv__sums,axiom,
% 9.10/5.30      ! [Z: real] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
% 9.10/5.30          @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7599_geometric__deriv__sums,axiom,
% 9.10/5.30      ! [Z: complex] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 9.10/5.30       => ( sums_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
% 9.10/5.30          @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7600_monoseq__def,axiom,
% 9.10/5.30      ( topolo6980174941875973593q_real
% 9.10/5.30      = ( ^ [X7: nat > real] :
% 9.10/5.30            ( ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 9.10/5.30            | ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_real @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7601_monoseq__def,axiom,
% 9.10/5.30      ( topolo3100542954746470799et_int
% 9.10/5.30      = ( ^ [X7: nat > set_int] :
% 9.10/5.30            ( ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_set_int @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 9.10/5.30            | ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_set_int @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7602_monoseq__def,axiom,
% 9.10/5.30      ( topolo4267028734544971653eq_rat
% 9.10/5.30      = ( ^ [X7: nat > rat] :
% 9.10/5.30            ( ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_rat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 9.10/5.30            | ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7603_monoseq__def,axiom,
% 9.10/5.30      ( topolo1459490580787246023eq_num
% 9.10/5.30      = ( ^ [X7: nat > num] :
% 9.10/5.30            ( ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_num @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 9.10/5.30            | ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_num @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7604_monoseq__def,axiom,
% 9.10/5.30      ( topolo4902158794631467389eq_nat
% 9.10/5.30      = ( ^ [X7: nat > nat] :
% 9.10/5.30            ( ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 9.10/5.30            | ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_nat @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7605_monoseq__def,axiom,
% 9.10/5.30      ( topolo4899668324122417113eq_int
% 9.10/5.30      = ( ^ [X7: nat > int] :
% 9.10/5.30            ( ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) )
% 9.10/5.30            | ! [M3: nat,N2: nat] :
% 9.10/5.30                ( ( ord_less_eq_nat @ M3 @ N2 )
% 9.10/5.30               => ( ord_less_eq_int @ ( X7 @ N2 ) @ ( X7 @ M3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7606_sin__cos__squared__add3,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ X3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ X3 ) ) )
% 9.10/5.30        = one_one_complex ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7607_sin__cos__squared__add3,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ X3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ X3 ) ) )
% 9.10/5.30        = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7608_sin__npi2,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.30        = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7609_sin__npi,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.10/5.30        = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7610_powser__sums__zero__iff,axiom,
% 9.10/5.30      ! [A: nat > complex,X3: complex] :
% 9.10/5.30        ( ( sums_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 9.10/5.30          @ X3 )
% 9.10/5.30        = ( ( A @ zero_zero_nat )
% 9.10/5.30          = X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7611_powser__sums__zero__iff,axiom,
% 9.10/5.30      ! [A: nat > real,X3: real] :
% 9.10/5.30        ( ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 9.10/5.30          @ X3 )
% 9.10/5.30        = ( ( A @ zero_zero_nat )
% 9.10/5.30          = X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7612_cos__pi__half,axiom,
% 9.10/5.30      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30      = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7613_sin__two__pi,axiom,
% 9.10/5.30      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30      = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7614_sin__pi__half,axiom,
% 9.10/5.30      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30      = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7615_cos__two__pi,axiom,
% 9.10/5.30      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30      = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7616_cos__periodic,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( cos_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.10/5.30        = ( cos_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7617_sin__periodic,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( sin_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.10/5.30        = ( sin_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7618_cos__2pi__minus,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X3 ) )
% 9.10/5.30        = ( cos_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7619_cos__npi2,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.30        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7620_cos__npi,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.10/5.30        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7621_sin__cos__squared__add,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( 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 ) ) ) )
% 9.10/5.30        = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7622_sin__cos__squared__add,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( 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 ) ) ) )
% 9.10/5.30        = one_one_complex ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7623_sin__cos__squared__add2,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( 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 ) ) ) )
% 9.10/5.30        = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7624_sin__cos__squared__add2,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( 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 ) ) ) )
% 9.10/5.30        = one_one_complex ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7625_sin__2npi,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 9.10/5.30        = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7626_cos__2npi,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 9.10/5.30        = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7627_sin__2pi__minus,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X3 ) )
% 9.10/5.30        = ( uminus_uminus_real @ ( sin_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7628_cos__3over2__pi,axiom,
% 9.10/5.30      ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 9.10/5.30      = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7629_sin__3over2__pi,axiom,
% 9.10/5.30      ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 9.10/5.30      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7630_sums__le,axiom,
% 9.10/5.30      ! [F: nat > real,G: nat > real,S: real,T: real] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.30       => ( ( sums_real @ F @ S )
% 9.10/5.30         => ( ( sums_real @ G @ T )
% 9.10/5.30           => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7631_sums__le,axiom,
% 9.10/5.30      ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.30       => ( ( sums_nat @ F @ S )
% 9.10/5.30         => ( ( sums_nat @ G @ T )
% 9.10/5.30           => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7632_sums__le,axiom,
% 9.10/5.30      ! [F: nat > int,G: nat > int,S: int,T: int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.30       => ( ( sums_int @ F @ S )
% 9.10/5.30         => ( ( sums_int @ G @ T )
% 9.10/5.30           => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7633_polar__Ex,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30      ? [R3: real,A3: real] :
% 9.10/5.30        ( ( X3
% 9.10/5.30          = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
% 9.10/5.30        & ( Y3
% 9.10/5.30          = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7634_sin__diff,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( sin_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.10/5.30        = ( minus_minus_complex @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7635_sin__diff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( sin_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7636_sin__add,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7637_sums__mult,axiom,
% 9.10/5.30      ! [F: nat > real,A: real,C: real] :
% 9.10/5.30        ( ( sums_real @ F @ A )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.10/5.30          @ ( times_times_real @ C @ A ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7638_sums__mult2,axiom,
% 9.10/5.30      ! [F: nat > real,A: real,C: real] :
% 9.10/5.30        ( ( sums_real @ F @ A )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 9.10/5.30          @ ( times_times_real @ A @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7639_sums__add,axiom,
% 9.10/5.30      ! [F: nat > real,A: real,G: nat > real,B2: real] :
% 9.10/5.30        ( ( sums_real @ F @ A )
% 9.10/5.30       => ( ( sums_real @ G @ B2 )
% 9.10/5.30         => ( sums_real
% 9.10/5.30            @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30            @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7640_sums__add,axiom,
% 9.10/5.30      ! [F: nat > nat,A: nat,G: nat > nat,B2: nat] :
% 9.10/5.30        ( ( sums_nat @ F @ A )
% 9.10/5.30       => ( ( sums_nat @ G @ B2 )
% 9.10/5.30         => ( sums_nat
% 9.10/5.30            @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30            @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7641_sums__add,axiom,
% 9.10/5.30      ! [F: nat > int,A: int,G: nat > int,B2: int] :
% 9.10/5.30        ( ( sums_int @ F @ A )
% 9.10/5.30       => ( ( sums_int @ G @ B2 )
% 9.10/5.30         => ( sums_int
% 9.10/5.30            @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30            @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7642_sums__diff,axiom,
% 9.10/5.30      ! [F: nat > complex,A: complex,G: nat > complex,B2: complex] :
% 9.10/5.30        ( ( sums_complex @ F @ A )
% 9.10/5.30       => ( ( sums_complex @ G @ B2 )
% 9.10/5.30         => ( sums_complex
% 9.10/5.30            @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30            @ ( minus_minus_complex @ A @ B2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7643_sums__diff,axiom,
% 9.10/5.30      ! [F: nat > real,A: real,G: nat > real,B2: real] :
% 9.10/5.30        ( ( sums_real @ F @ A )
% 9.10/5.30       => ( ( sums_real @ G @ B2 )
% 9.10/5.30         => ( sums_real
% 9.10/5.30            @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30            @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7644_sums__divide,axiom,
% 9.10/5.30      ! [F: nat > complex,A: complex,C: complex] :
% 9.10/5.30        ( ( sums_complex @ F @ A )
% 9.10/5.30       => ( sums_complex
% 9.10/5.30          @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
% 9.10/5.30          @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7645_sums__divide,axiom,
% 9.10/5.30      ! [F: nat > real,A: real,C: real] :
% 9.10/5.30        ( ( sums_real @ F @ A )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
% 9.10/5.30          @ ( divide_divide_real @ A @ C ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7646_cos__add,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.10/5.30        = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7647_cos__add,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7648_cos__diff,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.10/5.30        = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7649_cos__diff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7650_sin__double,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X3 ) ) @ ( cos_complex @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7651_sin__double,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X3 ) ) @ ( cos_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7652_sincos__principal__value,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30      ? [Y4: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 9.10/5.30        & ( ord_less_eq_real @ Y4 @ pi )
% 9.10/5.30        & ( ( sin_real @ Y4 )
% 9.10/5.30          = ( sin_real @ X3 ) )
% 9.10/5.30        & ( ( cos_real @ Y4 )
% 9.10/5.30          = ( cos_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7653_sin__x__le__x,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ ( sin_real @ X3 ) @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7654_sin__le__one,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( sin_real @ X3 ) @ one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7655_cos__le__one,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( cos_real @ X3 ) @ one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7656_abs__sin__x__le__abs__x,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ ( abs_abs_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7657_sums__mult__iff,axiom,
% 9.10/5.30      ! [C: complex,F: nat > complex,D: complex] :
% 9.10/5.30        ( ( C != zero_zero_complex )
% 9.10/5.30       => ( ( sums_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 9.10/5.30            @ ( times_times_complex @ C @ D ) )
% 9.10/5.30          = ( sums_complex @ F @ D ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7658_sums__mult__iff,axiom,
% 9.10/5.30      ! [C: real,F: nat > real,D: real] :
% 9.10/5.30        ( ( C != zero_zero_real )
% 9.10/5.30       => ( ( sums_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.10/5.30            @ ( times_times_real @ C @ D ) )
% 9.10/5.30          = ( sums_real @ F @ D ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7659_sums__mult2__iff,axiom,
% 9.10/5.30      ! [C: complex,F: nat > complex,D: complex] :
% 9.10/5.30        ( ( C != zero_zero_complex )
% 9.10/5.30       => ( ( sums_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 9.10/5.30            @ ( times_times_complex @ D @ C ) )
% 9.10/5.30          = ( sums_complex @ F @ D ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7660_sums__mult2__iff,axiom,
% 9.10/5.30      ! [C: real,F: nat > real,D: real] :
% 9.10/5.30        ( ( C != zero_zero_real )
% 9.10/5.30       => ( ( sums_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 9.10/5.30            @ ( times_times_real @ D @ C ) )
% 9.10/5.30          = ( sums_real @ F @ D ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7661_sin__cos__le1,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7662_cos__squared__eq,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7663_cos__squared__eq,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7664_sin__squared__eq,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7665_sin__squared__eq,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30        = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7666_sums__mult__D,axiom,
% 9.10/5.30      ! [C: complex,F: nat > complex,A: complex] :
% 9.10/5.30        ( ( sums_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 9.10/5.30          @ A )
% 9.10/5.30       => ( ( C != zero_zero_complex )
% 9.10/5.30         => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7667_sums__mult__D,axiom,
% 9.10/5.30      ! [C: real,F: nat > real,A: real] :
% 9.10/5.30        ( ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.10/5.30          @ A )
% 9.10/5.30       => ( ( C != zero_zero_real )
% 9.10/5.30         => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7668_sums__Suc__iff,axiom,
% 9.10/5.30      ! [F: nat > real,S: real] :
% 9.10/5.30        ( ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.10/5.30          @ S )
% 9.10/5.30        = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7669_sums__Suc,axiom,
% 9.10/5.30      ! [F: nat > real,L: real] :
% 9.10/5.30        ( ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.10/5.30          @ L )
% 9.10/5.30       => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7670_sums__Suc,axiom,
% 9.10/5.30      ! [F: nat > nat,L: nat] :
% 9.10/5.30        ( ( sums_nat
% 9.10/5.30          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.10/5.30          @ L )
% 9.10/5.30       => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7671_sums__Suc,axiom,
% 9.10/5.30      ! [F: nat > int,L: int] :
% 9.10/5.30        ( ( sums_int
% 9.10/5.30          @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.10/5.30          @ L )
% 9.10/5.30       => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7672_sin__gt__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ pi )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7673_sin__x__ge__neg__x,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ ( uminus_uminus_real @ X3 ) @ ( sin_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7674_sin__ge__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7675_sums__zero__iff__shift,axiom,
% 9.10/5.30      ! [N: nat,F: nat > complex,S: complex] :
% 9.10/5.30        ( ! [I3: nat] :
% 9.10/5.30            ( ( ord_less_nat @ I3 @ N )
% 9.10/5.30           => ( ( F @ I3 )
% 9.10/5.30              = zero_zero_complex ) )
% 9.10/5.30       => ( ( sums_complex
% 9.10/5.30            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30            @ S )
% 9.10/5.30          = ( sums_complex @ F @ S ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7676_sums__zero__iff__shift,axiom,
% 9.10/5.30      ! [N: nat,F: nat > real,S: real] :
% 9.10/5.30        ( ! [I3: nat] :
% 9.10/5.30            ( ( ord_less_nat @ I3 @ N )
% 9.10/5.30           => ( ( F @ I3 )
% 9.10/5.30              = zero_zero_real ) )
% 9.10/5.30       => ( ( sums_real
% 9.10/5.30            @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30            @ S )
% 9.10/5.30          = ( sums_real @ F @ S ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7677_sin__ge__minus__one,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7678_cos__monotone__0__pi__le,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.30           => ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7679_cos__mono__le__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30           => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.10/5.30             => ( ( ord_less_eq_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
% 9.10/5.30                = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7680_cos__inj__pi,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30           => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.10/5.30             => ( ( ( cos_real @ X3 )
% 9.10/5.30                  = ( cos_real @ Y3 ) )
% 9.10/5.30               => ( X3 = Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7681_cos__ge__minus__one,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7682_abs__sin__le__one,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X3 ) ) @ one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7683_abs__cos__le__one,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X3 ) ) @ one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7684_sin__times__sin,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 9.10/5.30        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7685_sin__times__sin,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 9.10/5.30        = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7686_sin__times__cos,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 9.10/5.30        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7687_sin__times__cos,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 9.10/5.30        = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7688_cos__times__sin,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 9.10/5.30        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7689_cos__times__sin,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 9.10/5.30        = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7690_sin__plus__sin,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 9.10/5.30        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7691_sin__plus__sin,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 9.10/5.30        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7692_sin__diff__sin,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 9.10/5.30        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7693_sin__diff__sin,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 9.10/5.30        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7694_cos__diff__cos,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 9.10/5.30        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7695_cos__diff__cos,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 9.10/5.30        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7696_cos__double,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30        = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7697_cos__double,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30        = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7698_cos__double__sin,axiom,
% 9.10/5.30      ! [W: complex] :
% 9.10/5.30        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 9.10/5.30        = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7699_cos__double__sin,axiom,
% 9.10/5.30      ! [W: real] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 9.10/5.30        = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7700_powser__sums__if,axiom,
% 9.10/5.30      ! [M: nat,Z: complex] :
% 9.10/5.30        ( sums_complex
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
% 9.10/5.30        @ ( power_power_complex @ Z @ M ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7701_powser__sums__if,axiom,
% 9.10/5.30      ! [M: nat,Z: real] :
% 9.10/5.30        ( sums_real
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
% 9.10/5.30        @ ( power_power_real @ Z @ M ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7702_powser__sums__if,axiom,
% 9.10/5.30      ! [M: nat,Z: int] :
% 9.10/5.30        ( sums_int
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
% 9.10/5.30        @ ( power_power_int @ Z @ M ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7703_powser__sums__zero,axiom,
% 9.10/5.30      ! [A: nat > complex] :
% 9.10/5.30        ( sums_complex
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 9.10/5.30        @ ( A @ zero_zero_nat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7704_powser__sums__zero,axiom,
% 9.10/5.30      ! [A: nat > real] :
% 9.10/5.30        ( sums_real
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 9.10/5.30        @ ( A @ zero_zero_nat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7705_cos__two__neq__zero,axiom,
% 9.10/5.30      ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30     != zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7706_cos__mono__less__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.30         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30           => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.10/5.30             => ( ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
% 9.10/5.30                = ( ord_less_real @ Y3 @ X3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7707_cos__monotone__0__pi,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30       => ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.30           => ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7708_sin__eq__0__pi,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ pi )
% 9.10/5.30         => ( ( ( sin_real @ X3 )
% 9.10/5.30              = zero_zero_real )
% 9.10/5.30           => ( X3 = zero_zero_real ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7709_sums__iff__shift,axiom,
% 9.10/5.30      ! [F: nat > real,N: nat,S: real] :
% 9.10/5.30        ( ( sums_real
% 9.10/5.30          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30          @ S )
% 9.10/5.30        = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7710_sums__split__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > complex,S: complex,N: nat] :
% 9.10/5.30        ( ( sums_complex @ F @ S )
% 9.10/5.30       => ( sums_complex
% 9.10/5.30          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30          @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7711_sums__split__initial__segment,axiom,
% 9.10/5.30      ! [F: nat > real,S: real,N: nat] :
% 9.10/5.30        ( ( sums_real @ F @ S )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7712_sums__iff__shift_H,axiom,
% 9.10/5.30      ! [F: nat > complex,N: nat,S: complex] :
% 9.10/5.30        ( ( sums_complex
% 9.10/5.30          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30          @ ( minus_minus_complex @ S @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 9.10/5.30        = ( sums_complex @ F @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7713_sums__iff__shift_H,axiom,
% 9.10/5.30      ! [F: nat > real,N: nat,S: real] :
% 9.10/5.30        ( ( sums_real
% 9.10/5.30          @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.10/5.30          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 9.10/5.30        = ( sums_real @ F @ S ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7714_sin__zero__pi__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ pi )
% 9.10/5.30       => ( ( ( sin_real @ X3 )
% 9.10/5.30            = zero_zero_real )
% 9.10/5.30          = ( X3 = zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7715_cos__monotone__minus__pi__0_H,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.30           => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7716_sums__If__finite__set_H,axiom,
% 9.10/5.30      ! [G: nat > complex,S2: complex,A4: set_nat,S5: complex,F: nat > complex] :
% 9.10/5.30        ( ( sums_complex @ G @ S2 )
% 9.10/5.30       => ( ( finite_finite_nat @ A4 )
% 9.10/5.30         => ( ( S5
% 9.10/5.30              = ( plus_plus_complex @ S2
% 9.10/5.30                @ ( groups2073611262835488442omplex
% 9.10/5.30                  @ ^ [N2: nat] : ( minus_minus_complex @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30                  @ A4 ) ) )
% 9.10/5.30           => ( sums_complex
% 9.10/5.30              @ ^ [N2: nat] : ( if_complex @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30              @ S5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7717_sums__If__finite__set_H,axiom,
% 9.10/5.30      ! [G: nat > real,S2: real,A4: set_nat,S5: real,F: nat > real] :
% 9.10/5.30        ( ( sums_real @ G @ S2 )
% 9.10/5.30       => ( ( finite_finite_nat @ A4 )
% 9.10/5.30         => ( ( S5
% 9.10/5.30              = ( plus_plus_real @ S2
% 9.10/5.30                @ ( groups6591440286371151544t_real
% 9.10/5.30                  @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30                  @ A4 ) ) )
% 9.10/5.30           => ( sums_real
% 9.10/5.30              @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A4 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.30              @ S5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7718_sincos__total__pi,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30       => ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30            = one_one_real )
% 9.10/5.30         => ? [T4: real] :
% 9.10/5.30              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.30              & ( ord_less_eq_real @ T4 @ pi )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( cos_real @ T4 ) )
% 9.10/5.30              & ( Y3
% 9.10/5.30                = ( sin_real @ T4 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7719_sin__expansion__lemma,axiom,
% 9.10/5.30      ! [X3: real,M: nat] :
% 9.10/5.30        ( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30        = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7720_cos__expansion__lemma,axiom,
% 9.10/5.30      ! [X3: real,M: nat] :
% 9.10/5.30        ( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30        = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7721_sin__gt__zero__02,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7722_cos__two__less__zero,axiom,
% 9.10/5.30      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7723_cos__is__zero,axiom,
% 9.10/5.30      ? [X5: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.10/5.30        & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30        & ( ( cos_real @ X5 )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30        & ! [Y5: real] :
% 9.10/5.30            ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 9.10/5.30              & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30              & ( ( cos_real @ Y5 )
% 9.10/5.30                = zero_zero_real ) )
% 9.10/5.30           => ( Y5 = X5 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7724_cos__two__le__zero,axiom,
% 9.10/5.30      ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7725_cos__monotone__minus__pi__0,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.30           => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7726_cos__total,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.30         => ? [X5: real] :
% 9.10/5.30              ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.10/5.30              & ( ord_less_eq_real @ X5 @ pi )
% 9.10/5.30              & ( ( cos_real @ X5 )
% 9.10/5.30                = Y3 )
% 9.10/5.30              & ! [Y5: real] :
% 9.10/5.30                  ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 9.10/5.30                    & ( ord_less_eq_real @ Y5 @ pi )
% 9.10/5.30                    & ( ( cos_real @ Y5 )
% 9.10/5.30                      = Y3 ) )
% 9.10/5.30                 => ( Y5 = X5 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7727_sincos__total__pi__half,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30              = one_one_real )
% 9.10/5.30           => ? [T4: real] :
% 9.10/5.30                ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.30                & ( ord_less_eq_real @ T4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30                & ( X3
% 9.10/5.30                  = ( cos_real @ T4 ) )
% 9.10/5.30                & ( Y3
% 9.10/5.30                  = ( sin_real @ T4 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7728_sincos__total__2pi__le,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30          = one_one_real )
% 9.10/5.30       => ? [T4: real] :
% 9.10/5.30            ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.30            & ( ord_less_eq_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30            & ( X3
% 9.10/5.30              = ( cos_real @ T4 ) )
% 9.10/5.30            & ( Y3
% 9.10/5.30              = ( sin_real @ T4 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7729_sincos__total__2pi,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30          = one_one_real )
% 9.10/5.30       => ~ ! [T4: real] :
% 9.10/5.30              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.30             => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30               => ( ( X3
% 9.10/5.30                    = ( cos_real @ T4 ) )
% 9.10/5.30                 => ( Y3
% 9.10/5.30                   != ( sin_real @ T4 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7730_sin__pi__divide__n__ge__0,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( N != zero_zero_nat )
% 9.10/5.30       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7731_cos__times__cos,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 9.10/5.30        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7732_cos__times__cos,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 9.10/5.30        = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7733_cos__plus__cos,axiom,
% 9.10/5.30      ! [W: complex,Z: complex] :
% 9.10/5.30        ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 9.10/5.30        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7734_cos__plus__cos,axiom,
% 9.10/5.30      ! [W: real,Z: real] :
% 9.10/5.30        ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 9.10/5.30        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7735_geometric__sums,axiom,
% 9.10/5.30      ! [C: real] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.10/5.30       => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7736_geometric__sums,axiom,
% 9.10/5.30      ! [C: complex] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.10/5.30       => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7737_power__half__series,axiom,
% 9.10/5.30      ( sums_real
% 9.10/5.30      @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
% 9.10/5.30      @ one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7738_sin__gt__zero2,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7739_sin__lt__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ pi @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30         => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7740_cos__double__less__one,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.30         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7741_sin__30,axiom,
% 9.10/5.30      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7742_cos__gt__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7743_sin__monotone__2pi__le,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30           => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7744_sin__mono__le__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30             => ( ( ord_less_eq_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
% 9.10/5.30                = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7745_sin__inj__pi,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30             => ( ( ( sin_real @ X3 )
% 9.10/5.30                  = ( sin_real @ Y3 ) )
% 9.10/5.30               => ( X3 = Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7746_cos__60,axiom,
% 9.10/5.30      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7747_cos__double__cos,axiom,
% 9.10/5.30      ! [W: complex] :
% 9.10/5.30        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 9.10/5.30        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7748_cos__double__cos,axiom,
% 9.10/5.30      ! [W: real] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 9.10/5.30        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7749_cos__treble__cos,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X3 ) )
% 9.10/5.30        = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7750_cos__treble__cos,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X3 ) )
% 9.10/5.30        = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7751_sums__if_H,axiom,
% 9.10/5.30      ! [G: nat > real,X3: real] :
% 9.10/5.30        ( ( sums_real @ G @ X3 )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30          @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7752_sums__if,axiom,
% 9.10/5.30      ! [G: nat > real,X3: real,F: nat > real,Y3: real] :
% 9.10/5.30        ( ( sums_real @ G @ X3 )
% 9.10/5.30       => ( ( sums_real @ F @ Y3 )
% 9.10/5.30         => ( sums_real
% 9.10/5.30            @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30            @ ( plus_plus_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7753_sin__le__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ pi @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30         => ( ord_less_eq_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7754_sin__less__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.30         => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7755_sin__monotone__2pi,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30           => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7756_sin__mono__less__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30             => ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
% 9.10/5.30                = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7757_sin__total,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.30         => ? [X5: real] :
% 9.10/5.30              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 9.10/5.30              & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30              & ( ( sin_real @ X5 )
% 9.10/5.30                = Y3 )
% 9.10/5.30              & ! [Y5: real] :
% 9.10/5.30                  ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 9.10/5.30                    & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30                    & ( ( sin_real @ Y5 )
% 9.10/5.30                      = Y3 ) )
% 9.10/5.30                 => ( Y5 = X5 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7758_cos__gt__zero__pi,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7759_cos__ge__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7760_cos__one__2pi,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30          = one_one_real )
% 9.10/5.30        = ( ? [X: nat] :
% 9.10/5.30              ( X3
% 9.10/5.30              = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 9.10/5.30          | ? [X: nat] :
% 9.10/5.30              ( X3
% 9.10/5.30              = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7761_accp__subset__induct,axiom,
% 9.10/5.30      ! [D4: product_prod_num_num > $o,R2: product_prod_num_num > product_prod_num_num > $o,X3: product_prod_num_num,P: product_prod_num_num > $o] :
% 9.10/5.30        ( ( ord_le2239182809043710856_num_o @ D4 @ ( accp_P3113834385874906142um_num @ R2 ) )
% 9.10/5.30       => ( ! [X5: product_prod_num_num,Z4: product_prod_num_num] :
% 9.10/5.30              ( ( D4 @ X5 )
% 9.10/5.30             => ( ( R2 @ Z4 @ X5 )
% 9.10/5.30               => ( D4 @ Z4 ) ) )
% 9.10/5.30         => ( ( D4 @ X3 )
% 9.10/5.30           => ( ! [X5: product_prod_num_num] :
% 9.10/5.30                  ( ( D4 @ X5 )
% 9.10/5.30                 => ( ! [Z3: product_prod_num_num] :
% 9.10/5.30                        ( ( R2 @ Z3 @ X5 )
% 9.10/5.30                       => ( P @ Z3 ) )
% 9.10/5.30                   => ( P @ X5 ) ) )
% 9.10/5.30             => ( P @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7762_accp__subset__induct,axiom,
% 9.10/5.30      ! [D4: product_prod_nat_nat > $o,R2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.10/5.30        ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R2 ) )
% 9.10/5.30       => ( ! [X5: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 9.10/5.30              ( ( D4 @ X5 )
% 9.10/5.30             => ( ( R2 @ Z4 @ X5 )
% 9.10/5.30               => ( D4 @ Z4 ) ) )
% 9.10/5.30         => ( ( D4 @ X3 )
% 9.10/5.30           => ( ! [X5: product_prod_nat_nat] :
% 9.10/5.30                  ( ( D4 @ X5 )
% 9.10/5.30                 => ( ! [Z3: product_prod_nat_nat] :
% 9.10/5.30                        ( ( R2 @ Z3 @ X5 )
% 9.10/5.30                       => ( P @ Z3 ) )
% 9.10/5.30                   => ( P @ X5 ) ) )
% 9.10/5.30             => ( P @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7763_accp__subset__induct,axiom,
% 9.10/5.30      ! [D4: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,P: product_prod_int_int > $o] :
% 9.10/5.30        ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R2 ) )
% 9.10/5.30       => ( ! [X5: product_prod_int_int,Z4: product_prod_int_int] :
% 9.10/5.30              ( ( D4 @ X5 )
% 9.10/5.30             => ( ( R2 @ Z4 @ X5 )
% 9.10/5.30               => ( D4 @ Z4 ) ) )
% 9.10/5.30         => ( ( D4 @ X3 )
% 9.10/5.30           => ( ! [X5: product_prod_int_int] :
% 9.10/5.30                  ( ( D4 @ X5 )
% 9.10/5.30                 => ( ! [Z3: product_prod_int_int] :
% 9.10/5.30                        ( ( R2 @ Z3 @ X5 )
% 9.10/5.30                       => ( P @ Z3 ) )
% 9.10/5.30                   => ( P @ X5 ) ) )
% 9.10/5.30             => ( P @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7764_accp__subset__induct,axiom,
% 9.10/5.30      ! [D4: list_nat > $o,R2: list_nat > list_nat > $o,X3: list_nat,P: list_nat > $o] :
% 9.10/5.30        ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R2 ) )
% 9.10/5.30       => ( ! [X5: list_nat,Z4: list_nat] :
% 9.10/5.30              ( ( D4 @ X5 )
% 9.10/5.30             => ( ( R2 @ Z4 @ X5 )
% 9.10/5.30               => ( D4 @ Z4 ) ) )
% 9.10/5.30         => ( ( D4 @ X3 )
% 9.10/5.30           => ( ! [X5: list_nat] :
% 9.10/5.30                  ( ( D4 @ X5 )
% 9.10/5.30                 => ( ! [Z3: list_nat] :
% 9.10/5.30                        ( ( R2 @ Z3 @ X5 )
% 9.10/5.30                       => ( P @ Z3 ) )
% 9.10/5.30                   => ( P @ X5 ) ) )
% 9.10/5.30             => ( P @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7765_accp__subset__induct,axiom,
% 9.10/5.30      ! [D4: nat > $o,R2: nat > nat > $o,X3: nat,P: nat > $o] :
% 9.10/5.30        ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R2 ) )
% 9.10/5.30       => ( ! [X5: nat,Z4: nat] :
% 9.10/5.30              ( ( D4 @ X5 )
% 9.10/5.30             => ( ( R2 @ Z4 @ X5 )
% 9.10/5.30               => ( D4 @ Z4 ) ) )
% 9.10/5.30         => ( ( D4 @ X3 )
% 9.10/5.30           => ( ! [X5: nat] :
% 9.10/5.30                  ( ( D4 @ X5 )
% 9.10/5.30                 => ( ! [Z3: nat] :
% 9.10/5.30                        ( ( R2 @ Z3 @ X5 )
% 9.10/5.30                       => ( P @ Z3 ) )
% 9.10/5.30                   => ( P @ X5 ) ) )
% 9.10/5.30             => ( P @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7766_sin__pi__divide__n__gt__0,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.30       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7767_sin__zero__lemma,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ( sin_real @ X3 )
% 9.10/5.30            = zero_zero_real )
% 9.10/5.30         => ? [N3: nat] :
% 9.10/5.30              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7768_sin__zero__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( sin_real @ X3 )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30        = ( ? [N2: nat] :
% 9.10/5.30              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.30          | ? [N2: nat] :
% 9.10/5.30              ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7769_cos__zero__lemma,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ( cos_real @ X3 )
% 9.10/5.30            = zero_zero_real )
% 9.10/5.30         => ? [N3: nat] :
% 9.10/5.30              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7770_cos__zero__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30        = ( ? [N2: nat] :
% 9.10/5.30              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.30          | ? [N2: nat] :
% 9.10/5.30              ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.10/5.30              & ( X3
% 9.10/5.30                = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7771_mono__SucI1,axiom,
% 9.10/5.30      ! [X9: nat > real] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_real @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 9.10/5.30       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7772_mono__SucI1,axiom,
% 9.10/5.30      ! [X9: nat > set_int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 9.10/5.30       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7773_mono__SucI1,axiom,
% 9.10/5.30      ! [X9: nat > rat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 9.10/5.30       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7774_mono__SucI1,axiom,
% 9.10/5.30      ! [X9: nat > num] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_num @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 9.10/5.30       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7775_mono__SucI1,axiom,
% 9.10/5.30      ! [X9: nat > nat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 9.10/5.30       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7776_mono__SucI1,axiom,
% 9.10/5.30      ! [X9: nat > int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_int @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
% 9.10/5.30       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7777_mono__SucI2,axiom,
% 9.10/5.30      ! [X9: nat > real] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_real @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 9.10/5.30       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7778_mono__SucI2,axiom,
% 9.10/5.30      ! [X9: nat > set_int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 9.10/5.30       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7779_mono__SucI2,axiom,
% 9.10/5.30      ! [X9: nat > rat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 9.10/5.30       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7780_mono__SucI2,axiom,
% 9.10/5.30      ! [X9: nat > num] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_num @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 9.10/5.30       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7781_mono__SucI2,axiom,
% 9.10/5.30      ! [X9: nat > nat] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 9.10/5.30       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7782_mono__SucI2,axiom,
% 9.10/5.30      ! [X9: nat > int] :
% 9.10/5.30        ( ! [N3: nat] : ( ord_less_eq_int @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
% 9.10/5.30       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7783_monoseq__Suc,axiom,
% 9.10/5.30      ( topolo6980174941875973593q_real
% 9.10/5.30      = ( ^ [X7: nat > real] :
% 9.10/5.30            ( ! [N2: nat] : ( ord_less_eq_real @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 9.10/5.30            | ! [N2: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7784_monoseq__Suc,axiom,
% 9.10/5.30      ( topolo3100542954746470799et_int
% 9.10/5.30      = ( ^ [X7: nat > set_int] :
% 9.10/5.30            ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 9.10/5.30            | ! [N2: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7785_monoseq__Suc,axiom,
% 9.10/5.30      ( topolo4267028734544971653eq_rat
% 9.10/5.30      = ( ^ [X7: nat > rat] :
% 9.10/5.30            ( ! [N2: nat] : ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 9.10/5.30            | ! [N2: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7786_monoseq__Suc,axiom,
% 9.10/5.30      ( topolo1459490580787246023eq_num
% 9.10/5.30      = ( ^ [X7: nat > num] :
% 9.10/5.30            ( ! [N2: nat] : ( ord_less_eq_num @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 9.10/5.30            | ! [N2: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7787_monoseq__Suc,axiom,
% 9.10/5.30      ( topolo4902158794631467389eq_nat
% 9.10/5.30      = ( ^ [X7: nat > nat] :
% 9.10/5.30            ( ! [N2: nat] : ( ord_less_eq_nat @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 9.10/5.30            | ! [N2: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7788_monoseq__Suc,axiom,
% 9.10/5.30      ( topolo4899668324122417113eq_int
% 9.10/5.30      = ( ^ [X7: nat > int] :
% 9.10/5.30            ( ! [N2: nat] : ( ord_less_eq_int @ ( X7 @ N2 ) @ ( X7 @ ( suc @ N2 ) ) )
% 9.10/5.30            | ! [N2: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N2 ) ) @ ( X7 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7789_monoI1,axiom,
% 9.10/5.30      ! [X9: nat > real] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_real @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 9.10/5.30       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7790_monoI1,axiom,
% 9.10/5.30      ! [X9: nat > set_int] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_set_int @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 9.10/5.30       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7791_monoI1,axiom,
% 9.10/5.30      ! [X9: nat > rat] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_rat @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 9.10/5.30       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7792_monoI1,axiom,
% 9.10/5.30      ! [X9: nat > num] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_num @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 9.10/5.30       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7793_monoI1,axiom,
% 9.10/5.30      ! [X9: nat > nat] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_nat @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 9.10/5.30       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7794_monoI1,axiom,
% 9.10/5.30      ! [X9: nat > int] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_int @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
% 9.10/5.30       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7795_monoI2,axiom,
% 9.10/5.30      ! [X9: nat > real] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_real @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 9.10/5.30       => ( topolo6980174941875973593q_real @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7796_monoI2,axiom,
% 9.10/5.30      ! [X9: nat > set_int] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_set_int @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 9.10/5.30       => ( topolo3100542954746470799et_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7797_monoI2,axiom,
% 9.10/5.30      ! [X9: nat > rat] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_rat @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 9.10/5.30       => ( topolo4267028734544971653eq_rat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7798_monoI2,axiom,
% 9.10/5.30      ! [X9: nat > num] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_num @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 9.10/5.30       => ( topolo1459490580787246023eq_num @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7799_monoI2,axiom,
% 9.10/5.30      ! [X9: nat > nat] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_nat @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 9.10/5.30       => ( topolo4902158794631467389eq_nat @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7800_monoI2,axiom,
% 9.10/5.30      ! [X9: nat > int] :
% 9.10/5.30        ( ! [M4: nat,N3: nat] :
% 9.10/5.30            ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.10/5.30           => ( ord_less_eq_int @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
% 9.10/5.30       => ( topolo4899668324122417113eq_int @ X9 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7801_vebt__maxt_Opelims,axiom,
% 9.10/5.30      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.10/5.30        ( ( ( vEBT_vebt_maxt @ X3 )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
% 9.10/5.30         => ( ! [A3: $o,B3: $o] :
% 9.10/5.30                ( ( X3
% 9.10/5.30                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.30               => ( ( ( B3
% 9.10/5.30                     => ( Y3
% 9.10/5.30                        = ( some_nat @ one_one_nat ) ) )
% 9.10/5.30                    & ( ~ B3
% 9.10/5.30                     => ( ( A3
% 9.10/5.30                         => ( Y3
% 9.10/5.30                            = ( some_nat @ zero_zero_nat ) ) )
% 9.10/5.30                        & ( ~ A3
% 9.10/5.30                         => ( Y3 = none_nat ) ) ) ) )
% 9.10/5.30                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 9.10/5.30           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.30                  ( ( X3
% 9.10/5.30                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.30                 => ( ( Y3 = none_nat )
% 9.10/5.30                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.10/5.30             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.30                    ( ( X3
% 9.10/5.30                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.10/5.30                   => ( ( Y3
% 9.10/5.30                        = ( some_nat @ Ma2 ) )
% 9.10/5.30                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7802_vebt__mint_Opelims,axiom,
% 9.10/5.30      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.10/5.30        ( ( ( vEBT_vebt_mint @ X3 )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
% 9.10/5.30         => ( ! [A3: $o,B3: $o] :
% 9.10/5.30                ( ( X3
% 9.10/5.30                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.10/5.30               => ( ( ( A3
% 9.10/5.30                     => ( Y3
% 9.10/5.30                        = ( some_nat @ zero_zero_nat ) ) )
% 9.10/5.30                    & ( ~ A3
% 9.10/5.30                     => ( ( B3
% 9.10/5.30                         => ( Y3
% 9.10/5.30                            = ( some_nat @ one_one_nat ) ) )
% 9.10/5.30                        & ( ~ B3
% 9.10/5.30                         => ( Y3 = none_nat ) ) ) ) )
% 9.10/5.30                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
% 9.10/5.30           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.10/5.30                  ( ( X3
% 9.10/5.30                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.10/5.30                 => ( ( Y3 = none_nat )
% 9.10/5.30                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.10/5.30             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.10/5.30                    ( ( X3
% 9.10/5.30                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.10/5.30                   => ( ( Y3
% 9.10/5.30                        = ( some_nat @ Mi2 ) )
% 9.10/5.30                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7803_Maclaurin__cos__expansion2,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30         => ? [T4: real] :
% 9.10/5.30              ( ( ord_less_real @ zero_zero_real @ T4 )
% 9.10/5.30              & ( ord_less_real @ T4 @ X3 )
% 9.10/5.30              & ( ( cos_real @ X3 )
% 9.10/5.30                = ( plus_plus_real
% 9.10/5.30                  @ ( groups6591440286371151544t_real
% 9.10/5.30                    @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7804_Maclaurin__minus__cos__expansion,axiom,
% 9.10/5.30      ! [N: nat,X3: real] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.30         => ? [T4: real] :
% 9.10/5.30              ( ( ord_less_real @ X3 @ T4 )
% 9.10/5.30              & ( ord_less_real @ T4 @ zero_zero_real )
% 9.10/5.30              & ( ( cos_real @ X3 )
% 9.10/5.30                = ( plus_plus_real
% 9.10/5.30                  @ ( groups6591440286371151544t_real
% 9.10/5.30                    @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7805_fact__Suc,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7806_fact__Suc,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7807_fact__Suc,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7808_fact__Suc,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7809_fact__2,axiom,
% 9.10/5.30      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7810_fact__2,axiom,
% 9.10/5.30      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7811_fact__2,axiom,
% 9.10/5.30      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7812_fact__2,axiom,
% 9.10/5.30      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7813_fact__2,axiom,
% 9.10/5.30      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7814_fact__ge__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7815_fact__ge__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7816_fact__ge__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7817_fact__ge__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7818_fact__not__neg,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7819_fact__not__neg,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7820_fact__not__neg,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7821_fact__not__neg,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7822_fact__gt__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7823_fact__gt__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7824_fact__gt__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7825_fact__gt__zero,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7826_fact__ge__1,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7827_fact__ge__1,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7828_fact__ge__1,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7829_fact__ge__1,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7830_fact__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7831_fact__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7832_fact__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7833_fact__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7834_fact__dvd,axiom,
% 9.10/5.30      ! [N: nat,M: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7835_fact__dvd,axiom,
% 9.10/5.30      ! [N: nat,M: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7836_fact__dvd,axiom,
% 9.10/5.30      ! [N: nat,M: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7837_fact__dvd,axiom,
% 9.10/5.30      ! [N: nat,M: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7838_fact__less__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.30       => ( ( ord_less_nat @ M @ N )
% 9.10/5.30         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7839_fact__less__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.30       => ( ( ord_less_nat @ M @ N )
% 9.10/5.30         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7840_fact__less__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.30       => ( ( ord_less_nat @ M @ N )
% 9.10/5.30         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7841_fact__less__mono,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.30       => ( ( ord_less_nat @ M @ N )
% 9.10/5.30         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7842_fact__mod,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 9.10/5.30          = zero_zero_int ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7843_fact__mod,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 9.10/5.30          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7844_fact__mod,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 9.10/5.30          = zero_zero_nat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7845_fact__le__power,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7846_fact__le__power,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7847_fact__le__power,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7848_fact__le__power,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7849_fact__numeral,axiom,
% 9.10/5.30      ! [K: num] :
% 9.10/5.30        ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 9.10/5.30        = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7850_fact__numeral,axiom,
% 9.10/5.30      ! [K: num] :
% 9.10/5.30        ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 9.10/5.30        = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7851_fact__numeral,axiom,
% 9.10/5.30      ! [K: num] :
% 9.10/5.30        ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 9.10/5.30        = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7852_fact__numeral,axiom,
% 9.10/5.30      ! [K: num] :
% 9.10/5.30        ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 9.10/5.30        = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7853_fact__numeral,axiom,
% 9.10/5.30      ! [K: num] :
% 9.10/5.30        ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 9.10/5.30        = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7854_square__fact__le__2__fact,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7855_fact__num__eq__if,axiom,
% 9.10/5.30      ( semiri5044797733671781792omplex
% 9.10/5.30      = ( ^ [M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M3 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7856_fact__num__eq__if,axiom,
% 9.10/5.30      ( semiri773545260158071498ct_rat
% 9.10/5.30      = ( ^ [M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M3 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7857_fact__num__eq__if,axiom,
% 9.10/5.30      ( semiri1406184849735516958ct_int
% 9.10/5.30      = ( ^ [M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7858_fact__num__eq__if,axiom,
% 9.10/5.30      ( semiri2265585572941072030t_real
% 9.10/5.30      = ( ^ [M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7859_fact__num__eq__if,axiom,
% 9.10/5.30      ( semiri1408675320244567234ct_nat
% 9.10/5.30      = ( ^ [M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7860_fact__reduce,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( semiri773545260158071498ct_rat @ N )
% 9.10/5.30          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7861_fact__reduce,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( semiri1406184849735516958ct_int @ N )
% 9.10/5.30          = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7862_fact__reduce,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( semiri2265585572941072030t_real @ N )
% 9.10/5.30          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7863_fact__reduce,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( semiri1408675320244567234ct_nat @ N )
% 9.10/5.30          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7864_Maclaurin__zero,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Diff: nat > complex > real] :
% 9.10/5.30        ( ( X3 = zero_zero_real )
% 9.10/5.30       => ( ( N != zero_zero_nat )
% 9.10/5.30         => ( ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7865_Maclaurin__zero,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Diff: nat > real > real] :
% 9.10/5.30        ( ( X3 = zero_zero_real )
% 9.10/5.30       => ( ( N != zero_zero_nat )
% 9.10/5.30         => ( ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7866_Maclaurin__zero,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Diff: nat > rat > real] :
% 9.10/5.30        ( ( X3 = zero_zero_real )
% 9.10/5.30       => ( ( N != zero_zero_nat )
% 9.10/5.30         => ( ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7867_Maclaurin__zero,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Diff: nat > nat > real] :
% 9.10/5.30        ( ( X3 = zero_zero_real )
% 9.10/5.30       => ( ( N != zero_zero_nat )
% 9.10/5.30         => ( ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7868_Maclaurin__zero,axiom,
% 9.10/5.30      ! [X3: real,N: nat,Diff: nat > int > real] :
% 9.10/5.30        ( ( X3 = zero_zero_real )
% 9.10/5.30       => ( ( N != zero_zero_nat )
% 9.10/5.30         => ( ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7869_Maclaurin__lemma,axiom,
% 9.10/5.30      ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.10/5.30       => ? [B9: real] :
% 9.10/5.30            ( ( F @ H2 )
% 9.10/5.30            = ( plus_plus_real
% 9.10/5.30              @ ( groups6591440286371151544t_real
% 9.10/5.30                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 9.10/5.30                @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30              @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7870_cos__paired,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( sums_real
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.10/5.30        @ ( cos_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7871_cos__coeff__def,axiom,
% 9.10/5.30      ( cos_coeff
% 9.10/5.30      = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7872_sin__paired,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( sums_real
% 9.10/5.30        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 9.10/5.30        @ ( sin_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7873_Maclaurin__cos__expansion,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30      ? [T4: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.30        & ( ( cos_real @ X3 )
% 9.10/5.30          = ( plus_plus_real
% 9.10/5.30            @ ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7874_Maclaurin__sin__expansion3,axiom,
% 9.10/5.30      ! [N: nat,X3: real] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30         => ? [T4: real] :
% 9.10/5.30              ( ( ord_less_real @ zero_zero_real @ T4 )
% 9.10/5.30              & ( ord_less_real @ T4 @ X3 )
% 9.10/5.30              & ( ( sin_real @ X3 )
% 9.10/5.30                = ( plus_plus_real
% 9.10/5.30                  @ ( groups6591440286371151544t_real
% 9.10/5.30                    @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7875_Maclaurin__sin__expansion4,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ? [T4: real] :
% 9.10/5.30            ( ( ord_less_real @ zero_zero_real @ T4 )
% 9.10/5.30            & ( ord_less_eq_real @ T4 @ X3 )
% 9.10/5.30            & ( ( sin_real @ X3 )
% 9.10/5.30              = ( plus_plus_real
% 9.10/5.30                @ ( groups6591440286371151544t_real
% 9.10/5.30                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30                  @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7876_Maclaurin__sin__expansion2,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30      ? [T4: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.30        & ( ( sin_real @ X3 )
% 9.10/5.30          = ( plus_plus_real
% 9.10/5.30            @ ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7877_Maclaurin__sin__expansion,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30      ? [T4: real] :
% 9.10/5.30        ( ( sin_real @ X3 )
% 9.10/5.30        = ( plus_plus_real
% 9.10/5.30          @ ( groups6591440286371151544t_real
% 9.10/5.30            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.30            @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T4 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7878_sin__coeff__def,axiom,
% 9.10/5.30      ( sin_coeff
% 9.10/5.30      = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7879_fact__ge__self,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7880_fact__mono__nat,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7881_fact__less__mono__nat,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.30       => ( ( ord_less_nat @ M @ N )
% 9.10/5.30         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7882_fact__ge__Suc__0__nat,axiom,
% 9.10/5.30      ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7883_dvd__fact,axiom,
% 9.10/5.30      ! [M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.10/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7884_fact__diff__Suc,axiom,
% 9.10/5.30      ! [N: nat,M: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.10/5.30       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 9.10/5.30          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7885_fact__div__fact__le__pow,axiom,
% 9.10/5.30      ! [R: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ R @ N )
% 9.10/5.30       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7886_choose__dvd,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7887_choose__dvd,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7888_choose__dvd,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7889_choose__dvd,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7890_choose__dvd,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7891_fold__atLeastAtMost__nat_Opsimps,axiom,
% 9.10/5.30      ! [F: nat > nat > nat,A: nat,B2: nat,Acc: nat] :
% 9.10/5.30        ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) )
% 9.10/5.30       => ( ( ( ord_less_nat @ B2 @ A )
% 9.10/5.30           => ( ( set_fo2584398358068434914at_nat @ F @ A @ B2 @ Acc )
% 9.10/5.30              = Acc ) )
% 9.10/5.30          & ( ~ ( ord_less_nat @ B2 @ A )
% 9.10/5.30           => ( ( set_fo2584398358068434914at_nat @ F @ A @ B2 @ Acc )
% 9.10/5.30              = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A @ one_one_nat ) @ B2 @ ( F @ A @ Acc ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7892_fold__atLeastAtMost__nat_Opelims,axiom,
% 9.10/5.30      ! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y3: nat] :
% 9.10/5.30        ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) )
% 9.10/5.30         => ~ ( ( ( ( ord_less_nat @ Xb2 @ Xa2 )
% 9.10/5.30                 => ( Y3 = Xc ) )
% 9.10/5.30                & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
% 9.10/5.30                 => ( Y3
% 9.10/5.30                    = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) )
% 9.10/5.30             => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X3 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb2 @ Xc ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7893_VEBT__internal_Ooption__shift_Opelims,axiom,
% 9.10/5.30      ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 9.10/5.30        ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb2 )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ Xa2 @ Xb2 ) ) )
% 9.10/5.30         => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 9.10/5.30             => ( ( Y3 = none_P5556105721700978146at_nat )
% 9.10/5.30               => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb2 ) ) ) ) )
% 9.10/5.30           => ( ! [V2: product_prod_nat_nat] :
% 9.10/5.30                  ( ( Xa2
% 9.10/5.30                    = ( some_P7363390416028606310at_nat @ V2 ) )
% 9.10/5.30                 => ( ( Xb2 = none_P5556105721700978146at_nat )
% 9.10/5.30                   => ( ( Y3 = none_P5556105721700978146at_nat )
% 9.10/5.30                     => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
% 9.10/5.30             => ~ ! [A3: product_prod_nat_nat] :
% 9.10/5.30                    ( ( Xa2
% 9.10/5.30                      = ( some_P7363390416028606310at_nat @ A3 ) )
% 9.10/5.30                   => ! [B3: product_prod_nat_nat] :
% 9.10/5.30                        ( ( Xb2
% 9.10/5.30                          = ( some_P7363390416028606310at_nat @ B3 ) )
% 9.10/5.30                       => ( ( Y3
% 9.10/5.30                            = ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) )
% 9.10/5.30                         => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7894_VEBT__internal_Ooption__shift_Opelims,axiom,
% 9.10/5.30      ! [X3: num > num > num,Xa2: option_num,Xb2: option_num,Y3: option_num] :
% 9.10/5.30        ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb2 )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ Xa2 @ Xb2 ) ) )
% 9.10/5.30         => ( ( ( Xa2 = none_num )
% 9.10/5.30             => ( ( Y3 = none_num )
% 9.10/5.30               => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ none_num @ Xb2 ) ) ) ) )
% 9.10/5.30           => ( ! [V2: num] :
% 9.10/5.30                  ( ( Xa2
% 9.10/5.30                    = ( some_num @ V2 ) )
% 9.10/5.30                 => ( ( Xb2 = none_num )
% 9.10/5.30                   => ( ( Y3 = none_num )
% 9.10/5.30                     => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
% 9.10/5.30             => ~ ! [A3: num] :
% 9.10/5.30                    ( ( Xa2
% 9.10/5.30                      = ( some_num @ A3 ) )
% 9.10/5.30                   => ! [B3: num] :
% 9.10/5.30                        ( ( Xb2
% 9.10/5.30                          = ( some_num @ B3 ) )
% 9.10/5.30                       => ( ( Y3
% 9.10/5.30                            = ( some_num @ ( X3 @ A3 @ B3 ) ) )
% 9.10/5.30                         => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X3 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7895_VEBT__internal_Ooption__shift_Opelims,axiom,
% 9.10/5.30      ! [X3: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y3: option_nat] :
% 9.10/5.30        ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb2 )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb2 ) ) )
% 9.10/5.30         => ( ( ( Xa2 = none_nat )
% 9.10/5.30             => ( ( Y3 = none_nat )
% 9.10/5.30               => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ none_nat @ Xb2 ) ) ) ) )
% 9.10/5.30           => ( ! [V2: nat] :
% 9.10/5.30                  ( ( Xa2
% 9.10/5.30                    = ( some_nat @ V2 ) )
% 9.10/5.30                 => ( ( Xb2 = none_nat )
% 9.10/5.30                   => ( ( Y3 = none_nat )
% 9.10/5.30                     => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
% 9.10/5.30             => ~ ! [A3: nat] :
% 9.10/5.30                    ( ( Xa2
% 9.10/5.30                      = ( some_nat @ A3 ) )
% 9.10/5.30                   => ! [B3: nat] :
% 9.10/5.30                        ( ( Xb2
% 9.10/5.30                          = ( some_nat @ B3 ) )
% 9.10/5.30                       => ( ( Y3
% 9.10/5.30                            = ( some_nat @ ( X3 @ A3 @ B3 ) ) )
% 9.10/5.30                         => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7896_fact__fact__dvd__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7897_fact__fact__dvd__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7898_fact__fact__dvd__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7899_fact__fact__dvd__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7900_fact__fact__dvd__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7901_fold__atLeastAtMost__nat_Oelims,axiom,
% 9.10/5.30      ! [X3: nat > nat > nat,Xa2: nat,Xb2: nat,Xc: nat,Y3: nat] :
% 9.10/5.30        ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb2 @ Xc )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( ( ( ord_less_nat @ Xb2 @ Xa2 )
% 9.10/5.30           => ( Y3 = Xc ) )
% 9.10/5.30          & ( ~ ( ord_less_nat @ Xb2 @ Xa2 )
% 9.10/5.30           => ( Y3
% 9.10/5.30              = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb2 @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7902_fold__atLeastAtMost__nat_Osimps,axiom,
% 9.10/5.30      ( set_fo2584398358068434914at_nat
% 9.10/5.30      = ( ^ [F3: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F3 @ A5 @ Acc2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7903_sum__atLeastAtMost__code,axiom,
% 9.10/5.30      ! [F: nat > complex,A: nat,B2: nat] :
% 9.10/5.30        ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.30        = ( set_fo1517530859248394432omplex
% 9.10/5.30          @ ^ [A5: nat] : ( plus_plus_complex @ ( F @ A5 ) )
% 9.10/5.30          @ A
% 9.10/5.30          @ B2
% 9.10/5.30          @ zero_zero_complex ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7904_sum__atLeastAtMost__code,axiom,
% 9.10/5.30      ! [F: nat > rat,A: nat,B2: nat] :
% 9.10/5.30        ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.30        = ( set_fo1949268297981939178at_rat
% 9.10/5.30          @ ^ [A5: nat] : ( plus_plus_rat @ ( F @ A5 ) )
% 9.10/5.30          @ A
% 9.10/5.30          @ B2
% 9.10/5.30          @ zero_zero_rat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7905_sum__atLeastAtMost__code,axiom,
% 9.10/5.30      ! [F: nat > int,A: nat,B2: nat] :
% 9.10/5.30        ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.30        = ( set_fo2581907887559384638at_int
% 9.10/5.30          @ ^ [A5: nat] : ( plus_plus_int @ ( F @ A5 ) )
% 9.10/5.30          @ A
% 9.10/5.30          @ B2
% 9.10/5.30          @ zero_zero_int ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7906_sum__atLeastAtMost__code,axiom,
% 9.10/5.30      ! [F: nat > nat,A: nat,B2: nat] :
% 9.10/5.30        ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.30        = ( set_fo2584398358068434914at_nat
% 9.10/5.30          @ ^ [A5: nat] : ( plus_plus_nat @ ( F @ A5 ) )
% 9.10/5.30          @ A
% 9.10/5.30          @ B2
% 9.10/5.30          @ zero_zero_nat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7907_sum__atLeastAtMost__code,axiom,
% 9.10/5.30      ! [F: nat > real,A: nat,B2: nat] :
% 9.10/5.30        ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.30        = ( set_fo3111899725591712190t_real
% 9.10/5.30          @ ^ [A5: nat] : ( plus_plus_real @ ( F @ A5 ) )
% 9.10/5.30          @ A
% 9.10/5.30          @ B2
% 9.10/5.30          @ zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7908_fact__code,axiom,
% 9.10/5.30      ( semiri1406184849735516958ct_int
% 9.10/5.30      = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7909_fact__code,axiom,
% 9.10/5.30      ( semiri2265585572941072030t_real
% 9.10/5.30      = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7910_fact__code,axiom,
% 9.10/5.30      ( semiri1408675320244567234ct_nat
% 9.10/5.30      = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7911_diffs__equiv,axiom,
% 9.10/5.30      ! [C: nat > complex,X3: complex] :
% 9.10/5.30        ( ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 9.10/5.30       => ( sums_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.10/5.30          @ ( suminf_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7912_diffs__equiv,axiom,
% 9.10/5.30      ! [C: nat > real,X3: real] :
% 9.10/5.30        ( ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 9.10/5.30       => ( sums_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.10/5.30          @ ( suminf_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7913_tan__double,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( ( cos_complex @ X3 )
% 9.10/5.30         != zero_zero_complex )
% 9.10/5.30       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30           != zero_zero_complex )
% 9.10/5.30         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30            = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7914_tan__double,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30         != zero_zero_real )
% 9.10/5.30       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30           != zero_zero_real )
% 9.10/5.30         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 9.10/5.30            = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7915_in__measure,axiom,
% 9.10/5.30      ! [X3: int,Y3: int,F: int > nat] :
% 9.10/5.30        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( measure_int @ F ) )
% 9.10/5.30        = ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7916_complex__unimodular__polar,axiom,
% 9.10/5.30      ! [Z: complex] :
% 9.10/5.30        ( ( ( real_V1022390504157884413omplex @ Z )
% 9.10/5.30          = one_one_real )
% 9.10/5.30       => ~ ! [T4: real] :
% 9.10/5.30              ( ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.30             => ( ( ord_less_real @ T4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.30               => ( Z
% 9.10/5.30                 != ( complex2 @ ( cos_real @ T4 ) @ ( sin_real @ T4 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7917_Maclaurin__exp__lt,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( X3 != zero_zero_real )
% 9.10/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30         => ? [T4: real] :
% 9.10/5.30              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
% 9.10/5.30              & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.30              & ( ( exp_real @ X3 )
% 9.10/5.30                = ( plus_plus_real
% 9.10/5.30                  @ ( groups6591440286371151544t_real
% 9.10/5.30                    @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 9.10/5.30                    @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7918_exp__less__mono,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.30       => ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7919_exp__less__cancel__iff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.10/5.30        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7920_exp__le__cancel__iff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.10/5.30        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7921_tan__npi,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.10/5.30        = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7922_one__less__exp__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) )
% 9.10/5.30        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7923_exp__less__one__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( exp_real @ X3 ) @ one_one_real )
% 9.10/5.30        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7924_one__le__exp__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X3 ) )
% 9.10/5.30        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7925_exp__le__one__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( exp_real @ X3 ) @ one_one_real )
% 9.10/5.30        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7926_tan__periodic__n,axiom,
% 9.10/5.30      ! [X3: real,N: num] :
% 9.10/5.30        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 9.10/5.30        = ( tan_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7927_tan__periodic__nat,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 9.10/5.30        = ( tan_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7928_exp__ln,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( exp_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.30          = X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7929_exp__ln__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( exp_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.30          = X3 )
% 9.10/5.30        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7930_tan__periodic,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.10/5.30        = ( tan_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7931_norm__exp,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X3 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7932_norm__exp,axiom,
% 9.10/5.30      ! [X3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X3 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7933_exp__less__cancel,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.10/5.30       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7934_exp__times__arg__commute,axiom,
% 9.10/5.30      ! [A4: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( exp_complex @ A4 ) @ A4 )
% 9.10/5.30        = ( times_times_complex @ A4 @ ( exp_complex @ A4 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7935_exp__times__arg__commute,axiom,
% 9.10/5.30      ! [A4: real] :
% 9.10/5.30        ( ( times_times_real @ ( exp_real @ A4 ) @ A4 )
% 9.10/5.30        = ( times_times_real @ A4 @ ( exp_real @ A4 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7936_exp__total,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.30       => ? [X5: real] :
% 9.10/5.30            ( ( exp_real @ X5 )
% 9.10/5.30            = Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7937_exp__gt__zero,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7938_not__exp__less__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ~ ( ord_less_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7939_exp__ge__zero,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7940_not__exp__le__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ~ ( ord_less_eq_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7941_mult__exp__exp,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) )
% 9.10/5.30        = ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7942_mult__exp__exp,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.10/5.30        = ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7943_exp__add__commuting,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( ( times_times_complex @ X3 @ Y3 )
% 9.10/5.30          = ( times_times_complex @ Y3 @ X3 ) )
% 9.10/5.30       => ( ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.10/5.30          = ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7944_exp__add__commuting,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( times_times_real @ X3 @ Y3 )
% 9.10/5.30          = ( times_times_real @ Y3 @ X3 ) )
% 9.10/5.30       => ( ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.30          = ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7945_exp__diff,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( exp_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.10/5.30        = ( divide1717551699836669952omplex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7946_exp__diff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( exp_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( divide_divide_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7947_Complex__eq__numeral,axiom,
% 9.10/5.30      ! [A: real,B2: real,W: num] :
% 9.10/5.30        ( ( ( complex2 @ A @ B2 )
% 9.10/5.30          = ( numera6690914467698888265omplex @ W ) )
% 9.10/5.30        = ( ( A
% 9.10/5.30            = ( numeral_numeral_real @ W ) )
% 9.10/5.30          & ( B2 = zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7948_exp__gt__one,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7949_exp__ge__add__one__self,axiom,
% 9.10/5.30      ! [X3: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7950_exp__minus__inverse,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) )
% 9.10/5.30        = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7951_exp__minus__inverse,axiom,
% 9.10/5.30      ! [X3: complex] :
% 9.10/5.30        ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) )
% 9.10/5.30        = one_one_complex ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7952_exp__of__nat2__mult,axiom,
% 9.10/5.30      ! [X3: complex,N: nat] :
% 9.10/5.30        ( ( exp_complex @ ( times_times_complex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) )
% 9.10/5.30        = ( power_power_complex @ ( exp_complex @ X3 ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7953_exp__of__nat2__mult,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( exp_real @ ( times_times_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.30        = ( power_power_real @ ( exp_real @ X3 ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7954_exp__of__nat__mult,axiom,
% 9.10/5.30      ! [N: nat,X3: complex] :
% 9.10/5.30        ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X3 ) )
% 9.10/5.30        = ( power_power_complex @ ( exp_complex @ X3 ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7955_exp__of__nat__mult,axiom,
% 9.10/5.30      ! [N: nat,X3: real] :
% 9.10/5.30        ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X3 ) )
% 9.10/5.30        = ( power_power_real @ ( exp_real @ X3 ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7956_Complex__eq__neg__numeral,axiom,
% 9.10/5.30      ! [A: real,B2: real,W: num] :
% 9.10/5.30        ( ( ( complex2 @ A @ B2 )
% 9.10/5.30          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.30        = ( ( A
% 9.10/5.30            = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 9.10/5.30          & ( B2 = zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7957_complex__mult,axiom,
% 9.10/5.30      ! [A: real,B2: real,C: real,D: real] :
% 9.10/5.30        ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ ( complex2 @ C @ D ) )
% 9.10/5.30        = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7958_tan__def,axiom,
% 9.10/5.30      ( tan_complex
% 9.10/5.30      = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7959_tan__def,axiom,
% 9.10/5.30      ( tan_real
% 9.10/5.30      = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7960_exp__ge__add__one__self__aux,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( exp_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7961_lemma__exp__total,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 9.10/5.30       => ? [X5: real] :
% 9.10/5.30            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.10/5.30            & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 9.10/5.30            & ( ( exp_real @ X5 )
% 9.10/5.30              = Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7962_ln__ge__iff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X3 ) )
% 9.10/5.30          = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7963_ln__x__over__x__mono,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.30         => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7964_diffs__def,axiom,
% 9.10/5.30      ( diffs_rat
% 9.10/5.30      = ( ^ [C3: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7965_diffs__def,axiom,
% 9.10/5.30      ( diffs_int
% 9.10/5.30      = ( ^ [C3: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7966_diffs__def,axiom,
% 9.10/5.30      ( diffs_real
% 9.10/5.30      = ( ^ [C3: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7967_exp__le,axiom,
% 9.10/5.30      ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7968_termdiff__converges__all,axiom,
% 9.10/5.30      ! [C: nat > complex,X3: complex] :
% 9.10/5.30        ( ! [X5: complex] :
% 9.10/5.30            ( summable_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) )
% 9.10/5.30       => ( summable_complex
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7969_termdiff__converges__all,axiom,
% 9.10/5.30      ! [C: nat > real,X3: real] :
% 9.10/5.30        ( ! [X5: real] :
% 9.10/5.30            ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) )
% 9.10/5.30       => ( summable_real
% 9.10/5.30          @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7970_exp__divide__power__eq,axiom,
% 9.10/5.30      ! [N: nat,X3: complex] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 9.10/5.30          = ( exp_complex @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7971_exp__divide__power__eq,axiom,
% 9.10/5.30      ! [N: nat,X3: real] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 9.10/5.30          = ( exp_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7972_tanh__altdef,axiom,
% 9.10/5.30      ( tanh_real
% 9.10/5.30      = ( ^ [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 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7973_tanh__altdef,axiom,
% 9.10/5.30      ( tanh_complex
% 9.10/5.30      = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7974_exp__half__le2,axiom,
% 9.10/5.30      ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7975_tan__45,axiom,
% 9.10/5.30      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30      = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7976_exp__double,axiom,
% 9.10/5.30      ! [Z: complex] :
% 9.10/5.30        ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 9.10/5.30        = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7977_exp__double,axiom,
% 9.10/5.30      ! [Z: real] :
% 9.10/5.30        ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 9.10/5.30        = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7978_lemma__tan__total,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.30       => ? [X5: real] :
% 9.10/5.30            ( ( ord_less_real @ zero_zero_real @ X5 )
% 9.10/5.30            & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30            & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7979_tan__gt__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7980_lemma__tan__total1,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30      ? [X5: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 9.10/5.30        & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30        & ( ( tan_real @ X5 )
% 9.10/5.30          = Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7981_tan__mono__lt__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30           => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30             => ( ( ord_less_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 9.10/5.30                = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7982_tan__monotone_H,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30           => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30             => ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.30                = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7983_tan__monotone,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30       => ( ( ord_less_real @ Y3 @ X3 )
% 9.10/5.30         => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30           => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7984_tan__total,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30      ? [X5: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 9.10/5.30        & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30        & ( ( tan_real @ X5 )
% 9.10/5.30          = Y3 )
% 9.10/5.30        & ! [Y5: real] :
% 9.10/5.30            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 9.10/5.30              & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30              & ( ( tan_real @ Y5 )
% 9.10/5.30                = Y3 ) )
% 9.10/5.30           => ( Y5 = X5 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7985_tan__minus__45,axiom,
% 9.10/5.30      ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.30      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7986_tan__inverse,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 9.10/5.30        = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7987_exp__bound__half,axiom,
% 9.10/5.30      ! [Z: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7988_exp__bound__half,axiom,
% 9.10/5.30      ! [Z: complex] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7989_add__tan__eq,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( ( cos_complex @ X3 )
% 9.10/5.30         != zero_zero_complex )
% 9.10/5.30       => ( ( ( cos_complex @ Y3 )
% 9.10/5.30           != zero_zero_complex )
% 9.10/5.30         => ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) )
% 9.10/5.30            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7990_add__tan__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30         != zero_zero_real )
% 9.10/5.30       => ( ( ( cos_real @ Y3 )
% 9.10/5.30           != zero_zero_real )
% 9.10/5.30         => ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 9.10/5.30            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7991_termdiff__converges,axiom,
% 9.10/5.30      ! [X3: real,K5: real,C: nat > real] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
% 9.10/5.30       => ( ! [X5: real] :
% 9.10/5.30              ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
% 9.10/5.30             => ( summable_real
% 9.10/5.30                @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 9.10/5.30         => ( summable_real
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7992_termdiff__converges,axiom,
% 9.10/5.30      ! [X3: complex,K5: real,C: nat > complex] :
% 9.10/5.30        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
% 9.10/5.30       => ( ! [X5: complex] :
% 9.10/5.30              ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
% 9.10/5.30             => ( summable_complex
% 9.10/5.30                @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) ) )
% 9.10/5.30         => ( summable_complex
% 9.10/5.30            @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7993_exp__bound,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.30         => ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7994_tan__pos__pi2__le,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7995_tan__total__pos,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30       => ? [X5: real] :
% 9.10/5.30            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.10/5.30            & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30            & ( ( tan_real @ X5 )
% 9.10/5.30              = Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7996_tan__less__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.30         => ( ord_less_real @ ( tan_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7997_tan__mono__le__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.10/5.30           => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30             => ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 9.10/5.30                = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7998_tan__mono__le,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.30         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30           => ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_7999_tan__bound__pi2,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X3 ) ) @ one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8000_arctan__unique,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( ( tan_real @ X3 )
% 9.10/5.30              = Y3 )
% 9.10/5.30           => ( ( arctan @ Y3 )
% 9.10/5.30              = X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8001_arctan__tan,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ( arctan @ ( tan_real @ X3 ) )
% 9.10/5.30            = X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8002_arctan,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 9.10/5.30        & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30        & ( ( tan_real @ ( arctan @ Y3 ) )
% 9.10/5.30          = Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8003_lemma__tan__add1,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( ( cos_complex @ X3 )
% 9.10/5.30         != zero_zero_complex )
% 9.10/5.30       => ( ( ( cos_complex @ Y3 )
% 9.10/5.30           != zero_zero_complex )
% 9.10/5.30         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) )
% 9.10/5.30            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8004_lemma__tan__add1,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30         != zero_zero_real )
% 9.10/5.30       => ( ( ( cos_real @ Y3 )
% 9.10/5.30           != zero_zero_real )
% 9.10/5.30         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) )
% 9.10/5.30            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8005_tan__diff,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( ( cos_complex @ X3 )
% 9.10/5.30         != zero_zero_complex )
% 9.10/5.30       => ( ( ( cos_complex @ Y3 )
% 9.10/5.30           != zero_zero_complex )
% 9.10/5.30         => ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.10/5.30             != zero_zero_complex )
% 9.10/5.30           => ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.10/5.30              = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8006_tan__diff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30         != zero_zero_real )
% 9.10/5.30       => ( ( ( cos_real @ Y3 )
% 9.10/5.30           != zero_zero_real )
% 9.10/5.30         => ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.10/5.30             != zero_zero_real )
% 9.10/5.30           => ( ( tan_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.10/5.30              = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8007_tan__add,axiom,
% 9.10/5.30      ! [X3: complex,Y3: complex] :
% 9.10/5.30        ( ( ( cos_complex @ X3 )
% 9.10/5.30         != zero_zero_complex )
% 9.10/5.30       => ( ( ( cos_complex @ Y3 )
% 9.10/5.30           != zero_zero_complex )
% 9.10/5.30         => ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.10/5.30             != zero_zero_complex )
% 9.10/5.30           => ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.10/5.30              = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8008_tan__add,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( cos_real @ X3 )
% 9.10/5.30         != zero_zero_real )
% 9.10/5.30       => ( ( ( cos_real @ Y3 )
% 9.10/5.30           != zero_zero_real )
% 9.10/5.30         => ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.30             != zero_zero_real )
% 9.10/5.30           => ( ( tan_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.10/5.30              = ( 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 ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8009_real__exp__bound__lemma,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30         => ( ord_less_eq_real @ ( exp_real @ X3 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8010_exp__ge__one__plus__x__over__n__power__n,axiom,
% 9.10/5.30      ! [N: nat,X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X3 )
% 9.10/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8011_exp__ge__one__minus__x__over__n__power__n,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.30       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8012_tan__total__pi4,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.30       => ? [Z4: real] :
% 9.10/5.30            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 9.10/5.30            & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30            & ( ( tan_real @ Z4 )
% 9.10/5.30              = X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8013_exp__bound__lemma,axiom,
% 9.10/5.30      ! [Z: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8014_exp__bound__lemma,axiom,
% 9.10/5.30      ! [Z: complex] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8015_Maclaurin__exp__le,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30      ? [T4: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.30        & ( ( exp_real @ X3 )
% 9.10/5.30          = ( plus_plus_real
% 9.10/5.30            @ ( groups6591440286371151544t_real
% 9.10/5.30              @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 9.10/5.30              @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.30            @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8016_exp__lower__Taylor__quadratic,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( divide_divide_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8017_tanh__real__altdef,axiom,
% 9.10/5.30      ( tanh_real
% 9.10/5.30      = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8018_tan__half,axiom,
% 9.10/5.30      ( tan_complex
% 9.10/5.30      = ( ^ [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 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8019_tan__half,axiom,
% 9.10/5.30      ( tan_real
% 9.10/5.30      = ( ^ [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 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8020_binomial__code,axiom,
% 9.10/5.30      ( binomial
% 9.10/5.30      = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8021_in__finite__psubset,axiom,
% 9.10/5.30      ! [A4: set_nat,B5: set_nat] :
% 9.10/5.30        ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A4 @ B5 ) @ finite_psubset_nat )
% 9.10/5.30        = ( ( ord_less_set_nat @ A4 @ B5 )
% 9.10/5.30          & ( finite_finite_nat @ B5 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8022_in__finite__psubset,axiom,
% 9.10/5.30      ! [A4: set_int,B5: set_int] :
% 9.10/5.30        ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A4 @ B5 ) @ finite_psubset_int )
% 9.10/5.30        = ( ( ord_less_set_int @ A4 @ B5 )
% 9.10/5.30          & ( finite_finite_int @ B5 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8023_in__finite__psubset,axiom,
% 9.10/5.30      ! [A4: set_complex,B5: set_complex] :
% 9.10/5.30        ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A4 @ B5 ) @ finite8643634255014194347omplex )
% 9.10/5.30        = ( ( ord_less_set_complex @ A4 @ B5 )
% 9.10/5.30          & ( finite3207457112153483333omplex @ B5 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8024_sin__tan,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ( sin_real @ X3 )
% 9.10/5.30          = ( divide_divide_real @ ( tan_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8025_cos__tan,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ( cos_real @ X3 )
% 9.10/5.30          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8026_pochhammer__double,axiom,
% 9.10/5.30      ! [Z: complex,N: nat] :
% 9.10/5.30        ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.30        = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8027_pochhammer__double,axiom,
% 9.10/5.30      ! [Z: rat,N: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8028_pochhammer__double,axiom,
% 9.10/5.30      ! [Z: real,N: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.30        = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8029_real__sqrt__eq__iff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( sqrt @ X3 )
% 9.10/5.30          = ( sqrt @ Y3 ) )
% 9.10/5.30        = ( X3 = Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8030_real__sqrt__zero,axiom,
% 9.10/5.30      ( ( sqrt @ zero_zero_real )
% 9.10/5.30      = zero_zero_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8031_real__sqrt__eq__zero__cancel__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( sqrt @ X3 )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30        = ( X3 = zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8032_real__sqrt__less__iff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
% 9.10/5.30        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8033_real__sqrt__le__iff,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
% 9.10/5.30        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8034_real__sqrt__one,axiom,
% 9.10/5.30      ( ( sqrt @ one_one_real )
% 9.10/5.30      = one_one_real ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8035_real__sqrt__eq__1__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( sqrt @ X3 )
% 9.10/5.30          = one_one_real )
% 9.10/5.30        = ( X3 = one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8036_real__sqrt__gt__0__iff,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 9.10/5.30        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8037_real__sqrt__lt__0__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( sqrt @ X3 ) @ zero_zero_real )
% 9.10/5.30        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8038_real__sqrt__ge__0__iff,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 9.10/5.30        = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8039_real__sqrt__le__0__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ zero_zero_real )
% 9.10/5.30        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8040_binomial__eq__0__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ( binomial @ N @ K )
% 9.10/5.30          = zero_zero_nat )
% 9.10/5.30        = ( ord_less_nat @ N @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8041_binomial__Suc__Suc,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 9.10/5.30        = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8042_real__sqrt__lt__1__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( sqrt @ X3 ) @ one_one_real )
% 9.10/5.30        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8043_real__sqrt__gt__1__iff,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 9.10/5.30        = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8044_real__sqrt__ge__1__iff,axiom,
% 9.10/5.30      ! [Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 9.10/5.30        = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8045_real__sqrt__le__1__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ one_one_real )
% 9.10/5.30        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8046_real__sqrt__mult__self,axiom,
% 9.10/5.30      ! [A: real] :
% 9.10/5.30        ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 9.10/5.30        = ( abs_abs_real @ A ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8047_real__sqrt__abs2,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( sqrt @ ( times_times_real @ X3 @ X3 ) )
% 9.10/5.30        = ( abs_abs_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8048_real__sqrt__four,axiom,
% 9.10/5.30      ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.10/5.30      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8049_zero__less__binomial__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 9.10/5.30        = ( ord_less_eq_nat @ K @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8050_real__sqrt__abs,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( sqrt @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30        = ( abs_abs_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8051_real__sqrt__pow2,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30          = X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8052_real__sqrt__pow2__iff,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ( power_power_real @ ( sqrt @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30          = X3 )
% 9.10/5.30        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8053_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real,Xa2: real,Ya: real] :
% 9.10/5.30        ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30        = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8054_real__sqrt__less__mono,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.30       => ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8055_real__sqrt__le__mono,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.30       => ( ord_less_eq_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8056_real__sqrt__minus,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( sqrt @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.30        = ( uminus_uminus_real @ ( sqrt @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8057_real__sqrt__power,axiom,
% 9.10/5.30      ! [X3: real,K: nat] :
% 9.10/5.30        ( ( sqrt @ ( power_power_real @ X3 @ K ) )
% 9.10/5.30        = ( power_power_real @ ( sqrt @ X3 ) @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8058_real__sqrt__mult,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( sqrt @ ( times_times_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( times_times_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8059_real__sqrt__divide,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( sqrt @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.10/5.30        = ( divide_divide_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8060_binomial__eq__0,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ K )
% 9.10/5.30       => ( ( binomial @ N @ K )
% 9.10/5.30          = zero_zero_nat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8061_Suc__times__binomial,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 9.10/5.30        = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8062_Suc__times__binomial__eq,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 9.10/5.30        = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8063_binomial__symmetric,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( binomial @ N @ K )
% 9.10/5.30          = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8064_real__sqrt__gt__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8065_real__sqrt__eq__zero__cancel,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ( sqrt @ X3 )
% 9.10/5.30            = zero_zero_real )
% 9.10/5.30         => ( X3 = zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8066_real__sqrt__ge__zero,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8067_choose__mult__lemma,axiom,
% 9.10/5.30      ! [M: nat,R: nat,K: nat] :
% 9.10/5.30        ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 9.10/5.30        = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8068_real__sqrt__ge__one,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8069_binomial__le__pow,axiom,
% 9.10/5.30      ! [R: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ R @ N )
% 9.10/5.30       => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8070_pochhammer__pos,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8071_pochhammer__pos,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.10/5.30       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8072_pochhammer__pos,axiom,
% 9.10/5.30      ! [X3: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.10/5.30       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8073_pochhammer__pos,axiom,
% 9.10/5.30      ! [X3: int,N: nat] :
% 9.10/5.30        ( ( ord_less_int @ zero_zero_int @ X3 )
% 9.10/5.30       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8074_pochhammer__neq__0__mono,axiom,
% 9.10/5.30      ! [A: complex,M: nat,N: nat] :
% 9.10/5.30        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.10/5.30         != zero_zero_complex )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30         => ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.10/5.30           != zero_zero_complex ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8075_pochhammer__neq__0__mono,axiom,
% 9.10/5.30      ! [A: real,M: nat,N: nat] :
% 9.10/5.30        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.10/5.30         != zero_zero_real )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30         => ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.10/5.30           != zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8076_pochhammer__neq__0__mono,axiom,
% 9.10/5.30      ! [A: rat,M: nat,N: nat] :
% 9.10/5.30        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.10/5.30         != zero_zero_rat )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30         => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.10/5.30           != zero_zero_rat ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8077_pochhammer__eq__0__mono,axiom,
% 9.10/5.30      ! [A: complex,N: nat,M: nat] :
% 9.10/5.30        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.10/5.30          = zero_zero_complex )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.10/5.30            = zero_zero_complex ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8078_pochhammer__eq__0__mono,axiom,
% 9.10/5.30      ! [A: real,N: nat,M: nat] :
% 9.10/5.30        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.10/5.30            = zero_zero_real ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8079_pochhammer__eq__0__mono,axiom,
% 9.10/5.30      ! [A: rat,N: nat,M: nat] :
% 9.10/5.30        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.10/5.30          = zero_zero_rat )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.30         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.10/5.30            = zero_zero_rat ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8080_zero__less__binomial,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8081_Suc__times__binomial__add,axiom,
% 9.10/5.30      ! [A: nat,B2: nat] :
% 9.10/5.30        ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ ( suc @ A ) ) )
% 9.10/5.30        = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B2 ) ) @ A ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8082_binomial__Suc__Suc__eq__times,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 9.10/5.30        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8083_choose__mult,axiom,
% 9.10/5.30      ! [K: nat,M: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ M )
% 9.10/5.30       => ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30         => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 9.10/5.30            = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8084_real__div__sqrt,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( divide_divide_real @ X3 @ ( sqrt @ X3 ) )
% 9.10/5.30          = ( sqrt @ X3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8085_binomial__absorb__comp,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 9.10/5.30        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8086_sqrt__add__le__add__sqrt,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8087_le__real__sqrt__sumsq,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8088_pochhammer__nonneg,axiom,
% 9.10/5.30      ! [X3: real,N: nat] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8089_pochhammer__nonneg,axiom,
% 9.10/5.30      ! [X3: rat,N: nat] :
% 9.10/5.30        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.10/5.30       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8090_pochhammer__nonneg,axiom,
% 9.10/5.30      ! [X3: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.10/5.30       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8091_pochhammer__nonneg,axiom,
% 9.10/5.30      ! [X3: int,N: nat] :
% 9.10/5.30        ( ( ord_less_int @ zero_zero_int @ X3 )
% 9.10/5.30       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8092_sqrt2__less__2,axiom,
% 9.10/5.30      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8093_binomial__absorption,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 9.10/5.30        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8094_binomial__fact__lemma,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 9.10/5.30          = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8095_pochhammer__rec,axiom,
% 9.10/5.30      ! [A: complex,N: nat] :
% 9.10/5.30        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8096_pochhammer__rec,axiom,
% 9.10/5.30      ! [A: real,N: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8097_pochhammer__rec,axiom,
% 9.10/5.30      ! [A: rat,N: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8098_pochhammer__rec,axiom,
% 9.10/5.30      ! [A: nat,N: nat] :
% 9.10/5.30        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8099_pochhammer__rec,axiom,
% 9.10/5.30      ! [A: int,N: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8100_pochhammer__Suc,axiom,
% 9.10/5.30      ! [A: rat,N: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8101_pochhammer__Suc,axiom,
% 9.10/5.30      ! [A: int,N: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8102_pochhammer__Suc,axiom,
% 9.10/5.30      ! [A: real,N: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8103_pochhammer__Suc,axiom,
% 9.10/5.30      ! [A: nat,N: nat] :
% 9.10/5.30        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8104_pochhammer__rec_H,axiom,
% 9.10/5.30      ! [Z: rat,N: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8105_pochhammer__rec_H,axiom,
% 9.10/5.30      ! [Z: int,N: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8106_pochhammer__rec_H,axiom,
% 9.10/5.30      ! [Z: real,N: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8107_pochhammer__rec_H,axiom,
% 9.10/5.30      ! [Z: nat,N: nat] :
% 9.10/5.30        ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 9.10/5.30        = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8108_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ K )
% 9.10/5.30       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.10/5.30          = zero_zero_complex ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8109_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ K )
% 9.10/5.30       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.10/5.30          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8110_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ K )
% 9.10/5.30       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.10/5.30          = zero_zero_rat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8111_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ K )
% 9.10/5.30       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.10/5.30          = zero_zero_int ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8112_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ N @ K )
% 9.10/5.30       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.10/5.30          = zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8113_pochhammer__of__nat__eq__0__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.10/5.30          = zero_zero_complex )
% 9.10/5.30        = ( ord_less_nat @ N @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8114_pochhammer__of__nat__eq__0__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.10/5.30          = zero_z3403309356797280102nteger )
% 9.10/5.30        = ( ord_less_nat @ N @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8115_pochhammer__of__nat__eq__0__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.10/5.30          = zero_zero_rat )
% 9.10/5.30        = ( ord_less_nat @ N @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8116_pochhammer__of__nat__eq__0__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.10/5.30          = zero_zero_int )
% 9.10/5.30        = ( ord_less_nat @ N @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8117_pochhammer__of__nat__eq__0__iff,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30        = ( ord_less_nat @ N @ K ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8118_pochhammer__eq__0__iff,axiom,
% 9.10/5.30      ! [A: complex,N: nat] :
% 9.10/5.30        ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 9.10/5.30          = zero_zero_complex )
% 9.10/5.30        = ( ? [K2: nat] :
% 9.10/5.30              ( ( ord_less_nat @ K2 @ N )
% 9.10/5.30              & ( A
% 9.10/5.30                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8119_pochhammer__eq__0__iff,axiom,
% 9.10/5.30      ! [A: rat,N: nat] :
% 9.10/5.30        ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 9.10/5.30          = zero_zero_rat )
% 9.10/5.30        = ( ? [K2: nat] :
% 9.10/5.30              ( ( ord_less_nat @ K2 @ N )
% 9.10/5.30              & ( A
% 9.10/5.30                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8120_pochhammer__eq__0__iff,axiom,
% 9.10/5.30      ! [A: real,N: nat] :
% 9.10/5.30        ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 9.10/5.30          = zero_zero_real )
% 9.10/5.30        = ( ? [K2: nat] :
% 9.10/5.30              ( ( ord_less_nat @ K2 @ N )
% 9.10/5.30              & ( A
% 9.10/5.30                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8121_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 9.10/5.30         != zero_zero_complex ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8122_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 9.10/5.30         != zero_z3403309356797280102nteger ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8123_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 9.10/5.30         != zero_zero_rat ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8124_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 9.10/5.30         != zero_zero_int ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8125_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 9.10/5.30         != zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8126_pochhammer__product_H,axiom,
% 9.10/5.30      ! [Z: rat,N: nat,M: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.10/5.30        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8127_pochhammer__product_H,axiom,
% 9.10/5.30      ! [Z: int,N: nat,M: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.10/5.30        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8128_pochhammer__product_H,axiom,
% 9.10/5.30      ! [Z: real,N: nat,M: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.10/5.30        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8129_pochhammer__product_H,axiom,
% 9.10/5.30      ! [Z: nat,N: nat,M: nat] :
% 9.10/5.30        ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 9.10/5.30        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8130_binomial__ge__n__over__k__pow__k,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8131_binomial__ge__n__over__k__pow__k,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8132_binomial__mono,axiom,
% 9.10/5.30      ! [K: nat,K6: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ K6 )
% 9.10/5.30       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 9.10/5.30         => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8133_binomial__maximum_H,axiom,
% 9.10/5.30      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8134_binomial__maximum,axiom,
% 9.10/5.30      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8135_binomial__antimono,axiom,
% 9.10/5.30      ! [K: nat,K6: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ K6 )
% 9.10/5.30       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 9.10/5.30         => ( ( ord_less_eq_nat @ K6 @ N )
% 9.10/5.30           => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8136_binomial__le__pow2,axiom,
% 9.10/5.30      ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8137_real__less__rsqrt,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 9.10/5.30       => ( ord_less_real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8138_choose__reduce__nat,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.30         => ( ( binomial @ N @ K )
% 9.10/5.30            = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8139_sqrt__le__D,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y3 )
% 9.10/5.30       => ( ord_less_eq_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8140_real__le__rsqrt,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 9.10/5.30       => ( ord_less_eq_real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8141_times__binomial__minus1__eq,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.30       => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 9.10/5.30          = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8142_tan__60,axiom,
% 9.10/5.30      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 9.10/5.30      = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8143_binomial__altdef__nat,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( binomial @ N @ K )
% 9.10/5.30          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8144_pochhammer__product,axiom,
% 9.10/5.30      ! [M: nat,N: nat,Z: rat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 9.10/5.30          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8145_pochhammer__product,axiom,
% 9.10/5.30      ! [M: nat,N: nat,Z: int] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 9.10/5.30          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8146_pochhammer__product,axiom,
% 9.10/5.30      ! [M: nat,N: nat,Z: real] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 9.10/5.30          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8147_pochhammer__product,axiom,
% 9.10/5.30      ! [M: nat,N: nat,Z: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.30       => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 9.10/5.30          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8148_binomial__less__binomial__Suc,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8149_binomial__strict__mono,axiom,
% 9.10/5.30      ! [K: nat,K6: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ K @ K6 )
% 9.10/5.30       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 9.10/5.30         => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8150_binomial__strict__antimono,axiom,
% 9.10/5.30      ! [K: nat,K6: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_nat @ K @ K6 )
% 9.10/5.30       => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 9.10/5.30         => ( ( ord_less_eq_nat @ K6 @ N )
% 9.10/5.30           => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8151_central__binomial__odd,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.30       => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30          = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8152_binomial__addition__formula,axiom,
% 9.10/5.30      ! [N: nat,K: nat] :
% 9.10/5.30        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.30       => ( ( binomial @ N @ ( suc @ K ) )
% 9.10/5.30          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8153_binomial__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) )
% 9.10/5.30          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8154_binomial__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) )
% 9.10/5.30          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8155_binomial__fact,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) )
% 9.10/5.30          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8156_fact__binomial,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K ) ) )
% 9.10/5.30          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8157_fact__binomial,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K ) ) )
% 9.10/5.30          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8158_fact__binomial,axiom,
% 9.10/5.30      ! [K: nat,N: nat] :
% 9.10/5.30        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.30       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K ) ) )
% 9.10/5.30          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8159_real__le__lsqrt,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ( ord_less_eq_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30           => ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8160_real__sqrt__unique,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] :
% 9.10/5.30        ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30          = X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ( sqrt @ X3 )
% 9.10/5.30            = Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8161_lemma__real__divide__sqrt__less,axiom,
% 9.10/5.30      ! [U: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ U )
% 9.10/5.30       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8162_real__sqrt__sum__squares__eq__cancel,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30          = X3 )
% 9.10/5.30       => ( Y3 = zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8163_real__sqrt__sum__squares__eq__cancel2,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30          = Y3 )
% 9.10/5.30       => ( X3 = zero_zero_real ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8164_real__sqrt__sum__squares__ge1,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8165_real__sqrt__sum__squares__ge2,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8166_real__sqrt__sum__squares__triangle__ineq,axiom,
% 9.10/5.30      ! [A: real,C: real,B2: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8167_sqrt__ge__absD,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ Y3 ) )
% 9.10/5.30       => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8168_cos__45,axiom,
% 9.10/5.30      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8169_sin__45,axiom,
% 9.10/5.30      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8170_pochhammer__absorb__comp,axiom,
% 9.10/5.30      ! [R: complex,K: nat] :
% 9.10/5.30        ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 9.10/5.30        = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8171_pochhammer__absorb__comp,axiom,
% 9.10/5.30      ! [R: code_integer,K: nat] :
% 9.10/5.30        ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 9.10/5.30        = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8172_pochhammer__absorb__comp,axiom,
% 9.10/5.30      ! [R: rat,K: nat] :
% 9.10/5.30        ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 9.10/5.30        = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8173_pochhammer__absorb__comp,axiom,
% 9.10/5.30      ! [R: int,K: nat] :
% 9.10/5.30        ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 9.10/5.30        = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8174_pochhammer__absorb__comp,axiom,
% 9.10/5.30      ! [R: real,K: nat] :
% 9.10/5.30        ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 9.10/5.30        = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8175_pochhammer__same,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ N )
% 9.10/5.30        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8176_pochhammer__same,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ N )
% 9.10/5.30        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8177_pochhammer__same,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ N )
% 9.10/5.30        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8178_pochhammer__same,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ N )
% 9.10/5.30        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8179_pochhammer__same,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ N )
% 9.10/5.30        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8180_choose__two,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.30        = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8181_real__less__lsqrt,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ( ord_less_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.30           => ( ord_less_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8182_sqrt__sum__squares__le__sum,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8183_tan__30,axiom,
% 9.10/5.30      ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8184_sqrt__even__pow2,axiom,
% 9.10/5.30      ! [N: nat] :
% 9.10/5.30        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.30       => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.30          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8185_real__sqrt__ge__abs1,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8186_real__sqrt__ge__abs2,axiom,
% 9.10/5.30      ! [Y3: real,X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8187_sqrt__sum__squares__le__sum__abs,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8188_ln__sqrt,axiom,
% 9.10/5.30      ! [X3: real] :
% 9.10/5.30        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ln_ln_real @ ( sqrt @ X3 ) )
% 9.10/5.30          = ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8189_cos__30,axiom,
% 9.10/5.30      ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8190_sin__60,axiom,
% 9.10/5.30      ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 9.10/5.30      = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8191_complex__norm,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( real_V1022390504157884413omplex @ ( complex2 @ X3 @ Y3 ) )
% 9.10/5.30        = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8192_pochhammer__minus,axiom,
% 9.10/5.30      ! [B2: complex,K: nat] :
% 9.10/5.30        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K )
% 9.10/5.30        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8193_pochhammer__minus,axiom,
% 9.10/5.30      ! [B2: code_integer,K: nat] :
% 9.10/5.30        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K )
% 9.10/5.30        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8194_pochhammer__minus,axiom,
% 9.10/5.30      ! [B2: rat,K: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K )
% 9.10/5.30        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8195_pochhammer__minus,axiom,
% 9.10/5.30      ! [B2: int,K: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K )
% 9.10/5.30        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8196_pochhammer__minus,axiom,
% 9.10/5.30      ! [B2: real,K: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K )
% 9.10/5.30        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8197_pochhammer__minus_H,axiom,
% 9.10/5.30      ! [B2: complex,K: nat] :
% 9.10/5.30        ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B2 @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 9.10/5.30        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B2 ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8198_pochhammer__minus_H,axiom,
% 9.10/5.30      ! [B2: code_integer,K: nat] :
% 9.10/5.30        ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B2 @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 9.10/5.30        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B2 ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8199_pochhammer__minus_H,axiom,
% 9.10/5.30      ! [B2: rat,K: nat] :
% 9.10/5.30        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B2 @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 9.10/5.30        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B2 ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8200_pochhammer__minus_H,axiom,
% 9.10/5.30      ! [B2: int,K: nat] :
% 9.10/5.30        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B2 @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 9.10/5.30        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B2 ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8201_pochhammer__minus_H,axiom,
% 9.10/5.30      ! [B2: real,K: nat] :
% 9.10/5.30        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 9.10/5.30        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B2 ) @ K ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8202_finite__psubset__def,axiom,
% 9.10/5.30      ( finite_psubset_nat
% 9.10/5.30      = ( collec6662362479098859352et_nat
% 9.10/5.30        @ ( produc6247414631856714078_nat_o
% 9.10/5.30          @ ^ [A6: set_nat,B6: set_nat] :
% 9.10/5.30              ( ( ord_less_set_nat @ A6 @ B6 )
% 9.10/5.30              & ( finite_finite_nat @ B6 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8203_finite__psubset__def,axiom,
% 9.10/5.30      ( finite_psubset_int
% 9.10/5.30      = ( collec957716948307931664et_int
% 9.10/5.30        @ ( produc4109468873575309990_int_o
% 9.10/5.30          @ ^ [A6: set_int,B6: set_int] :
% 9.10/5.30              ( ( ord_less_set_int @ A6 @ B6 )
% 9.10/5.30              & ( finite_finite_int @ B6 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8204_finite__psubset__def,axiom,
% 9.10/5.30      ( finite8643634255014194347omplex
% 9.10/5.30      = ( collec5108298041176329748omplex
% 9.10/5.30        @ ( produc3914248068834153634plex_o
% 9.10/5.30          @ ^ [A6: set_complex,B6: set_complex] :
% 9.10/5.30              ( ( ord_less_set_complex @ A6 @ B6 )
% 9.10/5.30              & ( finite3207457112153483333omplex @ B6 ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8205_arsinh__real__aux,axiom,
% 9.10/5.30      ! [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 ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8206_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 9.10/5.30      ! [X3: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8207_real__sqrt__power__even,axiom,
% 9.10/5.30      ! [N: nat,X3: real] :
% 9.10/5.30        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30         => ( ( power_power_real @ ( sqrt @ X3 ) @ N )
% 9.10/5.30            = ( power_power_real @ X3 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8208_arith__geo__mean__sqrt,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] :
% 9.10/5.30        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.30       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.30         => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X3 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.30  
% 9.10/5.30  thf(fact_8209_cos__x__y__le__one,axiom,
% 9.10/5.30      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8210_real__sqrt__sum__squares__less,axiom,
% 9.10/5.31      ! [X3: real,U: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.31       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.31         => ( 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 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8211_cos__arctan,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( cos_real @ ( arctan @ X3 ) )
% 9.10/5.31        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8212_sin__arctan,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( sin_real @ ( arctan @ X3 ) )
% 9.10/5.31        = ( divide_divide_real @ X3 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8213_pochhammer__code,axiom,
% 9.10/5.31      ( comm_s2602460028002588243omplex
% 9.10/5.31      = ( ^ [A5: complex,N2: nat] :
% 9.10/5.31            ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 9.10/5.31            @ ( set_fo1517530859248394432omplex
% 9.10/5.31              @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
% 9.10/5.31              @ zero_zero_nat
% 9.10/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.10/5.31              @ one_one_complex ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8214_pochhammer__code,axiom,
% 9.10/5.31      ( comm_s4028243227959126397er_rat
% 9.10/5.31      = ( ^ [A5: rat,N2: nat] :
% 9.10/5.31            ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 9.10/5.31            @ ( set_fo1949268297981939178at_rat
% 9.10/5.31              @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
% 9.10/5.31              @ zero_zero_nat
% 9.10/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.10/5.31              @ one_one_rat ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8215_pochhammer__code,axiom,
% 9.10/5.31      ( comm_s4660882817536571857er_int
% 9.10/5.31      = ( ^ [A5: int,N2: nat] :
% 9.10/5.31            ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 9.10/5.31            @ ( set_fo2581907887559384638at_int
% 9.10/5.31              @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
% 9.10/5.31              @ zero_zero_nat
% 9.10/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.10/5.31              @ one_one_int ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8216_pochhammer__code,axiom,
% 9.10/5.31      ( comm_s7457072308508201937r_real
% 9.10/5.31      = ( ^ [A5: real,N2: nat] :
% 9.10/5.31            ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 9.10/5.31            @ ( set_fo3111899725591712190t_real
% 9.10/5.31              @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
% 9.10/5.31              @ zero_zero_nat
% 9.10/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.10/5.31              @ one_one_real ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8217_pochhammer__code,axiom,
% 9.10/5.31      ( comm_s4663373288045622133er_nat
% 9.10/5.31      = ( ^ [A5: nat,N2: nat] :
% 9.10/5.31            ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 9.10/5.31            @ ( set_fo2584398358068434914at_nat
% 9.10/5.31              @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 9.10/5.31              @ zero_zero_nat
% 9.10/5.31              @ ( minus_minus_nat @ N2 @ one_one_nat )
% 9.10/5.31              @ one_one_nat ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8218_sqrt__sum__squares__half__less,axiom,
% 9.10/5.31      ! [X3: real,U: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.31       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.31         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.31             => ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8219_sin__cos__sqrt,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X3 ) )
% 9.10/5.31       => ( ( sin_real @ X3 )
% 9.10/5.31          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8220_arctan__half,axiom,
% 9.10/5.31      ( arctan
% 9.10/5.31      = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8221_fact__double,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.31        = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8222_fact__double,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.31        = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8223_fact__double,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.31        = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8224_central__binomial__lower__bound,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8225_arcosh__real__def,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.31       => ( ( arcosh_real @ X3 )
% 9.10/5.31          = ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8226_pochhammer__times__pochhammer__half,axiom,
% 9.10/5.31      ! [Z: complex,N: nat] :
% 9.10/5.31        ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 9.10/5.31        = ( groups6464643781859351333omplex
% 9.10/5.31          @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8227_pochhammer__times__pochhammer__half,axiom,
% 9.10/5.31      ! [Z: rat,N: nat] :
% 9.10/5.31        ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 9.10/5.31        = ( groups73079841787564623at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8228_pochhammer__times__pochhammer__half,axiom,
% 9.10/5.31      ! [Z: real,N: nat] :
% 9.10/5.31        ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 9.10/5.31        = ( groups129246275422532515t_real
% 9.10/5.31          @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8229_arsinh__real__def,axiom,
% 9.10/5.31      ( arsinh_real
% 9.10/5.31      = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8230_choose__odd__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] :
% 9.10/5.31                  ( if_complex
% 9.10/5.31                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.10/5.31                  @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) )
% 9.10/5.31                  @ zero_zero_complex )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8231_choose__odd__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] :
% 9.10/5.31                  ( if_rat
% 9.10/5.31                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.10/5.31                  @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) )
% 9.10/5.31                  @ zero_zero_rat )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8232_choose__odd__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] :
% 9.10/5.31                  ( if_int
% 9.10/5.31                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.10/5.31                  @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) )
% 9.10/5.31                  @ zero_zero_int )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8233_choose__odd__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] :
% 9.10/5.31                  ( if_real
% 9.10/5.31                  @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 )
% 9.10/5.31                  @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) )
% 9.10/5.31                  @ zero_zero_real )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8234_choose__even__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) @ zero_zero_complex )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8235_choose__even__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) @ zero_zero_rat )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8236_choose__even__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) @ zero_zero_int )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8237_choose__even__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) @ zero_zero_real )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8238_atMost__iff,axiom,
% 9.10/5.31      ! [I: real,K: real] :
% 9.10/5.31        ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
% 9.10/5.31        = ( ord_less_eq_real @ I @ K ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8239_atMost__iff,axiom,
% 9.10/5.31      ! [I: set_int,K: set_int] :
% 9.10/5.31        ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
% 9.10/5.31        = ( ord_less_eq_set_int @ I @ K ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8240_atMost__iff,axiom,
% 9.10/5.31      ! [I: rat,K: rat] :
% 9.10/5.31        ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
% 9.10/5.31        = ( ord_less_eq_rat @ I @ K ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8241_atMost__iff,axiom,
% 9.10/5.31      ! [I: num,K: num] :
% 9.10/5.31        ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
% 9.10/5.31        = ( ord_less_eq_num @ I @ K ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8242_atMost__iff,axiom,
% 9.10/5.31      ! [I: nat,K: nat] :
% 9.10/5.31        ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
% 9.10/5.31        = ( ord_less_eq_nat @ I @ K ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8243_atMost__iff,axiom,
% 9.10/5.31      ! [I: int,K: int] :
% 9.10/5.31        ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
% 9.10/5.31        = ( ord_less_eq_int @ I @ K ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8244_atMost__subset__iff,axiom,
% 9.10/5.31      ! [X3: set_int,Y3: set_int] :
% 9.10/5.31        ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X3 ) @ ( set_or58775011639299419et_int @ Y3 ) )
% 9.10/5.31        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8245_atMost__subset__iff,axiom,
% 9.10/5.31      ! [X3: rat,Y3: rat] :
% 9.10/5.31        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y3 ) )
% 9.10/5.31        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8246_atMost__subset__iff,axiom,
% 9.10/5.31      ! [X3: num,Y3: num] :
% 9.10/5.31        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y3 ) )
% 9.10/5.31        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8247_atMost__subset__iff,axiom,
% 9.10/5.31      ! [X3: nat,Y3: nat] :
% 9.10/5.31        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y3 ) )
% 9.10/5.31        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8248_atMost__subset__iff,axiom,
% 9.10/5.31      ! [X3: int,Y3: int] :
% 9.10/5.31        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y3 ) )
% 9.10/5.31        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8249_Icc__subset__Iic__iff,axiom,
% 9.10/5.31      ! [L: set_int,H2: set_int,H3: set_int] :
% 9.10/5.31        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 9.10/5.31        = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.10/5.31          | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8250_Icc__subset__Iic__iff,axiom,
% 9.10/5.31      ! [L: rat,H2: rat,H3: rat] :
% 9.10/5.31        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 9.10/5.31        = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.10/5.31          | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8251_Icc__subset__Iic__iff,axiom,
% 9.10/5.31      ! [L: num,H2: num,H3: num] :
% 9.10/5.31        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 9.10/5.31        = ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.10/5.31          | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8252_Icc__subset__Iic__iff,axiom,
% 9.10/5.31      ! [L: nat,H2: nat,H3: nat] :
% 9.10/5.31        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 9.10/5.31        = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.10/5.31          | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8253_Icc__subset__Iic__iff,axiom,
% 9.10/5.31      ! [L: int,H2: int,H3: int] :
% 9.10/5.31        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 9.10/5.31        = ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.10/5.31          | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8254_Icc__subset__Iic__iff,axiom,
% 9.10/5.31      ! [L: real,H2: real,H3: real] :
% 9.10/5.31        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 9.10/5.31        = ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.10/5.31          | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8255_sum_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8256_sum_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8257_sum_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8258_sum_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8259_prod_OlessThan__Suc,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8260_prod_OlessThan__Suc,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8261_prod_OlessThan__Suc,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8262_prod_OlessThan__Suc,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8263_prod_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8264_prod_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8265_prod_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8266_prod_OatMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8267_prod_Ocl__ivl__Suc,axiom,
% 9.10/5.31      ! [N: nat,M: nat,G: nat > complex] :
% 9.10/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = one_one_complex ) )
% 9.10/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8268_prod_Ocl__ivl__Suc,axiom,
% 9.10/5.31      ! [N: nat,M: nat,G: nat > real] :
% 9.10/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = one_one_real ) )
% 9.10/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8269_prod_Ocl__ivl__Suc,axiom,
% 9.10/5.31      ! [N: nat,M: nat,G: nat > rat] :
% 9.10/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = one_one_rat ) )
% 9.10/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8270_prod_Ocl__ivl__Suc,axiom,
% 9.10/5.31      ! [N: nat,M: nat,G: nat > nat] :
% 9.10/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = one_one_nat ) )
% 9.10/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8271_prod_Ocl__ivl__Suc,axiom,
% 9.10/5.31      ! [N: nat,M: nat,G: nat > int] :
% 9.10/5.31        ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = one_one_int ) )
% 9.10/5.31        & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.10/5.31         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8272_prod_Odistrib,axiom,
% 9.10/5.31      ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [X: nat] : ( times_times_nat @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.31          @ A4 )
% 9.10/5.31        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A4 ) @ ( groups708209901874060359at_nat @ H2 @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8273_prod_Odistrib,axiom,
% 9.10/5.31      ! [G: nat > int,H2: nat > int,A4: set_nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [X: nat] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.31          @ A4 )
% 9.10/5.31        = ( times_times_int @ ( groups705719431365010083at_int @ G @ A4 ) @ ( groups705719431365010083at_int @ H2 @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8274_prod_Odistrib,axiom,
% 9.10/5.31      ! [G: int > int,H2: int > int,A4: set_int] :
% 9.10/5.31        ( ( groups1705073143266064639nt_int
% 9.10/5.31          @ ^ [X: int] : ( times_times_int @ ( G @ X ) @ ( H2 @ X ) )
% 9.10/5.31          @ A4 )
% 9.10/5.31        = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A4 ) @ ( groups1705073143266064639nt_int @ H2 @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8275_prod__power__distrib,axiom,
% 9.10/5.31      ! [F: nat > nat,A4: set_nat,N: nat] :
% 9.10/5.31        ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ N )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8276_prod__power__distrib,axiom,
% 9.10/5.31      ! [F: nat > int,A4: set_nat,N: nat] :
% 9.10/5.31        ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A4 ) @ N )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8277_prod__power__distrib,axiom,
% 9.10/5.31      ! [F: int > int,A4: set_int,N: nat] :
% 9.10/5.31        ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ N )
% 9.10/5.31        = ( groups1705073143266064639nt_int
% 9.10/5.31          @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8278_mod__prod__eq,axiom,
% 9.10/5.31      ! [F: nat > nat,A: nat,A4: set_nat] :
% 9.10/5.31        ( ( modulo_modulo_nat
% 9.10/5.31          @ ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 9.10/5.31            @ A4 )
% 9.10/5.31          @ A )
% 9.10/5.31        = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ A ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8279_mod__prod__eq,axiom,
% 9.10/5.31      ! [F: nat > int,A: int,A4: set_nat] :
% 9.10/5.31        ( ( modulo_modulo_int
% 9.10/5.31          @ ( groups705719431365010083at_int
% 9.10/5.31            @ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 9.10/5.31            @ A4 )
% 9.10/5.31          @ A )
% 9.10/5.31        = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A4 ) @ A ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8280_mod__prod__eq,axiom,
% 9.10/5.31      ! [F: int > int,A: int,A4: set_int] :
% 9.10/5.31        ( ( modulo_modulo_int
% 9.10/5.31          @ ( groups1705073143266064639nt_int
% 9.10/5.31            @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 9.10/5.31            @ A4 )
% 9.10/5.31          @ A )
% 9.10/5.31        = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ A ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8281_prod_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_real @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups129246275422532515t_real
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8282_prod_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_rat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups73079841787564623at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8283_prod_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_nat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8284_prod_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_int @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups705719431365010083at_int
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8285_atMost__def,axiom,
% 9.10/5.31      ( set_ord_atMost_real
% 9.10/5.31      = ( ^ [U2: real] :
% 9.10/5.31            ( collect_real
% 9.10/5.31            @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8286_atMost__def,axiom,
% 9.10/5.31      ( set_or58775011639299419et_int
% 9.10/5.31      = ( ^ [U2: set_int] :
% 9.10/5.31            ( collect_set_int
% 9.10/5.31            @ ^ [X: set_int] : ( ord_less_eq_set_int @ X @ U2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8287_atMost__def,axiom,
% 9.10/5.31      ( set_ord_atMost_rat
% 9.10/5.31      = ( ^ [U2: rat] :
% 9.10/5.31            ( collect_rat
% 9.10/5.31            @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8288_atMost__def,axiom,
% 9.10/5.31      ( set_ord_atMost_num
% 9.10/5.31      = ( ^ [U2: num] :
% 9.10/5.31            ( collect_num
% 9.10/5.31            @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8289_atMost__def,axiom,
% 9.10/5.31      ( set_ord_atMost_nat
% 9.10/5.31      = ( ^ [U2: nat] :
% 9.10/5.31            ( collect_nat
% 9.10/5.31            @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8290_atMost__def,axiom,
% 9.10/5.31      ( set_ord_atMost_int
% 9.10/5.31      = ( ^ [U2: int] :
% 9.10/5.31            ( collect_int
% 9.10/5.31            @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8291_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > real,G: nat > real] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( member_nat @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8292_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > real,G: real > real] :
% 9.10/5.31        ( ! [I3: real] :
% 9.10/5.31            ( ( member_real @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8293_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > real,G: int > real] :
% 9.10/5.31        ( ! [I3: int] :
% 9.10/5.31            ( ( member_int @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8294_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 9.10/5.31        ( ! [I3: complex] :
% 9.10/5.31            ( ( member_complex @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8295_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( member_nat @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8296_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > rat,G: real > rat] :
% 9.10/5.31        ( ! [I3: real] :
% 9.10/5.31            ( ( member_real @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8297_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > rat,G: int > rat] :
% 9.10/5.31        ( ! [I3: int] :
% 9.10/5.31            ( ( member_int @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8298_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 9.10/5.31        ( ! [I3: complex] :
% 9.10/5.31            ( ( member_complex @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8299_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > nat,G: real > nat] :
% 9.10/5.31        ( ! [I3: real] :
% 9.10/5.31            ( ( member_real @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8300_prod__mono,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > nat,G: int > nat] :
% 9.10/5.31        ( ! [I3: int] :
% 9.10/5.31            ( ( member_int @ I3 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 9.10/5.31              & ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8301_prod__nonneg,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > nat] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8302_prod__nonneg,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > int] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8303_prod__nonneg,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > int] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8304_prod__pos,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > nat] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8305_prod__pos,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > int] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8306_prod__pos,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > int] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_int @ zero_zero_int @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8307_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > real] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8308_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > real] :
% 9.10/5.31        ( ! [X5: real] :
% 9.10/5.31            ( ( member_real @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8309_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > real] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8310_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > real] :
% 9.10/5.31        ( ! [X5: complex] :
% 9.10/5.31            ( ( member_complex @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8311_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > rat] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8312_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > rat] :
% 9.10/5.31        ( ! [X5: real] :
% 9.10/5.31            ( ( member_real @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8313_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > rat] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8314_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > rat] :
% 9.10/5.31        ( ! [X5: complex] :
% 9.10/5.31            ( ( member_complex @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8315_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > nat] :
% 9.10/5.31        ( ! [X5: real] :
% 9.10/5.31            ( ( member_real @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8316_prod__ge__1,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > nat] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ord_less_eq_nat @ one_one_nat @ ( F @ X5 ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8317_prod_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( times_times_real @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups129246275422532515t_real
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8318_prod_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( times_times_rat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups73079841787564623at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8319_prod_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( times_times_nat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8320_prod_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( times_times_int @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups705719431365010083at_int
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8321_not__Iic__le__Icc,axiom,
% 9.10/5.31      ! [H2: int,L3: int,H3: int] :
% 9.10/5.31        ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H2 ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8322_not__Iic__le__Icc,axiom,
% 9.10/5.31      ! [H2: real,L3: real,H3: real] :
% 9.10/5.31        ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H2 ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8323_power__sum,axiom,
% 9.10/5.31      ! [C: real,F: nat > nat,A4: set_nat] :
% 9.10/5.31        ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 9.10/5.31        = ( groups129246275422532515t_real
% 9.10/5.31          @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8324_power__sum,axiom,
% 9.10/5.31      ! [C: complex,F: nat > nat,A4: set_nat] :
% 9.10/5.31        ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 9.10/5.31        = ( groups6464643781859351333omplex
% 9.10/5.31          @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8325_power__sum,axiom,
% 9.10/5.31      ! [C: nat,F: nat > nat,A4: set_nat] :
% 9.10/5.31        ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8326_power__sum,axiom,
% 9.10/5.31      ! [C: int,F: nat > nat,A4: set_nat] :
% 9.10/5.31        ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8327_power__sum,axiom,
% 9.10/5.31      ! [C: int,F: int > nat,A4: set_int] :
% 9.10/5.31        ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
% 9.10/5.31        = ( groups1705073143266064639nt_int
% 9.10/5.31          @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
% 9.10/5.31          @ A4 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8328_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 9.10/5.31      ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8329_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 9.10/5.31      ! [G: nat > int,M: nat,K: nat,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8330_finite__nat__iff__bounded__le,axiom,
% 9.10/5.31      ( finite_finite_nat
% 9.10/5.31      = ( ^ [S4: set_nat] :
% 9.10/5.31          ? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8331_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > real] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ one_one_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8332_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > real] :
% 9.10/5.31        ( ! [X5: real] :
% 9.10/5.31            ( ( member_real @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ one_one_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8333_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > real] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ one_one_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8334_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > real] :
% 9.10/5.31        ( ! [X5: complex] :
% 9.10/5.31            ( ( member_complex @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
% 9.10/5.31       => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ one_one_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8335_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > rat] :
% 9.10/5.31        ( ! [X5: nat] :
% 9.10/5.31            ( ( member_nat @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8336_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > rat] :
% 9.10/5.31        ( ! [X5: real] :
% 9.10/5.31            ( ( member_real @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8337_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > rat] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8338_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > rat] :
% 9.10/5.31        ( ! [X5: complex] :
% 9.10/5.31            ( ( member_complex @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
% 9.10/5.31       => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ one_one_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8339_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > nat] :
% 9.10/5.31        ( ! [X5: real] :
% 9.10/5.31            ( ( member_real @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ one_one_nat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8340_prod__le__1,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > nat] :
% 9.10/5.31        ( ! [X5: int] :
% 9.10/5.31            ( ( member_int @ X5 @ A4 )
% 9.10/5.31           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) )
% 9.10/5.31              & ( ord_less_eq_nat @ ( F @ X5 ) @ one_one_nat ) ) )
% 9.10/5.31       => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ one_one_nat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8341_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.10/5.31        ( ( R2 @ one_one_complex @ one_one_complex )
% 9.10/5.31       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_nat @ S2 )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8342_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 9.10/5.31        ( ( R2 @ one_one_complex @ one_one_complex )
% 9.10/5.31       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_int @ S2 )
% 9.10/5.31           => ( ! [X5: int] :
% 9.10/5.31                  ( ( member_int @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8343_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 9.10/5.31        ( ( R2 @ one_one_complex @ one_one_complex )
% 9.10/5.31       => ( ! [X15: complex,Y15: complex,X22: complex,Y22: complex] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite3207457112153483333omplex @ S2 )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8344_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
% 9.10/5.31        ( ( R2 @ one_one_real @ one_one_real )
% 9.10/5.31       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_nat @ S2 )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8345_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 9.10/5.31        ( ( R2 @ one_one_real @ one_one_real )
% 9.10/5.31       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_int @ S2 )
% 9.10/5.31           => ( ! [X5: int] :
% 9.10/5.31                  ( ( member_int @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8346_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.10/5.31        ( ( R2 @ one_one_real @ one_one_real )
% 9.10/5.31       => ( ! [X15: real,Y15: real,X22: real,Y22: real] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite3207457112153483333omplex @ S2 )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8347_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 9.10/5.31        ( ( R2 @ one_one_rat @ one_one_rat )
% 9.10/5.31       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_nat @ S2 )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8348_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 9.10/5.31        ( ( R2 @ one_one_rat @ one_one_rat )
% 9.10/5.31       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_int @ S2 )
% 9.10/5.31           => ( ! [X5: int] :
% 9.10/5.31                  ( ( member_int @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8349_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.10/5.31        ( ( R2 @ one_one_rat @ one_one_rat )
% 9.10/5.31       => ( ! [X15: rat,Y15: rat,X22: rat,Y22: rat] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite3207457112153483333omplex @ S2 )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8350_prod_Orelated,axiom,
% 9.10/5.31      ! [R2: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 9.10/5.31        ( ( R2 @ one_one_nat @ one_one_nat )
% 9.10/5.31       => ( ! [X15: nat,Y15: nat,X22: nat,Y22: nat] :
% 9.10/5.31              ( ( ( R2 @ X15 @ X22 )
% 9.10/5.31                & ( R2 @ Y15 @ Y22 ) )
% 9.10/5.31             => ( R2 @ ( times_times_nat @ X15 @ Y15 ) @ ( times_times_nat @ X22 @ Y22 ) ) )
% 9.10/5.31         => ( ( finite_finite_int @ S2 )
% 9.10/5.31           => ( ! [X5: int] :
% 9.10/5.31                  ( ( member_int @ X5 @ S2 )
% 9.10/5.31                 => ( R2 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
% 9.10/5.31             => ( R2 @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8351_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8352_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8353_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > code_integer] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8354_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > code_integer] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8355_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > nat] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8356_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > code_integer] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8357_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > nat] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8358_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > int] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8359_prod__dvd__prod__subset,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > int] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8360_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_real,A4: set_real,F: real > nat,G: real > nat] :
% 9.10/5.31        ( ( finite_finite_real @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: real] :
% 9.10/5.31                ( ( member_real @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8361_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: complex] :
% 9.10/5.31                ( ( member_complex @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8362_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_real,A4: set_real,F: real > int,G: real > int] :
% 9.10/5.31        ( ( finite_finite_real @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: real] :
% 9.10/5.31                ( ( member_real @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8363_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: complex] :
% 9.10/5.31                ( ( member_complex @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8364_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_real,A4: set_real,F: real > code_integer,G: real > code_integer] :
% 9.10/5.31        ( ( finite_finite_real @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: real] :
% 9.10/5.31                ( ( member_real @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A4 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8365_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: nat] :
% 9.10/5.31                ( ( member_nat @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8366_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: complex] :
% 9.10/5.31                ( ( member_complex @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8367_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > nat,G: int > nat] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: int] :
% 9.10/5.31                ( ( member_int @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8368_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > code_integer,G: int > code_integer] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: int] :
% 9.10/5.31                ( ( member_int @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8369_prod__dvd__prod__subset2,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > nat,G: nat > nat] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( ! [A3: nat] :
% 9.10/5.31                ( ( member_nat @ A3 @ A4 )
% 9.10/5.31               => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 9.10/5.31           => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8370_prod_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups129246275422532515t_real
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8371_prod_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups73079841787564623at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8372_prod_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8373_prod_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8374_prod_Onat__diff__reindex,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.31          @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.31        = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8375_prod_Onat__diff__reindex,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.10/5.31          @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.31        = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8376_Iic__subset__Iio__iff,axiom,
% 9.10/5.31      ! [A: rat,B2: rat] :
% 9.10/5.31        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B2 ) )
% 9.10/5.31        = ( ord_less_rat @ A @ B2 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8377_Iic__subset__Iio__iff,axiom,
% 9.10/5.31      ! [A: num,B2: num] :
% 9.10/5.31        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B2 ) )
% 9.10/5.31        = ( ord_less_num @ A @ B2 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8378_Iic__subset__Iio__iff,axiom,
% 9.10/5.31      ! [A: nat,B2: nat] :
% 9.10/5.31        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B2 ) )
% 9.10/5.31        = ( ord_less_nat @ A @ B2 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8379_Iic__subset__Iio__iff,axiom,
% 9.10/5.31      ! [A: int,B2: int] :
% 9.10/5.31        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B2 ) )
% 9.10/5.31        = ( ord_less_int @ A @ B2 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8380_Iic__subset__Iio__iff,axiom,
% 9.10/5.31      ! [A: real,B2: real] :
% 9.10/5.31        ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B2 ) )
% 9.10/5.31        = ( ord_less_real @ A @ B2 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8381_prod_OatLeastAtMost__rev,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat,M: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8382_prod_OatLeastAtMost__rev,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat,M: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8383_prod_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups6464643781859351333omplex
% 9.10/5.31              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ one_one_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups6464643781859351333omplex
% 9.10/5.31              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8384_prod_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups129246275422532515t_real
% 9.10/5.31              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ one_one_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups129246275422532515t_real
% 9.10/5.31              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8385_prod_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups73079841787564623at_rat
% 9.10/5.31              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ one_one_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups73079841787564623at_rat
% 9.10/5.31              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8386_prod_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups708209901874060359at_nat
% 9.10/5.31              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ one_one_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups708209901874060359at_nat
% 9.10/5.31              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8387_prod_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups705719431365010083at_int
% 9.10/5.31              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ one_one_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups705719431365010083at_int
% 9.10/5.31              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8388_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_real,I: real,F: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ( member_real @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: real] :
% 9.10/5.31                  ( ( member_real @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8389_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_nat,I: nat,F: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ I5 )
% 9.10/5.31       => ( ( member_nat @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: nat] :
% 9.10/5.31                  ( ( member_nat @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8390_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_int,I: int,F: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ I5 )
% 9.10/5.31       => ( ( member_int @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: int] :
% 9.10/5.31                  ( ( member_int @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8391_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_complex,I: complex,F: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ( member_complex @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: complex] :
% 9.10/5.31                  ( ( member_complex @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8392_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_real,I: real,F: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ( member_real @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: real] :
% 9.10/5.31                  ( ( member_real @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8393_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_nat,I: nat,F: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ I5 )
% 9.10/5.31       => ( ( member_nat @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: nat] :
% 9.10/5.31                  ( ( member_nat @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8394_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_int,I: int,F: int > rat] :
% 9.10/5.31        ( ( finite_finite_int @ I5 )
% 9.10/5.31       => ( ( member_int @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: int] :
% 9.10/5.31                  ( ( member_int @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8395_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_complex,I: complex,F: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ( member_complex @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: complex] :
% 9.10/5.31                  ( ( member_complex @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8396_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_real,I: real,F: real > int] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ( member_real @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: real] :
% 9.10/5.31                  ( ( member_real @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8397_less__1__prod2,axiom,
% 9.10/5.31      ! [I5: set_complex,I: complex,F: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ( member_complex @ I @ I5 )
% 9.10/5.31         => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 9.10/5.31           => ( ! [I3: complex] :
% 9.10/5.31                  ( ( member_complex @ I3 @ I5 )
% 9.10/5.31                 => ( ord_less_eq_int @ one_one_int @ ( F @ I3 ) ) )
% 9.10/5.31             => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8398_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_complex,F: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_complex )
% 9.10/5.31         => ( ! [I3: complex] :
% 9.10/5.31                ( ( member_complex @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8399_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_nat,F: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_nat )
% 9.10/5.31         => ( ! [I3: nat] :
% 9.10/5.31                ( ( member_nat @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8400_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_int,F: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_int )
% 9.10/5.31         => ( ! [I3: int] :
% 9.10/5.31                ( ( member_int @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8401_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_real,F: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_real )
% 9.10/5.31         => ( ! [I3: real] :
% 9.10/5.31                ( ( member_real @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8402_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_complex,F: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_complex )
% 9.10/5.31         => ( ! [I3: complex] :
% 9.10/5.31                ( ( member_complex @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8403_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_nat,F: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_nat )
% 9.10/5.31         => ( ! [I3: nat] :
% 9.10/5.31                ( ( member_nat @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8404_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_int,F: int > rat] :
% 9.10/5.31        ( ( finite_finite_int @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_int )
% 9.10/5.31         => ( ! [I3: int] :
% 9.10/5.31                ( ( member_int @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8405_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_real,F: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_real )
% 9.10/5.31         => ( ! [I3: real] :
% 9.10/5.31                ( ( member_real @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8406_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_complex,F: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_complex )
% 9.10/5.31         => ( ! [I3: complex] :
% 9.10/5.31                ( ( member_complex @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8407_less__1__prod,axiom,
% 9.10/5.31      ! [I5: set_real,F: real > int] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ( I5 != bot_bot_set_real )
% 9.10/5.31         => ( ! [I3: real] :
% 9.10/5.31                ( ( member_real @ I3 @ I5 )
% 9.10/5.31               => ( ord_less_int @ one_one_int @ ( F @ I3 ) ) )
% 9.10/5.31           => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8408_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,G: nat > real] :
% 9.10/5.31        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.31       => ( ( finite_finite_nat @ A4 )
% 9.10/5.31         => ( ( groups129246275422532515t_real @ G @ A4 )
% 9.10/5.31            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8409_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,G: complex > real] :
% 9.10/5.31        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.31       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31         => ( ( groups766887009212190081x_real @ G @ A4 )
% 9.10/5.31            = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8410_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,G: nat > rat] :
% 9.10/5.31        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.31       => ( ( finite_finite_nat @ A4 )
% 9.10/5.31         => ( ( groups73079841787564623at_rat @ G @ A4 )
% 9.10/5.31            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8411_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,G: complex > rat] :
% 9.10/5.31        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.31       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31         => ( ( groups225925009352817453ex_rat @ G @ A4 )
% 9.10/5.31            = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8412_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,G: complex > nat] :
% 9.10/5.31        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.31       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31         => ( ( groups861055069439313189ex_nat @ G @ A4 )
% 9.10/5.31            = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8413_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,G: complex > int] :
% 9.10/5.31        ( ( ord_le211207098394363844omplex @ B5 @ A4 )
% 9.10/5.31       => ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31         => ( ( groups858564598930262913ex_int @ G @ A4 )
% 9.10/5.31            = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8414_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,G: int > real] :
% 9.10/5.31        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.31       => ( ( finite_finite_int @ A4 )
% 9.10/5.31         => ( ( groups2316167850115554303t_real @ G @ A4 )
% 9.10/5.31            = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8415_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,G: int > rat] :
% 9.10/5.31        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.31       => ( ( finite_finite_int @ A4 )
% 9.10/5.31         => ( ( groups1072433553688619179nt_rat @ G @ A4 )
% 9.10/5.31            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8416_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,G: int > nat] :
% 9.10/5.31        ( ( ord_less_eq_set_int @ B5 @ A4 )
% 9.10/5.31       => ( ( finite_finite_int @ A4 )
% 9.10/5.31         => ( ( groups1707563613775114915nt_nat @ G @ A4 )
% 9.10/5.31            = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8417_prod_Osubset__diff,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,G: nat > nat] :
% 9.10/5.31        ( ( ord_less_eq_set_nat @ B5 @ A4 )
% 9.10/5.31       => ( ( finite_finite_nat @ A4 )
% 9.10/5.31         => ( ( groups708209901874060359at_nat @ G @ A4 )
% 9.10/5.31            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8418_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_complex ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_complex ) )
% 9.10/5.31               => ( ( ( groups713298508707869441omplex @ G @ A4 )
% 9.10/5.31                    = ( groups713298508707869441omplex @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups713298508707869441omplex @ G @ C4 )
% 9.10/5.31                    = ( groups713298508707869441omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8419_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.31        ( ( finite_finite_nat @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_complex ) )
% 9.10/5.31             => ( ! [B3: nat] :
% 9.10/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_complex ) )
% 9.10/5.31               => ( ( ( groups6464643781859351333omplex @ G @ A4 )
% 9.10/5.31                    = ( groups6464643781859351333omplex @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups6464643781859351333omplex @ G @ C4 )
% 9.10/5.31                    = ( groups6464643781859351333omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8420_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_complex ) )
% 9.10/5.31             => ( ! [B3: complex] :
% 9.10/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_complex ) )
% 9.10/5.31               => ( ( ( groups3708469109370488835omplex @ G @ A4 )
% 9.10/5.31                    = ( groups3708469109370488835omplex @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups3708469109370488835omplex @ G @ C4 )
% 9.10/5.31                    = ( groups3708469109370488835omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8421_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_real ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_real ) )
% 9.10/5.31               => ( ( ( groups1681761925125756287l_real @ G @ A4 )
% 9.10/5.31                    = ( groups1681761925125756287l_real @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups1681761925125756287l_real @ G @ C4 )
% 9.10/5.31                    = ( groups1681761925125756287l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8422_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_real ) )
% 9.10/5.31             => ( ! [B3: nat] :
% 9.10/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_real ) )
% 9.10/5.31               => ( ( ( groups129246275422532515t_real @ G @ A4 )
% 9.10/5.31                    = ( groups129246275422532515t_real @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups129246275422532515t_real @ G @ C4 )
% 9.10/5.31                    = ( groups129246275422532515t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8423_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_real ) )
% 9.10/5.31             => ( ! [B3: complex] :
% 9.10/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_real ) )
% 9.10/5.31               => ( ( ( groups766887009212190081x_real @ G @ A4 )
% 9.10/5.31                    = ( groups766887009212190081x_real @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups766887009212190081x_real @ G @ C4 )
% 9.10/5.31                    = ( groups766887009212190081x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8424_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_rat ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_rat ) )
% 9.10/5.31               => ( ( ( groups4061424788464935467al_rat @ G @ A4 )
% 9.10/5.31                    = ( groups4061424788464935467al_rat @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups4061424788464935467al_rat @ G @ C4 )
% 9.10/5.31                    = ( groups4061424788464935467al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8425_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_rat ) )
% 9.10/5.31             => ( ! [B3: nat] :
% 9.10/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_rat ) )
% 9.10/5.31               => ( ( ( groups73079841787564623at_rat @ G @ A4 )
% 9.10/5.31                    = ( groups73079841787564623at_rat @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups73079841787564623at_rat @ G @ C4 )
% 9.10/5.31                    = ( groups73079841787564623at_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8426_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_rat ) )
% 9.10/5.31             => ( ! [B3: complex] :
% 9.10/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_rat ) )
% 9.10/5.31               => ( ( ( groups225925009352817453ex_rat @ G @ A4 )
% 9.10/5.31                    = ( groups225925009352817453ex_rat @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups225925009352817453ex_rat @ G @ C4 )
% 9.10/5.31                    = ( groups225925009352817453ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8427_prod_Osame__carrier,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_nat ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_nat ) )
% 9.10/5.31               => ( ( ( groups4696554848551431203al_nat @ G @ A4 )
% 9.10/5.31                    = ( groups4696554848551431203al_nat @ H2 @ B5 ) )
% 9.10/5.31                  = ( ( groups4696554848551431203al_nat @ G @ C4 )
% 9.10/5.31                    = ( groups4696554848551431203al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8428_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_complex ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_complex ) )
% 9.10/5.31               => ( ( ( groups713298508707869441omplex @ G @ C4 )
% 9.10/5.31                    = ( groups713298508707869441omplex @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups713298508707869441omplex @ G @ A4 )
% 9.10/5.31                    = ( groups713298508707869441omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8429_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.31        ( ( finite_finite_nat @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_complex ) )
% 9.10/5.31             => ( ! [B3: nat] :
% 9.10/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_complex ) )
% 9.10/5.31               => ( ( ( groups6464643781859351333omplex @ G @ C4 )
% 9.10/5.31                    = ( groups6464643781859351333omplex @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups6464643781859351333omplex @ G @ A4 )
% 9.10/5.31                    = ( groups6464643781859351333omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8430_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > complex,H2: complex > complex] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_complex ) )
% 9.10/5.31             => ( ! [B3: complex] :
% 9.10/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_complex ) )
% 9.10/5.31               => ( ( ( groups3708469109370488835omplex @ G @ C4 )
% 9.10/5.31                    = ( groups3708469109370488835omplex @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups3708469109370488835omplex @ G @ A4 )
% 9.10/5.31                    = ( groups3708469109370488835omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8431_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > real,H2: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_real ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_real ) )
% 9.10/5.31               => ( ( ( groups1681761925125756287l_real @ G @ C4 )
% 9.10/5.31                    = ( groups1681761925125756287l_real @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups1681761925125756287l_real @ G @ A4 )
% 9.10/5.31                    = ( groups1681761925125756287l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8432_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > real,H2: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_real ) )
% 9.10/5.31             => ( ! [B3: nat] :
% 9.10/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_real ) )
% 9.10/5.31               => ( ( ( groups129246275422532515t_real @ G @ C4 )
% 9.10/5.31                    = ( groups129246275422532515t_real @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups129246275422532515t_real @ G @ A4 )
% 9.10/5.31                    = ( groups129246275422532515t_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8433_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_real ) )
% 9.10/5.31             => ( ! [B3: complex] :
% 9.10/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_real ) )
% 9.10/5.31               => ( ( ( groups766887009212190081x_real @ G @ C4 )
% 9.10/5.31                    = ( groups766887009212190081x_real @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups766887009212190081x_real @ G @ A4 )
% 9.10/5.31                    = ( groups766887009212190081x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8434_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_rat ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_rat ) )
% 9.10/5.31               => ( ( ( groups4061424788464935467al_rat @ G @ C4 )
% 9.10/5.31                    = ( groups4061424788464935467al_rat @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups4061424788464935467al_rat @ G @ A4 )
% 9.10/5.31                    = ( groups4061424788464935467al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8435_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_nat,A4: set_nat,B5: set_nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_rat ) )
% 9.10/5.31             => ( ! [B3: nat] :
% 9.10/5.31                    ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_rat ) )
% 9.10/5.31               => ( ( ( groups73079841787564623at_rat @ G @ C4 )
% 9.10/5.31                    = ( groups73079841787564623at_rat @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups73079841787564623at_rat @ G @ A4 )
% 9.10/5.31                    = ( groups73079841787564623at_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8436_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_complex,A4: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ C4 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_rat ) )
% 9.10/5.31             => ( ! [B3: complex] :
% 9.10/5.31                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_rat ) )
% 9.10/5.31               => ( ( ( groups225925009352817453ex_rat @ G @ C4 )
% 9.10/5.31                    = ( groups225925009352817453ex_rat @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups225925009352817453ex_rat @ G @ A4 )
% 9.10/5.31                    = ( groups225925009352817453ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8437_prod_Osame__carrierI,axiom,
% 9.10/5.31      ! [C4: set_real,A4: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 9.10/5.31        ( ( finite_finite_real @ C4 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ C4 )
% 9.10/5.31         => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
% 9.10/5.31                 => ( ( G @ A3 )
% 9.10/5.31                    = one_one_nat ) )
% 9.10/5.31             => ( ! [B3: real] :
% 9.10/5.31                    ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 9.10/5.31                   => ( ( H2 @ B3 )
% 9.10/5.31                      = one_one_nat ) )
% 9.10/5.31               => ( ( ( groups4696554848551431203al_nat @ G @ C4 )
% 9.10/5.31                    = ( groups4696554848551431203al_nat @ H2 @ C4 ) )
% 9.10/5.31                 => ( ( groups4696554848551431203al_nat @ G @ A4 )
% 9.10/5.31                    = ( groups4696554848551431203al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8438_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ( groups6464643781859351333omplex @ G @ S2 )
% 9.10/5.31              = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8439_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > complex] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ( groups3708469109370488835omplex @ G @ S2 )
% 9.10/5.31              = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8440_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ( groups129246275422532515t_real @ G @ S2 )
% 9.10/5.31              = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8441_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ( groups766887009212190081x_real @ G @ S2 )
% 9.10/5.31              = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8442_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ( groups73079841787564623at_rat @ G @ S2 )
% 9.10/5.31              = ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8443_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ( groups225925009352817453ex_rat @ G @ S2 )
% 9.10/5.31              = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8444_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_nat ) )
% 9.10/5.31           => ( ( groups861055069439313189ex_nat @ G @ S2 )
% 9.10/5.31              = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8445_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_int ) )
% 9.10/5.31           => ( ( groups858564598930262913ex_int @ G @ S2 )
% 9.10/5.31              = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8446_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_int,S2: set_int,G: int > complex] :
% 9.10/5.31        ( ( finite_finite_int @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: int] :
% 9.10/5.31                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ( groups7440179247065528705omplex @ G @ S2 )
% 9.10/5.31              = ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8447_prod_Omono__neutral__left,axiom,
% 9.10/5.31      ! [T3: set_int,S2: set_int,G: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: int] :
% 9.10/5.31                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ( groups2316167850115554303t_real @ G @ S2 )
% 9.10/5.31              = ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8448_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ( groups6464643781859351333omplex @ G @ T3 )
% 9.10/5.31              = ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8449_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > complex] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ( groups3708469109370488835omplex @ G @ T3 )
% 9.10/5.31              = ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8450_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ( groups129246275422532515t_real @ G @ T3 )
% 9.10/5.31              = ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8451_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ( groups766887009212190081x_real @ G @ T3 )
% 9.10/5.31              = ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8452_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ( groups73079841787564623at_rat @ G @ T3 )
% 9.10/5.31              = ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8453_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 9.10/5.31              = ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8454_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_nat ) )
% 9.10/5.31           => ( ( groups861055069439313189ex_nat @ G @ T3 )
% 9.10/5.31              = ( groups861055069439313189ex_nat @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8455_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_int ) )
% 9.10/5.31           => ( ( groups858564598930262913ex_int @ G @ T3 )
% 9.10/5.31              = ( groups858564598930262913ex_int @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8456_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_int,S2: set_int,G: int > complex] :
% 9.10/5.31        ( ( finite_finite_int @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: int] :
% 9.10/5.31                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ( groups7440179247065528705omplex @ G @ T3 )
% 9.10/5.31              = ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8457_prod_Omono__neutral__right,axiom,
% 9.10/5.31      ! [T3: set_int,S2: set_int,G: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: int] :
% 9.10/5.31                ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ( groups2316167850115554303t_real @ G @ T3 )
% 9.10/5.31              = ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8458_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups713298508707869441omplex @ G @ S2 )
% 9.10/5.31                = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8459_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups6464643781859351333omplex @ G @ S2 )
% 9.10/5.31                = ( groups6464643781859351333omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8460_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups3708469109370488835omplex @ G @ S2 )
% 9.10/5.31                = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8461_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups1681761925125756287l_real @ G @ S2 )
% 9.10/5.31                = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8462_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,H2: nat > real,G: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups129246275422532515t_real @ G @ S2 )
% 9.10/5.31                = ( groups129246275422532515t_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8463_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups766887009212190081x_real @ G @ S2 )
% 9.10/5.31                = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8464_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups4061424788464935467al_rat @ G @ S2 )
% 9.10/5.31                = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8465_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups73079841787564623at_rat @ G @ S2 )
% 9.10/5.31                = ( groups73079841787564623at_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8466_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups225925009352817453ex_rat @ G @ S2 )
% 9.10/5.31                = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8467_prod_Omono__neutral__cong__left,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( H2 @ X5 )
% 9.10/5.31                  = one_one_nat ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups4696554848551431203al_nat @ G @ S2 )
% 9.10/5.31                = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8468_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups713298508707869441omplex @ G @ T3 )
% 9.10/5.31                = ( groups713298508707869441omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8469_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups6464643781859351333omplex @ G @ T3 )
% 9.10/5.31                = ( groups6464643781859351333omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8470_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > complex,H2: complex > complex] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_complex ) )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups3708469109370488835omplex @ G @ T3 )
% 9.10/5.31                = ( groups3708469109370488835omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8471_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups1681761925125756287l_real @ G @ T3 )
% 9.10/5.31                = ( groups1681761925125756287l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8472_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > real,H2: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups129246275422532515t_real @ G @ T3 )
% 9.10/5.31                = ( groups129246275422532515t_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8473_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_real ) )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups766887009212190081x_real @ G @ T3 )
% 9.10/5.31                = ( groups766887009212190081x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8474_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups4061424788464935467al_rat @ G @ T3 )
% 9.10/5.31                = ( groups4061424788464935467al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8475_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_nat,S2: set_nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: nat] :
% 9.10/5.31                ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ! [X5: nat] :
% 9.10/5.31                  ( ( member_nat @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups73079841787564623at_rat @ G @ T3 )
% 9.10/5.31                = ( groups73079841787564623at_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8476_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ T3 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: complex] :
% 9.10/5.31                ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_rat ) )
% 9.10/5.31           => ( ! [X5: complex] :
% 9.10/5.31                  ( ( member_complex @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 9.10/5.31                = ( groups225925009352817453ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8477_prod_Omono__neutral__cong__right,axiom,
% 9.10/5.31      ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 9.10/5.31        ( ( finite_finite_real @ T3 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 9.10/5.31         => ( ! [X5: real] :
% 9.10/5.31                ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 9.10/5.31               => ( ( G @ X5 )
% 9.10/5.31                  = one_one_nat ) )
% 9.10/5.31           => ( ! [X5: real] :
% 9.10/5.31                  ( ( member_real @ X5 @ S2 )
% 9.10/5.31                 => ( ( G @ X5 )
% 9.10/5.31                    = ( H2 @ X5 ) ) )
% 9.10/5.31             => ( ( groups4696554848551431203al_nat @ G @ T3 )
% 9.10/5.31                = ( groups4696554848551431203al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8478_prod_OatLeast0__atMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8479_prod_OatLeast0__atMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8480_prod_OatLeast0__atMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8481_prod_OatLeast0__atMost__Suc,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8482_prod_OatLeast__Suc__atMost,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8483_prod_OatLeast__Suc__atMost,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8484_prod_OatLeast__Suc__atMost,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8485_prod_OatLeast__Suc__atMost,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8486_prod_Onat__ivl__Suc_H,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.31       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8487_prod_Onat__ivl__Suc_H,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.31       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8488_prod_Onat__ivl__Suc_H,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.31       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8489_prod_Onat__ivl__Suc_H,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.31       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8490_prod_OlessThan__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_real @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups129246275422532515t_real
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8491_prod_OlessThan__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_rat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups73079841787564623at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8492_prod_OlessThan__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_nat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8493_prod_OlessThan__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( times_times_int @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups705719431365010083at_int
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8494_prod_OSuc__reindex__ivl,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_real @ ( G @ M )
% 9.10/5.31            @ ( groups129246275422532515t_real
% 9.10/5.31              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8495_prod_OSuc__reindex__ivl,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_rat @ ( G @ M )
% 9.10/5.31            @ ( groups73079841787564623at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8496_prod_OSuc__reindex__ivl,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_nat @ ( G @ M )
% 9.10/5.31            @ ( groups708209901874060359at_nat
% 9.10/5.31              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8497_prod_OSuc__reindex__ivl,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 9.10/5.31          = ( times_times_int @ ( G @ M )
% 9.10/5.31            @ ( groups705719431365010083at_int
% 9.10/5.31              @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31              @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8498_sum_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups2906978787729119204at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8499_sum_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups3539618377306564664at_int
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8500_sum_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups3542108847815614940at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8501_sum_OatMost__Suc__shift,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
% 9.10/5.31        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups6591440286371151544t_real
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8502_sum__telescope,axiom,
% 9.10/5.31      ! [F: nat > complex,I: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ I ) )
% 9.10/5.31        = ( minus_minus_complex @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8503_sum__telescope,axiom,
% 9.10/5.31      ! [F: nat > rat,I: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ I ) )
% 9.10/5.31        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8504_sum__telescope,axiom,
% 9.10/5.31      ! [F: nat > int,I: nat] :
% 9.10/5.31        ( ( groups3539618377306564664at_int
% 9.10/5.31          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ I ) )
% 9.10/5.31        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8505_sum__telescope,axiom,
% 9.10/5.31      ! [F: nat > real,I: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ ( F @ ( suc @ I4 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ I ) )
% 9.10/5.31        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8506_polyfun__eq__coeffs,axiom,
% 9.10/5.31      ! [C: nat > complex,N: nat,D: nat > complex] :
% 9.10/5.31        ( ( ! [X: complex] :
% 9.10/5.31              ( ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31              = ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) ) )
% 9.10/5.31        = ( ! [I4: nat] :
% 9.10/5.31              ( ( ord_less_eq_nat @ I4 @ N )
% 9.10/5.31             => ( ( C @ I4 )
% 9.10/5.31                = ( D @ I4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8507_polyfun__eq__coeffs,axiom,
% 9.10/5.31      ! [C: nat > real,N: nat,D: nat > real] :
% 9.10/5.31        ( ( ! [X: real] :
% 9.10/5.31              ( ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31              = ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) ) )
% 9.10/5.31        = ( ! [I4: nat] :
% 9.10/5.31              ( ( ord_less_eq_nat @ I4 @ N )
% 9.10/5.31             => ( ( C @ I4 )
% 9.10/5.31                = ( D @ I4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8508_bounded__imp__summable,axiom,
% 9.10/5.31      ! [A: nat > int,B5: int] :
% 9.10/5.31        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
% 9.10/5.31       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
% 9.10/5.31         => ( summable_int @ A ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8509_bounded__imp__summable,axiom,
% 9.10/5.31      ! [A: nat > nat,B5: nat] :
% 9.10/5.31        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
% 9.10/5.31       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
% 9.10/5.31         => ( summable_nat @ A ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8510_bounded__imp__summable,axiom,
% 9.10/5.31      ! [A: nat > real,B5: real] :
% 9.10/5.31        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.31       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B5 )
% 9.10/5.31         => ( summable_real @ A ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8511_prod__atLeastAtMost__code,axiom,
% 9.10/5.31      ! [F: nat > complex,A: nat,B2: nat] :
% 9.10/5.31        ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.31        = ( set_fo1517530859248394432omplex
% 9.10/5.31          @ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
% 9.10/5.31          @ A
% 9.10/5.31          @ B2
% 9.10/5.31          @ one_one_complex ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8512_prod__atLeastAtMost__code,axiom,
% 9.10/5.31      ! [F: nat > real,A: nat,B2: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.31        = ( set_fo3111899725591712190t_real
% 9.10/5.31          @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
% 9.10/5.31          @ A
% 9.10/5.31          @ B2
% 9.10/5.31          @ one_one_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8513_prod__atLeastAtMost__code,axiom,
% 9.10/5.31      ! [F: nat > rat,A: nat,B2: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.31        = ( set_fo1949268297981939178at_rat
% 9.10/5.31          @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
% 9.10/5.31          @ A
% 9.10/5.31          @ B2
% 9.10/5.31          @ one_one_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8514_prod__atLeastAtMost__code,axiom,
% 9.10/5.31      ! [F: nat > nat,A: nat,B2: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.31        = ( set_fo2584398358068434914at_nat
% 9.10/5.31          @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
% 9.10/5.31          @ A
% 9.10/5.31          @ B2
% 9.10/5.31          @ one_one_nat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8515_prod__atLeastAtMost__code,axiom,
% 9.10/5.31      ! [F: nat > int,A: nat,B2: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B2 ) )
% 9.10/5.31        = ( set_fo2581907887559384638at_int
% 9.10/5.31          @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
% 9.10/5.31          @ A
% 9.10/5.31          @ B2
% 9.10/5.31          @ one_one_int ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8516_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > real,G: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ! [I3: complex] :
% 9.10/5.31              ( ( member_complex @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_complex )
% 9.10/5.31           => ( ord_less_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8517_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > real,G: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ A4 )
% 9.10/5.31       => ( ! [I3: nat] :
% 9.10/5.31              ( ( member_nat @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_nat )
% 9.10/5.31           => ( ord_less_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8518_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > real,G: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ! [I3: int] :
% 9.10/5.31              ( ( member_int @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_int )
% 9.10/5.31           => ( ord_less_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8519_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > real,G: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ A4 )
% 9.10/5.31       => ( ! [I3: real] :
% 9.10/5.31              ( ( member_real @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_real )
% 9.10/5.31           => ( ord_less_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8520_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > rat,G: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ! [I3: complex] :
% 9.10/5.31              ( ( member_complex @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_complex )
% 9.10/5.31           => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8521_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > rat,G: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ A4 )
% 9.10/5.31       => ( ! [I3: nat] :
% 9.10/5.31              ( ( member_nat @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_nat )
% 9.10/5.31           => ( ord_less_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8522_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > rat,G: int > rat] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ! [I3: int] :
% 9.10/5.31              ( ( member_int @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_int )
% 9.10/5.31           => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8523_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_real,F: real > rat,G: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ A4 )
% 9.10/5.31       => ( ! [I3: real] :
% 9.10/5.31              ( ( member_real @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_real )
% 9.10/5.31           => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8524_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > nat,G: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ! [I3: complex] :
% 9.10/5.31              ( ( member_complex @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_complex )
% 9.10/5.31           => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8525_prod__mono__strict,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > nat,G: int > nat] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ! [I3: int] :
% 9.10/5.31              ( ( member_int @ I3 @ A4 )
% 9.10/5.31             => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
% 9.10/5.31                & ( ord_less_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
% 9.10/5.31         => ( ( A4 != bot_bot_set_int )
% 9.10/5.31           => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ A4 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8526_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > code_integer] :
% 9.10/5.31        ( ( finite_finite_nat @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3455450783089532116nteger @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: nat] :
% 9.10/5.31                ( ( member_nat @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8527_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > code_integer] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups3827104343326376752nteger @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: int] :
% 9.10/5.31                ( ( member_int @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8528_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > code_integer] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: complex] :
% 9.10/5.31                ( ( member_complex @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8529_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > nat] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: int] :
% 9.10/5.31                ( ( member_int @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8530_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: complex] :
% 9.10/5.31                ( ( member_complex @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8531_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: complex] :
% 9.10/5.31                ( ( member_complex @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8532_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > nat] :
% 9.10/5.31        ( ( finite_finite_nat @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups708209901874060359at_nat @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: nat] :
% 9.10/5.31                ( ( member_nat @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8533_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > int] :
% 9.10/5.31        ( ( finite_finite_nat @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups705719431365010083at_int @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: nat] :
% 9.10/5.31                ( ( member_nat @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8534_even__prod__iff,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > int] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups1705073143266064639nt_int @ F @ A4 ) )
% 9.10/5.31          = ( ? [X: int] :
% 9.10/5.31                ( ( member_int @ X @ A4 )
% 9.10/5.31                & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8535_sum__choose__lower,axiom,
% 9.10/5.31      ! [R: nat,N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8536_choose__rising__sum_I2_J,axiom,
% 9.10/5.31      ! [N: nat,M: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8537_choose__rising__sum_I1_J,axiom,
% 9.10/5.31      ! [N: nat,M: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8538_prod_Oub__add__nat,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > real,P4: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.31       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.31          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8539_prod_Oub__add__nat,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > rat,P4: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.31       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.31          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8540_prod_Oub__add__nat,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > nat,P4: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.31       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.31          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8541_prod_Oub__add__nat,axiom,
% 9.10/5.31      ! [M: nat,N: nat,G: nat > int,P4: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.10/5.31       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P4 ) ) )
% 9.10/5.31          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P4 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8542_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_real,Z: real > real,W: real > real] :
% 9.10/5.31        ( ! [I3: real] :
% 9.10/5.31            ( ( member_real @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: real] :
% 9.10/5.31              ( ( member_real @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups1681761925125756287l_real @ Z @ I5 ) @ ( groups1681761925125756287l_real @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups8097168146408367636l_real
% 9.10/5.31              @ ^ [I4: real] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8543_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_int,Z: int > real,W: int > real] :
% 9.10/5.31        ( ! [I3: int] :
% 9.10/5.31            ( ( member_int @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: int] :
% 9.10/5.31              ( ( member_int @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups2316167850115554303t_real @ Z @ I5 ) @ ( groups2316167850115554303t_real @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups8778361861064173332t_real
% 9.10/5.31              @ ^ [I4: int] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8544_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_complex,Z: complex > real,W: complex > real] :
% 9.10/5.31        ( ! [I3: complex] :
% 9.10/5.31            ( ( member_complex @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: complex] :
% 9.10/5.31              ( ( member_complex @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups766887009212190081x_real @ Z @ I5 ) @ ( groups766887009212190081x_real @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups5808333547571424918x_real
% 9.10/5.31              @ ^ [I4: complex] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8545_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_real,Z: real > complex,W: real > complex] :
% 9.10/5.31        ( ! [I3: real] :
% 9.10/5.31            ( ( member_real @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: real] :
% 9.10/5.31              ( ( member_real @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups713298508707869441omplex @ Z @ I5 ) @ ( groups713298508707869441omplex @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups8097168146408367636l_real
% 9.10/5.31              @ ^ [I4: real] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8546_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_int,Z: int > complex,W: int > complex] :
% 9.10/5.31        ( ! [I3: int] :
% 9.10/5.31            ( ( member_int @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: int] :
% 9.10/5.31              ( ( member_int @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups7440179247065528705omplex @ Z @ I5 ) @ ( groups7440179247065528705omplex @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups8778361861064173332t_real
% 9.10/5.31              @ ^ [I4: int] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8547_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_complex,Z: complex > complex,W: complex > complex] :
% 9.10/5.31        ( ! [I3: complex] :
% 9.10/5.31            ( ( member_complex @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: complex] :
% 9.10/5.31              ( ( member_complex @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups3708469109370488835omplex @ Z @ I5 ) @ ( groups3708469109370488835omplex @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups5808333547571424918x_real
% 9.10/5.31              @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8548_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_nat,Z: nat > real,W: nat > real] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( member_nat @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: nat] :
% 9.10/5.31              ( ( member_nat @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( groups129246275422532515t_real @ Z @ I5 ) @ ( groups129246275422532515t_real @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8549_norm__prod__diff,axiom,
% 9.10/5.31      ! [I5: set_nat,Z: nat > complex,W: nat > complex] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( member_nat @ I3 @ I5 )
% 9.10/5.31           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( Z @ I3 ) ) @ one_one_real ) )
% 9.10/5.31       => ( ! [I3: nat] :
% 9.10/5.31              ( ( member_nat @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( W @ I3 ) ) @ one_one_real ) )
% 9.10/5.31         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( groups6464643781859351333omplex @ Z @ I5 ) @ ( groups6464643781859351333omplex @ W @ I5 ) ) )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( Z @ I4 ) @ ( W @ I4 ) ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8550_polyfun__eq__0,axiom,
% 9.10/5.31      ! [C: nat > complex,N: nat] :
% 9.10/5.31        ( ( ! [X: complex] :
% 9.10/5.31              ( ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31              = zero_zero_complex ) )
% 9.10/5.31        = ( ! [I4: nat] :
% 9.10/5.31              ( ( ord_less_eq_nat @ I4 @ N )
% 9.10/5.31             => ( ( C @ I4 )
% 9.10/5.31                = zero_zero_complex ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8551_polyfun__eq__0,axiom,
% 9.10/5.31      ! [C: nat > real,N: nat] :
% 9.10/5.31        ( ( ! [X: real] :
% 9.10/5.31              ( ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31              = zero_zero_real ) )
% 9.10/5.31        = ( ! [I4: nat] :
% 9.10/5.31              ( ( ord_less_eq_nat @ I4 @ N )
% 9.10/5.31             => ( ( C @ I4 )
% 9.10/5.31                = zero_zero_real ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8552_zero__polynom__imp__zero__coeffs,axiom,
% 9.10/5.31      ! [C: nat > complex,N: nat,K: nat] :
% 9.10/5.31        ( ! [W2: complex] :
% 9.10/5.31            ( ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ W2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_zero_complex )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.31         => ( ( C @ K )
% 9.10/5.31            = zero_zero_complex ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8553_zero__polynom__imp__zero__coeffs,axiom,
% 9.10/5.31      ! [C: nat > real,N: nat,K: nat] :
% 9.10/5.31        ( ! [W2: real] :
% 9.10/5.31            ( ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ W2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_zero_real )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.31         => ( ( C @ K )
% 9.10/5.31            = zero_zero_real ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8554_sum_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups2906978787729119204at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8555_sum_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups3539618377306564664at_int
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8556_sum_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups3542108847815614940at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8557_sum_OatMost__shift,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.10/5.31          @ ( groups6591440286371151544t_real
% 9.10/5.31            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 9.10/5.31            @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8558_sum__up__index__split,axiom,
% 9.10/5.31      ! [F: nat > rat,M: nat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.31        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8559_sum__up__index__split,axiom,
% 9.10/5.31      ! [F: nat > int,M: nat,N: nat] :
% 9.10/5.31        ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.31        = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8560_sum__up__index__split,axiom,
% 9.10/5.31      ! [F: nat > nat,M: nat,N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.31        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8561_sum__up__index__split,axiom,
% 9.10/5.31      ! [F: nat > real,M: nat,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
% 9.10/5.31        = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8562_sum_Otriangle__reindex__eq,axiom,
% 9.10/5.31      ! [G: nat > nat > nat,N: nat] :
% 9.10/5.31        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups3542108847815614940at_nat
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8563_sum_Otriangle__reindex__eq,axiom,
% 9.10/5.31      ! [G: nat > nat > real,N: nat] :
% 9.10/5.31        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8564_fact__eq__fact__times,axiom,
% 9.10/5.31      ! [N: nat,M: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.31       => ( ( semiri1408675320244567234ct_nat @ M )
% 9.10/5.31          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 9.10/5.31            @ ( groups708209901874060359at_nat
% 9.10/5.31              @ ^ [X: nat] : X
% 9.10/5.31              @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8565_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_real,A4: set_real,F: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: real] :
% 9.10/5.31                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( groups1681761925125756287l_real @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8566_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: nat] :
% 9.10/5.31                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( groups129246275422532515t_real @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8567_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: complex] :
% 9.10/5.31                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( groups766887009212190081x_real @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8568_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_real,A4: set_real,F: real > rat] :
% 9.10/5.31        ( ( finite_finite_real @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: real] :
% 9.10/5.31                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A4 ) @ ( groups4061424788464935467al_rat @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8569_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_nat,A4: set_nat,F: nat > rat] :
% 9.10/5.31        ( ( finite_finite_nat @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: nat] :
% 9.10/5.31                ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: nat] :
% 9.10/5.31                  ( ( member_nat @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A4 ) @ ( groups73079841787564623at_rat @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8570_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > rat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: complex] :
% 9.10/5.31                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A4 ) @ ( groups225925009352817453ex_rat @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8571_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_real,A4: set_real,F: real > int] :
% 9.10/5.31        ( ( finite_finite_real @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_real @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: real] :
% 9.10/5.31                ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: real] :
% 9.10/5.31                  ( ( member_real @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8572_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_complex,A4: set_complex,F: complex > int] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ B5 )
% 9.10/5.31       => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: complex] :
% 9.10/5.31                ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: complex] :
% 9.10/5.31                  ( ( member_complex @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8573_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: int] :
% 9.10/5.31                ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: int] :
% 9.10/5.31                  ( ( member_int @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( groups2316167850115554303t_real @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8574_prod__mono2,axiom,
% 9.10/5.31      ! [B5: set_int,A4: set_int,F: int > rat] :
% 9.10/5.31        ( ( finite_finite_int @ B5 )
% 9.10/5.31       => ( ( ord_less_eq_set_int @ A4 @ B5 )
% 9.10/5.31         => ( ! [B3: int] :
% 9.10/5.31                ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
% 9.10/5.31               => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
% 9.10/5.31           => ( ! [A3: int] :
% 9.10/5.31                  ( ( member_int @ A3 @ A4 )
% 9.10/5.31                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
% 9.10/5.31             => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A4 ) @ ( groups1072433553688619179nt_rat @ F @ B5 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8575_sum__choose__diagonal,axiom,
% 9.10/5.31      ! [M: nat,N: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups3542108847815614940at_nat
% 9.10/5.31            @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31          = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8576_vandermonde,axiom,
% 9.10/5.31      ! [M: nat,N: nat,R: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ R ) )
% 9.10/5.31        = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8577_sum__gp__basic,axiom,
% 9.10/5.31      ! [X3: complex,N: nat] :
% 9.10/5.31        ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31        = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8578_sum__gp__basic,axiom,
% 9.10/5.31      ! [X3: rat,N: nat] :
% 9.10/5.31        ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31        = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8579_sum__gp__basic,axiom,
% 9.10/5.31      ! [X3: int,N: nat] :
% 9.10/5.31        ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31        = ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8580_sum__gp__basic,axiom,
% 9.10/5.31      ! [X3: real,N: nat] :
% 9.10/5.31        ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31        = ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8581_polyfun__finite__roots,axiom,
% 9.10/5.31      ! [C: nat > complex,N: nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex
% 9.10/5.31          @ ( collect_complex
% 9.10/5.31            @ ^ [X: complex] :
% 9.10/5.31                ( ( groups2073611262835488442omplex
% 9.10/5.31                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.10/5.31                  @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                = zero_zero_complex ) ) )
% 9.10/5.31        = ( ? [I4: nat] :
% 9.10/5.31              ( ( ord_less_eq_nat @ I4 @ N )
% 9.10/5.31              & ( ( C @ I4 )
% 9.10/5.31               != zero_zero_complex ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8582_polyfun__finite__roots,axiom,
% 9.10/5.31      ! [C: nat > real,N: nat] :
% 9.10/5.31        ( ( finite_finite_real
% 9.10/5.31          @ ( collect_real
% 9.10/5.31            @ ^ [X: real] :
% 9.10/5.31                ( ( groups6591440286371151544t_real
% 9.10/5.31                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.10/5.31                  @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                = zero_zero_real ) ) )
% 9.10/5.31        = ( ? [I4: nat] :
% 9.10/5.31              ( ( ord_less_eq_nat @ I4 @ N )
% 9.10/5.31              & ( ( C @ I4 )
% 9.10/5.31               != zero_zero_real ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8583_polyfun__roots__finite,axiom,
% 9.10/5.31      ! [C: nat > complex,K: nat,N: nat] :
% 9.10/5.31        ( ( ( C @ K )
% 9.10/5.31         != zero_zero_complex )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.31         => ( finite3207457112153483333omplex
% 9.10/5.31            @ ( collect_complex
% 9.10/5.31              @ ^ [Z2: complex] :
% 9.10/5.31                  ( ( groups2073611262835488442omplex
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z2 @ I4 ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                  = zero_zero_complex ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8584_polyfun__roots__finite,axiom,
% 9.10/5.31      ! [C: nat > real,K: nat,N: nat] :
% 9.10/5.31        ( ( ( C @ K )
% 9.10/5.31         != zero_zero_real )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.31         => ( finite_finite_real
% 9.10/5.31            @ ( collect_real
% 9.10/5.31              @ ^ [Z2: real] :
% 9.10/5.31                  ( ( groups6591440286371151544t_real
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z2 @ I4 ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                  = zero_zero_real ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8585_pochhammer__Suc__prod,axiom,
% 9.10/5.31      ! [A: rat,N: nat] :
% 9.10/5.31        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups73079841787564623at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8586_pochhammer__Suc__prod,axiom,
% 9.10/5.31      ! [A: real,N: nat] :
% 9.10/5.31        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups129246275422532515t_real
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8587_pochhammer__Suc__prod,axiom,
% 9.10/5.31      ! [A: nat,N: nat] :
% 9.10/5.31        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8588_pochhammer__Suc__prod,axiom,
% 9.10/5.31      ! [A: int,N: nat] :
% 9.10/5.31        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8589_polyfun__linear__factor__root,axiom,
% 9.10/5.31      ! [C: nat > complex,A: complex,N: nat] :
% 9.10/5.31        ( ( ( groups2073611262835488442omplex
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_complex )
% 9.10/5.31       => ~ ! [B3: nat > complex] :
% 9.10/5.31              ~ ! [Z3: complex] :
% 9.10/5.31                  ( ( groups2073611262835488442omplex
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                  = ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
% 9.10/5.31                    @ ( groups2073611262835488442omplex
% 9.10/5.31                      @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 9.10/5.31                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8590_polyfun__linear__factor__root,axiom,
% 9.10/5.31      ! [C: nat > rat,A: rat,N: nat] :
% 9.10/5.31        ( ( ( groups2906978787729119204at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_rat )
% 9.10/5.31       => ~ ! [B3: nat > rat] :
% 9.10/5.31              ~ ! [Z3: rat] :
% 9.10/5.31                  ( ( groups2906978787729119204at_rat
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                  = ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
% 9.10/5.31                    @ ( groups2906978787729119204at_rat
% 9.10/5.31                      @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 9.10/5.31                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8591_polyfun__linear__factor__root,axiom,
% 9.10/5.31      ! [C: nat > int,A: int,N: nat] :
% 9.10/5.31        ( ( ( groups3539618377306564664at_int
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_int )
% 9.10/5.31       => ~ ! [B3: nat > int] :
% 9.10/5.31              ~ ! [Z3: int] :
% 9.10/5.31                  ( ( groups3539618377306564664at_int
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                  = ( times_times_int @ ( minus_minus_int @ Z3 @ A )
% 9.10/5.31                    @ ( groups3539618377306564664at_int
% 9.10/5.31                      @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 9.10/5.31                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8592_polyfun__linear__factor__root,axiom,
% 9.10/5.31      ! [C: nat > real,A: real,N: nat] :
% 9.10/5.31        ( ( ( groups6591440286371151544t_real
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_real )
% 9.10/5.31       => ~ ! [B3: nat > real] :
% 9.10/5.31              ~ ! [Z3: real] :
% 9.10/5.31                  ( ( groups6591440286371151544t_real
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31                  = ( times_times_real @ ( minus_minus_real @ Z3 @ A )
% 9.10/5.31                    @ ( groups6591440286371151544t_real
% 9.10/5.31                      @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 9.10/5.31                      @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8593_polyfun__linear__factor,axiom,
% 9.10/5.31      ! [C: nat > complex,N: nat,A: complex] :
% 9.10/5.31      ? [B3: nat > complex] :
% 9.10/5.31      ! [Z3: complex] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_complex
% 9.10/5.31          @ ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) )
% 9.10/5.31          @ ( groups2073611262835488442omplex
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8594_polyfun__linear__factor,axiom,
% 9.10/5.31      ! [C: nat > rat,N: nat,A: rat] :
% 9.10/5.31      ? [B3: nat > rat] :
% 9.10/5.31      ! [Z3: rat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_rat
% 9.10/5.31          @ ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) )
% 9.10/5.31          @ ( groups2906978787729119204at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8595_polyfun__linear__factor,axiom,
% 9.10/5.31      ! [C: nat > int,N: nat,A: int] :
% 9.10/5.31      ? [B3: nat > int] :
% 9.10/5.31      ! [Z3: int] :
% 9.10/5.31        ( ( groups3539618377306564664at_int
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_int
% 9.10/5.31          @ ( times_times_int @ ( minus_minus_int @ Z3 @ A )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) )
% 9.10/5.31          @ ( groups3539618377306564664at_int
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8596_polyfun__linear__factor,axiom,
% 9.10/5.31      ! [C: nat > real,N: nat,A: real] :
% 9.10/5.31      ? [B3: nat > real] :
% 9.10/5.31      ! [Z3: real] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( plus_plus_real
% 9.10/5.31          @ ( times_times_real @ ( minus_minus_real @ Z3 @ A )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) )
% 9.10/5.31          @ ( groups6591440286371151544t_real
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ A @ I4 ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8597_pochhammer__prod__rev,axiom,
% 9.10/5.31      ( comm_s4028243227959126397er_rat
% 9.10/5.31      = ( ^ [A5: rat,N2: nat] :
% 9.10/5.31            ( groups73079841787564623at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8598_pochhammer__prod__rev,axiom,
% 9.10/5.31      ( comm_s7457072308508201937r_real
% 9.10/5.31      = ( ^ [A5: real,N2: nat] :
% 9.10/5.31            ( groups129246275422532515t_real
% 9.10/5.31            @ ^ [I4: nat] : ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8599_pochhammer__prod__rev,axiom,
% 9.10/5.31      ( comm_s4663373288045622133er_nat
% 9.10/5.31      = ( ^ [A5: nat,N2: nat] :
% 9.10/5.31            ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8600_pochhammer__prod__rev,axiom,
% 9.10/5.31      ( comm_s4660882817536571857er_int
% 9.10/5.31      = ( ^ [A5: int,N2: nat] :
% 9.10/5.31            ( groups705719431365010083at_int
% 9.10/5.31            @ ^ [I4: nat] : ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ I4 ) ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8601_sum__power__shift,axiom,
% 9.10/5.31      ! [M: nat,N: nat,X3: complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8602_sum__power__shift,axiom,
% 9.10/5.31      ! [M: nat,N: nat,X3: rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8603_sum__power__shift,axiom,
% 9.10/5.31      ! [M: nat,N: nat,X3: int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8604_sum__power__shift,axiom,
% 9.10/5.31      ! [M: nat,N: nat,X3: real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.31       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.10/5.31          = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8605_sum_Otriangle__reindex,axiom,
% 9.10/5.31      ! [G: nat > nat > nat,N: nat] :
% 9.10/5.31        ( ( groups977919841031483927at_nat @ ( produc6842872674320459806at_nat @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups3542108847815614940at_nat
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8606_sum_Otriangle__reindex,axiom,
% 9.10/5.31      ! [G: nat > nat > real,N: nat] :
% 9.10/5.31        ( ( groups4567486121110086003t_real @ ( produc1703576794950452218t_real @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8607_fact__div__fact,axiom,
% 9.10/5.31      ! [N: nat,M: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.31       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.10/5.31          = ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [X: nat] : X
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8608_summable__Cauchy__product,axiom,
% 9.10/5.31      ! [A: nat > complex,B2: nat > complex] :
% 9.10/5.31        ( ( summable_real
% 9.10/5.31          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
% 9.10/5.31       => ( ( summable_real
% 9.10/5.31            @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
% 9.10/5.31         => ( summable_complex
% 9.10/5.31            @ ^ [K2: nat] :
% 9.10/5.31                ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8609_summable__Cauchy__product,axiom,
% 9.10/5.31      ! [A: nat > real,B2: nat > real] :
% 9.10/5.31        ( ( summable_real
% 9.10/5.31          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
% 9.10/5.31       => ( ( summable_real
% 9.10/5.31            @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
% 9.10/5.31         => ( summable_real
% 9.10/5.31            @ ^ [K2: nat] :
% 9.10/5.31                ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8610_choose__row__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8611_Cauchy__product,axiom,
% 9.10/5.31      ! [A: nat > complex,B2: nat > complex] :
% 9.10/5.31        ( ( summable_real
% 9.10/5.31          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
% 9.10/5.31       => ( ( summable_real
% 9.10/5.31            @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
% 9.10/5.31         => ( ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B2 ) )
% 9.10/5.31            = ( suminf_complex
% 9.10/5.31              @ ^ [K2: nat] :
% 9.10/5.31                  ( groups2073611262835488442omplex
% 9.10/5.31                  @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 9.10/5.31                  @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8612_Cauchy__product,axiom,
% 9.10/5.31      ! [A: nat > real,B2: nat > real] :
% 9.10/5.31        ( ( summable_real
% 9.10/5.31          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
% 9.10/5.31       => ( ( summable_real
% 9.10/5.31            @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
% 9.10/5.31         => ( ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B2 ) )
% 9.10/5.31            = ( suminf_real
% 9.10/5.31              @ ^ [K2: nat] :
% 9.10/5.31                  ( groups6591440286371151544t_real
% 9.10/5.31                  @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 9.10/5.31                  @ ( set_ord_atMost_nat @ K2 ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8613_binomial,axiom,
% 9.10/5.31      ! [A: nat,B2: nat,N: nat] :
% 9.10/5.31        ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8614_prod_Oin__pairs,axiom,
% 9.10/5.31      ! [G: nat > real,M: nat,N: nat] :
% 9.10/5.31        ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups129246275422532515t_real
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8615_prod_Oin__pairs,axiom,
% 9.10/5.31      ! [G: nat > rat,M: nat,N: nat] :
% 9.10/5.31        ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups73079841787564623at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8616_prod_Oin__pairs,axiom,
% 9.10/5.31      ! [G: nat > nat,M: nat,N: nat] :
% 9.10/5.31        ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8617_prod_Oin__pairs,axiom,
% 9.10/5.31      ! [G: nat > int,M: nat,N: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8618_sum_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > rat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8619_sum_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > int,N: nat] :
% 9.10/5.31        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups3539618377306564664at_int
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8620_sum_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > nat,N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8621_sum_Oin__pairs__0,axiom,
% 9.10/5.31      ! [G: nat > real,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8622_polynomial__product,axiom,
% 9.10/5.31      ! [M: nat,A: nat > complex,N: nat,B2: nat > complex,X3: complex] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( ord_less_nat @ M @ I3 )
% 9.10/5.31           => ( ( A @ I3 )
% 9.10/5.31              = zero_zero_complex ) )
% 9.10/5.31       => ( ! [J2: nat] :
% 9.10/5.31              ( ( ord_less_nat @ N @ J2 )
% 9.10/5.31             => ( ( B2 @ J2 )
% 9.10/5.31                = zero_zero_complex ) )
% 9.10/5.31         => ( ( times_times_complex
% 9.10/5.31              @ ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31              @ ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [J3: nat] : ( times_times_complex @ ( B2 @ J3 ) @ ( power_power_complex @ X3 @ J3 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31            = ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [R5: nat] :
% 9.10/5.31                  ( times_times_complex
% 9.10/5.31                  @ ( groups2073611262835488442omplex
% 9.10/5.31                    @ ^ [K2: nat] : ( times_times_complex @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.10/5.31                  @ ( power_power_complex @ X3 @ R5 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8623_polynomial__product,axiom,
% 9.10/5.31      ! [M: nat,A: nat > rat,N: nat,B2: nat > rat,X3: rat] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( ord_less_nat @ M @ I3 )
% 9.10/5.31           => ( ( A @ I3 )
% 9.10/5.31              = zero_zero_rat ) )
% 9.10/5.31       => ( ! [J2: nat] :
% 9.10/5.31              ( ( ord_less_nat @ N @ J2 )
% 9.10/5.31             => ( ( B2 @ J2 )
% 9.10/5.31                = zero_zero_rat ) )
% 9.10/5.31         => ( ( times_times_rat
% 9.10/5.31              @ ( groups2906978787729119204at_rat
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31              @ ( groups2906978787729119204at_rat
% 9.10/5.31                @ ^ [J3: nat] : ( times_times_rat @ ( B2 @ J3 ) @ ( power_power_rat @ X3 @ J3 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31            = ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [R5: nat] :
% 9.10/5.31                  ( times_times_rat
% 9.10/5.31                  @ ( groups2906978787729119204at_rat
% 9.10/5.31                    @ ^ [K2: nat] : ( times_times_rat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.10/5.31                  @ ( power_power_rat @ X3 @ R5 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8624_polynomial__product,axiom,
% 9.10/5.31      ! [M: nat,A: nat > int,N: nat,B2: nat > int,X3: int] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( ord_less_nat @ M @ I3 )
% 9.10/5.31           => ( ( A @ I3 )
% 9.10/5.31              = zero_zero_int ) )
% 9.10/5.31       => ( ! [J2: nat] :
% 9.10/5.31              ( ( ord_less_nat @ N @ J2 )
% 9.10/5.31             => ( ( B2 @ J2 )
% 9.10/5.31                = zero_zero_int ) )
% 9.10/5.31         => ( ( times_times_int
% 9.10/5.31              @ ( groups3539618377306564664at_int
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31              @ ( groups3539618377306564664at_int
% 9.10/5.31                @ ^ [J3: nat] : ( times_times_int @ ( B2 @ J3 ) @ ( power_power_int @ X3 @ J3 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31            = ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [R5: nat] :
% 9.10/5.31                  ( times_times_int
% 9.10/5.31                  @ ( groups3539618377306564664at_int
% 9.10/5.31                    @ ^ [K2: nat] : ( times_times_int @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.10/5.31                  @ ( power_power_int @ X3 @ R5 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8625_polynomial__product,axiom,
% 9.10/5.31      ! [M: nat,A: nat > real,N: nat,B2: nat > real,X3: real] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( ord_less_nat @ M @ I3 )
% 9.10/5.31           => ( ( A @ I3 )
% 9.10/5.31              = zero_zero_real ) )
% 9.10/5.31       => ( ! [J2: nat] :
% 9.10/5.31              ( ( ord_less_nat @ N @ J2 )
% 9.10/5.31             => ( ( B2 @ J2 )
% 9.10/5.31                = zero_zero_real ) )
% 9.10/5.31         => ( ( times_times_real
% 9.10/5.31              @ ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31              @ ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [J3: nat] : ( times_times_real @ ( B2 @ J3 ) @ ( power_power_real @ X3 @ J3 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31            = ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [R5: nat] :
% 9.10/5.31                  ( times_times_real
% 9.10/5.31                  @ ( groups6591440286371151544t_real
% 9.10/5.31                    @ ^ [K2: nat] : ( times_times_real @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.10/5.31                  @ ( power_power_real @ X3 @ R5 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8626_pochhammer__Suc__prod__rev,axiom,
% 9.10/5.31      ! [A: rat,N: nat] :
% 9.10/5.31        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups73079841787564623at_rat
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I4 ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8627_pochhammer__Suc__prod__rev,axiom,
% 9.10/5.31      ! [A: real,N: nat] :
% 9.10/5.31        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups129246275422532515t_real
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I4 ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8628_pochhammer__Suc__prod__rev,axiom,
% 9.10/5.31      ! [A: nat,N: nat] :
% 9.10/5.31        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I4 ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8629_pochhammer__Suc__prod__rev,axiom,
% 9.10/5.31      ! [A: int,N: nat] :
% 9.10/5.31        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [I4: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I4 ) ) )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8630_polyfun__eq__const,axiom,
% 9.10/5.31      ! [C: nat > complex,N: nat,K: complex] :
% 9.10/5.31        ( ( ! [X: complex] :
% 9.10/5.31              ( ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31              = K ) )
% 9.10/5.31        = ( ( ( C @ zero_zero_nat )
% 9.10/5.31            = K )
% 9.10/5.31          & ! [X: nat] :
% 9.10/5.31              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 9.10/5.31             => ( ( C @ X )
% 9.10/5.31                = zero_zero_complex ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8631_polyfun__eq__const,axiom,
% 9.10/5.31      ! [C: nat > real,N: nat,K: real] :
% 9.10/5.31        ( ( ! [X: real] :
% 9.10/5.31              ( ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31              = K ) )
% 9.10/5.31        = ( ( ( C @ zero_zero_nat )
% 9.10/5.31            = K )
% 9.10/5.31          & ! [X: nat] :
% 9.10/5.31              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
% 9.10/5.31             => ( ( C @ X )
% 9.10/5.31                = zero_zero_real ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8632_binomial__ring,axiom,
% 9.10/5.31      ! [A: complex,B2: complex,N: nat] :
% 9.10/5.31        ( ( power_power_complex @ ( plus_plus_complex @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) ) @ ( power_power_complex @ A @ K2 ) ) @ ( power_power_complex @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8633_binomial__ring,axiom,
% 9.10/5.31      ! [A: rat,B2: rat,N: nat] :
% 9.10/5.31        ( ( power_power_rat @ ( plus_plus_rat @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( power_power_rat @ A @ K2 ) ) @ ( power_power_rat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8634_binomial__ring,axiom,
% 9.10/5.31      ! [A: int,B2: int,N: nat] :
% 9.10/5.31        ( ( power_power_int @ ( plus_plus_int @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups3539618377306564664at_int
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( power_power_int @ A @ K2 ) ) @ ( power_power_int @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8635_binomial__ring,axiom,
% 9.10/5.31      ! [A: nat,B2: nat,N: nat] :
% 9.10/5.31        ( ( power_power_nat @ ( plus_plus_nat @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8636_binomial__ring,axiom,
% 9.10/5.31      ! [A: real,B2: real,N: nat] :
% 9.10/5.31        ( ( power_power_real @ ( plus_plus_real @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( power_power_real @ A @ K2 ) ) @ ( power_power_real @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8637_pochhammer__binomial__sum,axiom,
% 9.10/5.31      ! [A: rat,B2: rat,N: nat] :
% 9.10/5.31        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8638_pochhammer__binomial__sum,axiom,
% 9.10/5.31      ! [A: int,B2: int,N: nat] :
% 9.10/5.31        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups3539618377306564664at_int
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K2 ) ) @ ( comm_s4660882817536571857er_int @ A @ K2 ) ) @ ( comm_s4660882817536571857er_int @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8639_pochhammer__binomial__sum,axiom,
% 9.10/5.31      ! [A: real,B2: real,N: nat] :
% 9.10/5.31        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B2 ) @ N )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) @ ( comm_s7457072308508201937r_real @ A @ K2 ) ) @ ( comm_s7457072308508201937r_real @ B2 @ ( minus_minus_nat @ N @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8640_polynomial__product__nat,axiom,
% 9.10/5.31      ! [M: nat,A: nat > nat,N: nat,B2: nat > nat,X3: nat] :
% 9.10/5.31        ( ! [I3: nat] :
% 9.10/5.31            ( ( ord_less_nat @ M @ I3 )
% 9.10/5.31           => ( ( A @ I3 )
% 9.10/5.31              = zero_zero_nat ) )
% 9.10/5.31       => ( ! [J2: nat] :
% 9.10/5.31              ( ( ord_less_nat @ N @ J2 )
% 9.10/5.31             => ( ( B2 @ J2 )
% 9.10/5.31                = zero_zero_nat ) )
% 9.10/5.31         => ( ( times_times_nat
% 9.10/5.31              @ ( groups3542108847815614940at_nat
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X3 @ I4 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31              @ ( groups3542108847815614940at_nat
% 9.10/5.31                @ ^ [J3: nat] : ( times_times_nat @ ( B2 @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31            = ( groups3542108847815614940at_nat
% 9.10/5.31              @ ^ [R5: nat] :
% 9.10/5.31                  ( times_times_nat
% 9.10/5.31                  @ ( groups3542108847815614940at_nat
% 9.10/5.31                    @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 9.10/5.31                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.10/5.31                  @ ( power_power_nat @ X3 @ R5 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8641_choose__square__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8642_Cauchy__product__sums,axiom,
% 9.10/5.31      ! [A: nat > complex,B2: nat > complex] :
% 9.10/5.31        ( ( summable_real
% 9.10/5.31          @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( A @ K2 ) ) )
% 9.10/5.31       => ( ( summable_real
% 9.10/5.31            @ ^ [K2: nat] : ( real_V1022390504157884413omplex @ ( B2 @ K2 ) ) )
% 9.10/5.31         => ( sums_complex
% 9.10/5.31            @ ^ [K2: nat] :
% 9.10/5.31                ( groups2073611262835488442omplex
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31            @ ( times_times_complex @ ( suminf_complex @ A ) @ ( suminf_complex @ B2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8643_Cauchy__product__sums,axiom,
% 9.10/5.31      ! [A: nat > real,B2: nat > real] :
% 9.10/5.31        ( ( summable_real
% 9.10/5.31          @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( A @ K2 ) ) )
% 9.10/5.31       => ( ( summable_real
% 9.10/5.31            @ ^ [K2: nat] : ( real_V7735802525324610683m_real @ ( B2 @ K2 ) ) )
% 9.10/5.31         => ( sums_real
% 9.10/5.31            @ ^ [K2: nat] :
% 9.10/5.31                ( groups6591440286371151544t_real
% 9.10/5.31                @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( B2 @ ( minus_minus_nat @ K2 @ I4 ) ) )
% 9.10/5.31                @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31            @ ( times_times_real @ ( suminf_real @ A ) @ ( suminf_real @ B2 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8644_sum_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > complex,H2: nat > complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8645_sum_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > rat,H2: nat > rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8646_sum_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > int,H2: nat > int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8647_sum_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > nat,H2: nat > nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups3542108847815614940at_nat
% 9.10/5.31              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups3542108847815614940at_nat
% 9.10/5.31              @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8648_sum_Ozero__middle,axiom,
% 9.10/5.31      ! [P4: nat,K: nat,G: nat > real,H2: nat > real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ P4 )
% 9.10/5.31       => ( ( ord_less_eq_nat @ K @ P4 )
% 9.10/5.31         => ( ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H2 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ P4 ) )
% 9.10/5.31            = ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P4 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8649_root__polyfun,axiom,
% 9.10/5.31      ! [N: nat,Z: int,A: int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( ( power_power_int @ Z @ N )
% 9.10/5.31            = A )
% 9.10/5.31          = ( ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_int @ ( if_int @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I4 = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_zero_int ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8650_root__polyfun,axiom,
% 9.10/5.31      ! [N: nat,Z: complex,A: complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( ( power_power_complex @ Z @ N )
% 9.10/5.31            = A )
% 9.10/5.31          = ( ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( if_complex @ ( I4 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I4 = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_zero_complex ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8651_root__polyfun,axiom,
% 9.10/5.31      ! [N: nat,Z: code_integer,A: code_integer] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( ( power_8256067586552552935nteger @ Z @ N )
% 9.10/5.31            = A )
% 9.10/5.31          = ( ( groups7501900531339628137nteger
% 9.10/5.31              @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I4 = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_z3403309356797280102nteger ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8652_root__polyfun,axiom,
% 9.10/5.31      ! [N: nat,Z: rat,A: rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( ( power_power_rat @ Z @ N )
% 9.10/5.31            = A )
% 9.10/5.31          = ( ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_rat @ ( if_rat @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I4 = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_zero_rat ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8653_root__polyfun,axiom,
% 9.10/5.31      ! [N: nat,Z: real,A: real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( ( power_power_real @ Z @ N )
% 9.10/5.31            = A )
% 9.10/5.31          = ( ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( if_real @ ( I4 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I4 = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = zero_zero_real ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8654_sum__gp0,axiom,
% 9.10/5.31      ! [X3: complex,N: nat] :
% 9.10/5.31        ( ( ( X3 = one_one_complex )
% 9.10/5.31         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
% 9.10/5.31        & ( ( X3 != one_one_complex )
% 9.10/5.31         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8655_sum__gp0,axiom,
% 9.10/5.31      ! [X3: rat,N: nat] :
% 9.10/5.31        ( ( ( X3 = one_one_rat )
% 9.10/5.31         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
% 9.10/5.31        & ( ( X3 != one_one_rat )
% 9.10/5.31         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8656_sum__gp0,axiom,
% 9.10/5.31      ! [X3: real,N: nat] :
% 9.10/5.31        ( ( ( X3 = one_one_real )
% 9.10/5.31         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
% 9.10/5.31        & ( ( X3 != one_one_real )
% 9.10/5.31         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8657_choose__alternating__linear__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( N != one_one_nat )
% 9.10/5.31       => ( ( groups2073611262835488442omplex
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ I4 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_complex ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8658_choose__alternating__linear__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( N != one_one_nat )
% 9.10/5.31       => ( ( groups7501900531339628137nteger
% 9.10/5.31            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ I4 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8659_choose__alternating__linear__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( N != one_one_nat )
% 9.10/5.31       => ( ( groups2906978787729119204at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ I4 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8660_choose__alternating__linear__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( N != one_one_nat )
% 9.10/5.31       => ( ( groups3539618377306564664at_int
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ I4 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_int ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8661_choose__alternating__linear__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( N != one_one_nat )
% 9.10/5.31       => ( ( groups6591440286371151544t_real
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ I4 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8662_polyfun__diff__alt,axiom,
% 9.10/5.31      ! [N: nat,A: nat > complex,X3: complex,Y3: complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_complex
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( groups2073611262835488442omplex
% 9.10/5.31                  @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_complex @ Y3 @ K2 ) ) @ ( power_power_complex @ X3 @ J3 ) )
% 9.10/5.31                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8663_polyfun__diff__alt,axiom,
% 9.10/5.31      ! [N: nat,A: nat > rat,X3: rat,Y3: rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_rat
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( groups2906978787729119204at_rat
% 9.10/5.31                  @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_rat @ Y3 @ K2 ) ) @ ( power_power_rat @ X3 @ J3 ) )
% 9.10/5.31                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8664_polyfun__diff__alt,axiom,
% 9.10/5.31      ! [N: nat,A: nat > int,X3: int,Y3: int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_int
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( groups3539618377306564664at_int
% 9.10/5.31                  @ ^ [K2: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_int @ Y3 @ K2 ) ) @ ( power_power_int @ X3 @ J3 ) )
% 9.10/5.31                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8665_polyfun__diff__alt,axiom,
% 9.10/5.31      ! [N: nat,A: nat > real,X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_real
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( groups6591440286371151544t_real
% 9.10/5.31                  @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K2 ) @ one_one_nat ) ) @ ( power_power_real @ Y3 @ K2 ) ) @ ( power_power_real @ X3 @ J3 ) )
% 9.10/5.31                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8666_binomial__r__part__sum,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8667_choose__linear__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( groups3542108847815614940at_nat
% 9.10/5.31          @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8668_choose__alternating__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( groups2073611262835488442omplex
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I4 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_complex ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8669_choose__alternating__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( groups7501900531339628137nteger
% 9.10/5.31            @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I4 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_z3403309356797280102nteger ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8670_choose__alternating__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( groups2906978787729119204at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I4 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_rat ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8671_choose__alternating__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( groups3539618377306564664at_int
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I4 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_int ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8672_choose__alternating__sum,axiom,
% 9.10/5.31      ! [N: nat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.31       => ( ( groups6591440286371151544t_real
% 9.10/5.31            @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I4 ) ) )
% 9.10/5.31            @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31          = zero_zero_real ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8673_polyfun__extremal__lemma,axiom,
% 9.10/5.31      ! [E: real,C: nat > complex,N: nat] :
% 9.10/5.31        ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.31       => ? [M8: real] :
% 9.10/5.31          ! [Z3: complex] :
% 9.10/5.31            ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z3 ) )
% 9.10/5.31           => ( ord_less_eq_real
% 9.10/5.31              @ ( real_V1022390504157884413omplex
% 9.10/5.31                @ ( groups2073611262835488442omplex
% 9.10/5.31                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z3 @ I4 ) )
% 9.10/5.31                  @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31              @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8674_polyfun__extremal__lemma,axiom,
% 9.10/5.31      ! [E: real,C: nat > real,N: nat] :
% 9.10/5.31        ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.31       => ? [M8: real] :
% 9.10/5.31          ! [Z3: real] :
% 9.10/5.31            ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z3 ) )
% 9.10/5.31           => ( ord_less_eq_real
% 9.10/5.31              @ ( real_V7735802525324610683m_real
% 9.10/5.31                @ ( groups6591440286371151544t_real
% 9.10/5.31                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z3 @ I4 ) )
% 9.10/5.31                  @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31              @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z3 ) @ ( suc @ N ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8675_polyfun__diff,axiom,
% 9.10/5.31      ! [N: nat,A: nat > complex,X3: complex,Y3: complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_complex
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 9.10/5.31            @ ( groups2073611262835488442omplex
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( times_times_complex
% 9.10/5.31                  @ ( groups2073611262835488442omplex
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.10/5.31                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.10/5.31                  @ ( power_power_complex @ X3 @ J3 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8676_polyfun__diff,axiom,
% 9.10/5.31      ! [N: nat,A: nat > rat,X3: rat,Y3: rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_rat
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 9.10/5.31            @ ( groups2906978787729119204at_rat
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( times_times_rat
% 9.10/5.31                  @ ( groups2906978787729119204at_rat
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.10/5.31                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.10/5.31                  @ ( power_power_rat @ X3 @ J3 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8677_polyfun__diff,axiom,
% 9.10/5.31      ! [N: nat,A: nat > int,X3: int,Y3: int] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_int
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 9.10/5.31            @ ( groups3539618377306564664at_int
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( times_times_int
% 9.10/5.31                  @ ( groups3539618377306564664at_int
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_int @ ( A @ I4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.10/5.31                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.10/5.31                  @ ( power_power_int @ X3 @ J3 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8678_polyfun__diff,axiom,
% 9.10/5.31      ! [N: nat,A: nat > real,X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.10/5.31       => ( ( minus_minus_real
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ X3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ N ) ) )
% 9.10/5.31          = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 9.10/5.31            @ ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [J3: nat] :
% 9.10/5.31                  ( times_times_real
% 9.10/5.31                  @ ( groups6591440286371151544t_real
% 9.10/5.31                    @ ^ [I4: nat] : ( times_times_real @ ( A @ I4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ ( minus_minus_nat @ I4 @ J3 ) @ one_one_nat ) ) )
% 9.10/5.31                    @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
% 9.10/5.31                  @ ( power_power_real @ X3 @ J3 ) )
% 9.10/5.31              @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8679_gbinomial__partial__row__sum,axiom,
% 9.10/5.31      ! [A: complex,M: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( 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 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8680_gbinomial__partial__row__sum,axiom,
% 9.10/5.31      ! [A: rat,M: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( 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 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8681_gbinomial__partial__row__sum,axiom,
% 9.10/5.31      ! [A: real,M: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( 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 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8682_gbinomial__r__part__sum,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8683_gbinomial__r__part__sum,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8684_gbinomial__r__part__sum,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8685_cos__arcsin,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.31         => ( ( cos_real @ ( arcsin @ X3 ) )
% 9.10/5.31            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8686_sin__arccos__abs,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31       => ( ( sin_real @ ( arccos @ Y3 ) )
% 9.10/5.31          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8687_sin__arccos,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.31         => ( ( sin_real @ ( arccos @ X3 ) )
% 9.10/5.31            = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8688_of__nat__id,axiom,
% 9.10/5.31      ( semiri1316708129612266289at_nat
% 9.10/5.31      = ( ^ [N2: nat] : N2 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8689_prod__pos__nat__iff,axiom,
% 9.10/5.31      ! [A4: set_int,F: int > nat] :
% 9.10/5.31        ( ( finite_finite_int @ A4 )
% 9.10/5.31       => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) )
% 9.10/5.31          = ( ! [X: int] :
% 9.10/5.31                ( ( member_int @ X @ A4 )
% 9.10/5.31               => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8690_prod__pos__nat__iff,axiom,
% 9.10/5.31      ! [A4: set_complex,F: complex > nat] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ A4 )
% 9.10/5.31       => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) )
% 9.10/5.31          = ( ! [X: complex] :
% 9.10/5.31                ( ( member_complex @ X @ A4 )
% 9.10/5.31               => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8691_prod__pos__nat__iff,axiom,
% 9.10/5.31      ! [A4: set_nat,F: nat > nat] :
% 9.10/5.31        ( ( finite_finite_nat @ A4 )
% 9.10/5.31       => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A4 ) )
% 9.10/5.31          = ( ! [X: nat] :
% 9.10/5.31                ( ( member_nat @ X @ A4 )
% 9.10/5.31               => ( ord_less_nat @ zero_zero_nat @ ( F @ X ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8692_cos__arccos,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ( cos_real @ ( arccos @ Y3 ) )
% 9.10/5.31            = Y3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8693_sin__arcsin,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ( sin_real @ ( arcsin @ Y3 ) )
% 9.10/5.31            = Y3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8694_arccos__0,axiom,
% 9.10/5.31      ( ( arccos @ zero_zero_real )
% 9.10/5.31      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8695_arcsin__1,axiom,
% 9.10/5.31      ( ( arcsin @ one_one_real )
% 9.10/5.31      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8696_arcsin__minus__1,axiom,
% 9.10/5.31      ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.31      = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8697_gbinomial__Suc__Suc,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.10/5.31        = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8698_gbinomial__Suc__Suc,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.10/5.31        = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8699_gbinomial__Suc__Suc,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.10/5.31        = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8700_gbinomial__of__nat__symmetric,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ K @ N )
% 9.10/5.31       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 9.10/5.31          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8701_arccos__le__arccos,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.31         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31           => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8702_arccos__eq__iff,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31          & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 9.10/5.31       => ( ( ( arccos @ X3 )
% 9.10/5.31            = ( arccos @ Y3 ) )
% 9.10/5.31          = ( X3 = Y3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8703_arccos__le__mono,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31         => ( ( ord_less_eq_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
% 9.10/5.31            = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8704_arcsin__minus,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.31         => ( ( arcsin @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.31            = ( uminus_uminus_real @ ( arcsin @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8705_arcsin__le__arcsin,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.31         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31           => ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8706_arcsin__eq__iff,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31         => ( ( ( arcsin @ X3 )
% 9.10/5.31              = ( arcsin @ Y3 ) )
% 9.10/5.31            = ( X3 = Y3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8707_arcsin__le__mono,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31         => ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
% 9.10/5.31            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8708_gbinomial__addition__formula,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 9.10/5.31        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8709_gbinomial__addition__formula,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 9.10/5.31        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8710_gbinomial__addition__formula,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 9.10/5.31        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8711_gbinomial__absorb__comp,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
% 9.10/5.31        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8712_gbinomial__absorb__comp,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
% 9.10/5.31        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8713_gbinomial__absorb__comp,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
% 9.10/5.31        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8714_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.10/5.31      ! [K: nat,A: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 9.10/5.31       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8715_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.10/5.31      ! [K: nat,A: rat] :
% 9.10/5.31        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 9.10/5.31       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8716_gbinomial__mult__1,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 9.10/5.31        = ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8717_gbinomial__mult__1,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 9.10/5.31        = ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8718_gbinomial__mult__1_H,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 9.10/5.31        = ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8719_gbinomial__mult__1_H,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 9.10/5.31        = ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8720_prod__int__plus__eq,axiom,
% 9.10/5.31      ! [I: nat,J: nat] :
% 9.10/5.31        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 9.10/5.31        = ( groups1705073143266064639nt_int
% 9.10/5.31          @ ^ [X: int] : X
% 9.10/5.31          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8721_arccos__lbound,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8722_arccos__less__arccos,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.31         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31           => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8723_arccos__less__mono,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31         => ( ( ord_less_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
% 9.10/5.31            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8724_arccos__ubound,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8725_arccos__cos,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.31         => ( ( arccos @ ( cos_real @ X3 ) )
% 9.10/5.31            = X3 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8726_arcsin__less__arcsin,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.31         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31           => ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8727_arcsin__less__mono,axiom,
% 9.10/5.31      ! [X3: real,Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31         => ( ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
% 9.10/5.31            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8728_cos__arccos__abs,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.10/5.31       => ( ( cos_real @ ( arccos @ Y3 ) )
% 9.10/5.31          = Y3 ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8729_arccos__cos__eq__abs,axiom,
% 9.10/5.31      ! [Theta: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 9.10/5.31       => ( ( arccos @ ( cos_real @ Theta ) )
% 9.10/5.31          = ( abs_abs_real @ Theta ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8730_Suc__times__gbinomial,axiom,
% 9.10/5.31      ! [K: nat,A: complex] :
% 9.10/5.31        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 9.10/5.31        = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8731_Suc__times__gbinomial,axiom,
% 9.10/5.31      ! [K: nat,A: rat] :
% 9.10/5.31        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 9.10/5.31        = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8732_Suc__times__gbinomial,axiom,
% 9.10/5.31      ! [K: nat,A: real] :
% 9.10/5.31        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 9.10/5.31        = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8733_gbinomial__absorption,axiom,
% 9.10/5.31      ! [K: nat,A: complex] :
% 9.10/5.31        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 9.10/5.31        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8734_gbinomial__absorption,axiom,
% 9.10/5.31      ! [K: nat,A: rat] :
% 9.10/5.31        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 9.10/5.31        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8735_gbinomial__absorption,axiom,
% 9.10/5.31      ! [K: nat,A: real] :
% 9.10/5.31        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 9.10/5.31        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8736_gbinomial__trinomial__revision,axiom,
% 9.10/5.31      ! [K: nat,M: nat,A: complex] :
% 9.10/5.31        ( ( ord_less_eq_nat @ K @ M )
% 9.10/5.31       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 9.10/5.31          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8737_gbinomial__trinomial__revision,axiom,
% 9.10/5.31      ! [K: nat,M: nat,A: rat] :
% 9.10/5.31        ( ( ord_less_eq_nat @ K @ M )
% 9.10/5.31       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 9.10/5.31          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8738_gbinomial__trinomial__revision,axiom,
% 9.10/5.31      ! [K: nat,M: nat,A: real] :
% 9.10/5.31        ( ( ord_less_eq_nat @ K @ M )
% 9.10/5.31       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 9.10/5.31          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8739_ln__prod,axiom,
% 9.10/5.31      ! [I5: set_real,F: real > real] :
% 9.10/5.31        ( ( finite_finite_real @ I5 )
% 9.10/5.31       => ( ! [I3: real] :
% 9.10/5.31              ( ( member_real @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.31         => ( ( ln_ln_real @ ( groups1681761925125756287l_real @ F @ I5 ) )
% 9.10/5.31            = ( groups8097168146408367636l_real
% 9.10/5.31              @ ^ [X: real] : ( ln_ln_real @ ( F @ X ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8740_ln__prod,axiom,
% 9.10/5.31      ! [I5: set_int,F: int > real] :
% 9.10/5.31        ( ( finite_finite_int @ I5 )
% 9.10/5.31       => ( ! [I3: int] :
% 9.10/5.31              ( ( member_int @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.31         => ( ( ln_ln_real @ ( groups2316167850115554303t_real @ F @ I5 ) )
% 9.10/5.31            = ( groups8778361861064173332t_real
% 9.10/5.31              @ ^ [X: int] : ( ln_ln_real @ ( F @ X ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8741_ln__prod,axiom,
% 9.10/5.31      ! [I5: set_complex,F: complex > real] :
% 9.10/5.31        ( ( finite3207457112153483333omplex @ I5 )
% 9.10/5.31       => ( ! [I3: complex] :
% 9.10/5.31              ( ( member_complex @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.31         => ( ( ln_ln_real @ ( groups766887009212190081x_real @ F @ I5 ) )
% 9.10/5.31            = ( groups5808333547571424918x_real
% 9.10/5.31              @ ^ [X: complex] : ( ln_ln_real @ ( F @ X ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8742_ln__prod,axiom,
% 9.10/5.31      ! [I5: set_nat,F: nat > real] :
% 9.10/5.31        ( ( finite_finite_nat @ I5 )
% 9.10/5.31       => ( ! [I3: nat] :
% 9.10/5.31              ( ( member_nat @ I3 @ I5 )
% 9.10/5.31             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.10/5.31         => ( ( ln_ln_real @ ( groups129246275422532515t_real @ F @ I5 ) )
% 9.10/5.31            = ( groups6591440286371151544t_real
% 9.10/5.31              @ ^ [X: nat] : ( ln_ln_real @ ( F @ X ) )
% 9.10/5.31              @ I5 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8743_gbinomial__parallel__sum,axiom,
% 9.10/5.31      ! [A: complex,N: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ K2 )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ N ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8744_gbinomial__parallel__sum,axiom,
% 9.10/5.31      ! [A: rat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ N ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8745_gbinomial__parallel__sum,axiom,
% 9.10/5.31      ! [A: real,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) )
% 9.10/5.31        = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ N ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8746_arccos__lt__bounded,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.10/5.31            & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8747_arccos__bounded,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.10/5.31            & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8748_gbinomial__rec,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.10/5.31        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8749_gbinomial__rec,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.10/5.31        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8750_gbinomial__rec,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.10/5.31        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8751_gbinomial__factors,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.10/5.31        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8752_gbinomial__factors,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.10/5.31        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8753_gbinomial__factors,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.10/5.31        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8754_gbinomial__index__swap,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K ) )
% 9.10/5.31        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8755_gbinomial__index__swap,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K ) )
% 9.10/5.31        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8756_gbinomial__index__swap,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K ) )
% 9.10/5.31        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8757_gbinomial__negated__upper,axiom,
% 9.10/5.31      ( gbinomial_complex
% 9.10/5.31      = ( ^ [A5: complex,K2: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A5 ) @ one_one_complex ) @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8758_gbinomial__negated__upper,axiom,
% 9.10/5.31      ( gbinomial_rat
% 9.10/5.31      = ( ^ [A5: rat,K2: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A5 ) @ one_one_rat ) @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8759_gbinomial__negated__upper,axiom,
% 9.10/5.31      ( gbinomial_real
% 9.10/5.31      = ( ^ [A5: real,K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A5 ) @ one_one_real ) @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8760_sin__arccos__nonzero,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.31         => ( ( sin_real @ ( arccos @ X3 ) )
% 9.10/5.31           != zero_zero_real ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8761_arccos__cos2,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.31       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 9.10/5.31         => ( ( arccos @ ( cos_real @ X3 ) )
% 9.10/5.31            = ( uminus_uminus_real @ X3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8762_arccos__minus,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.31         => ( ( arccos @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.31            = ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8763_cos__arcsin__nonzero,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.31       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.31         => ( ( cos_real @ ( arcsin @ X3 ) )
% 9.10/5.31           != zero_zero_real ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8764_prod_Otriangle__reindex__eq,axiom,
% 9.10/5.31      ! [G: nat > nat > nat,N: nat] :
% 9.10/5.31        ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups708209901874060359at_nat
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8765_prod_Otriangle__reindex__eq,axiom,
% 9.10/5.31      ! [G: nat > nat > int,N: nat] :
% 9.10/5.31        ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups705719431365010083at_int
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8766_gbinomial__minus,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 9.10/5.31        = ( 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 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8767_gbinomial__minus,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 9.10/5.31        = ( 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 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8768_gbinomial__minus,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 9.10/5.31        = ( 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 ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8769_gbinomial__reduce__nat,axiom,
% 9.10/5.31      ! [K: nat,A: complex] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.31       => ( ( gbinomial_complex @ A @ K )
% 9.10/5.31          = ( 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 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8770_gbinomial__reduce__nat,axiom,
% 9.10/5.31      ! [K: nat,A: real] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.31       => ( ( gbinomial_real @ A @ K )
% 9.10/5.31          = ( 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 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8771_gbinomial__reduce__nat,axiom,
% 9.10/5.31      ! [K: nat,A: rat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.31       => ( ( gbinomial_rat @ A @ K )
% 9.10/5.31          = ( 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 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8772_arccos,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.10/5.31            & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 9.10/5.31            & ( ( cos_real @ ( arccos @ Y3 ) )
% 9.10/5.31              = Y3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8773_prod_Otriangle__reindex,axiom,
% 9.10/5.31      ! [G: nat > nat > nat,N: nat] :
% 9.10/5.31        ( ( groups4077766827762148844at_nat @ ( produc6842872674320459806at_nat @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups708209901874060359at_nat
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups708209901874060359at_nat
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8774_prod_Otriangle__reindex,axiom,
% 9.10/5.31      ! [G: nat > nat > int,N: nat] :
% 9.10/5.31        ( ( groups4075276357253098568at_int @ ( produc6840382203811409530at_int @ G )
% 9.10/5.31          @ ( collec3392354462482085612at_nat
% 9.10/5.31            @ ( produc6081775807080527818_nat_o
% 9.10/5.31              @ ^ [I4: nat,J3: nat] : ( ord_less_nat @ ( plus_plus_nat @ I4 @ J3 ) @ N ) ) ) )
% 9.10/5.31        = ( groups705719431365010083at_int
% 9.10/5.31          @ ^ [K2: nat] :
% 9.10/5.31              ( groups705719431365010083at_int
% 9.10/5.31              @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K2 @ I4 ) )
% 9.10/5.31              @ ( set_ord_atMost_nat @ K2 ) )
% 9.10/5.31          @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8775_gbinomial__pochhammer,axiom,
% 9.10/5.31      ( gbinomial_complex
% 9.10/5.31      = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K2 ) ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8776_gbinomial__pochhammer,axiom,
% 9.10/5.31      ( gbinomial_rat
% 9.10/5.31      = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A5 ) @ K2 ) ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8777_gbinomial__pochhammer,axiom,
% 9.10/5.31      ( gbinomial_real
% 9.10/5.31      = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A5 ) @ K2 ) ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8778_gbinomial__pochhammer_H,axiom,
% 9.10/5.31      ( gbinomial_complex
% 9.10/5.31      = ( ^ [A5: complex,K2: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8779_gbinomial__pochhammer_H,axiom,
% 9.10/5.31      ( gbinomial_rat
% 9.10/5.31      = ( ^ [A5: rat,K2: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8780_gbinomial__pochhammer_H,axiom,
% 9.10/5.31      ( gbinomial_real
% 9.10/5.31      = ( ^ [A5: real,K2: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8781_arccos__minus__abs,axiom,
% 9.10/5.31      ! [X3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.31       => ( ( arccos @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.31          = ( minus_minus_real @ pi @ ( arccos @ X3 ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8782_gbinomial__sum__lower__neg,axiom,
% 9.10/5.31      ! [A: complex,M: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8783_gbinomial__sum__lower__neg,axiom,
% 9.10/5.31      ! [A: rat,M: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8784_gbinomial__sum__lower__neg,axiom,
% 9.10/5.31      ! [A: real,M: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8785_gbinomial__partial__sum__poly,axiom,
% 9.10/5.31      ! [M: nat,A: complex,X3: complex,Y3: complex] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8786_gbinomial__partial__sum__poly,axiom,
% 9.10/5.31      ! [M: nat,A: rat,X3: rat,Y3: rat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8787_gbinomial__partial__sum__poly,axiom,
% 9.10/5.31      ! [M: nat,A: real,X3: real,Y3: real] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 ) @ ( power_power_real @ ( uminus_uminus_real @ X3 ) @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8788_gbinomial__sum__up__index,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.31        = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8789_gbinomial__sum__up__index,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.31        = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8790_gbinomial__sum__up__index,axiom,
% 9.10/5.31      ! [K: nat,N: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 9.10/5.31          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.10/5.31        = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8791_gbinomial__Suc,axiom,
% 9.10/5.31      ! [A: complex,K: nat] :
% 9.10/5.31        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 9.10/5.31        = ( divide1717551699836669952omplex
% 9.10/5.31          @ ( groups6464643781859351333omplex
% 9.10/5.31            @ ^ [I4: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I4 ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 9.10/5.31          @ ( semiri5044797733671781792omplex @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8792_gbinomial__Suc,axiom,
% 9.10/5.31      ! [A: rat,K: nat] :
% 9.10/5.31        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 9.10/5.31        = ( divide_divide_rat
% 9.10/5.31          @ ( groups73079841787564623at_rat
% 9.10/5.31            @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 9.10/5.31          @ ( semiri773545260158071498ct_rat @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8793_gbinomial__Suc,axiom,
% 9.10/5.31      ! [A: real,K: nat] :
% 9.10/5.31        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 9.10/5.31        = ( divide_divide_real
% 9.10/5.31          @ ( groups129246275422532515t_real
% 9.10/5.31            @ ^ [I4: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 9.10/5.31          @ ( semiri2265585572941072030t_real @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8794_gbinomial__Suc,axiom,
% 9.10/5.31      ! [A: nat,K: nat] :
% 9.10/5.31        ( ( gbinomial_nat @ A @ ( suc @ K ) )
% 9.10/5.31        = ( divide_divide_nat
% 9.10/5.31          @ ( groups708209901874060359at_nat
% 9.10/5.31            @ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 9.10/5.31          @ ( semiri1408675320244567234ct_nat @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8795_gbinomial__Suc,axiom,
% 9.10/5.31      ! [A: int,K: nat] :
% 9.10/5.31        ( ( gbinomial_int @ A @ ( suc @ K ) )
% 9.10/5.31        = ( divide_divide_int
% 9.10/5.31          @ ( groups705719431365010083at_int
% 9.10/5.31            @ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
% 9.10/5.31            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) )
% 9.10/5.31          @ ( semiri1406184849735516958ct_int @ ( suc @ K ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8796_gbinomial__absorption_H,axiom,
% 9.10/5.31      ! [K: nat,A: complex] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.31       => ( ( gbinomial_complex @ A @ K )
% 9.10/5.31          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8797_gbinomial__absorption_H,axiom,
% 9.10/5.31      ! [K: nat,A: rat] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.31       => ( ( gbinomial_rat @ A @ K )
% 9.10/5.31          = ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8798_gbinomial__absorption_H,axiom,
% 9.10/5.31      ! [K: nat,A: real] :
% 9.10/5.31        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.31       => ( ( gbinomial_real @ A @ K )
% 9.10/5.31          = ( 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 ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8799_arccos__le__pi2,axiom,
% 9.10/5.31      ! [Y3: real] :
% 9.10/5.31        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.31       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.31         => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8800_gbinomial__sum__nat__pow2,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8801_gbinomial__sum__nat__pow2,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8802_gbinomial__sum__nat__pow2,axiom,
% 9.10/5.31      ! [M: nat] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K2 ) ) @ K2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K2 ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8803_gbinomial__partial__sum__poly__xpos,axiom,
% 9.10/5.31      ! [M: nat,A: complex,X3: complex,Y3: complex] :
% 9.10/5.31        ( ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( groups2073611262835488442omplex
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K2 ) @ A ) @ one_one_complex ) @ K2 ) @ ( power_power_complex @ X3 @ K2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8804_gbinomial__partial__sum__poly__xpos,axiom,
% 9.10/5.31      ! [M: nat,A: rat,X3: rat,Y3: rat] :
% 9.10/5.31        ( ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( groups2906978787729119204at_rat
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A ) @ one_one_rat ) @ K2 ) @ ( power_power_rat @ X3 @ K2 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 9.10/5.31  
% 9.10/5.31  thf(fact_8805_gbinomial__partial__sum__poly__xpos,axiom,
% 9.10/5.31      ! [M: nat,A: real,X3: real,Y3: real] :
% 9.10/5.31        ( ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.31          @ ( set_ord_atMost_nat @ M ) )
% 9.10/5.31        = ( groups6591440286371151544t_real
% 9.10/5.31          @ ^ [K2: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K2 ) @ A ) @ one_one_real ) @ K2 ) @ ( power_power_real @ X3 @ K2 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_nat @ M @ K2 ) ) )
% 9.10/5.32          @ ( set_ord_atMost_nat @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8806_arcsin__lt__bounded,axiom,
% 9.10/5.32      ! [Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.32       => ( ( ord_less_real @ Y3 @ one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.10/5.32            & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8807_arcsin__lbound,axiom,
% 9.10/5.32      ! [Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.32         => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8808_arcsin__ubound,axiom,
% 9.10/5.32      ! [Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.32         => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8809_arcsin__bounded,axiom,
% 9.10/5.32      ! [Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.10/5.32            & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8810_arcsin__sin,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32         => ( ( arcsin @ ( sin_real @ X3 ) )
% 9.10/5.32            = X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8811_arcsin,axiom,
% 9.10/5.32      ! [Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.10/5.32            & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32            & ( ( sin_real @ ( arcsin @ Y3 ) )
% 9.10/5.32              = Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8812_arcsin__pi,axiom,
% 9.10/5.32      ! [Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.10/5.32            & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 9.10/5.32            & ( ( sin_real @ ( arcsin @ Y3 ) )
% 9.10/5.32              = Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8813_arcsin__le__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 9.10/5.32           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32             => ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ Y3 )
% 9.10/5.32                = ( ord_less_eq_real @ X3 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8814_le__arcsin__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 9.10/5.32           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32             => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X3 ) )
% 9.10/5.32                = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8815_gbinomial__code,axiom,
% 9.10/5.32      ( gbinomial_complex
% 9.10/5.32      = ( ^ [A5: complex,K2: nat] :
% 9.10/5.32            ( if_complex @ ( K2 = zero_zero_nat ) @ one_one_complex
% 9.10/5.32            @ ( divide1717551699836669952omplex
% 9.10/5.32              @ ( set_fo1517530859248394432omplex
% 9.10/5.32                @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L2 ) ) )
% 9.10/5.32                @ zero_zero_nat
% 9.10/5.32                @ ( minus_minus_nat @ K2 @ one_one_nat )
% 9.10/5.32                @ one_one_complex )
% 9.10/5.32              @ ( semiri5044797733671781792omplex @ K2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8816_gbinomial__code,axiom,
% 9.10/5.32      ( gbinomial_rat
% 9.10/5.32      = ( ^ [A5: rat,K2: nat] :
% 9.10/5.32            ( if_rat @ ( K2 = zero_zero_nat ) @ one_one_rat
% 9.10/5.32            @ ( divide_divide_rat
% 9.10/5.32              @ ( set_fo1949268297981939178at_rat
% 9.10/5.32                @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
% 9.10/5.32                @ zero_zero_nat
% 9.10/5.32                @ ( minus_minus_nat @ K2 @ one_one_nat )
% 9.10/5.32                @ one_one_rat )
% 9.10/5.32              @ ( semiri773545260158071498ct_rat @ K2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8817_gbinomial__code,axiom,
% 9.10/5.32      ( gbinomial_real
% 9.10/5.32      = ( ^ [A5: real,K2: nat] :
% 9.10/5.32            ( if_real @ ( K2 = zero_zero_nat ) @ one_one_real
% 9.10/5.32            @ ( divide_divide_real
% 9.10/5.32              @ ( set_fo3111899725591712190t_real
% 9.10/5.32                @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L2 ) ) )
% 9.10/5.32                @ zero_zero_nat
% 9.10/5.32                @ ( minus_minus_nat @ K2 @ one_one_nat )
% 9.10/5.32                @ one_one_real )
% 9.10/5.32              @ ( semiri2265585572941072030t_real @ K2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8818_gchoose__row__sum__weighted,axiom,
% 9.10/5.32      ! [R: complex,M: nat] :
% 9.10/5.32        ( ( groups2073611262835488442omplex
% 9.10/5.32          @ ^ [K2: nat] : ( times_times_complex @ ( gbinomial_complex @ R @ K2 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K2 ) ) )
% 9.10/5.32          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.10/5.32        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R @ ( suc @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8819_gchoose__row__sum__weighted,axiom,
% 9.10/5.32      ! [R: rat,M: nat] :
% 9.10/5.32        ( ( groups2906978787729119204at_rat
% 9.10/5.32          @ ^ [K2: nat] : ( times_times_rat @ ( gbinomial_rat @ R @ K2 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K2 ) ) )
% 9.10/5.32          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.10/5.32        = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R @ ( suc @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8820_gchoose__row__sum__weighted,axiom,
% 9.10/5.32      ! [R: real,M: nat] :
% 9.10/5.32        ( ( groups6591440286371151544t_real
% 9.10/5.32          @ ^ [K2: nat] : ( times_times_real @ ( gbinomial_real @ R @ K2 ) @ ( minus_minus_real @ ( divide_divide_real @ R @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
% 9.10/5.32          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.10/5.32        = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R @ ( suc @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8821_of__nat__code,axiom,
% 9.10/5.32      ( semiri8010041392384452111omplex
% 9.10/5.32      = ( ^ [N2: nat] :
% 9.10/5.32            ( semiri2816024913162550771omplex
% 9.10/5.32            @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 9.10/5.32            @ N2
% 9.10/5.32            @ zero_zero_complex ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8822_of__nat__code,axiom,
% 9.10/5.32      ( semiri681578069525770553at_rat
% 9.10/5.32      = ( ^ [N2: nat] :
% 9.10/5.32            ( semiri7787848453975740701ux_rat
% 9.10/5.32            @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 9.10/5.32            @ N2
% 9.10/5.32            @ zero_zero_rat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8823_of__nat__code,axiom,
% 9.10/5.32      ( semiri1314217659103216013at_int
% 9.10/5.32      = ( ^ [N2: nat] :
% 9.10/5.32            ( semiri8420488043553186161ux_int
% 9.10/5.32            @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 9.10/5.32            @ N2
% 9.10/5.32            @ zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8824_of__nat__code,axiom,
% 9.10/5.32      ( semiri5074537144036343181t_real
% 9.10/5.32      = ( ^ [N2: nat] :
% 9.10/5.32            ( semiri7260567687927622513x_real
% 9.10/5.32            @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 9.10/5.32            @ N2
% 9.10/5.32            @ zero_zero_real ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8825_of__nat__code,axiom,
% 9.10/5.32      ( semiri1316708129612266289at_nat
% 9.10/5.32      = ( ^ [N2: nat] :
% 9.10/5.32            ( semiri8422978514062236437ux_nat
% 9.10/5.32            @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 9.10/5.32            @ N2
% 9.10/5.32            @ zero_zero_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8826_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 9.10/5.32      ! [X3: vEBT_VEBT,Y3: $o] :
% 9.10/5.32        ( ( ( vEBT_VEBT_minNull @ X3 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 9.10/5.32         => ( ( ( X3
% 9.10/5.32                = ( vEBT_Leaf @ $false @ $false ) )
% 9.10/5.32             => ( Y3
% 9.10/5.32               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 9.10/5.32           => ( ! [Uv2: $o] :
% 9.10/5.32                  ( ( X3
% 9.10/5.32                    = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.10/5.32                 => ( ~ Y3
% 9.10/5.32                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 9.10/5.32             => ( ! [Uu2: $o] :
% 9.10/5.32                    ( ( X3
% 9.10/5.32                      = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.10/5.32                   => ( ~ Y3
% 9.10/5.32                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 9.10/5.32               => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.10/5.32                      ( ( X3
% 9.10/5.32                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.10/5.32                     => ( Y3
% 9.10/5.32                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 9.10/5.32                 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.32                        ( ( X3
% 9.10/5.32                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.10/5.32                       => ( ~ Y3
% 9.10/5.32                         => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8827_sin__x__sin__y,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( sums_real
% 9.10/5.32        @ ^ [P5: nat] :
% 9.10/5.32            ( groups6591440286371151544t_real
% 9.10/5.32            @ ^ [N2: nat] :
% 9.10/5.32                ( if_real
% 9.10/5.32                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
% 9.10/5.32                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.10/5.32                @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) )
% 9.10/5.32                @ zero_zero_real )
% 9.10/5.32            @ ( set_ord_atMost_nat @ P5 ) )
% 9.10/5.32        @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8828_sin__x__sin__y,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( sums_complex
% 9.10/5.32        @ ^ [P5: nat] :
% 9.10/5.32            ( groups2073611262835488442omplex
% 9.10/5.32            @ ^ [N2: nat] :
% 9.10/5.32                ( if_complex
% 9.10/5.32                @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
% 9.10/5.32                  & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.10/5.32                @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X3 @ N2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) )
% 9.10/5.32                @ zero_zero_complex )
% 9.10/5.32            @ ( set_ord_atMost_nat @ P5 ) )
% 9.10/5.32        @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8829_Maclaurin__sin__bound,axiom,
% 9.10/5.32      ! [X3: real,N: nat] :
% 9.10/5.32        ( ord_less_eq_real
% 9.10/5.32        @ ( abs_abs_real
% 9.10/5.32          @ ( minus_minus_real @ ( sin_real @ X3 )
% 9.10/5.32            @ ( groups6591440286371151544t_real
% 9.10/5.32              @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.32              @ ( set_ord_lessThan_nat @ N ) ) ) )
% 9.10/5.32        @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X3 ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8830_sums__cos__x__plus__y,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( sums_real
% 9.10/5.32        @ ^ [P5: nat] :
% 9.10/5.32            ( groups6591440286371151544t_real
% 9.10/5.32            @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X3 @ N2 ) ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_real )
% 9.10/5.32            @ ( set_ord_atMost_nat @ P5 ) )
% 9.10/5.32        @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8831_sums__cos__x__plus__y,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( sums_complex
% 9.10/5.32        @ ^ [P5: nat] :
% 9.10/5.32            ( groups2073611262835488442omplex
% 9.10/5.32            @ ^ [N2: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N2 ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X3 @ N2 ) ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ P5 @ N2 ) ) ) @ zero_zero_complex )
% 9.10/5.32            @ ( set_ord_atMost_nat @ P5 ) )
% 9.10/5.32        @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8832_mult__scaleR__left,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( times_times_real @ ( real_V1485227260804924795R_real @ A @ X3 ) @ Y3 )
% 9.10/5.32        = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8833_mult__scaleR__left,axiom,
% 9.10/5.32      ! [A: real,X3: complex,Y3: complex] :
% 9.10/5.32        ( ( times_times_complex @ ( real_V2046097035970521341omplex @ A @ X3 ) @ Y3 )
% 9.10/5.32        = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8834_mult__scaleR__right,axiom,
% 9.10/5.32      ! [X3: real,A: real,Y3: real] :
% 9.10/5.32        ( ( times_times_real @ X3 @ ( real_V1485227260804924795R_real @ A @ Y3 ) )
% 9.10/5.32        = ( real_V1485227260804924795R_real @ A @ ( times_times_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8835_mult__scaleR__right,axiom,
% 9.10/5.32      ! [X3: complex,A: real,Y3: complex] :
% 9.10/5.32        ( ( times_times_complex @ X3 @ ( real_V2046097035970521341omplex @ A @ Y3 ) )
% 9.10/5.32        = ( real_V2046097035970521341omplex @ A @ ( times_times_complex @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8836_inverse__mult__distrib,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( inverse_inverse_real @ ( times_times_real @ A @ B2 ) )
% 9.10/5.32        = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8837_inverse__mult__distrib,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B2 ) )
% 9.10/5.32        = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8838_inverse__mult__distrib,axiom,
% 9.10/5.32      ! [A: rat,B2: rat] :
% 9.10/5.32        ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B2 ) )
% 9.10/5.32        = ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8839_inverse__divide,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B2 ) )
% 9.10/5.32        = ( divide_divide_real @ B2 @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8840_inverse__divide,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.32        = ( divide1717551699836669952omplex @ B2 @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8841_inverse__divide,axiom,
% 9.10/5.32      ! [A: rat,B2: rat] :
% 9.10/5.32        ( ( inverse_inverse_rat @ ( divide_divide_rat @ A @ B2 ) )
% 9.10/5.32        = ( divide_divide_rat @ B2 @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8842_scaleR__scaleR,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: real] :
% 9.10/5.32        ( ( real_V1485227260804924795R_real @ A @ ( real_V1485227260804924795R_real @ B2 @ X3 ) )
% 9.10/5.32        = ( real_V1485227260804924795R_real @ ( times_times_real @ A @ B2 ) @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8843_scaleR__scaleR,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: complex] :
% 9.10/5.32        ( ( real_V2046097035970521341omplex @ A @ ( real_V2046097035970521341omplex @ B2 @ X3 ) )
% 9.10/5.32        = ( real_V2046097035970521341omplex @ ( times_times_real @ A @ B2 ) @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8844_inverse__nonpositive__iff__nonpositive,axiom,
% 9.10/5.32      ! [A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8845_inverse__nonpositive__iff__nonpositive,axiom,
% 9.10/5.32      ! [A: rat] :
% 9.10/5.32        ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.10/5.32        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8846_inverse__nonnegative__iff__nonnegative,axiom,
% 9.10/5.32      ! [A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.10/5.32        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8847_inverse__nonnegative__iff__nonnegative,axiom,
% 9.10/5.32      ! [A: rat] :
% 9.10/5.32        ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.10/5.32        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8848_inverse__positive__iff__positive,axiom,
% 9.10/5.32      ! [A: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
% 9.10/5.32        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8849_inverse__positive__iff__positive,axiom,
% 9.10/5.32      ! [A: rat] :
% 9.10/5.32        ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
% 9.10/5.32        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8850_inverse__negative__iff__negative,axiom,
% 9.10/5.32      ! [A: real] :
% 9.10/5.32        ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8851_inverse__negative__iff__negative,axiom,
% 9.10/5.32      ! [A: rat] :
% 9.10/5.32        ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
% 9.10/5.32        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8852_inverse__less__iff__less__neg,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.32       => ( ( ord_less_real @ B2 @ zero_zero_real )
% 9.10/5.32         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 9.10/5.32            = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8853_inverse__less__iff__less__neg,axiom,
% 9.10/5.32      ! [A: rat,B2: rat] :
% 9.10/5.32        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.10/5.32       => ( ( ord_less_rat @ B2 @ zero_zero_rat )
% 9.10/5.32         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 9.10/5.32            = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8854_inverse__less__iff__less,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.32         => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B2 ) )
% 9.10/5.32            = ( ord_less_real @ B2 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8855_inverse__less__iff__less,axiom,
% 9.10/5.32      ! [A: rat,B2: rat] :
% 9.10/5.32        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.10/5.32       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
% 9.10/5.32         => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B2 ) )
% 9.10/5.32            = ( ord_less_rat @ B2 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8856_of__int__le__iff,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 9.10/5.32        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8857_of__int__le__iff,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 9.10/5.32        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8858_of__int__le__iff,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 9.10/5.32        = ( ord_less_eq_int @ W @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8859_of__int__eq__numeral__iff,axiom,
% 9.10/5.32      ! [Z: int,N: num] :
% 9.10/5.32        ( ( ( ring_17405671764205052669omplex @ Z )
% 9.10/5.32          = ( numera6690914467698888265omplex @ N ) )
% 9.10/5.32        = ( Z
% 9.10/5.32          = ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8860_of__int__eq__numeral__iff,axiom,
% 9.10/5.32      ! [Z: int,N: num] :
% 9.10/5.32        ( ( ( ring_1_of_int_real @ Z )
% 9.10/5.32          = ( numeral_numeral_real @ N ) )
% 9.10/5.32        = ( Z
% 9.10/5.32          = ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8861_of__int__eq__numeral__iff,axiom,
% 9.10/5.32      ! [Z: int,N: num] :
% 9.10/5.32        ( ( ( ring_1_of_int_rat @ Z )
% 9.10/5.32          = ( numeral_numeral_rat @ N ) )
% 9.10/5.32        = ( Z
% 9.10/5.32          = ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8862_of__int__eq__numeral__iff,axiom,
% 9.10/5.32      ! [Z: int,N: num] :
% 9.10/5.32        ( ( ( ring_1_of_int_int @ Z )
% 9.10/5.32          = ( numeral_numeral_int @ N ) )
% 9.10/5.32        = ( Z
% 9.10/5.32          = ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8863_of__int__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 9.10/5.32        = ( numera6690914467698888265omplex @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8864_of__int__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 9.10/5.32        = ( numeral_numeral_real @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8865_of__int__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 9.10/5.32        = ( numeral_numeral_rat @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8866_of__int__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 9.10/5.32        = ( numeral_numeral_int @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8867_scaleR__eq__iff,axiom,
% 9.10/5.32      ! [B2: complex,U: real,A: complex] :
% 9.10/5.32        ( ( ( plus_plus_complex @ B2 @ ( real_V2046097035970521341omplex @ U @ A ) )
% 9.10/5.32          = ( plus_plus_complex @ A @ ( real_V2046097035970521341omplex @ U @ B2 ) ) )
% 9.10/5.32        = ( ( A = B2 )
% 9.10/5.32          | ( U = one_one_real ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8868_sin__npi__int,axiom,
% 9.10/5.32      ! [N: int] :
% 9.10/5.32        ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 9.10/5.32        = zero_zero_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8869_tan__periodic__int,axiom,
% 9.10/5.32      ! [X3: real,I: int] :
% 9.10/5.32        ( ( tan_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 9.10/5.32        = ( tan_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8870_sin__int__2pin,axiom,
% 9.10/5.32      ! [N: int] :
% 9.10/5.32        ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 9.10/5.32        = zero_zero_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8871_cos__int__2pin,axiom,
% 9.10/5.32      ! [N: int] :
% 9.10/5.32        ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 9.10/5.32        = one_one_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8872_cos__npi__int,axiom,
% 9.10/5.32      ! [N: int] :
% 9.10/5.32        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.10/5.32         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 9.10/5.32            = one_one_real ) )
% 9.10/5.32        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.10/5.32         => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 9.10/5.32            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8873_real__scaleR__def,axiom,
% 9.10/5.32      real_V1485227260804924795R_real = times_times_real ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8874_real__sqrt__inverse,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( sqrt @ ( inverse_inverse_real @ X3 ) )
% 9.10/5.32        = ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8875_divide__real__def,axiom,
% 9.10/5.32      ( divide_divide_real
% 9.10/5.32      = ( ^ [X: real,Y: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8876_complex__scaleR,axiom,
% 9.10/5.32      ! [R: real,A: real,B2: real] :
% 9.10/5.32        ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B2 ) )
% 9.10/5.32        = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8877_real__of__int__div4,axiom,
% 9.10/5.32      ! [N: int,X3: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8878_real__of__int__div,axiom,
% 9.10/5.32      ! [D: int,N: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ D @ N )
% 9.10/5.32       => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 9.10/5.32          = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8879_forall__pos__mono__1,axiom,
% 9.10/5.32      ! [P: real > $o,E: real] :
% 9.10/5.32        ( ! [D3: real,E2: real] :
% 9.10/5.32            ( ( ord_less_real @ D3 @ E2 )
% 9.10/5.32           => ( ( P @ D3 )
% 9.10/5.32             => ( P @ E2 ) ) )
% 9.10/5.32       => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.32           => ( P @ E ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8880_forall__pos__mono,axiom,
% 9.10/5.32      ! [P: real > $o,E: real] :
% 9.10/5.32        ( ! [D3: real,E2: real] :
% 9.10/5.32            ( ( ord_less_real @ D3 @ E2 )
% 9.10/5.32           => ( ( P @ D3 )
% 9.10/5.32             => ( P @ E2 ) ) )
% 9.10/5.32       => ( ! [N3: nat] :
% 9.10/5.32              ( ( N3 != zero_zero_nat )
% 9.10/5.32             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.32           => ( P @ E ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8881_real__arch__inverse,axiom,
% 9.10/5.32      ! [E: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ E )
% 9.10/5.32        = ( ? [N2: nat] :
% 9.10/5.32              ( ( N2 != zero_zero_nat )
% 9.10/5.32              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.10/5.32              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8882_int__le__real__less,axiom,
% 9.10/5.32      ( ord_less_eq_int
% 9.10/5.32      = ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8883_sqrt__divide__self__eq,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( divide_divide_real @ ( sqrt @ X3 ) @ X3 )
% 9.10/5.32          = ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8884_int__less__real__le,axiom,
% 9.10/5.32      ( ord_less_int
% 9.10/5.32      = ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8885_ln__inverse,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
% 9.10/5.32          = ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8886_sin__zero__iff__int2,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ( sin_real @ X3 )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32        = ( ? [I4: int] :
% 9.10/5.32              ( X3
% 9.10/5.32              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8887_real__of__int__div__aux,axiom,
% 9.10/5.32      ! [X3: int,D: int] :
% 9.10/5.32        ( ( divide_divide_real @ ( ring_1_of_int_real @ X3 ) @ ( ring_1_of_int_real @ D ) )
% 9.10/5.32        = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X3 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X3 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8888_real__of__int__div2,axiom,
% 9.10/5.32      ! [N: int,X3: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8889_real__of__int__div3,axiom,
% 9.10/5.32      ! [N: int,X3: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X3 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X3 ) ) ) @ one_one_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8890_exp__plus__inverse__exp,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8891_plus__inverse__ge__2,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8892_real__inv__sqrt__pow2,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.32          = ( inverse_inverse_real @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8893_cos__one__2pi__int,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ( cos_real @ X3 )
% 9.10/5.32          = one_one_real )
% 9.10/5.32        = ( ? [X: int] :
% 9.10/5.32              ( X3
% 9.10/5.32              = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8894_tan__cot,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) )
% 9.10/5.32        = ( inverse_inverse_real @ ( tan_real @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8895_real__le__x__sinh,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ord_less_eq_real @ X3 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8896_real__le__abs__sinh,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( inverse_inverse_real @ ( exp_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8897_arccos__cos__eq__abs__2pi,axiom,
% 9.10/5.32      ! [Theta: real] :
% 9.10/5.32        ~ ! [K3: int] :
% 9.10/5.32            ( ( arccos @ ( cos_real @ Theta ) )
% 9.10/5.32           != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8898_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 9.10/5.32      ! [X3: vEBT_VEBT] :
% 9.10/5.32        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 9.10/5.32       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 9.10/5.32         => ( ! [Uv2: $o] :
% 9.10/5.32                ( ( X3
% 9.10/5.32                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.10/5.32               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 9.10/5.32           => ( ! [Uu2: $o] :
% 9.10/5.32                  ( ( X3
% 9.10/5.32                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.10/5.32                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 9.10/5.32             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.10/5.32                    ( ( X3
% 9.10/5.32                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.10/5.32                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8899_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 9.10/5.32      ! [X3: vEBT_VEBT] :
% 9.10/5.32        ( ( vEBT_VEBT_minNull @ X3 )
% 9.10/5.32       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 9.10/5.32         => ( ( ( X3
% 9.10/5.32                = ( vEBT_Leaf @ $false @ $false ) )
% 9.10/5.32             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 9.10/5.32           => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.10/5.32                  ( ( X3
% 9.10/5.32                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.10/5.32                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8900_cos__zero__iff__int,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ( cos_real @ X3 )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32        = ( ? [I4: int] :
% 9.10/5.32              ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 9.10/5.32              & ( X3
% 9.10/5.32                = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8901_sin__zero__iff__int,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ( sin_real @ X3 )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32        = ( ? [I4: int] :
% 9.10/5.32              ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 9.10/5.32              & ( X3
% 9.10/5.32                = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8902_sinh__real__less__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
% 9.10/5.32        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8903_sinh__real__le__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
% 9.10/5.32        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8904_sinh__real__neg__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8905_sinh__real__pos__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
% 9.10/5.32        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8906_sinh__real__nonneg__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X3 ) )
% 9.10/5.32        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8907_sinh__real__nonpos__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( sinh_real @ X3 ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8908_divide__complex__def,axiom,
% 9.10/5.32      ( divide1717551699836669952omplex
% 9.10/5.32      = ( ^ [X: complex,Y: complex] : ( times_times_complex @ X @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8909_complex__inverse,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B2 ) )
% 9.10/5.32        = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B2 ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8910_sinh__ln__real,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( sinh_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.32          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8911_divmod__BitM__2__eq,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 9.10/5.32        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8912_cot__less__zero,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.32         => ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8913_i__even__power,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8914_complex__i__mult__minus,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X3 ) )
% 9.10/5.32        = ( uminus1482373934393186551omplex @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8915_pred__numeral__simps_I2_J,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( pred_numeral @ ( bit0 @ K ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8916_divide__i,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( divide1717551699836669952omplex @ X3 @ imaginary_unit )
% 9.10/5.32        = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8917_i__squared,axiom,
% 9.10/5.32      ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 9.10/5.32      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8918_cot__npi,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 9.10/5.32        = zero_zero_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8919_divide__numeral__i,axiom,
% 9.10/5.32      ! [Z: complex,N: num] :
% 9.10/5.32        ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 9.10/5.32        = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8920_cot__periodic,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( cot_real @ ( plus_plus_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.10/5.32        = ( cot_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8921_power2__i,axiom,
% 9.10/5.32      ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.32      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8922_sinh__less__cosh__real,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8923_sinh__le__cosh__real,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_eq_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8924_semiring__norm_I26_J,axiom,
% 9.10/5.32      ( ( bitM @ one )
% 9.10/5.32      = one ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8925_complex__i__not__numeral,axiom,
% 9.10/5.32      ! [W: num] :
% 9.10/5.32        ( imaginary_unit
% 9.10/5.32       != ( numera6690914467698888265omplex @ W ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8926_cosh__real__pos,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8927_cosh__real__nonneg,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8928_cosh__real__nonneg__le__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 9.10/5.32            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8929_cosh__real__nonpos__le__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 9.10/5.32            = ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8930_cosh__real__ge__1,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8931_semiring__norm_I27_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bitM @ ( bit0 @ N ) )
% 9.10/5.32        = ( bit1 @ ( bitM @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8932_semiring__norm_I28_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bitM @ ( bit1 @ N ) )
% 9.10/5.32        = ( bit1 @ ( bit0 @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8933_i__times__eq__iff,axiom,
% 9.10/5.32      ! [W: complex,Z: complex] :
% 9.10/5.32        ( ( ( times_times_complex @ imaginary_unit @ W )
% 9.10/5.32          = Z )
% 9.10/5.32        = ( W
% 9.10/5.32          = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8934_complex__i__not__neg__numeral,axiom,
% 9.10/5.32      ! [W: num] :
% 9.10/5.32        ( imaginary_unit
% 9.10/5.32       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8935_cosh__real__strict__mono,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.32         => ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8936_cosh__real__nonneg__less__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.32         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 9.10/5.32            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8937_cosh__real__nonpos__less__iff,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.32       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.10/5.32         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 9.10/5.32            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8938_arcosh__cosh__real,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( arcosh_real @ ( cosh_real @ X3 ) )
% 9.10/5.32          = X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8939_eval__nat__numeral_I2_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 9.10/5.32        = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8940_one__plus__BitM,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 9.10/5.32        = ( bit0 @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8941_BitM__plus__one,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 9.10/5.32        = ( bit0 @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8942_Complex__mult__i,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( times_times_complex @ ( complex2 @ A @ B2 ) @ imaginary_unit )
% 9.10/5.32        = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8943_i__mult__Complex,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B2 ) )
% 9.10/5.32        = ( complex2 @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8944_cosh__ln__real,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( cosh_real @ ( ln_ln_real @ X3 ) )
% 9.10/5.32          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8945_cot__gt__zero,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8946_tan__cot_H,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) )
% 9.10/5.32        = ( cot_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8947_Arg__minus__ii,axiom,
% 9.10/5.32      ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 9.10/5.32      = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8948_Arg__ii,axiom,
% 9.10/5.32      ( ( arg @ imaginary_unit )
% 9.10/5.32      = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8949_cis__minus__pi__half,axiom,
% 9.10/5.32      ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.32      = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8950_log__base__10__eq1,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.32          = ( 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 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8951_log__eq__one,axiom,
% 9.10/5.32      ! [A: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( log @ A @ A )
% 9.10/5.32            = one_one_real ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8952_log__less__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.32           => ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 9.10/5.32              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8953_log__less__one__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
% 9.10/5.32            = ( ord_less_real @ X3 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8954_one__less__log__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
% 9.10/5.32            = ( ord_less_real @ A @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8955_log__less__zero__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 9.10/5.32            = ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8956_zero__less__log__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 9.10/5.32            = ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8957_zero__le__log__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 9.10/5.32            = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8958_log__le__zero__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 9.10/5.32            = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8959_one__le__log__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
% 9.10/5.32            = ( ord_less_eq_real @ A @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8960_log__le__one__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
% 9.10/5.32            = ( ord_less_eq_real @ X3 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8961_log__le__cancel__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.32           => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 9.10/5.32              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8962_log__pow__cancel,axiom,
% 9.10/5.32      ! [A: real,B2: nat] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( log @ A @ ( power_power_real @ A @ B2 ) )
% 9.10/5.32            = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8963_cis__pi__half,axiom,
% 9.10/5.32      ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32      = imaginary_unit ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8964_cis__2pi,axiom,
% 9.10/5.32      ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.10/5.32      = one_one_complex ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8965_cis__mult,axiom,
% 9.10/5.32      ! [A: real,B2: real] :
% 9.10/5.32        ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B2 ) )
% 9.10/5.32        = ( cis @ ( plus_plus_real @ A @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8966_log__base__change,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( log @ B2 @ X3 )
% 9.10/5.32            = ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8967_log__of__power__eq,axiom,
% 9.10/5.32      ! [M: nat,B2: real,N: nat] :
% 9.10/5.32        ( ( ( semiri5074537144036343181t_real @ M )
% 9.10/5.32          = ( power_power_real @ B2 @ N ) )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ( semiri5074537144036343181t_real @ N )
% 9.10/5.32            = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8968_less__log__of__power,axiom,
% 9.10/5.32      ! [B2: real,N: nat,M: real] :
% 9.10/5.32        ( ( ord_less_real @ ( power_power_real @ B2 @ N ) @ M )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8969_DeMoivre,axiom,
% 9.10/5.32      ! [A: real,N: nat] :
% 9.10/5.32        ( ( power_power_complex @ ( cis @ A ) @ N )
% 9.10/5.32        = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8970_log__mult,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.32             => ( ( log @ A @ ( times_times_real @ X3 @ Y3 ) )
% 9.10/5.32                = ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8971_log__divide,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.32             => ( ( log @ A @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.10/5.32                = ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8972_le__log__of__power,axiom,
% 9.10/5.32      ! [B2: real,N: nat,M: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N ) @ M )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8973_log__base__pow,axiom,
% 9.10/5.32      ! [A: real,N: nat,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( log @ ( power_power_real @ A @ N ) @ X3 )
% 9.10/5.32          = ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8974_log__nat__power,axiom,
% 9.10/5.32      ! [X3: real,B2: real,N: nat] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( log @ B2 @ ( power_power_real @ X3 @ N ) )
% 9.10/5.32          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8975_log__inverse,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
% 9.10/5.32              = ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8976_log2__of__power__eq,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( M
% 9.10/5.32          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32       => ( ( semiri5074537144036343181t_real @ N )
% 9.10/5.32          = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8977_log__of__power__less,axiom,
% 9.10/5.32      ! [M: nat,B2: real,N: nat] :
% 9.10/5.32        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.32           => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8978_log__eq__div__ln__mult__log,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.32           => ( ( B2 != one_one_real )
% 9.10/5.32             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32               => ( ( log @ A @ X3 )
% 9.10/5.32                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A ) ) @ ( log @ B2 @ X3 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8979_Arg__bounded,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 9.10/5.32        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8980_log__of__power__le,axiom,
% 9.10/5.32      ! [M: nat,B2: real,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N ) )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.32           => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8981_less__log2__of__power,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.10/5.32       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8982_le__log2__of__power,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.10/5.32       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8983_log2__of__power__less,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.32         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8984_log2__of__power__le,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.32         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8985_log__base__10__eq2,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.10/5.32          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8986_ceiling__log__nat__eq__powr__iff,axiom,
% 9.10/5.32      ! [B2: nat,K: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.32       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.32         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.10/5.32              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 9.10/5.32            = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 9.10/5.32              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8987_bij__betw__roots__unity,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.32       => ( bij_betw_nat_complex
% 9.10/5.32          @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.32          @ ( set_ord_lessThan_nat @ N )
% 9.10/5.32          @ ( collect_complex
% 9.10/5.32            @ ^ [Z2: complex] :
% 9.10/5.32                ( ( power_power_complex @ Z2 @ N )
% 9.10/5.32                = one_one_complex ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8988_ceiling__log__nat__eq__if,axiom,
% 9.10/5.32      ! [B2: nat,N: nat,K: nat] :
% 9.10/5.32        ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 9.10/5.32       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.10/5.32         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.32           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.10/5.32              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8989_ceiling__log2__div2,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.32       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.32          = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8990_ceiling__divide__eq__div__numeral,axiom,
% 9.10/5.32      ! [A: num,B2: num] :
% 9.10/5.32        ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8991_ceiling__minus__divide__eq__div__numeral,axiom,
% 9.10/5.32      ! [A: num,B2: num] :
% 9.10/5.32        ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8992_ceiling__log__eq__powr__iff,axiom,
% 9.10/5.32      ! [X3: real,B2: real,K: nat] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X3 ) )
% 9.10/5.32              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 9.10/5.32            = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X3 )
% 9.10/5.32              & ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8993_floor__log__nat__eq__powr__iff,axiom,
% 9.10/5.32      ! [B2: nat,K: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.32       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.10/5.32         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.10/5.32              = ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.32            = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 9.10/5.32              & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8994_int__ge__less__than2__def,axiom,
% 9.10/5.32      ( int_ge_less_than2
% 9.10/5.32      = ( ^ [D2: int] :
% 9.10/5.32            ( collec213857154873943460nt_int
% 9.10/5.32            @ ( produc4947309494688390418_int_o
% 9.10/5.32              @ ^ [Z6: int,Z2: int] :
% 9.10/5.32                  ( ( ord_less_eq_int @ D2 @ Z2 )
% 9.10/5.32                  & ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8995_int__ge__less__than__def,axiom,
% 9.10/5.32      ( int_ge_less_than
% 9.10/5.32      = ( ^ [D2: int] :
% 9.10/5.32            ( collec213857154873943460nt_int
% 9.10/5.32            @ ( produc4947309494688390418_int_o
% 9.10/5.32              @ ^ [Z6: int,Z2: int] :
% 9.10/5.32                  ( ( ord_less_eq_int @ D2 @ Z6 )
% 9.10/5.32                  & ( ord_less_int @ Z6 @ Z2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8996_powr__gt__zero,axiom,
% 9.10/5.32      ! [X3: real,A: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
% 9.10/5.32        = ( X3 != zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8997_powr__nonneg__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( powr_real @ A @ X3 ) @ zero_zero_real )
% 9.10/5.32        = ( A = zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8998_powr__less__cancel__iff,axiom,
% 9.10/5.32      ! [X3: real,A: real,B2: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
% 9.10/5.32          = ( ord_less_real @ A @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_8999_powr__eq__one__iff,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ A )
% 9.10/5.32       => ( ( ( powr_real @ A @ X3 )
% 9.10/5.32            = one_one_real )
% 9.10/5.32          = ( X3 = zero_zero_real ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9000_powr__one,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( powr_real @ X3 @ one_one_real )
% 9.10/5.32          = X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9001_powr__one__gt__zero__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ( powr_real @ X3 @ one_one_real )
% 9.10/5.32          = X3 )
% 9.10/5.32        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9002_powr__le__cancel__iff,axiom,
% 9.10/5.32      ! [X3: real,A: real,B2: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
% 9.10/5.32          = ( ord_less_eq_real @ A @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9003_numeral__powr__numeral__real,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.10/5.32        = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9004_powr__log__cancel,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( powr_real @ A @ ( log @ A @ X3 ) )
% 9.10/5.32              = X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9005_log__powr__cancel,axiom,
% 9.10/5.32      ! [A: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 9.10/5.32            = Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9006_floor__divide__eq__div__numeral,axiom,
% 9.10/5.32      ! [A: num,B2: num] :
% 9.10/5.32        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) )
% 9.10/5.32        = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9007_powr__numeral,axiom,
% 9.10/5.32      ! [X3: real,N: num] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( powr_real @ X3 @ ( numeral_numeral_real @ N ) )
% 9.10/5.32          = ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9008_floor__one__divide__eq__div__numeral,axiom,
% 9.10/5.32      ! [B2: num] :
% 9.10/5.32        ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) )
% 9.10/5.32        = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9009_floor__minus__divide__eq__div__numeral,axiom,
% 9.10/5.32      ! [A: num,B2: num] :
% 9.10/5.32        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B2 ) ) ) )
% 9.10/5.32        = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9010_square__powr__half,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( powr_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32        = ( abs_abs_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9011_floor__minus__one__divide__eq__div__numeral,axiom,
% 9.10/5.32      ! [B2: num] :
% 9.10/5.32        ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) ) )
% 9.10/5.32        = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9012_powr__powr,axiom,
% 9.10/5.32      ! [X3: real,A: real,B2: real] :
% 9.10/5.32        ( ( powr_real @ ( powr_real @ X3 @ A ) @ B2 )
% 9.10/5.32        = ( powr_real @ X3 @ ( times_times_real @ A @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9013_powr__less__mono2__neg,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ A @ zero_zero_real )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.32           => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9014_powr__non__neg,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9015_powr__ge__pzero,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X3 @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9016_powr__mono2,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.32           => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9017_powr__less__cancel,axiom,
% 9.10/5.32      ! [X3: real,A: real,B2: real] :
% 9.10/5.32        ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.32         => ( ord_less_real @ A @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9018_powr__less__mono,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ A @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.32         => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9019_powr__mono,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.32       => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.32         => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9020_powr__less__mono2,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.32           => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9021_powr__mono2_H,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.32           => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9022_powr__inj,axiom,
% 9.10/5.32      ! [A: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( A != one_one_real )
% 9.10/5.32         => ( ( ( powr_real @ A @ X3 )
% 9.10/5.32              = ( powr_real @ A @ Y3 ) )
% 9.10/5.32            = ( X3 = Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9023_gr__one__powr,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.10/5.32         => ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9024_powr__le1,axiom,
% 9.10/5.32      ! [A: real,X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.32           => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ one_one_real ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9025_powr__mono__both,axiom,
% 9.10/5.32      ! [A: real,B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.32       => ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.32         => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.32           => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.32             => ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ B2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9026_ge__one__powr__ge__zero,axiom,
% 9.10/5.32      ! [X3: real,A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.10/5.32         => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X3 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9027_powr__divide,axiom,
% 9.10/5.32      ! [X3: real,Y3: real,A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.32         => ( ( powr_real @ ( divide_divide_real @ X3 @ Y3 ) @ A )
% 9.10/5.32            = ( divide_divide_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9028_powr__mult,axiom,
% 9.10/5.32      ! [X3: real,Y3: real,A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.32         => ( ( powr_real @ ( times_times_real @ X3 @ Y3 ) @ A )
% 9.10/5.32            = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9029_inverse__powr,axiom,
% 9.10/5.32      ! [Y3: real,A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.32       => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 9.10/5.32          = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9030_divide__powr__uminus,axiom,
% 9.10/5.32      ! [A: real,B2: real,C: real] :
% 9.10/5.32        ( ( divide_divide_real @ A @ ( powr_real @ B2 @ C ) )
% 9.10/5.32        = ( times_times_real @ A @ ( powr_real @ B2 @ ( uminus_uminus_real @ C ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9031_ln__powr,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( X3 != zero_zero_real )
% 9.10/5.32       => ( ( ln_ln_real @ ( powr_real @ X3 @ Y3 ) )
% 9.10/5.32          = ( times_times_real @ Y3 @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9032_log__powr,axiom,
% 9.10/5.32      ! [X3: real,B2: real,Y3: real] :
% 9.10/5.32        ( ( X3 != zero_zero_real )
% 9.10/5.32       => ( ( log @ B2 @ ( powr_real @ X3 @ Y3 ) )
% 9.10/5.32          = ( times_times_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9033_floor__log__eq__powr__iff,axiom,
% 9.10/5.32      ! [X3: real,B2: real,K: int] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32         => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X3 ) )
% 9.10/5.32              = K )
% 9.10/5.32            = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X3 )
% 9.10/5.32              & ( ord_less_real @ X3 @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9034_powr__realpow,axiom,
% 9.10/5.32      ! [X3: real,N: nat] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N ) )
% 9.10/5.32          = ( power_power_real @ X3 @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9035_powr__less__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X3 )
% 9.10/5.32            = ( ord_less_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9036_less__powr__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ X3 @ ( powr_real @ B2 @ Y3 ) )
% 9.10/5.32            = ( ord_less_real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9037_log__less__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ ( log @ B2 @ X3 ) @ Y3 )
% 9.10/5.32            = ( ord_less_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9038_less__log__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_real @ Y3 @ ( log @ B2 @ X3 ) )
% 9.10/5.32            = ( ord_less_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9039_floor__eq,axiom,
% 9.10/5.32      ! [N: int,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.10/5.32         => ( ( archim6058952711729229775r_real @ X3 )
% 9.10/5.32            = N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9040_real__of__int__floor__add__one__gt,axiom,
% 9.10/5.32      ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9041_real__of__int__floor__add__one__ge,axiom,
% 9.10/5.32      ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9042_real__of__int__floor__gt__diff__one,axiom,
% 9.10/5.32      ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9043_real__of__int__floor__ge__diff__one,axiom,
% 9.10/5.32      ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9044_powr__neg__one,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 9.10/5.32          = ( divide_divide_real @ one_one_real @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9045_powr__mult__base,axiom,
% 9.10/5.32      ! [X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( times_times_real @ X3 @ ( powr_real @ X3 @ Y3 ) )
% 9.10/5.32          = ( powr_real @ X3 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9046_le__log__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X3 ) )
% 9.10/5.32            = ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9047_log__le__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( log @ B2 @ X3 ) @ Y3 )
% 9.10/5.32            = ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9048_le__powr__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ X3 @ ( powr_real @ B2 @ Y3 ) )
% 9.10/5.32            = ( ord_less_eq_real @ ( log @ B2 @ X3 ) @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9049_powr__le__iff,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y3 ) @ X3 )
% 9.10/5.32            = ( ord_less_eq_real @ Y3 @ ( log @ B2 @ X3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9050_floor__eq2,axiom,
% 9.10/5.32      ! [N: int,X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.10/5.32         => ( ( archim6058952711729229775r_real @ X3 )
% 9.10/5.32            = N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9051_floor__divide__real__eq__div,axiom,
% 9.10/5.32      ! [B2: int,A: real] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ B2 )
% 9.10/5.32       => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B2 ) ) )
% 9.10/5.32          = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9052_ln__powr__bound,axiom,
% 9.10/5.32      ! [X3: real,A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32         => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9053_ln__powr__bound2,axiom,
% 9.10/5.32      ! [X3: real,A: real] :
% 9.10/5.32        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.32         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9054_log__add__eq__powr,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.32       => ( ( B2 != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( plus_plus_real @ ( log @ B2 @ X3 ) @ Y3 )
% 9.10/5.32              = ( log @ B2 @ ( times_times_real @ X3 @ ( powr_real @ B2 @ Y3 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9055_add__log__eq__powr,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.32       => ( ( B2 != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( plus_plus_real @ Y3 @ ( log @ B2 @ X3 ) )
% 9.10/5.32              = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9056_minus__log__eq__powr,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.32       => ( ( B2 != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( minus_minus_real @ Y3 @ ( log @ B2 @ X3 ) )
% 9.10/5.32              = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9057_log__minus__eq__powr,axiom,
% 9.10/5.32      ! [B2: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.32       => ( ( B2 != one_one_real )
% 9.10/5.32         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32           => ( ( minus_minus_real @ ( log @ B2 @ X3 ) @ Y3 )
% 9.10/5.32              = ( log @ B2 @ ( times_times_real @ X3 @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9058_powr__half__sqrt,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32          = ( sqrt @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9059_powr__neg__numeral,axiom,
% 9.10/5.32      ! [X3: real,N: num] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.10/5.32          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9060_floor__log2__div2,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.32       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.10/5.32          = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9061_floor__log__nat__eq__if,axiom,
% 9.10/5.32      ! [B2: nat,N: nat,K: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
% 9.10/5.32       => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.10/5.32         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
% 9.10/5.32           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.10/5.32              = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9062_upto_Opinduct,axiom,
% 9.10/5.32      ! [A0: int,A12: int,P: int > int > $o] :
% 9.10/5.32        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 9.10/5.32       => ( ! [I3: int,J2: int] :
% 9.10/5.32              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 9.10/5.32             => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 9.10/5.32                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 9.10/5.32               => ( P @ I3 @ J2 ) ) )
% 9.10/5.32         => ( P @ A0 @ A12 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9063_exp__pi__i,axiom,
% 9.10/5.32      ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 9.10/5.32      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9064_exp__pi__i_H,axiom,
% 9.10/5.32      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 9.10/5.32      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9065_exp__two__pi__i,axiom,
% 9.10/5.32      ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 9.10/5.32      = one_one_complex ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9066_exp__two__pi__i_H,axiom,
% 9.10/5.32      ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 9.10/5.32      = one_one_complex ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9067_complex__exp__exists,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32      ? [A3: complex,R3: real] :
% 9.10/5.32        ( Z
% 9.10/5.32        = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( exp_complex @ A3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9068_complex__of__real__mult__Complex,axiom,
% 9.10/5.32      ! [R: real,X3: real,Y3: real] :
% 9.10/5.32        ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X3 @ Y3 ) )
% 9.10/5.32        = ( complex2 @ ( times_times_real @ R @ X3 ) @ ( times_times_real @ R @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9069_Complex__mult__complex__of__real,axiom,
% 9.10/5.32      ! [X3: real,Y3: real,R: real] :
% 9.10/5.32        ( ( times_times_complex @ ( complex2 @ X3 @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
% 9.10/5.32        = ( complex2 @ ( times_times_real @ X3 @ R ) @ ( times_times_real @ Y3 @ R ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9070_cis__conv__exp,axiom,
% 9.10/5.32      ( cis
% 9.10/5.32      = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9071_i__complex__of__real,axiom,
% 9.10/5.32      ! [R: real] :
% 9.10/5.32        ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
% 9.10/5.32        = ( complex2 @ zero_zero_real @ R ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9072_complex__of__real__i,axiom,
% 9.10/5.32      ! [R: real] :
% 9.10/5.32        ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
% 9.10/5.32        = ( complex2 @ zero_zero_real @ R ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9073_Complex__eq,axiom,
% 9.10/5.32      ( complex2
% 9.10/5.32      = ( ^ [A5: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A5 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9074_complex__split__polar,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32      ? [R3: real,A3: real] :
% 9.10/5.32        ( Z
% 9.10/5.32        = ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9075_cmod__unit__one,axiom,
% 9.10/5.32      ! [A: real] :
% 9.10/5.32        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 9.10/5.32        = one_one_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9076_cmod__complex__polar,axiom,
% 9.10/5.32      ! [R: real,A: real] :
% 9.10/5.32        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 9.10/5.32        = ( abs_abs_real @ R ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9077_csqrt__ii,axiom,
% 9.10/5.32      ( ( csqrt @ imaginary_unit )
% 9.10/5.32      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9078_arctan__def,axiom,
% 9.10/5.32      ( arctan
% 9.10/5.32      = ( ^ [Y: real] :
% 9.10/5.32            ( the_real
% 9.10/5.32            @ ^ [X: real] :
% 9.10/5.32                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.10/5.32                & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32                & ( ( tan_real @ X )
% 9.10/5.32                  = Y ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9079_arcsin__def,axiom,
% 9.10/5.32      ( arcsin
% 9.10/5.32      = ( ^ [Y: real] :
% 9.10/5.32            ( the_real
% 9.10/5.32            @ ^ [X: real] :
% 9.10/5.32                ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.10/5.32                & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32                & ( ( sin_real @ X )
% 9.10/5.32                  = Y ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9080_even__set__encode__iff,axiom,
% 9.10/5.32      ! [A4: set_nat] :
% 9.10/5.32        ( ( finite_finite_nat @ A4 )
% 9.10/5.32       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
% 9.10/5.32          = ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9081_power2__csqrt,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.32        = Z ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9082_ln__neg__is__const,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.32       => ( ( ln_ln_real @ X3 )
% 9.10/5.32          = ( the_real
% 9.10/5.32            @ ^ [X: real] : $false ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9083_of__real__sqrt,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( real_V4546457046886955230omplex @ ( sqrt @ X3 ) )
% 9.10/5.32          = ( csqrt @ ( real_V4546457046886955230omplex @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9084_arccos__def,axiom,
% 9.10/5.32      ( arccos
% 9.10/5.32      = ( ^ [Y: real] :
% 9.10/5.32            ( the_real
% 9.10/5.32            @ ^ [X: real] :
% 9.10/5.32                ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.10/5.32                & ( ord_less_eq_real @ X @ pi )
% 9.10/5.32                & ( ( cos_real @ X )
% 9.10/5.32                  = Y ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9085_set__encode__def,axiom,
% 9.10/5.32      ( nat_set_encode
% 9.10/5.32      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9086_pi__half,axiom,
% 9.10/5.32      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.32      = ( the_real
% 9.10/5.32        @ ^ [X: real] :
% 9.10/5.32            ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.10/5.32            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.32            & ( ( cos_real @ X )
% 9.10/5.32              = zero_zero_real ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9087_pi__def,axiom,
% 9.10/5.32      ( pi
% 9.10/5.32      = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.10/5.32        @ ( the_real
% 9.10/5.32          @ ^ [X: real] :
% 9.10/5.32              ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.10/5.32              & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.10/5.32              & ( ( cos_real @ X )
% 9.10/5.32                = zero_zero_real ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9088_modulo__int__unfold,axiom,
% 9.10/5.32      ! [L: int,K: int,N: nat,M: nat] :
% 9.10/5.32        ( ( ( ( ( sgn_sgn_int @ L )
% 9.10/5.32              = zero_zero_int )
% 9.10/5.32            | ( ( sgn_sgn_int @ K )
% 9.10/5.32              = zero_zero_int )
% 9.10/5.32            | ( N = zero_zero_nat ) )
% 9.10/5.32         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.32            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 9.10/5.32        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.10/5.32                = zero_zero_int )
% 9.10/5.32              | ( ( sgn_sgn_int @ K )
% 9.10/5.32                = zero_zero_int )
% 9.10/5.32              | ( N = zero_zero_nat ) )
% 9.10/5.32         => ( ( ( ( sgn_sgn_int @ K )
% 9.10/5.32                = ( sgn_sgn_int @ L ) )
% 9.10/5.32             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.32                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 9.10/5.32            & ( ( ( sgn_sgn_int @ K )
% 9.10/5.32               != ( sgn_sgn_int @ L ) )
% 9.10/5.32             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.32                = ( times_times_int @ ( sgn_sgn_int @ L )
% 9.10/5.32                  @ ( minus_minus_int
% 9.10/5.32                    @ ( semiri1314217659103216013at_int
% 9.10/5.32                      @ ( times_times_nat @ N
% 9.10/5.32                        @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32                          @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 9.10/5.32                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9089_num_Osize__gen_I3_J,axiom,
% 9.10/5.32      ! [X32: num] :
% 9.10/5.32        ( ( size_num @ ( bit1 @ X32 ) )
% 9.10/5.32        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9090_powr__int,axiom,
% 9.10/5.32      ! [X3: real,I: int] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.10/5.32           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 9.10/5.32              = ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
% 9.10/5.32          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 9.10/5.32           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 9.10/5.32              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9091_mask__nat__positive__iff,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.10/5.32        = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9092_nat__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 9.10/5.32        = ( numeral_numeral_nat @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9093_nat__0__iff,axiom,
% 9.10/5.32      ! [I: int] :
% 9.10/5.32        ( ( ( nat2 @ I )
% 9.10/5.32          = zero_zero_nat )
% 9.10/5.32        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9094_nat__le__0,axiom,
% 9.10/5.32      ! [Z: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 9.10/5.32       => ( ( nat2 @ Z )
% 9.10/5.32          = zero_zero_nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9095_zless__nat__conj,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.10/5.32        = ( ( ord_less_int @ zero_zero_int @ Z )
% 9.10/5.32          & ( ord_less_int @ W @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9096_nat__neg__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.10/5.32        = zero_zero_nat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9097_int__nat__eq,axiom,
% 9.10/5.32      ! [Z: int] :
% 9.10/5.32        ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.10/5.32            = Z ) )
% 9.10/5.32        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.10/5.32            = zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9098_sgn__mult__dvd__iff,axiom,
% 9.10/5.32      ! [R: int,L: int,K: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L ) @ K )
% 9.10/5.32        = ( ( dvd_dvd_int @ L @ K )
% 9.10/5.32          & ( ( R = zero_zero_int )
% 9.10/5.32           => ( K = zero_zero_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9099_mult__sgn__dvd__iff,axiom,
% 9.10/5.32      ! [L: int,R: int,K: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R ) ) @ K )
% 9.10/5.32        = ( ( dvd_dvd_int @ L @ K )
% 9.10/5.32          & ( ( R = zero_zero_int )
% 9.10/5.32           => ( K = zero_zero_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9100_dvd__sgn__mult__iff,axiom,
% 9.10/5.32      ! [L: int,R: int,K: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
% 9.10/5.32        = ( ( dvd_dvd_int @ L @ K )
% 9.10/5.32          | ( R = zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9101_dvd__mult__sgn__iff,axiom,
% 9.10/5.32      ! [L: int,K: int,R: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
% 9.10/5.32        = ( ( dvd_dvd_int @ L @ K )
% 9.10/5.32          | ( R = zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9102_zero__less__nat__eq,axiom,
% 9.10/5.32      ! [Z: int] :
% 9.10/5.32        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 9.10/5.32        = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9103_diff__nat__numeral,axiom,
% 9.10/5.32      ! [V: num,V3: num] :
% 9.10/5.32        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 9.10/5.32        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9104_nat__eq__numeral__power__cancel__iff,axiom,
% 9.10/5.32      ! [Y3: int,X3: num,N: nat] :
% 9.10/5.32        ( ( ( nat2 @ Y3 )
% 9.10/5.32          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 9.10/5.32        = ( Y3
% 9.10/5.32          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9105_numeral__power__eq__nat__cancel__iff,axiom,
% 9.10/5.32      ! [X3: num,N: nat,Y3: int] :
% 9.10/5.32        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
% 9.10/5.32          = ( nat2 @ Y3 ) )
% 9.10/5.32        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
% 9.10/5.32          = Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9106_nat__ceiling__le__eq,axiom,
% 9.10/5.32      ! [X3: real,A: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
% 9.10/5.32        = ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9107_one__less__nat__eq,axiom,
% 9.10/5.32      ! [Z: int] :
% 9.10/5.32        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 9.10/5.32        = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9108_nat__numeral__diff__1,axiom,
% 9.10/5.32      ! [V: num] :
% 9.10/5.32        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 9.10/5.32        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9109_nat__less__numeral__power__cancel__iff,axiom,
% 9.10/5.32      ! [A: int,X3: num,N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 9.10/5.32        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9110_numeral__power__less__nat__cancel__iff,axiom,
% 9.10/5.32      ! [X3: num,N: nat,A: int] :
% 9.10/5.32        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
% 9.10/5.32        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9111_nat__le__numeral__power__cancel__iff,axiom,
% 9.10/5.32      ! [A: int,X3: num,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
% 9.10/5.32        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9112_numeral__power__le__nat__cancel__iff,axiom,
% 9.10/5.32      ! [X3: num,N: nat,A: int] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) @ ( nat2 @ A ) )
% 9.10/5.32        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9113_less__eq__mask,axiom,
% 9.10/5.32      ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9114_nat__mask__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 9.10/5.32        = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9115_nat__zero__as__int,axiom,
% 9.10/5.32      ( zero_zero_nat
% 9.10/5.32      = ( nat2 @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9116_nat__numeral__as__int,axiom,
% 9.10/5.32      ( numeral_numeral_nat
% 9.10/5.32      = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9117_nat__mono,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.10/5.32       => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9118_int__sgnE,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ~ ! [N3: nat,L4: int] :
% 9.10/5.32            ( K
% 9.10/5.32           != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9119_ex__nat,axiom,
% 9.10/5.32      ( ( ^ [P2: nat > $o] :
% 9.10/5.32          ? [X4: nat] : ( P2 @ X4 ) )
% 9.10/5.32      = ( ^ [P3: nat > $o] :
% 9.10/5.32          ? [X: int] :
% 9.10/5.32            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.10/5.32            & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9120_all__nat,axiom,
% 9.10/5.32      ( ( ^ [P2: nat > $o] :
% 9.10/5.32          ! [X4: nat] : ( P2 @ X4 ) )
% 9.10/5.32      = ( ^ [P3: nat > $o] :
% 9.10/5.32          ! [X: int] :
% 9.10/5.32            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.10/5.32           => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9121_eq__nat__nat__iff,axiom,
% 9.10/5.32      ! [Z: int,Z7: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 9.10/5.32         => ( ( ( nat2 @ Z )
% 9.10/5.32              = ( nat2 @ Z7 ) )
% 9.10/5.32            = ( Z = Z7 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9122_nat__one__as__int,axiom,
% 9.10/5.32      ( one_one_nat
% 9.10/5.32      = ( nat2 @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9123_unset__bit__nat__def,axiom,
% 9.10/5.32      ( bit_se4205575877204974255it_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9124_mask__nonnegative__int,axiom,
% 9.10/5.32      ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9125_not__mask__negative__int,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9126_nat__mono__iff,axiom,
% 9.10/5.32      ! [Z: int,W: int] :
% 9.10/5.32        ( ( ord_less_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.10/5.32          = ( ord_less_int @ W @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9127_zless__nat__eq__int__zless,axiom,
% 9.10/5.32      ! [M: nat,Z: int] :
% 9.10/5.32        ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 9.10/5.32        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9128_nat__le__iff,axiom,
% 9.10/5.32      ! [X3: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N )
% 9.10/5.32        = ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9129_nat__0__le,axiom,
% 9.10/5.32      ! [Z: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.10/5.32          = Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9130_int__eq__iff,axiom,
% 9.10/5.32      ! [M: nat,Z: int] :
% 9.10/5.32        ( ( ( semiri1314217659103216013at_int @ M )
% 9.10/5.32          = Z )
% 9.10/5.32        = ( ( M
% 9.10/5.32            = ( nat2 @ Z ) )
% 9.10/5.32          & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9131_nat__int__add,axiom,
% 9.10/5.32      ! [A: nat,B2: nat] :
% 9.10/5.32        ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
% 9.10/5.32        = ( plus_plus_nat @ A @ B2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9132_sgn__mod,axiom,
% 9.10/5.32      ! [L: int,K: int] :
% 9.10/5.32        ( ( L != zero_zero_int )
% 9.10/5.32       => ( ~ ( dvd_dvd_int @ L @ K )
% 9.10/5.32         => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
% 9.10/5.32            = ( sgn_sgn_int @ L ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9133_int__minus,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 9.10/5.32        = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9134_nat__abs__mult__distrib,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 9.10/5.32        = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9135_nat__plus__as__int,axiom,
% 9.10/5.32      ( plus_plus_nat
% 9.10/5.32      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9136_nat__times__as__int,axiom,
% 9.10/5.32      ( times_times_nat
% 9.10/5.32      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9137_real__nat__ceiling__ge,axiom,
% 9.10/5.32      ! [X3: real] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9138_less__mask,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.32       => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9139_nat__minus__as__int,axiom,
% 9.10/5.32      ( minus_minus_nat
% 9.10/5.32      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9140_nat__div__as__int,axiom,
% 9.10/5.32      ( divide_divide_nat
% 9.10/5.32      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9141_nat__mod__as__int,axiom,
% 9.10/5.32      ( modulo_modulo_nat
% 9.10/5.32      = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9142_zsgn__def,axiom,
% 9.10/5.32      ( sgn_sgn_int
% 9.10/5.32      = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9143_nat__less__eq__zless,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.10/5.32       => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.10/5.32          = ( ord_less_int @ W @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9144_nat__le__eq__zle,axiom,
% 9.10/5.32      ! [W: int,Z: int] :
% 9.10/5.32        ( ( ( ord_less_int @ zero_zero_int @ W )
% 9.10/5.32          | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 9.10/5.32       => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 9.10/5.32          = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9145_nat__eq__iff,axiom,
% 9.10/5.32      ! [W: int,M: nat] :
% 9.10/5.32        ( ( ( nat2 @ W )
% 9.10/5.32          = M )
% 9.10/5.32        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.10/5.32           => ( W
% 9.10/5.32              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.10/5.32          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 9.10/5.32           => ( M = zero_zero_nat ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9146_nat__eq__iff2,axiom,
% 9.10/5.32      ! [M: nat,W: int] :
% 9.10/5.32        ( ( M
% 9.10/5.32          = ( nat2 @ W ) )
% 9.10/5.32        = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.10/5.32           => ( W
% 9.10/5.32              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.10/5.32          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 9.10/5.32           => ( M = zero_zero_nat ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9147_split__nat,axiom,
% 9.10/5.32      ! [P: nat > $o,I: int] :
% 9.10/5.32        ( ( P @ ( nat2 @ I ) )
% 9.10/5.32        = ( ! [N2: nat] :
% 9.10/5.32              ( ( I
% 9.10/5.32                = ( semiri1314217659103216013at_int @ N2 ) )
% 9.10/5.32             => ( P @ N2 ) )
% 9.10/5.32          & ( ( ord_less_int @ I @ zero_zero_int )
% 9.10/5.32           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9148_le__nat__iff,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32       => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 9.10/5.32          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9149_nat__add__distrib,axiom,
% 9.10/5.32      ! [Z: int,Z7: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 9.10/5.32         => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 9.10/5.32            = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9150_div__sgn__abs__cancel,axiom,
% 9.10/5.32      ! [V: int,K: int,L: int] :
% 9.10/5.32        ( ( V != zero_zero_int )
% 9.10/5.32       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
% 9.10/5.32          = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9151_nat__mult__distrib,axiom,
% 9.10/5.32      ! [Z: int,Z7: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 9.10/5.32          = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9152_Suc__as__int,axiom,
% 9.10/5.32      ( suc
% 9.10/5.32      = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9153_nat__diff__distrib_H,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32         => ( ( nat2 @ ( minus_minus_int @ X3 @ Y3 ) )
% 9.10/5.32            = ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9154_nat__diff__distrib,axiom,
% 9.10/5.32      ! [Z7: int,Z: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 9.10/5.32       => ( ( ord_less_eq_int @ Z7 @ Z )
% 9.10/5.32         => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 9.10/5.32            = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9155_nat__abs__triangle__ineq,axiom,
% 9.10/5.32      ! [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 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9156_nat__div__distrib_H,axiom,
% 9.10/5.32      ! [Y3: int,X3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ( nat2 @ ( divide_divide_int @ X3 @ Y3 ) )
% 9.10/5.32          = ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9157_nat__div__distrib,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( nat2 @ ( divide_divide_int @ X3 @ Y3 ) )
% 9.10/5.32          = ( divide_divide_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9158_div__dvd__sgn__abs,axiom,
% 9.10/5.32      ! [L: int,K: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ L @ K )
% 9.10/5.32       => ( ( divide_divide_int @ K @ L )
% 9.10/5.32          = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9159_nat__power__eq,axiom,
% 9.10/5.32      ! [Z: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 9.10/5.32          = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9160_nat__floor__neg,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.10/5.32       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.10/5.32          = zero_zero_nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9161_nat__mod__distrib,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32         => ( ( nat2 @ ( modulo_modulo_int @ X3 @ Y3 ) )
% 9.10/5.32            = ( modulo_modulo_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9162_div__abs__eq__div__nat,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
% 9.10/5.32        = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9163_floor__eq3,axiom,
% 9.10/5.32      ! [N: nat,X3: real] :
% 9.10/5.32        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.10/5.32         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.10/5.32            = N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9164_le__nat__floor,axiom,
% 9.10/5.32      ! [X3: nat,A: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
% 9.10/5.32       => ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9165_divide__int__def,axiom,
% 9.10/5.32      ( divide_divide_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 9.10/5.32            @ ( if_int
% 9.10/5.32              @ ( ( sgn_sgn_int @ K2 )
% 9.10/5.32                = ( sgn_sgn_int @ L2 ) )
% 9.10/5.32              @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 9.10/5.32              @ ( uminus_uminus_int
% 9.10/5.32                @ ( semiri1314217659103216013at_int
% 9.10/5.32                  @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 9.10/5.32                    @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32                      @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9166_modulo__int__def,axiom,
% 9.10/5.32      ( modulo_modulo_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( if_int @ ( L2 = zero_zero_int ) @ K2
% 9.10/5.32            @ ( if_int
% 9.10/5.32              @ ( ( sgn_sgn_int @ K2 )
% 9.10/5.32                = ( sgn_sgn_int @ L2 ) )
% 9.10/5.32              @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
% 9.10/5.32              @ ( times_times_int @ ( sgn_sgn_int @ L2 )
% 9.10/5.32                @ ( minus_minus_int
% 9.10/5.32                  @ ( times_times_int @ ( abs_abs_int @ L2 )
% 9.10/5.32                    @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                      @ ~ ( dvd_dvd_int @ L2 @ K2 ) ) )
% 9.10/5.32                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9167_nat__2,axiom,
% 9.10/5.32      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.32      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9168_Suc__nat__eq__nat__zadd1,axiom,
% 9.10/5.32      ! [Z: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32       => ( ( suc @ ( nat2 @ Z ) )
% 9.10/5.32          = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9169_nat__less__iff,axiom,
% 9.10/5.32      ! [W: int,M: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ W )
% 9.10/5.32       => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 9.10/5.32          = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9170_nat__mult__distrib__neg,axiom,
% 9.10/5.32      ! [Z: int,Z7: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 9.10/5.32       => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 9.10/5.32          = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9171_nat__abs__int__diff,axiom,
% 9.10/5.32      ! [A: nat,B2: nat] :
% 9.10/5.32        ( ( ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.32         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 9.10/5.32            = ( minus_minus_nat @ B2 @ A ) ) )
% 9.10/5.32        & ( ~ ( ord_less_eq_nat @ A @ B2 )
% 9.10/5.32         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
% 9.10/5.32            = ( minus_minus_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9172_floor__eq4,axiom,
% 9.10/5.32      ! [N: nat,X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X3 )
% 9.10/5.32       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.10/5.32         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.10/5.32            = N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9173_num_Osize__gen_I1_J,axiom,
% 9.10/5.32      ( ( size_num @ one )
% 9.10/5.32      = zero_zero_nat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9174_diff__nat__eq__if,axiom,
% 9.10/5.32      ! [Z7: int,Z: int] :
% 9.10/5.32        ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 9.10/5.32         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 9.10/5.32            = ( nat2 @ Z ) ) )
% 9.10/5.32        & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 9.10/5.32         => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 9.10/5.32            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9175_Suc__mask__eq__exp,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.10/5.32        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9176_mask__nat__less__exp,axiom,
% 9.10/5.32      ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9177_nat__dvd__iff,axiom,
% 9.10/5.32      ! [Z: int,M: nat] :
% 9.10/5.32        ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 9.10/5.32        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32           => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.10/5.32          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.10/5.32           => ( M = zero_zero_nat ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9178_eucl__rel__int__remainderI,axiom,
% 9.10/5.32      ! [R: int,L: int,K: int,Q2: int] :
% 9.10/5.32        ( ( ( sgn_sgn_int @ R )
% 9.10/5.32          = ( sgn_sgn_int @ L ) )
% 9.10/5.32       => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L ) )
% 9.10/5.32         => ( ( K
% 9.10/5.32              = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R ) )
% 9.10/5.32           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9179_mask__nat__def,axiom,
% 9.10/5.32      ( bit_se2002935070580805687sk_nat
% 9.10/5.32      = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9180_mask__half__int,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.32        = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9181_mask__int__def,axiom,
% 9.10/5.32      ( bit_se2000444600071755411sk_int
% 9.10/5.32      = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9182_eucl__rel__int_Osimps,axiom,
% 9.10/5.32      ( eucl_rel_int
% 9.10/5.32      = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
% 9.10/5.32            ( ? [K2: int] :
% 9.10/5.32                ( ( A1 = K2 )
% 9.10/5.32                & ( A22 = zero_zero_int )
% 9.10/5.32                & ( A32
% 9.10/5.32                  = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 9.10/5.32            | ? [L2: int,K2: int,Q4: int] :
% 9.10/5.32                ( ( A1 = K2 )
% 9.10/5.32                & ( A22 = L2 )
% 9.10/5.32                & ( A32
% 9.10/5.32                  = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 9.10/5.32                & ( L2 != zero_zero_int )
% 9.10/5.32                & ( K2
% 9.10/5.32                  = ( times_times_int @ Q4 @ L2 ) ) )
% 9.10/5.32            | ? [R5: int,L2: int,K2: int,Q4: int] :
% 9.10/5.32                ( ( A1 = K2 )
% 9.10/5.32                & ( A22 = L2 )
% 9.10/5.32                & ( A32
% 9.10/5.32                  = ( product_Pair_int_int @ Q4 @ R5 ) )
% 9.10/5.32                & ( ( sgn_sgn_int @ R5 )
% 9.10/5.32                  = ( sgn_sgn_int @ L2 ) )
% 9.10/5.32                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 9.10/5.32                & ( K2
% 9.10/5.32                  = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9183_eucl__rel__int_Ocases,axiom,
% 9.10/5.32      ! [A12: int,A23: int,A33: product_prod_int_int] :
% 9.10/5.32        ( ( eucl_rel_int @ A12 @ A23 @ A33 )
% 9.10/5.32       => ( ( ( A23 = zero_zero_int )
% 9.10/5.32           => ( A33
% 9.10/5.32             != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 9.10/5.32         => ( ! [Q3: int] :
% 9.10/5.32                ( ( A33
% 9.10/5.32                  = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 9.10/5.32               => ( ( A23 != zero_zero_int )
% 9.10/5.32                 => ( A12
% 9.10/5.32                   != ( times_times_int @ Q3 @ A23 ) ) ) )
% 9.10/5.32           => ~ ! [R3: int,Q3: int] :
% 9.10/5.32                  ( ( A33
% 9.10/5.32                    = ( product_Pair_int_int @ Q3 @ R3 ) )
% 9.10/5.32                 => ( ( ( sgn_sgn_int @ R3 )
% 9.10/5.32                      = ( sgn_sgn_int @ A23 ) )
% 9.10/5.32                   => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 9.10/5.32                     => ( A12
% 9.10/5.32                       != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9184_even__nat__iff,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 9.10/5.32          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9185_powr__real__of__int,axiom,
% 9.10/5.32      ! [X3: real,N: int] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.32       => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.10/5.32           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
% 9.10/5.32              = ( power_power_real @ X3 @ ( nat2 @ N ) ) ) )
% 9.10/5.32          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 9.10/5.32           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
% 9.10/5.32              = ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9186_num_Osize__gen_I2_J,axiom,
% 9.10/5.32      ! [X2: num] :
% 9.10/5.32        ( ( size_num @ ( bit0 @ X2 ) )
% 9.10/5.32        = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9187_divide__int__unfold,axiom,
% 9.10/5.32      ! [L: int,K: int,N: nat,M: nat] :
% 9.10/5.32        ( ( ( ( ( sgn_sgn_int @ L )
% 9.10/5.32              = zero_zero_int )
% 9.10/5.32            | ( ( sgn_sgn_int @ K )
% 9.10/5.32              = zero_zero_int )
% 9.10/5.32            | ( N = zero_zero_nat ) )
% 9.10/5.32         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.32            = zero_zero_int ) )
% 9.10/5.32        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.10/5.32                = zero_zero_int )
% 9.10/5.32              | ( ( sgn_sgn_int @ K )
% 9.10/5.32                = zero_zero_int )
% 9.10/5.32              | ( N = zero_zero_nat ) )
% 9.10/5.32         => ( ( ( ( sgn_sgn_int @ K )
% 9.10/5.32                = ( sgn_sgn_int @ L ) )
% 9.10/5.32             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.32                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 9.10/5.32            & ( ( ( sgn_sgn_int @ K )
% 9.10/5.32               != ( sgn_sgn_int @ L ) )
% 9.10/5.32             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.10/5.32                = ( uminus_uminus_int
% 9.10/5.32                  @ ( semiri1314217659103216013at_int
% 9.10/5.32                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 9.10/5.32                      @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9188_and__int__unfold,axiom,
% 9.10/5.32      ( bit_se725231765392027082nd_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( if_int
% 9.10/5.32            @ ( ( K2 = zero_zero_int )
% 9.10/5.32              | ( L2 = zero_zero_int ) )
% 9.10/5.32            @ zero_zero_int
% 9.10/5.32            @ ( if_int
% 9.10/5.32              @ ( K2
% 9.10/5.32                = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32              @ L2
% 9.10/5.32              @ ( if_int
% 9.10/5.32                @ ( L2
% 9.10/5.32                  = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32                @ K2
% 9.10/5.32                @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9189_zero__le__sgn__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X3 ) )
% 9.10/5.32        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9190_sgn__le__0__iff,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( sgn_sgn_real @ X3 ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9191_concat__bit__of__zero__2,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 9.10/5.32        = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9192_and__nonnegative__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.10/5.32        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32          | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9193_and__negative__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 9.10/5.32        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.32          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9194_take__bit__of__Suc__0,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9195_and__minus__numerals_I2_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = one_one_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9196_and__minus__numerals_I6_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 9.10/5.32        = one_one_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9197_and__minus__numerals_I5_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 9.10/5.32        = zero_zero_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9198_and__minus__numerals_I1_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = zero_zero_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9199_take__bit__nat__eq,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32       => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 9.10/5.32          = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9200_nat__take__bit__eq,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32       => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.10/5.32          = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9201_take__bit__nat__less__eq__self,axiom,
% 9.10/5.32      ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9202_take__bit__tightened__less__eq__nat,axiom,
% 9.10/5.32      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9203_take__bit__diff,axiom,
% 9.10/5.32      ! [N: nat,K: int,L: int] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 9.10/5.32        = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9204_take__bit__mult,axiom,
% 9.10/5.32      ! [N: nat,K: int,L: int] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
% 9.10/5.32        = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9205_take__bit__minus,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 9.10/5.32        = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9206_concat__bit__take__bit__eq,axiom,
% 9.10/5.32      ! [N: nat,B2: int] :
% 9.10/5.32        ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B2 ) )
% 9.10/5.32        = ( bit_concat_bit @ N @ B2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9207_concat__bit__eq__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int,L: int,R: int,S: int] :
% 9.10/5.32        ( ( ( bit_concat_bit @ N @ K @ L )
% 9.10/5.32          = ( bit_concat_bit @ N @ R @ S ) )
% 9.10/5.32        = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32            = ( bit_se2923211474154528505it_int @ N @ R ) )
% 9.10/5.32          & ( L = S ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9208_take__bit__tightened__less__eq__int,axiom,
% 9.10/5.32      ! [M: nat,N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9209_AND__lower,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9210_AND__upper1,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9211_AND__upper2,axiom,
% 9.10/5.32      ! [Y3: int,X3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9212_AND__upper1_H,axiom,
% 9.10/5.32      ! [Y3: int,Z: int,Ya: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.10/5.32         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9213_AND__upper2_H,axiom,
% 9.10/5.32      ! [Y3: int,Z: int,X3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.10/5.32         => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9214_take__bit__nonnegative,axiom,
% 9.10/5.32      ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9215_take__bit__int__less__eq__self__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.10/5.32        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9216_not__take__bit__negative,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9217_take__bit__int__greater__self__iff,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.10/5.32        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9218_pow_Osimps_I1_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( pow @ X3 @ one )
% 9.10/5.32        = X3 ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9219_AND__upper2_H_H,axiom,
% 9.10/5.32      ! [Y3: int,Z: int,X3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ( ord_less_int @ Y3 @ Z )
% 9.10/5.32         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9220_AND__upper1_H_H,axiom,
% 9.10/5.32      ! [Y3: int,Z: int,Ya: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ( ord_less_int @ Y3 @ Z )
% 9.10/5.32         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9221_and__less__eq,axiom,
% 9.10/5.32      ! [L: int,K: int] :
% 9.10/5.32        ( ( ord_less_int @ L @ zero_zero_int )
% 9.10/5.32       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9222_take__bit__eq__mask__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32          = ( bit_se2000444600071755411sk_int @ N ) )
% 9.10/5.32        = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 9.10/5.32          = zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9223_take__bit__decr__eq,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32         != zero_zero_int )
% 9.10/5.32       => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 9.10/5.32          = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9224_sgn__real__def,axiom,
% 9.10/5.32      ( sgn_sgn_real
% 9.10/5.32      = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9225_even__and__iff__int,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.10/5.32        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.10/5.32          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9226_take__bit__nat__eq__self,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32       => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.10/5.32          = M ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9227_take__bit__nat__less__exp,axiom,
% 9.10/5.32      ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9228_take__bit__nat__eq__self__iff,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.10/5.32          = M )
% 9.10/5.32        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9229_take__bit__nat__def,axiom,
% 9.10/5.32      ( bit_se2925701944663578781it_nat
% 9.10/5.32      = ( ^ [N2: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9230_sgn__power__injE,axiom,
% 9.10/5.32      ! [A: real,N: nat,X3: real,B2: real] :
% 9.10/5.32        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 9.10/5.32          = X3 )
% 9.10/5.32       => ( ( X3
% 9.10/5.32            = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N ) ) )
% 9.10/5.32         => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.32           => ( A = B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9231_take__bit__int__less__exp,axiom,
% 9.10/5.32      ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9232_take__bit__int__def,axiom,
% 9.10/5.32      ( bit_se2923211474154528505it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9233_take__bit__nat__less__self__iff,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 9.10/5.32        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9234_take__bit__Suc__minus__bit0,axiom,
% 9.10/5.32      ! [N: nat,K: num] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.10/5.32        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9235_take__bit__int__less__self__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.10/5.32        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9236_take__bit__int__greater__eq__self__iff,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.10/5.32        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9237_cis__Arg__unique,axiom,
% 9.10/5.32      ! [Z: complex,X3: real] :
% 9.10/5.32        ( ( ( sgn_sgn_complex @ Z )
% 9.10/5.32          = ( cis @ X3 ) )
% 9.10/5.32       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_real @ X3 @ pi )
% 9.10/5.32           => ( ( arg @ Z )
% 9.10/5.32              = X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9238_take__bit__int__eq__self,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32         => ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32            = K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9239_take__bit__int__eq__self__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32          = K )
% 9.10/5.32        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9240_take__bit__numeral__minus__bit0,axiom,
% 9.10/5.32      ! [L: num,K: num] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.10/5.32        = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9241_take__bit__incr__eq,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32         != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 9.10/5.32       => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 9.10/5.32          = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9242_take__bit__eq__mask__iff__exp__dvd,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.10/5.32          = ( bit_se2000444600071755411sk_int @ N ) )
% 9.10/5.32        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9243_Arg__correct,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( Z != zero_zero_complex )
% 9.10/5.32       => ( ( ( sgn_sgn_complex @ Z )
% 9.10/5.32            = ( cis @ ( arg @ Z ) ) )
% 9.10/5.32          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 9.10/5.32          & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9244_take__bit__int__less__eq,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.10/5.32       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.32         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9245_take__bit__int__greater__eq,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.32       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9246_signed__take__bit__eq__take__bit__shift,axiom,
% 9.10/5.32      ( bit_ri631733984087533419it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9247_and__int__rec,axiom,
% 9.10/5.32      ( bit_se725231765392027082nd_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( plus_plus_int
% 9.10/5.32            @ ( zero_n2684676970156552555ol_int
% 9.10/5.32              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.10/5.32                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 9.10/5.32            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9248_take__bit__minus__small__eq,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( ord_less_int @ zero_zero_int @ K )
% 9.10/5.32       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32         => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 9.10/5.32            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9249_arctan__inverse,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( X3 != zero_zero_real )
% 9.10/5.32       => ( ( arctan @ ( divide_divide_real @ one_one_real @ X3 ) )
% 9.10/5.32          = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X3 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9250_take__bit__numeral__minus__bit1,axiom,
% 9.10/5.32      ! [L: num,K: num] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.10/5.32        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9251_and__int_Osimps,axiom,
% 9.10/5.32      ( bit_se725231765392027082nd_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( if_int
% 9.10/5.32            @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32            @ ( uminus_uminus_int
% 9.10/5.32              @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.10/5.32                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 9.10/5.32            @ ( plus_plus_int
% 9.10/5.32              @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.10/5.32                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 9.10/5.32              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9252_and__int_Oelims,axiom,
% 9.10/5.32      ! [X3: int,Xa2: int,Y3: int] :
% 9.10/5.32        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32           => ( Y3
% 9.10/5.32              = ( uminus_uminus_int
% 9.10/5.32                @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.10/5.32                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.10/5.32          & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32                & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32           => ( Y3
% 9.10/5.32              = ( plus_plus_int
% 9.10/5.32                @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.10/5.32                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.10/5.32                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9253_take__bit__Suc__minus__bit1,axiom,
% 9.10/5.32      ! [N: nat,K: num] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.10/5.32        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9254_pred__numeral__inc,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( pred_numeral @ ( inc @ K ) )
% 9.10/5.32        = ( numeral_numeral_nat @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9255_and__nat__numerals_I1_J,axiom,
% 9.10/5.32      ! [Y3: num] :
% 9.10/5.32        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.10/5.32        = zero_zero_nat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9256_and__nat__numerals_I3_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = zero_zero_nat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9257_and__nat__numerals_I4_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = one_one_nat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9258_and__nat__numerals_I2_J,axiom,
% 9.10/5.32      ! [Y3: num] :
% 9.10/5.32        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.10/5.32        = one_one_nat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9259_and__Suc__0__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9260_Suc__0__and__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.32        = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9261_num__induct,axiom,
% 9.10/5.32      ! [P: num > $o,X3: num] :
% 9.10/5.32        ( ( P @ one )
% 9.10/5.32       => ( ! [X5: num] :
% 9.10/5.32              ( ( P @ X5 )
% 9.10/5.32             => ( P @ ( inc @ X5 ) ) )
% 9.10/5.32         => ( P @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9262_add__inc,axiom,
% 9.10/5.32      ! [X3: num,Y3: num] :
% 9.10/5.32        ( ( plus_plus_num @ X3 @ ( inc @ Y3 ) )
% 9.10/5.32        = ( inc @ ( plus_plus_num @ X3 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9263_inc_Osimps_I1_J,axiom,
% 9.10/5.32      ( ( inc @ one )
% 9.10/5.32      = ( bit0 @ one ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9264_inc_Osimps_I2_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( inc @ ( bit0 @ X3 ) )
% 9.10/5.32        = ( bit1 @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9265_inc_Osimps_I3_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( inc @ ( bit1 @ X3 ) )
% 9.10/5.32        = ( bit0 @ ( inc @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9266_add__One,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( plus_plus_num @ X3 @ one )
% 9.10/5.32        = ( inc @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9267_inc__BitM__eq,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( inc @ ( bitM @ N ) )
% 9.10/5.32        = ( bit0 @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9268_BitM__inc__eq,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bitM @ ( inc @ N ) )
% 9.10/5.32        = ( bit1 @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9269_and__nat__def,axiom,
% 9.10/5.32      ( bit_se727722235901077358nd_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9270_mult__inc,axiom,
% 9.10/5.32      ! [X3: num,Y3: num] :
% 9.10/5.32        ( ( times_times_num @ X3 @ ( inc @ Y3 ) )
% 9.10/5.32        = ( plus_plus_num @ ( times_times_num @ X3 @ Y3 ) @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9271_atLeastAtMostPlus1__int__conv,axiom,
% 9.10/5.32      ! [M: int,N: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 9.10/5.32       => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 9.10/5.32          = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9272_simp__from__to,axiom,
% 9.10/5.32      ( set_or1266510415728281911st_int
% 9.10/5.32      = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9273_and__nat__unfold,axiom,
% 9.10/5.32      ( bit_se727722235901077358nd_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.32            ( if_nat
% 9.10/5.32            @ ( ( M3 = zero_zero_nat )
% 9.10/5.32              | ( N2 = zero_zero_nat ) )
% 9.10/5.32            @ zero_zero_nat
% 9.10/5.32            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9274_and__nat__rec,axiom,
% 9.10/5.32      ( bit_se727722235901077358nd_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.32            ( plus_plus_nat
% 9.10/5.32            @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 9.10/5.32                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.10/5.32            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9275_and__int_Opsimps,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 9.10/5.32       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.10/5.32              = ( uminus_uminus_int
% 9.10/5.32                @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.10/5.32                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 9.10/5.32          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.10/5.32              = ( plus_plus_int
% 9.10/5.32                @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.10/5.32                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 9.10/5.32                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9276_and__int_Opelims,axiom,
% 9.10/5.32      ! [X3: int,Xa2: int,Y3: int] :
% 9.10/5.32        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 9.10/5.32         => ~ ( ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                    = ( uminus_uminus_int
% 9.10/5.32                      @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.10/5.32                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.10/5.32                & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32                      & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                    = ( plus_plus_int
% 9.10/5.32                      @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.10/5.32                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.10/5.32                      @ ( 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 ) ) ) ) ) ) ) ) )
% 9.10/5.32             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9277_and__int_Opinduct,axiom,
% 9.10/5.32      ! [A0: int,A12: int,P: int > int > $o] :
% 9.10/5.32        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 9.10/5.32       => ( ! [K3: int,L4: int] :
% 9.10/5.32              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
% 9.10/5.32             => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.10/5.32                      & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.10/5.32                 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.10/5.32               => ( P @ K3 @ L4 ) ) )
% 9.10/5.32         => ( P @ A0 @ A12 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9278_Arg__def,axiom,
% 9.10/5.32      ( arg
% 9.10/5.32      = ( ^ [Z2: complex] :
% 9.10/5.32            ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
% 9.10/5.32            @ ( fChoice_real
% 9.10/5.32              @ ^ [A5: real] :
% 9.10/5.32                  ( ( ( sgn_sgn_complex @ Z2 )
% 9.10/5.32                    = ( cis @ A5 ) )
% 9.10/5.32                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
% 9.10/5.32                  & ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9279_set__encode__insert,axiom,
% 9.10/5.32      ! [A4: set_nat,N: nat] :
% 9.10/5.32        ( ( finite_finite_nat @ A4 )
% 9.10/5.32       => ( ~ ( member_nat @ N @ A4 )
% 9.10/5.32         => ( ( nat_set_encode @ ( insert_nat @ N @ A4 ) )
% 9.10/5.32            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9280_atLeastAtMost__insertL,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.10/5.32          = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9281_atLeastAtMostSuc__conv,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.10/5.32       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 9.10/5.32          = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9282_Icc__eq__insert__lb__nat,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32       => ( ( set_or1269000886237332187st_nat @ M @ N )
% 9.10/5.32          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9283_lessThan__nat__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 9.10/5.32        = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9284_atMost__nat__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 9.10/5.32        = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9285_set__decode__plus__power__2,axiom,
% 9.10/5.32      ! [N: nat,Z: nat] :
% 9.10/5.32        ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 9.10/5.32       => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 9.10/5.32          = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9286_signed__take__bit__eq__take__bit__minus,axiom,
% 9.10/5.32      ( bit_ri631733984087533419it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9287_or__int__unfold,axiom,
% 9.10/5.32      ( bit_se1409905431419307370or_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( if_int
% 9.10/5.32            @ ( ( K2
% 9.10/5.32                = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32              | ( L2
% 9.10/5.32                = ( uminus_uminus_int @ one_one_int ) ) )
% 9.10/5.32            @ ( uminus_uminus_int @ one_one_int )
% 9.10/5.32            @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9288_cis__multiple__2pi,axiom,
% 9.10/5.32      ! [N: real] :
% 9.10/5.32        ( ( member_real @ N @ ring_1_Ints_real )
% 9.10/5.32       => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 9.10/5.32          = one_one_complex ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9289_take__bit__Suc__from__most,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 9.10/5.32        = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9290_or__nonnegative__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 9.10/5.32        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32          & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9291_or__negative__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 9.10/5.32        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.32          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9292_signed__take__bit__nonnegative__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.10/5.32        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9293_signed__take__bit__negative__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 9.10/5.32        = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9294_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 9.10/5.32      ! [W: num,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 9.10/5.32        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9295_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 9.10/5.32      ! [W: num,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 9.10/5.32        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9296_or__minus__numerals_I6_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9297_or__minus__numerals_I2_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9298_bit__minus__numeral__int_I1_J,axiom,
% 9.10/5.32      ! [W: num,N: num] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.32        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9299_bit__minus__numeral__int_I2_J,axiom,
% 9.10/5.32      ! [W: num,N: num] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.32        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9300_bit__or__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
% 9.10/5.32        = ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.10/5.32          | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9301_bit__and__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
% 9.10/5.32        = ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.10/5.32          & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9302_or__greater__eq,axiom,
% 9.10/5.32      ! [L: int,K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ L )
% 9.10/5.32       => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9303_OR__lower,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9304_plus__and__or,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) )
% 9.10/5.32        = ( plus_plus_int @ X3 @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9305_bit__not__int__iff_H,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 9.10/5.32        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9306_bit__imp__take__bit__positive,axiom,
% 9.10/5.32      ! [N: nat,M: nat,K: int] :
% 9.10/5.32        ( ( ord_less_nat @ N @ M )
% 9.10/5.32       => ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.10/5.32         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9307_bit__concat__bit__iff,axiom,
% 9.10/5.32      ! [M: nat,K: int,L: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 9.10/5.32        = ( ( ( ord_less_nat @ N @ M )
% 9.10/5.32            & ( bit_se1146084159140164899it_int @ K @ N ) )
% 9.10/5.32          | ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9308_sin__times__pi__eq__0,axiom,
% 9.10/5.32      ! [X3: real] :
% 9.10/5.32        ( ( ( sin_real @ ( times_times_real @ X3 @ pi ) )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32        = ( member_real @ X3 @ ring_1_Ints_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9309_signed__take__bit__eq__concat__bit,axiom,
% 9.10/5.32      ( bit_ri631733984087533419it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9310_int__bit__bound,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ~ ! [N3: nat] :
% 9.10/5.32            ( ! [M2: nat] :
% 9.10/5.32                ( ( ord_less_eq_nat @ N3 @ M2 )
% 9.10/5.32               => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 9.10/5.32                  = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 9.10/5.32           => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.10/5.32               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 9.10/5.32                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9311_bit__int__def,axiom,
% 9.10/5.32      ( bit_se1146084159140164899it_int
% 9.10/5.32      = ( ^ [K2: int,N2: nat] :
% 9.10/5.32            ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9312_OR__upper,axiom,
% 9.10/5.32      ! [X3: int,N: nat,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9313_sin__integer__2pi,axiom,
% 9.10/5.32      ! [N: real] :
% 9.10/5.32        ( ( member_real @ N @ ring_1_Ints_real )
% 9.10/5.32       => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 9.10/5.32          = zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9314_cos__integer__2pi,axiom,
% 9.10/5.32      ! [N: real] :
% 9.10/5.32        ( ( member_real @ N @ ring_1_Ints_real )
% 9.10/5.32       => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 9.10/5.32          = one_one_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9315_or__int__rec,axiom,
% 9.10/5.32      ( bit_se1409905431419307370or_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( plus_plus_int
% 9.10/5.32            @ ( zero_n2684676970156552555ol_int
% 9.10/5.32              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 9.10/5.32                | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 9.10/5.32            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9316_set__bit__eq,axiom,
% 9.10/5.32      ( bit_se7879613467334960850it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] :
% 9.10/5.32            ( plus_plus_int @ K2
% 9.10/5.32            @ ( times_times_int
% 9.10/5.32              @ ( zero_n2684676970156552555ol_int
% 9.10/5.32                @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
% 9.10/5.32              @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9317_unset__bit__eq,axiom,
% 9.10/5.32      ( bit_se4203085406695923979it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9318_or__minus__numerals_I1_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9319_or__minus__numerals_I5_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9320_xor__Suc__0__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.32          @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9321_or__nat__numerals_I2_J,axiom,
% 9.10/5.32      ! [Y3: num] :
% 9.10/5.32        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9322_or__nat__numerals_I4_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9323_or__nat__numerals_I3_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9324_or__nat__numerals_I1_J,axiom,
% 9.10/5.32      ! [Y3: num] :
% 9.10/5.32        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9325_xor__nat__numerals_I4_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9326_xor__nat__numerals_I3_J,axiom,
% 9.10/5.32      ! [X3: num] :
% 9.10/5.32        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9327_xor__nat__numerals_I2_J,axiom,
% 9.10/5.32      ! [Y3: num] :
% 9.10/5.32        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9328_xor__nat__numerals_I1_J,axiom,
% 9.10/5.32      ! [Y3: num] :
% 9.10/5.32        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.10/5.32        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9329_or__minus__numerals_I4_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9330_or__minus__numerals_I8_J,axiom,
% 9.10/5.32      ! [N: num,M: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9331_or__minus__numerals_I3_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9332_or__minus__numerals_I7_J,axiom,
% 9.10/5.32      ! [N: num,M: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9333_or__not__num__neg_Osimps_I1_J,axiom,
% 9.10/5.32      ( ( bit_or_not_num_neg @ one @ one )
% 9.10/5.32      = one ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9334_bit__Suc__0__iff,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.32        = ( N = zero_zero_nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9335_not__bit__Suc__0__Suc,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9336_or__not__num__neg_Osimps_I4_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 9.10/5.32        = ( bit0 @ one ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9337_or__not__num__neg_Osimps_I6_J,axiom,
% 9.10/5.32      ! [N: num,M: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 9.10/5.32        = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9338_or__not__num__neg_Osimps_I7_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 9.10/5.32        = one ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9339_or__not__num__neg_Osimps_I3_J,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 9.10/5.32        = ( bit1 @ M ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9340_or__not__num__neg_Osimps_I5_J,axiom,
% 9.10/5.32      ! [N: num,M: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 9.10/5.32        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9341_not__bit__Suc__0__numeral,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9342_or__not__num__neg_Osimps_I9_J,axiom,
% 9.10/5.32      ! [N: num,M: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 9.10/5.32        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9343_or__nat__def,axiom,
% 9.10/5.32      ( bit_se1412395901928357646or_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9344_or__not__num__neg_Osimps_I2_J,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 9.10/5.32        = ( bit1 @ M ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9345_or__not__num__neg_Osimps_I8_J,axiom,
% 9.10/5.32      ! [N: num,M: num] :
% 9.10/5.32        ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 9.10/5.32        = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9346_bit__nat__iff,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 9.10/5.32        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32          & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9347_or__not__num__neg_Oelims,axiom,
% 9.10/5.32      ! [X3: num,Xa2: num,Y3: num] :
% 9.10/5.32        ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( ( X3 = one )
% 9.10/5.32           => ( ( Xa2 = one )
% 9.10/5.32             => ( Y3 != one ) ) )
% 9.10/5.32         => ( ( ( X3 = one )
% 9.10/5.32             => ! [M4: num] :
% 9.10/5.32                  ( ( Xa2
% 9.10/5.32                    = ( bit0 @ M4 ) )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                   != ( bit1 @ M4 ) ) ) )
% 9.10/5.32           => ( ( ( X3 = one )
% 9.10/5.32               => ! [M4: num] :
% 9.10/5.32                    ( ( Xa2
% 9.10/5.32                      = ( bit1 @ M4 ) )
% 9.10/5.32                   => ( Y3
% 9.10/5.32                     != ( bit1 @ M4 ) ) ) )
% 9.10/5.32             => ( ( ? [N3: num] :
% 9.10/5.32                      ( X3
% 9.10/5.32                      = ( bit0 @ N3 ) )
% 9.10/5.32                 => ( ( Xa2 = one )
% 9.10/5.32                   => ( Y3
% 9.10/5.32                     != ( bit0 @ one ) ) ) )
% 9.10/5.32               => ( ! [N3: num] :
% 9.10/5.32                      ( ( X3
% 9.10/5.32                        = ( bit0 @ N3 ) )
% 9.10/5.32                     => ! [M4: num] :
% 9.10/5.32                          ( ( Xa2
% 9.10/5.32                            = ( bit0 @ M4 ) )
% 9.10/5.32                         => ( Y3
% 9.10/5.32                           != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
% 9.10/5.32                 => ( ! [N3: num] :
% 9.10/5.32                        ( ( X3
% 9.10/5.32                          = ( bit0 @ N3 ) )
% 9.10/5.32                       => ! [M4: num] :
% 9.10/5.32                            ( ( Xa2
% 9.10/5.32                              = ( bit1 @ M4 ) )
% 9.10/5.32                           => ( Y3
% 9.10/5.32                             != ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
% 9.10/5.32                   => ( ( ? [N3: num] :
% 9.10/5.32                            ( X3
% 9.10/5.32                            = ( bit1 @ N3 ) )
% 9.10/5.32                       => ( ( Xa2 = one )
% 9.10/5.32                         => ( Y3 != one ) ) )
% 9.10/5.32                     => ( ! [N3: num] :
% 9.10/5.32                            ( ( X3
% 9.10/5.32                              = ( bit1 @ N3 ) )
% 9.10/5.32                           => ! [M4: num] :
% 9.10/5.32                                ( ( Xa2
% 9.10/5.32                                  = ( bit0 @ M4 ) )
% 9.10/5.32                               => ( Y3
% 9.10/5.32                                 != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
% 9.10/5.32                       => ~ ! [N3: num] :
% 9.10/5.32                              ( ( X3
% 9.10/5.32                                = ( bit1 @ N3 ) )
% 9.10/5.32                             => ! [M4: num] :
% 9.10/5.32                                  ( ( Xa2
% 9.10/5.32                                    = ( bit1 @ M4 ) )
% 9.10/5.32                                 => ( Y3
% 9.10/5.32                                   != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9348_bit__nat__def,axiom,
% 9.10/5.32      ( bit_se1148574629649215175it_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.32            ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9349_Suc__0__or__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.32        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9350_or__Suc__0__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9351_or__nat__rec,axiom,
% 9.10/5.32      ( bit_se1412395901928357646or_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.32            ( plus_plus_nat
% 9.10/5.32            @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 9.10/5.32                | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.10/5.32            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9352_xor__nat__unfold,axiom,
% 9.10/5.32      ( bit_se6528837805403552850or_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9353_xor__nat__rec,axiom,
% 9.10/5.32      ( bit_se6528837805403552850or_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.32            ( plus_plus_nat
% 9.10/5.32            @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32              @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.10/5.32               != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.10/5.32            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9354_or__nat__unfold,axiom,
% 9.10/5.32      ( bit_se1412395901928357646or_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9355_Suc__0__xor__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.10/5.32        = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.32          @ ( zero_n2687167440665602831ol_nat
% 9.10/5.32            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9356_or__not__num__neg_Opelims,axiom,
% 9.10/5.32      ! [X3: num,Xa2: num,Y3: num] :
% 9.10/5.32        ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 9.10/5.32         => ( ( ( X3 = one )
% 9.10/5.32             => ( ( Xa2 = one )
% 9.10/5.32               => ( ( Y3 = one )
% 9.10/5.32                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.10/5.32           => ( ( ( X3 = one )
% 9.10/5.32               => ! [M4: num] :
% 9.10/5.32                    ( ( Xa2
% 9.10/5.32                      = ( bit0 @ M4 ) )
% 9.10/5.32                   => ( ( Y3
% 9.10/5.32                        = ( bit1 @ M4 ) )
% 9.10/5.32                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 9.10/5.32             => ( ( ( X3 = one )
% 9.10/5.32                 => ! [M4: num] :
% 9.10/5.32                      ( ( Xa2
% 9.10/5.32                        = ( bit1 @ M4 ) )
% 9.10/5.32                     => ( ( Y3
% 9.10/5.32                          = ( bit1 @ M4 ) )
% 9.10/5.32                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 9.10/5.32               => ( ! [N3: num] :
% 9.10/5.32                      ( ( X3
% 9.10/5.32                        = ( bit0 @ N3 ) )
% 9.10/5.32                     => ( ( Xa2 = one )
% 9.10/5.32                       => ( ( Y3
% 9.10/5.32                            = ( bit0 @ one ) )
% 9.10/5.32                         => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
% 9.10/5.32                 => ( ! [N3: num] :
% 9.10/5.32                        ( ( X3
% 9.10/5.32                          = ( bit0 @ N3 ) )
% 9.10/5.32                       => ! [M4: num] :
% 9.10/5.32                            ( ( Xa2
% 9.10/5.32                              = ( bit0 @ M4 ) )
% 9.10/5.32                           => ( ( Y3
% 9.10/5.32                                = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 9.10/5.32                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 9.10/5.32                   => ( ! [N3: num] :
% 9.10/5.32                          ( ( X3
% 9.10/5.32                            = ( bit0 @ N3 ) )
% 9.10/5.32                         => ! [M4: num] :
% 9.10/5.32                              ( ( Xa2
% 9.10/5.32                                = ( bit1 @ M4 ) )
% 9.10/5.32                             => ( ( Y3
% 9.10/5.32                                  = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 9.10/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 9.10/5.32                     => ( ! [N3: num] :
% 9.10/5.32                            ( ( X3
% 9.10/5.32                              = ( bit1 @ N3 ) )
% 9.10/5.32                           => ( ( Xa2 = one )
% 9.10/5.32                             => ( ( Y3 = one )
% 9.10/5.32                               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
% 9.10/5.32                       => ( ! [N3: num] :
% 9.10/5.32                              ( ( X3
% 9.10/5.32                                = ( bit1 @ N3 ) )
% 9.10/5.32                             => ! [M4: num] :
% 9.10/5.32                                  ( ( Xa2
% 9.10/5.32                                    = ( bit0 @ M4 ) )
% 9.10/5.32                                 => ( ( Y3
% 9.10/5.32                                      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 9.10/5.32                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 9.10/5.32                         => ~ ! [N3: num] :
% 9.10/5.32                                ( ( X3
% 9.10/5.32                                  = ( bit1 @ N3 ) )
% 9.10/5.32                               => ! [M4: num] :
% 9.10/5.32                                    ( ( Xa2
% 9.10/5.32                                      = ( bit1 @ M4 ) )
% 9.10/5.32                                   => ( ( Y3
% 9.10/5.32                                        = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
% 9.10/5.32                                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9357_horner__sum__of__bool__2__less,axiom,
% 9.10/5.32      ! [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 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9358_push__bit__nonnegative__int__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 9.10/5.32        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9359_push__bit__negative__int__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 9.10/5.32        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9360_concat__bit__of__zero__1,axiom,
% 9.10/5.32      ! [N: nat,L: int] :
% 9.10/5.32        ( ( bit_concat_bit @ N @ zero_zero_int @ L )
% 9.10/5.32        = ( bit_se545348938243370406it_int @ N @ L ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9361_xor__nonnegative__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 9.10/5.32        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.10/5.32          = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9362_xor__negative__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 9.10/5.32        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.10/5.32         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9363_push__bit__of__Suc__0,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9364_bit__xor__int__iff,axiom,
% 9.10/5.32      ! [K: int,L: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
% 9.10/5.32        = ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.10/5.32         != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9365_flip__bit__int__def,axiom,
% 9.10/5.32      ( bit_se2159334234014336723it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9366_push__bit__nat__eq,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 9.10/5.32        = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9367_XOR__lower,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32         => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9368_flip__bit__nat__def,axiom,
% 9.10/5.32      ( bit_se2161824704523386999it_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9369_set__bit__nat__def,axiom,
% 9.10/5.32      ( bit_se7882103937844011126it_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9370_bit__push__bit__iff__int,axiom,
% 9.10/5.32      ! [M: nat,K: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 9.10/5.32        = ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9371_xor__nat__def,axiom,
% 9.10/5.32      ( bit_se6528837805403552850or_nat
% 9.10/5.32      = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9372_bit__push__bit__iff__nat,axiom,
% 9.10/5.32      ! [M: nat,Q2: nat,N: nat] :
% 9.10/5.32        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 9.10/5.32        = ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32          & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9373_concat__bit__eq,axiom,
% 9.10/5.32      ( bit_concat_bit
% 9.10/5.32      = ( ^ [N2: nat,K2: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9374_concat__bit__def,axiom,
% 9.10/5.32      ( bit_concat_bit
% 9.10/5.32      = ( ^ [N2: nat,K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9375_set__bit__int__def,axiom,
% 9.10/5.32      ( bit_se7879613467334960850it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9376_push__bit__int__def,axiom,
% 9.10/5.32      ( bit_se545348938243370406it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9377_push__bit__nat__def,axiom,
% 9.10/5.32      ( bit_se547839408752420682it_nat
% 9.10/5.32      = ( ^ [N2: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9378_push__bit__minus__one,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9379_XOR__upper,axiom,
% 9.10/5.32      ! [X3: int,N: nat,Y3: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.10/5.32           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9380_xor__int__rec,axiom,
% 9.10/5.32      ( bit_se6526347334894502574or_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( plus_plus_int
% 9.10/5.32            @ ( zero_n2684676970156552555ol_int
% 9.10/5.32              @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 9.10/5.32               != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 9.10/5.32            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9381_xor__int__unfold,axiom,
% 9.10/5.32      ( bit_se6526347334894502574or_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] :
% 9.10/5.32            ( if_int
% 9.10/5.32            @ ( K2
% 9.10/5.32              = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32            @ ( bit_ri7919022796975470100ot_int @ L2 )
% 9.10/5.32            @ ( if_int
% 9.10/5.32              @ ( L2
% 9.10/5.32                = ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32              @ ( bit_ri7919022796975470100ot_int @ K2 )
% 9.10/5.32              @ ( if_int @ ( K2 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9382_Sum__Ico__nat,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( groups3542108847815614940at_nat
% 9.10/5.32          @ ^ [X: nat] : X
% 9.10/5.32          @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.10/5.32        = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9383_Cauchy__iff2,axiom,
% 9.10/5.32      ( topolo4055970368930404560y_real
% 9.10/5.32      = ( ^ [X7: nat > real] :
% 9.10/5.32          ! [J3: nat] :
% 9.10/5.32          ? [M9: nat] :
% 9.10/5.32          ! [M3: nat] :
% 9.10/5.32            ( ( ord_less_eq_nat @ M9 @ M3 )
% 9.10/5.32           => ! [N2: nat] :
% 9.10/5.32                ( ( ord_less_eq_nat @ M9 @ N2 )
% 9.10/5.32               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X7 @ M3 ) @ ( X7 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9384_not__nonnegative__int__iff,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.10/5.32        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9385_not__negative__int__iff,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 9.10/5.32        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9386_and__minus__minus__numerals,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9387_or__minus__minus__numerals,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9388_bit__not__int__iff,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 9.10/5.32        = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9389_ex__nat__less__eq,axiom,
% 9.10/5.32      ! [N: nat,P: nat > $o] :
% 9.10/5.32        ( ( ? [M3: nat] :
% 9.10/5.32              ( ( ord_less_nat @ M3 @ N )
% 9.10/5.32              & ( P @ M3 ) ) )
% 9.10/5.32        = ( ? [X: nat] :
% 9.10/5.32              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.10/5.32              & ( P @ X ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9390_all__nat__less__eq,axiom,
% 9.10/5.32      ! [N: nat,P: nat > $o] :
% 9.10/5.32        ( ( ! [M3: nat] :
% 9.10/5.32              ( ( ord_less_nat @ M3 @ N )
% 9.10/5.32             => ( P @ M3 ) ) )
% 9.10/5.32        = ( ! [X: nat] :
% 9.10/5.32              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.10/5.32             => ( P @ X ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9391_or__int__def,axiom,
% 9.10/5.32      ( bit_se1409905431419307370or_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9392_not__int__def,axiom,
% 9.10/5.32      ( bit_ri7919022796975470100ot_int
% 9.10/5.32      = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9393_and__not__numerals_I1_J,axiom,
% 9.10/5.32      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.10/5.32      = zero_zero_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9394_or__not__numerals_I1_J,axiom,
% 9.10/5.32      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.10/5.32      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9395_unset__bit__int__def,axiom,
% 9.10/5.32      ( bit_se4203085406695923979it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9396_xor__int__def,axiom,
% 9.10/5.32      ( bit_se6526347334894502574or_int
% 9.10/5.32      = ( ^ [K2: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9397_subset__eq__atLeast0__lessThan__finite,axiom,
% 9.10/5.32      ! [N4: set_nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.10/5.32       => ( finite_finite_nat @ N4 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9398_not__int__div__2,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9399_even__not__iff__int,axiom,
% 9.10/5.32      ! [K: int] :
% 9.10/5.32        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.10/5.32        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9400_and__not__numerals_I2_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = one_one_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9401_and__not__numerals_I4_J,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.10/5.32        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9402_or__not__numerals_I2_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9403_or__not__numerals_I4_J,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9404_bit__minus__int__iff,axiom,
% 9.10/5.32      ! [K: int,N: nat] :
% 9.10/5.32        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 9.10/5.32        = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9405_numeral__or__not__num__eq,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9406_int__numeral__not__or__num__neg,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9407_int__numeral__or__not__num__neg,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9408_atLeastLessThanSuc,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.10/5.32            = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 9.10/5.32        & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.10/5.32         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.10/5.32            = bot_bot_set_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9409_and__not__numerals_I5_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9410_and__not__numerals_I7_J,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.10/5.32        = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9411_or__not__numerals_I3_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9412_and__not__numerals_I3_J,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = zero_zero_int ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9413_or__not__numerals_I7_J,axiom,
% 9.10/5.32      ! [M: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 9.10/5.32        = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9414_atLeastLessThan__nat__numeral,axiom,
% 9.10/5.32      ! [M: nat,K: num] :
% 9.10/5.32        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.10/5.32         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.10/5.32            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 9.10/5.32        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.10/5.32         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.10/5.32            = bot_bot_set_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9415_and__not__numerals_I9_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9416_and__not__numerals_I6_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9417_or__not__numerals_I6_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9418_or__not__numerals_I5_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9419_and__not__numerals_I8_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9420_or__not__numerals_I8_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.32        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9421_or__not__numerals_I9_J,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.32        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9422_not__int__rec,axiom,
% 9.10/5.32      ( bit_ri7919022796975470100ot_int
% 9.10/5.32      = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9423_sum__power2,axiom,
% 9.10/5.32      ! [K: nat] :
% 9.10/5.32        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 9.10/5.32        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9424_Chebyshev__sum__upper__nat,axiom,
% 9.10/5.32      ! [N: nat,A: nat > nat,B2: nat > nat] :
% 9.10/5.32        ( ! [I3: nat,J2: nat] :
% 9.10/5.32            ( ( ord_less_eq_nat @ I3 @ J2 )
% 9.10/5.32           => ( ( ord_less_nat @ J2 @ N )
% 9.10/5.32             => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 9.10/5.32       => ( ! [I3: nat,J2: nat] :
% 9.10/5.32              ( ( ord_less_eq_nat @ I3 @ J2 )
% 9.10/5.32             => ( ( ord_less_nat @ J2 @ N )
% 9.10/5.32               => ( ord_less_eq_nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
% 9.10/5.32         => ( ord_less_eq_nat
% 9.10/5.32            @ ( times_times_nat @ N
% 9.10/5.32              @ ( groups3542108847815614940at_nat
% 9.10/5.32                @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B2 @ I4 ) )
% 9.10/5.32                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 9.10/5.32            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9425_valid__eq,axiom,
% 9.10/5.32      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9426_valid__eq1,axiom,
% 9.10/5.32      ! [T: vEBT_VEBT,D: nat] :
% 9.10/5.32        ( ( vEBT_invar_vebt @ T @ D )
% 9.10/5.32       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9427_valid__eq2,axiom,
% 9.10/5.32      ! [T: vEBT_VEBT,D: nat] :
% 9.10/5.32        ( ( vEBT_VEBT_valid @ T @ D )
% 9.10/5.32       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9428_Code__Target__Int_Opositive__def,axiom,
% 9.10/5.32      code_Target_positive = numeral_numeral_int ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9429_divmod__step__integer__def,axiom,
% 9.10/5.32      ( unique4921790084139445826nteger
% 9.10/5.32      = ( ^ [L2: num] :
% 9.10/5.32            ( produc6916734918728496179nteger
% 9.10/5.32            @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9430_times__integer__code_I2_J,axiom,
% 9.10/5.32      ! [L: code_integer] :
% 9.10/5.32        ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
% 9.10/5.32        = zero_z3403309356797280102nteger ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9431_times__integer__code_I1_J,axiom,
% 9.10/5.32      ! [K: code_integer] :
% 9.10/5.32        ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 9.10/5.32        = zero_z3403309356797280102nteger ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9432_exhaustive__integer_H_Ocases,axiom,
% 9.10/5.32      ! [X3: produc8763457246119570046nteger] :
% 9.10/5.32        ~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
% 9.10/5.32            ( X3
% 9.10/5.32           != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9433_full__exhaustive__integer_H_Ocases,axiom,
% 9.10/5.32      ! [X3: produc1908205239877642774nteger] :
% 9.10/5.32        ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I3: code_integer] :
% 9.10/5.32            ( X3
% 9.10/5.32           != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9434_less__eq__integer__code_I1_J,axiom,
% 9.10/5.32      ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9435_divmod__integer_H__def,axiom,
% 9.10/5.32      ( unique3479559517661332726nteger
% 9.10/5.32      = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9436_sgn__integer__code,axiom,
% 9.10/5.32      ( sgn_sgn_Code_integer
% 9.10/5.32      = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9437_integer__of__int__code,axiom,
% 9.10/5.32      ( code_integer_of_int
% 9.10/5.32      = ( ^ [K2: int] :
% 9.10/5.32            ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 9.10/5.32            @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.10/5.32              @ ( if_Code_integer
% 9.10/5.32                @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.10/5.32                  = zero_zero_int )
% 9.10/5.32                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.10/5.32                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9438_Code__Numeral_Opositive__def,axiom,
% 9.10/5.32      code_positive = numera6620942414471956472nteger ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9439_abs__integer__code,axiom,
% 9.10/5.32      ( abs_abs_Code_integer
% 9.10/5.32      = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9440_less__integer_Oabs__eq,axiom,
% 9.10/5.32      ! [Xa2: int,X3: int] :
% 9.10/5.32        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 9.10/5.32        = ( ord_less_int @ Xa2 @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9441_less__integer__code_I1_J,axiom,
% 9.10/5.32      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9442_times__integer_Oabs__eq,axiom,
% 9.10/5.32      ! [Xa2: int,X3: int] :
% 9.10/5.32        ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 9.10/5.32        = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9443_less__eq__integer_Oabs__eq,axiom,
% 9.10/5.32      ! [Xa2: int,X3: int] :
% 9.10/5.32        ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 9.10/5.32        = ( ord_less_eq_int @ Xa2 @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9444_int__of__integer__code,axiom,
% 9.10/5.32      ( code_int_of_integer
% 9.10/5.32      = ( ^ [K2: code_integer] :
% 9.10/5.32            ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 9.10/5.32            @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 9.10/5.32              @ ( produc1553301316500091796er_int
% 9.10/5.32                @ ^ [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 ) )
% 9.10/5.32                @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9445_bit__cut__integer__def,axiom,
% 9.10/5.32      ( code_bit_cut_integer
% 9.10/5.32      = ( ^ [K2: code_integer] :
% 9.10/5.32            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.10/5.32            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9446_int__of__integer__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 9.10/5.32        = ( numeral_numeral_int @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9447_times__integer_Orep__eq,axiom,
% 9.10/5.32      ! [X3: code_integer,Xa2: code_integer] :
% 9.10/5.32        ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X3 @ Xa2 ) )
% 9.10/5.32        = ( times_times_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9448_less__integer_Orep__eq,axiom,
% 9.10/5.32      ( ord_le6747313008572928689nteger
% 9.10/5.32      = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9449_integer__less__iff,axiom,
% 9.10/5.32      ( ord_le6747313008572928689nteger
% 9.10/5.32      = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9450_integer__less__eq__iff,axiom,
% 9.10/5.32      ( ord_le3102999989581377725nteger
% 9.10/5.32      = ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9451_less__eq__integer_Orep__eq,axiom,
% 9.10/5.32      ( ord_le3102999989581377725nteger
% 9.10/5.32      = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9452_divmod__integer__def,axiom,
% 9.10/5.32      ( code_divmod_integer
% 9.10/5.32      = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L2 ) @ ( modulo364778990260209775nteger @ K2 @ L2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9453_num__of__integer__code,axiom,
% 9.10/5.32      ( code_num_of_integer
% 9.10/5.32      = ( ^ [K2: code_integer] :
% 9.10/5.32            ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 9.10/5.32            @ ( produc7336495610019696514er_num
% 9.10/5.32              @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 9.10/5.32              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9454_bit__cut__integer__code,axiom,
% 9.10/5.32      ( code_bit_cut_integer
% 9.10/5.32      = ( ^ [K2: code_integer] :
% 9.10/5.32            ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 9.10/5.32            @ ( produc9125791028180074456eger_o
% 9.10/5.32              @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
% 9.10/5.32              @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9455_nat__of__integer__code,axiom,
% 9.10/5.32      ( code_nat_of_integer
% 9.10/5.32      = ( ^ [K2: code_integer] :
% 9.10/5.32            ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 9.10/5.32            @ ( produc1555791787009142072er_nat
% 9.10/5.32              @ ^ [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 ) )
% 9.10/5.32              @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9456_nat__of__integer__non__positive,axiom,
% 9.10/5.32      ! [K: code_integer] :
% 9.10/5.32        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 9.10/5.32       => ( ( code_nat_of_integer @ K )
% 9.10/5.32          = zero_zero_nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9457_nat__of__integer__code__post_I3_J,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 9.10/5.32        = ( numeral_numeral_nat @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9458_divmod__abs__code_I6_J,axiom,
% 9.10/5.32      ! [J: code_integer] :
% 9.10/5.32        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 9.10/5.32        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9459_divmod__abs__code_I5_J,axiom,
% 9.10/5.32      ! [J: code_integer] :
% 9.10/5.32        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 9.10/5.32        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9460_divmod__abs__def,axiom,
% 9.10/5.32      ( code_divmod_abs
% 9.10/5.32      = ( ^ [K2: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9461_divmod__integer__code,axiom,
% 9.10/5.32      ( code_divmod_integer
% 9.10/5.32      = ( ^ [K2: code_integer,L2: code_integer] :
% 9.10/5.32            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.10/5.32            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 9.10/5.32              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
% 9.10/5.32                @ ( produc6916734918728496179nteger
% 9.10/5.32                  @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
% 9.10/5.32                  @ ( code_divmod_abs @ K2 @ L2 ) ) )
% 9.10/5.32              @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 9.10/5.32                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 9.10/5.32                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L2 )
% 9.10/5.32                    @ ( produc6916734918728496179nteger
% 9.10/5.32                      @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
% 9.10/5.32                      @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9462_csqrt_Osimps_I1_J,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( re @ ( csqrt @ Z ) )
% 9.10/5.32        = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9463_card__Collect__less__nat,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( finite_card_nat
% 9.10/5.32          @ ( collect_nat
% 9.10/5.32            @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 9.10/5.32        = N ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9464_card__atLeastLessThan,axiom,
% 9.10/5.32      ! [L: nat,U: nat] :
% 9.10/5.32        ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 9.10/5.32        = ( minus_minus_nat @ U @ L ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9465_card__Collect__le__nat,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( finite_card_nat
% 9.10/5.32          @ ( collect_nat
% 9.10/5.32            @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 9.10/5.32        = ( suc @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9466_card__atLeastAtMost,axiom,
% 9.10/5.32      ! [L: nat,U: nat] :
% 9.10/5.32        ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.10/5.32        = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9467_complex__Re__numeral,axiom,
% 9.10/5.32      ! [V: num] :
% 9.10/5.32        ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 9.10/5.32        = ( numeral_numeral_real @ V ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9468_Re__divide__numeral,axiom,
% 9.10/5.32      ! [Z: complex,W: num] :
% 9.10/5.32        ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.32        = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9469_complex__Re__le__cmod,axiom,
% 9.10/5.32      ! [X3: complex] : ( ord_less_eq_real @ ( re @ X3 ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9470_scaleR__complex_Osimps_I1_J,axiom,
% 9.10/5.32      ! [R: real,X3: complex] :
% 9.10/5.32        ( ( re @ ( real_V2046097035970521341omplex @ R @ X3 ) )
% 9.10/5.32        = ( times_times_real @ R @ ( re @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9471_abs__Re__le__cmod,axiom,
% 9.10/5.32      ! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9472_Re__csqrt,axiom,
% 9.10/5.32      ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9473_card__less__Suc2,axiom,
% 9.10/5.32      ! [M7: set_nat,I: nat] :
% 9.10/5.32        ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 9.10/5.32       => ( ( finite_card_nat
% 9.10/5.32            @ ( collect_nat
% 9.10/5.32              @ ^ [K2: nat] :
% 9.10/5.32                  ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 9.10/5.32                  & ( ord_less_nat @ K2 @ I ) ) ) )
% 9.10/5.32          = ( finite_card_nat
% 9.10/5.32            @ ( collect_nat
% 9.10/5.32              @ ^ [K2: nat] :
% 9.10/5.32                  ( ( member_nat @ K2 @ M7 )
% 9.10/5.32                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9474_card__less__Suc,axiom,
% 9.10/5.32      ! [M7: set_nat,I: nat] :
% 9.10/5.32        ( ( member_nat @ zero_zero_nat @ M7 )
% 9.10/5.32       => ( ( suc
% 9.10/5.32            @ ( finite_card_nat
% 9.10/5.32              @ ( collect_nat
% 9.10/5.32                @ ^ [K2: nat] :
% 9.10/5.32                    ( ( member_nat @ ( suc @ K2 ) @ M7 )
% 9.10/5.32                    & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 9.10/5.32          = ( finite_card_nat
% 9.10/5.32            @ ( collect_nat
% 9.10/5.32              @ ^ [K2: nat] :
% 9.10/5.32                  ( ( member_nat @ K2 @ M7 )
% 9.10/5.32                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9475_card__less,axiom,
% 9.10/5.32      ! [M7: set_nat,I: nat] :
% 9.10/5.32        ( ( member_nat @ zero_zero_nat @ M7 )
% 9.10/5.32       => ( ( finite_card_nat
% 9.10/5.32            @ ( collect_nat
% 9.10/5.32              @ ^ [K2: nat] :
% 9.10/5.32                  ( ( member_nat @ K2 @ M7 )
% 9.10/5.32                  & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 9.10/5.32         != zero_zero_nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9476_subset__card__intvl__is__intvl,axiom,
% 9.10/5.32      ! [A4: set_nat,K: nat] :
% 9.10/5.32        ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) )
% 9.10/5.32       => ( A4
% 9.10/5.32          = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9477_subset__eq__atLeast0__lessThan__card,axiom,
% 9.10/5.32      ! [N4: set_nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.10/5.32       => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9478_card__sum__le__nat__sum,axiom,
% 9.10/5.32      ! [S2: set_nat] :
% 9.10/5.32        ( ord_less_eq_nat
% 9.10/5.32        @ ( groups3542108847815614940at_nat
% 9.10/5.32          @ ^ [X: nat] : X
% 9.10/5.32          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 9.10/5.32        @ ( groups3542108847815614940at_nat
% 9.10/5.32          @ ^ [X: nat] : X
% 9.10/5.32          @ S2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9479_card__nth__roots,axiom,
% 9.10/5.32      ! [C: complex,N: nat] :
% 9.10/5.32        ( ( C != zero_zero_complex )
% 9.10/5.32       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.32         => ( ( finite_card_complex
% 9.10/5.32              @ ( collect_complex
% 9.10/5.32                @ ^ [Z2: complex] :
% 9.10/5.32                    ( ( power_power_complex @ Z2 @ N )
% 9.10/5.32                    = C ) ) )
% 9.10/5.32            = N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9480_card__roots__unity__eq,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.32       => ( ( finite_card_complex
% 9.10/5.32            @ ( collect_complex
% 9.10/5.32              @ ^ [Z2: complex] :
% 9.10/5.32                  ( ( power_power_complex @ Z2 @ N )
% 9.10/5.32                  = one_one_complex ) ) )
% 9.10/5.32          = N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9481_cmod__plus__Re__le__0__iff,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 9.10/5.32        = ( ( re @ Z )
% 9.10/5.32          = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9482_cos__n__Re__cis__pow__n,axiom,
% 9.10/5.32      ! [N: nat,A: real] :
% 9.10/5.32        ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 9.10/5.32        = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9483_csqrt_Ocode,axiom,
% 9.10/5.32      ( csqrt
% 9.10/5.32      = ( ^ [Z2: complex] :
% 9.10/5.32            ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.32            @ ( times_times_real
% 9.10/5.32              @ ( if_real
% 9.10/5.32                @ ( ( im @ Z2 )
% 9.10/5.32                  = zero_zero_real )
% 9.10/5.32                @ one_one_real
% 9.10/5.32                @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
% 9.10/5.32              @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9484_csqrt_Osimps_I2_J,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( im @ ( csqrt @ Z ) )
% 9.10/5.32        = ( times_times_real
% 9.10/5.32          @ ( if_real
% 9.10/5.32            @ ( ( im @ Z )
% 9.10/5.32              = zero_zero_real )
% 9.10/5.32            @ one_one_real
% 9.10/5.32            @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 9.10/5.32          @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9485_csqrt__of__real__nonpos,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( ( im @ X3 )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32       => ( ( ord_less_eq_real @ ( re @ X3 ) @ zero_zero_real )
% 9.10/5.32         => ( ( csqrt @ X3 )
% 9.10/5.32            = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X3 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9486_complex__Im__numeral,axiom,
% 9.10/5.32      ! [V: num] :
% 9.10/5.32        ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 9.10/5.32        = zero_zero_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9487_Im__i__times,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 9.10/5.32        = ( re @ Z ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9488_Re__i__times,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 9.10/5.32        = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9489_Im__divide__numeral,axiom,
% 9.10/5.32      ! [Z: complex,W: num] :
% 9.10/5.32        ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.32        = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9490_csqrt__of__real__nonneg,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( ( im @ X3 )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32       => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) )
% 9.10/5.32         => ( ( csqrt @ X3 )
% 9.10/5.32            = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9491_csqrt__minus,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( ( ord_less_real @ ( im @ X3 ) @ zero_zero_real )
% 9.10/5.32          | ( ( ( im @ X3 )
% 9.10/5.32              = zero_zero_real )
% 9.10/5.32            & ( ord_less_eq_real @ zero_zero_real @ ( re @ X3 ) ) ) )
% 9.10/5.32       => ( ( csqrt @ ( uminus1482373934393186551omplex @ X3 ) )
% 9.10/5.32          = ( times_times_complex @ imaginary_unit @ ( csqrt @ X3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9492_scaleR__complex_Osimps_I2_J,axiom,
% 9.10/5.32      ! [R: real,X3: complex] :
% 9.10/5.32        ( ( im @ ( real_V2046097035970521341omplex @ R @ X3 ) )
% 9.10/5.32        = ( times_times_real @ R @ ( im @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9493_abs__Im__le__cmod,axiom,
% 9.10/5.32      ! [X3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( real_V1022390504157884413omplex @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9494_times__complex_Osimps_I2_J,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( im @ ( times_times_complex @ X3 @ Y3 ) )
% 9.10/5.32        = ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9495_cmod__Im__le__iff,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( ( re @ X3 )
% 9.10/5.32          = ( re @ Y3 ) )
% 9.10/5.32       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 9.10/5.32          = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X3 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9496_cmod__Re__le__iff,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( ( im @ X3 )
% 9.10/5.32          = ( im @ Y3 ) )
% 9.10/5.32       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 9.10/5.32          = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X3 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9497_times__complex_Osimps_I1_J,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( re @ ( times_times_complex @ X3 @ Y3 ) )
% 9.10/5.32        = ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9498_scaleR__complex_Ocode,axiom,
% 9.10/5.32      ( real_V2046097035970521341omplex
% 9.10/5.32      = ( ^ [R5: real,X: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X ) ) @ ( times_times_real @ R5 @ ( im @ X ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9499_csqrt__principal,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 9.10/5.32        | ( ( ( re @ ( csqrt @ Z ) )
% 9.10/5.32            = zero_zero_real )
% 9.10/5.32          & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9500_cmod__le,axiom,
% 9.10/5.32      ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9501_sin__n__Im__cis__pow__n,axiom,
% 9.10/5.32      ! [N: nat,A: real] :
% 9.10/5.32        ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 9.10/5.32        = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9502_Re__exp,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( re @ ( exp_complex @ Z ) )
% 9.10/5.32        = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9503_Im__exp,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( im @ ( exp_complex @ Z ) )
% 9.10/5.32        = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9504_complex__eq,axiom,
% 9.10/5.32      ! [A: complex] :
% 9.10/5.32        ( A
% 9.10/5.32        = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9505_times__complex_Ocode,axiom,
% 9.10/5.32      ( times_times_complex
% 9.10/5.32      = ( ^ [X: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9506_exp__eq__polar,axiom,
% 9.10/5.32      ( exp_complex
% 9.10/5.32      = ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9507_cmod__power2,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.32        = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9508_Im__power2,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( im @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X3 ) ) @ ( im @ X3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9509_Re__power2,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( re @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32        = ( minus_minus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9510_complex__eq__0,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( Z = zero_zero_complex )
% 9.10/5.32        = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32          = zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9511_norm__complex__def,axiom,
% 9.10/5.32      ( real_V1022390504157884413omplex
% 9.10/5.32      = ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9512_inverse__complex_Osimps_I1_J,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( re @ ( invers8013647133539491842omplex @ X3 ) )
% 9.10/5.32        = ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9513_complex__neq__0,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( Z != zero_zero_complex )
% 9.10/5.32        = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9514_Re__divide,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( re @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 9.10/5.32        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9515_csqrt__unique,axiom,
% 9.10/5.32      ! [W: complex,Z: complex] :
% 9.10/5.32        ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.10/5.32          = Z )
% 9.10/5.32       => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 9.10/5.32            | ( ( ( re @ W )
% 9.10/5.32                = zero_zero_real )
% 9.10/5.32              & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 9.10/5.32         => ( ( csqrt @ Z )
% 9.10/5.32            = W ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9516_csqrt__square,axiom,
% 9.10/5.32      ! [B2: complex] :
% 9.10/5.32        ( ( ( ord_less_real @ zero_zero_real @ ( re @ B2 ) )
% 9.10/5.32          | ( ( ( re @ B2 )
% 9.10/5.32              = zero_zero_real )
% 9.10/5.32            & ( ord_less_eq_real @ zero_zero_real @ ( im @ B2 ) ) ) )
% 9.10/5.32       => ( ( csqrt @ ( power_power_complex @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32          = B2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9517_inverse__complex_Osimps_I2_J,axiom,
% 9.10/5.32      ! [X3: complex] :
% 9.10/5.32        ( ( im @ ( invers8013647133539491842omplex @ X3 ) )
% 9.10/5.32        = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9518_Im__divide,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( im @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 9.10/5.32        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9519_complex__abs__le__norm,axiom,
% 9.10/5.32      ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9520_complex__unit__circle,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( Z != zero_zero_complex )
% 9.10/5.32       => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32          = one_one_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9521_inverse__complex_Ocode,axiom,
% 9.10/5.32      ( invers8013647133539491842omplex
% 9.10/5.32      = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9522_Complex__divide,axiom,
% 9.10/5.32      ( divide1717551699836669952omplex
% 9.10/5.32      = ( ^ [X: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9523_Im__Reals__divide,axiom,
% 9.10/5.32      ! [R: complex,Z: complex] :
% 9.10/5.32        ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 9.10/5.32       => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
% 9.10/5.32          = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9524_imaginary__eq__real__iff,axiom,
% 9.10/5.32      ! [Y3: complex,X3: complex] :
% 9.10/5.32        ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 9.10/5.32       => ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
% 9.10/5.32         => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 9.10/5.32              = X3 )
% 9.10/5.32            = ( ( X3 = zero_zero_complex )
% 9.10/5.32              & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9525_real__eq__imaginary__iff,axiom,
% 9.10/5.32      ! [Y3: complex,X3: complex] :
% 9.10/5.32        ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 9.10/5.32       => ( ( member_complex @ X3 @ real_V2521375963428798218omplex )
% 9.10/5.32         => ( ( X3
% 9.10/5.32              = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 9.10/5.32            = ( ( X3 = zero_zero_complex )
% 9.10/5.32              & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9526_Re__Reals__divide,axiom,
% 9.10/5.32      ! [R: complex,Z: complex] :
% 9.10/5.32        ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 9.10/5.32       => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
% 9.10/5.32          = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9527_complex__mult__cnj,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 9.10/5.32        = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9528_cnj__add__mult__eq__Re,axiom,
% 9.10/5.32      ! [Z: complex,W: complex] :
% 9.10/5.32        ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 9.10/5.32        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9529_complex__div__cnj,axiom,
% 9.10/5.32      ( divide1717551699836669952omplex
% 9.10/5.32      = ( ^ [A5: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9530_complex__cnj__mult,axiom,
% 9.10/5.32      ! [X3: complex,Y3: complex] :
% 9.10/5.32        ( ( cnj @ ( times_times_complex @ X3 @ Y3 ) )
% 9.10/5.32        = ( times_times_complex @ ( cnj @ X3 ) @ ( cnj @ Y3 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9531_complex__cnj__numeral,axiom,
% 9.10/5.32      ! [W: num] :
% 9.10/5.32        ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 9.10/5.32        = ( numera6690914467698888265omplex @ W ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9532_complex__cnj__neg__numeral,axiom,
% 9.10/5.32      ! [W: num] :
% 9.10/5.32        ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 9.10/5.32        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9533_complex__In__mult__cnj__zero,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 9.10/5.32        = zero_zero_real ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9534_Re__complex__div__eq__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ( re @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32        = ( ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 9.10/5.32          = zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9535_Im__complex__div__eq__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ( im @ ( divide1717551699836669952omplex @ A @ B2 ) )
% 9.10/5.32          = zero_zero_real )
% 9.10/5.32        = ( ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) )
% 9.10/5.32          = zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9536_complex__mod__sqrt__Re__mult__cnj,axiom,
% 9.10/5.32      ( real_V1022390504157884413omplex
% 9.10/5.32      = ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9537_Re__complex__div__lt__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9538_Re__complex__div__gt__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.32        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9539_Re__complex__div__le__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9540_Re__complex__div__ge__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.32        = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9541_Im__complex__div__lt__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9542_Im__complex__div__gt__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.32        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9543_Im__complex__div__le__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) @ zero_zero_real )
% 9.10/5.32        = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9544_Im__complex__div__ge__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.32        = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9545_complex__mod__mult__cnj,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 9.10/5.32        = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9546_complex__div__gt__0,axiom,
% 9.10/5.32      ! [A: complex,B2: complex] :
% 9.10/5.32        ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.32          = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) )
% 9.10/5.32        & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B2 ) ) )
% 9.10/5.32          = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9547_complex__norm__square,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.10/5.32        = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9548_complex__add__cnj,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 9.10/5.32        = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9549_complex__diff__cnj,axiom,
% 9.10/5.32      ! [Z: complex] :
% 9.10/5.32        ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 9.10/5.32        = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9550_nat_Odisc__eq__case_I2_J,axiom,
% 9.10/5.32      ! [Nat: nat] :
% 9.10/5.32        ( ( Nat != zero_zero_nat )
% 9.10/5.32        = ( case_nat_o @ $false
% 9.10/5.32          @ ^ [Uu3: nat] : $true
% 9.10/5.32          @ Nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9551_nat_Odisc__eq__case_I1_J,axiom,
% 9.10/5.32      ! [Nat: nat] :
% 9.10/5.32        ( ( Nat = zero_zero_nat )
% 9.10/5.32        = ( case_nat_o @ $true
% 9.10/5.32          @ ^ [Uu3: nat] : $false
% 9.10/5.32          @ Nat ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9552_less__eq__nat_Osimps_I2_J,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.10/5.32        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9553_max__Suc2,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_max_nat @ M @ ( suc @ N ) )
% 9.10/5.32        = ( case_nat_nat @ ( suc @ N )
% 9.10/5.32          @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
% 9.10/5.32          @ M ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9554_max__Suc1,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( ord_max_nat @ ( suc @ N ) @ M )
% 9.10/5.32        = ( case_nat_nat @ ( suc @ N )
% 9.10/5.32          @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
% 9.10/5.32          @ M ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9555_diff__Suc,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.10/5.32        = ( case_nat_nat @ zero_zero_nat
% 9.10/5.32          @ ^ [K2: nat] : K2
% 9.10/5.32          @ ( minus_minus_nat @ M @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9556_pred__def,axiom,
% 9.10/5.32      ( pred
% 9.10/5.32      = ( case_nat_nat @ zero_zero_nat
% 9.10/5.32        @ ^ [X23: nat] : X23 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9557_floor__real__def,axiom,
% 9.10/5.32      ( archim6058952711729229775r_real
% 9.10/5.32      = ( ^ [X: real] :
% 9.10/5.32            ( the_int
% 9.10/5.32            @ ^ [Z2: int] :
% 9.10/5.32                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 9.10/5.32                & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9558_floor__rat__def,axiom,
% 9.10/5.32      ( archim3151403230148437115or_rat
% 9.10/5.32      = ( ^ [X: rat] :
% 9.10/5.32            ( the_int
% 9.10/5.32            @ ^ [Z2: int] :
% 9.10/5.32                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 9.10/5.32                & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9559_bezw__0,axiom,
% 9.10/5.32      ! [X3: nat] :
% 9.10/5.32        ( ( bezw @ X3 @ zero_zero_nat )
% 9.10/5.32        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9560_prod__decode__aux_Oelims,axiom,
% 9.10/5.32      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 9.10/5.32        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.10/5.32           => ( Y3
% 9.10/5.32              = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 9.10/5.32          & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.10/5.32           => ( Y3
% 9.10/5.32              = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9561_less__eq__rat__def,axiom,
% 9.10/5.32      ( ord_less_eq_rat
% 9.10/5.32      = ( ^ [X: rat,Y: rat] :
% 9.10/5.32            ( ( ord_less_rat @ X @ Y )
% 9.10/5.32            | ( X = Y ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9562_obtain__pos__sum,axiom,
% 9.10/5.32      ! [R: rat] :
% 9.10/5.32        ( ( ord_less_rat @ zero_zero_rat @ R )
% 9.10/5.32       => ~ ! [S3: rat] :
% 9.10/5.32              ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 9.10/5.32             => ! [T4: rat] :
% 9.10/5.32                  ( ( ord_less_rat @ zero_zero_rat @ T4 )
% 9.10/5.32                 => ( R
% 9.10/5.32                   != ( plus_plus_rat @ S3 @ T4 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9563_sgn__rat__def,axiom,
% 9.10/5.32      ( sgn_sgn_rat
% 9.10/5.32      = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9564_abs__rat__def,axiom,
% 9.10/5.32      ( abs_abs_rat
% 9.10/5.32      = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9565_prod__decode__aux_Osimps,axiom,
% 9.10/5.32      ( nat_prod_decode_aux
% 9.10/5.32      = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9566_prod__decode__aux_Opelims,axiom,
% 9.10/5.32      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 9.10/5.32        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 9.10/5.32         => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                    = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 9.10/5.32                & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                    = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
% 9.10/5.32             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9567_rat__inverse__code,axiom,
% 9.10/5.32      ! [P4: rat] :
% 9.10/5.32        ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9568_normalize__negative,axiom,
% 9.10/5.32      ! [Q2: int,P4: int] :
% 9.10/5.32        ( ( ord_less_int @ Q2 @ zero_zero_int )
% 9.10/5.32       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.32          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9569_quotient__of__number_I3_J,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 9.10/5.32        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9570_normalize__denom__zero,axiom,
% 9.10/5.32      ! [P4: int] :
% 9.10/5.32        ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 9.10/5.32        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9571_rat__one__code,axiom,
% 9.10/5.32      ( ( quotient_of @ one_one_rat )
% 9.10/5.32      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9572_rat__zero__code,axiom,
% 9.10/5.32      ( ( quotient_of @ zero_zero_rat )
% 9.10/5.32      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9573_quotient__of__number_I5_J,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.10/5.32        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9574_quotient__of__number_I4_J,axiom,
% 9.10/5.32      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.10/5.32      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9575_divide__rat__def,axiom,
% 9.10/5.32      ( divide_divide_rat
% 9.10/5.32      = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9576_rat__divide__code,axiom,
% 9.10/5.32      ! [P4: rat,Q2: rat] :
% 9.10/5.32        ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int,C3: int] :
% 9.10/5.32              ( produc4245557441103728435nt_int
% 9.10/5.32              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
% 9.10/5.32              @ ( quotient_of @ Q2 ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9577_rat__times__code,axiom,
% 9.10/5.32      ! [P4: rat,Q2: rat] :
% 9.10/5.32        ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int,C3: int] :
% 9.10/5.32              ( produc4245557441103728435nt_int
% 9.10/5.32              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
% 9.10/5.32              @ ( quotient_of @ Q2 ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9578_quotient__of__div,axiom,
% 9.10/5.32      ! [R: rat,N: int,D: int] :
% 9.10/5.32        ( ( ( quotient_of @ R )
% 9.10/5.32          = ( product_Pair_int_int @ N @ D ) )
% 9.10/5.32       => ( R
% 9.10/5.32          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9579_rat__plus__code,axiom,
% 9.10/5.32      ! [P4: rat,Q2: rat] :
% 9.10/5.32        ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int,C3: int] :
% 9.10/5.32              ( produc4245557441103728435nt_int
% 9.10/5.32              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 9.10/5.32              @ ( quotient_of @ Q2 ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9580_rat__minus__code,axiom,
% 9.10/5.32      ! [P4: rat,Q2: rat] :
% 9.10/5.32        ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int,C3: int] :
% 9.10/5.32              ( produc4245557441103728435nt_int
% 9.10/5.32              @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 9.10/5.32              @ ( quotient_of @ Q2 ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9581_quotient__of__denom__pos,axiom,
% 9.10/5.32      ! [R: rat,P4: int,Q2: int] :
% 9.10/5.32        ( ( ( quotient_of @ R )
% 9.10/5.32          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.32       => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9582_rat__uminus__code,axiom,
% 9.10/5.32      ! [P4: rat] :
% 9.10/5.32        ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9583_rat__abs__code,axiom,
% 9.10/5.32      ! [P4: rat] :
% 9.10/5.32        ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 9.10/5.32        = ( produc4245557441103728435nt_int
% 9.10/5.32          @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
% 9.10/5.32          @ ( quotient_of @ P4 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9584_normalize__denom__pos,axiom,
% 9.10/5.32      ! [R: product_prod_int_int,P4: int,Q2: int] :
% 9.10/5.32        ( ( ( normalize @ R )
% 9.10/5.32          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.32       => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9585_normalize__crossproduct,axiom,
% 9.10/5.32      ! [Q2: int,S: int,P4: int,R: int] :
% 9.10/5.32        ( ( Q2 != zero_zero_int )
% 9.10/5.32       => ( ( S != zero_zero_int )
% 9.10/5.32         => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.32              = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
% 9.10/5.32           => ( ( times_times_int @ P4 @ S )
% 9.10/5.32              = ( times_times_int @ R @ Q2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9586_rat__less__code,axiom,
% 9.10/5.32      ( ord_less_rat
% 9.10/5.32      = ( ^ [P5: rat,Q4: rat] :
% 9.10/5.32            ( produc4947309494688390418_int_o
% 9.10/5.32            @ ^ [A5: int,C3: int] :
% 9.10/5.32                ( produc4947309494688390418_int_o
% 9.10/5.32                @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 9.10/5.32                @ ( quotient_of @ Q4 ) )
% 9.10/5.32            @ ( quotient_of @ P5 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9587_rat__less__eq__code,axiom,
% 9.10/5.32      ( ord_less_eq_rat
% 9.10/5.32      = ( ^ [P5: rat,Q4: rat] :
% 9.10/5.32            ( produc4947309494688390418_int_o
% 9.10/5.32            @ ^ [A5: int,C3: int] :
% 9.10/5.32                ( produc4947309494688390418_int_o
% 9.10/5.32                @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 9.10/5.32                @ ( quotient_of @ Q4 ) )
% 9.10/5.32            @ ( quotient_of @ P5 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9588_quotient__of__int,axiom,
% 9.10/5.32      ! [A: int] :
% 9.10/5.32        ( ( quotient_of @ ( of_int @ A ) )
% 9.10/5.32        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9589_drop__bit__numeral__minus__bit1,axiom,
% 9.10/5.32      ! [L: num,K: num] :
% 9.10/5.32        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.10/5.32        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9590_Suc__0__div__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.32        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9591_drop__bit__nonnegative__int__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 9.10/5.32        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9592_drop__bit__negative__int__iff,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 9.10/5.32        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9593_drop__bit__minus__one,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 9.10/5.32        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9594_fst__divmod__nat,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 9.10/5.32        = ( divide_divide_nat @ M @ N ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9595_drop__bit__Suc__minus__bit0,axiom,
% 9.10/5.32      ! [N: nat,K: num] :
% 9.10/5.32        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.10/5.32        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9596_drop__bit__numeral__minus__bit0,axiom,
% 9.10/5.32      ! [L: num,K: num] :
% 9.10/5.32        ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.10/5.32        = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9597_drop__bit__Suc__minus__bit1,axiom,
% 9.10/5.32      ! [N: nat,K: num] :
% 9.10/5.32        ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.10/5.32        = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9598_drop__bit__push__bit__int,axiom,
% 9.10/5.32      ! [M: nat,N: nat,K: int] :
% 9.10/5.32        ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 9.10/5.32        = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9599_drop__bit__int__def,axiom,
% 9.10/5.32      ( bit_se8568078237143864401it_int
% 9.10/5.32      = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9600_Frct__code__post_I5_J,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 9.10/5.32        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9601_Frct__code__post_I6_J,axiom,
% 9.10/5.32      ! [K: num,L: num] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 9.10/5.32        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9602_Suc__0__mod__numeral,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.32        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9603_drop__bit__of__Suc__0,axiom,
% 9.10/5.32      ! [N: nat] :
% 9.10/5.32        ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.10/5.32        = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9604_drop__bit__nat__eq,axiom,
% 9.10/5.32      ! [N: nat,K: int] :
% 9.10/5.32        ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 9.10/5.32        = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9605_drop__bit__nat__def,axiom,
% 9.10/5.32      ( bit_se8570568707652914677it_nat
% 9.10/5.32      = ( ^ [N2: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9606_Frct__code__post_I1_J,axiom,
% 9.10/5.32      ! [A: int] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 9.10/5.32        = zero_zero_rat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9607_Frct__code__post_I2_J,axiom,
% 9.10/5.32      ! [A: int] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 9.10/5.32        = zero_zero_rat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9608_Frct__code__post_I8_J,axiom,
% 9.10/5.32      ! [A: int,B2: int] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B2 ) ) )
% 9.10/5.32        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9609_Frct__code__post_I7_J,axiom,
% 9.10/5.32      ! [A: int,B2: int] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B2 ) )
% 9.10/5.32        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9610_Frct__code__post_I3_J,axiom,
% 9.10/5.32      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 9.10/5.32      = one_one_rat ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9611_rat__sgn__code,axiom,
% 9.10/5.32      ! [P4: rat] :
% 9.10/5.32        ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 9.10/5.32        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9612_Frct__code__post_I4_J,axiom,
% 9.10/5.32      ! [K: num] :
% 9.10/5.32        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 9.10/5.32        = ( numeral_numeral_rat @ K ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9613_quotient__of__denom__pos_H,axiom,
% 9.10/5.32      ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9614_bezw__non__0,axiom,
% 9.10/5.32      ! [Y3: nat,X3: nat] :
% 9.10/5.32        ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 9.10/5.32       => ( ( bezw @ X3 @ Y3 )
% 9.10/5.32          = ( 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 ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9615_bezw_Osimps,axiom,
% 9.10/5.32      ( bezw
% 9.10/5.32      = ( ^ [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 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9616_bezw_Oelims,axiom,
% 9.10/5.32      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 9.10/5.32        ( ( ( bezw @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.32           => ( Y3
% 9.10/5.32              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.10/5.32          & ( ( Xa2 != zero_zero_nat )
% 9.10/5.32           => ( Y3
% 9.10/5.32              = ( 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 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9617_bezw_Opelims,axiom,
% 9.10/5.32      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 9.10/5.32        ( ( ( bezw @ X3 @ Xa2 )
% 9.10/5.32          = Y3 )
% 9.10/5.32       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 9.10/5.32         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.10/5.32                & ( ( Xa2 != zero_zero_nat )
% 9.10/5.32                 => ( Y3
% 9.10/5.32                    = ( 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 ) ) ) ) ) ) ) )
% 9.10/5.32             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9618_minus__one__mod__numeral,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.32        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9619_one__mod__minus__numeral,axiom,
% 9.10/5.32      ! [N: num] :
% 9.10/5.32        ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9620_minus__numeral__mod__numeral,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.32        = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9621_numeral__mod__minus__numeral,axiom,
% 9.10/5.32      ! [M: num,N: num] :
% 9.10/5.32        ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.32        = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9622_normalize__def,axiom,
% 9.10/5.32      ( normalize
% 9.10/5.32      = ( ^ [P5: product_prod_int_int] :
% 9.10/5.32            ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
% 9.10/5.32            @ ( if_Pro3027730157355071871nt_int
% 9.10/5.32              @ ( ( product_snd_int_int @ P5 )
% 9.10/5.32                = zero_zero_int )
% 9.10/5.32              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.10/5.32              @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9623_gcd__pos__int,axiom,
% 9.10/5.32      ! [M: int,N: int] :
% 9.10/5.32        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 9.10/5.32        = ( ( M != zero_zero_int )
% 9.10/5.32          | ( N != zero_zero_int ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9624_gcd__neg__numeral__2__int,axiom,
% 9.10/5.32      ! [X3: int,N: num] :
% 9.10/5.32        ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.32        = ( gcd_gcd_int @ X3 @ ( numeral_numeral_int @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9625_gcd__neg__numeral__1__int,axiom,
% 9.10/5.32      ! [N: num,X3: int] :
% 9.10/5.32        ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X3 )
% 9.10/5.32        = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9626_gcd__ge__0__int,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9627_bezout__int,axiom,
% 9.10/5.32      ! [X3: int,Y3: int] :
% 9.10/5.32      ? [U3: int,V2: int] :
% 9.10/5.32        ( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y3 ) )
% 9.10/5.32        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9628_gcd__mult__distrib__int,axiom,
% 9.10/5.32      ! [K: int,M: int,N: int] :
% 9.10/5.32        ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 9.10/5.32        = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9629_gcd__le2__int,axiom,
% 9.10/5.32      ! [B2: int,A: int] :
% 9.10/5.32        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.32       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ B2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9630_gcd__le1__int,axiom,
% 9.10/5.32      ! [A: int,B2: int] :
% 9.10/5.32        ( ( ord_less_int @ zero_zero_int @ A )
% 9.10/5.32       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B2 ) @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9631_gcd__cases__int,axiom,
% 9.10/5.32      ! [X3: int,Y3: int,P: int > $o] :
% 9.10/5.32        ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32           => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) )
% 9.10/5.32       => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.10/5.32           => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.10/5.32             => ( P @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.10/5.32         => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 9.10/5.32             => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.10/5.32               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 ) ) ) )
% 9.10/5.32           => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 9.10/5.32               => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.10/5.32                 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.10/5.32             => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9632_gcd__unique__int,axiom,
% 9.10/5.32      ! [D: int,A: int,B2: int] :
% 9.10/5.32        ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 9.10/5.32          & ( dvd_dvd_int @ D @ A )
% 9.10/5.32          & ( dvd_dvd_int @ D @ B2 )
% 9.10/5.32          & ! [E3: int] :
% 9.10/5.32              ( ( ( dvd_dvd_int @ E3 @ A )
% 9.10/5.32                & ( dvd_dvd_int @ E3 @ B2 ) )
% 9.10/5.32             => ( dvd_dvd_int @ E3 @ D ) ) )
% 9.10/5.32        = ( D
% 9.10/5.32          = ( gcd_gcd_int @ A @ B2 ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9633_gcd__non__0__int,axiom,
% 9.10/5.32      ! [Y3: int,X3: int] :
% 9.10/5.32        ( ( ord_less_int @ zero_zero_int @ Y3 )
% 9.10/5.32       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 9.10/5.32          = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9634_nat__descend__induct,axiom,
% 9.10/5.32      ! [N: nat,P: nat > $o,M: nat] :
% 9.10/5.32        ( ! [K3: nat] :
% 9.10/5.32            ( ( ord_less_nat @ N @ K3 )
% 9.10/5.32           => ( P @ K3 ) )
% 9.10/5.32       => ( ! [K3: nat] :
% 9.10/5.32              ( ( ord_less_eq_nat @ K3 @ N )
% 9.10/5.32             => ( ! [I2: nat] :
% 9.10/5.32                    ( ( ord_less_nat @ K3 @ I2 )
% 9.10/5.32                   => ( P @ I2 ) )
% 9.10/5.32               => ( P @ K3 ) ) )
% 9.10/5.32         => ( P @ M ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9635_finite__enumerate,axiom,
% 9.10/5.32      ! [S2: set_nat] :
% 9.10/5.32        ( ( finite_finite_nat @ S2 )
% 9.10/5.32       => ? [R3: nat > nat] :
% 9.10/5.32            ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 9.10/5.32            & ! [N9: nat] :
% 9.10/5.32                ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S2 ) )
% 9.10/5.32               => ( member_nat @ ( R3 @ N9 ) @ S2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9636_divmod__integer__eq__cases,axiom,
% 9.10/5.32      ( code_divmod_integer
% 9.10/5.32      = ( ^ [K2: code_integer,L2: code_integer] :
% 9.10/5.32            ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.10/5.32            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 9.10/5.32              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 9.10/5.32                @ ( if_Pro6119634080678213985nteger
% 9.10/5.32                  @ ( ( sgn_sgn_Code_integer @ K2 )
% 9.10/5.32                    = ( sgn_sgn_Code_integer @ L2 ) )
% 9.10/5.32                  @ ( code_divmod_abs @ K2 @ L2 )
% 9.10/5.32                  @ ( produc6916734918728496179nteger
% 9.10/5.32                    @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
% 9.10/5.32                    @ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9637_gcd__pos__nat,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 9.10/5.32        = ( ( M != zero_zero_nat )
% 9.10/5.32          | ( N != zero_zero_nat ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9638_gcd__mult__distrib__nat,axiom,
% 9.10/5.32      ! [K: nat,M: nat,N: nat] :
% 9.10/5.32        ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 9.10/5.32        = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9639_gcd__le2__nat,axiom,
% 9.10/5.32      ! [B2: nat,A: nat] :
% 9.10/5.32        ( ( B2 != zero_zero_nat )
% 9.10/5.32       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ B2 ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9640_gcd__le1__nat,axiom,
% 9.10/5.32      ! [A: nat,B2: nat] :
% 9.10/5.32        ( ( A != zero_zero_nat )
% 9.10/5.32       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B2 ) @ A ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9641_gcd__diff1__nat,axiom,
% 9.10/5.32      ! [N: nat,M: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ N @ M )
% 9.10/5.32       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 9.10/5.32          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9642_gcd__diff2__nat,axiom,
% 9.10/5.32      ! [M: nat,N: nat] :
% 9.10/5.32        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.32       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 9.10/5.32          = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9643_bezout__nat,axiom,
% 9.10/5.32      ! [A: nat,B2: nat] :
% 9.10/5.32        ( ( A != zero_zero_nat )
% 9.10/5.32       => ? [X5: nat,Y4: nat] :
% 9.10/5.32            ( ( times_times_nat @ A @ X5 )
% 9.10/5.32            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.32  
% 9.10/5.32  thf(fact_9644_bezout__gcd__nat_H,axiom,
% 9.10/5.32      ! [B2: nat,A: nat] :
% 9.10/5.32      ? [X5: nat,Y4: nat] :
% 9.10/5.32        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
% 9.10/5.33          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B2 @ Y4 ) )
% 9.10/5.33            = ( gcd_gcd_nat @ A @ B2 ) ) )
% 9.10/5.33        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B2 @ X5 ) )
% 9.10/5.33          & ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
% 9.10/5.33            = ( gcd_gcd_nat @ A @ B2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9645_bezw__aux,axiom,
% 9.10/5.33      ! [X3: nat,Y3: nat] :
% 9.10/5.33        ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y3 ) )
% 9.10/5.33        = ( 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 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9646_gcd__nat_Opelims,axiom,
% 9.10/5.33      ! [X3: nat,Xa2: nat,Y3: nat] :
% 9.10/5.33        ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 9.10/5.33         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.10/5.33                 => ( Y3 = X3 ) )
% 9.10/5.33                & ( ( Xa2 != zero_zero_nat )
% 9.10/5.33                 => ( Y3
% 9.10/5.33                    = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
% 9.10/5.33             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9647_Code__Numeral_Onegative__def,axiom,
% 9.10/5.33      ( code_negative
% 9.10/5.33      = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9648_Code__Target__Int_Onegative__def,axiom,
% 9.10/5.33      ( code_Target_negative
% 9.10/5.33      = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9649_root__powr__inverse,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( ( root @ N @ X3 )
% 9.10/5.33            = ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9650_real__root__zero,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( root @ N @ zero_zero_real )
% 9.10/5.33        = zero_zero_real ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9651_real__root__Suc__0,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9652_root__0,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( root @ zero_zero_nat @ X3 )
% 9.10/5.33        = zero_zero_real ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9653_real__root__eq__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ( root @ N @ X3 )
% 9.10/5.33            = ( root @ N @ Y3 ) )
% 9.10/5.33          = ( X3 = Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9654_real__root__eq__0__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ( root @ N @ X3 )
% 9.10/5.33            = zero_zero_real )
% 9.10/5.33          = ( X3 = zero_zero_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9655_real__root__less__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
% 9.10/5.33          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9656_real__root__le__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) )
% 9.10/5.33          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9657_real__root__one,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( root @ N @ one_one_real )
% 9.10/5.33          = one_one_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9658_real__root__eq__1__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ( root @ N @ X3 )
% 9.10/5.33            = one_one_real )
% 9.10/5.33          = ( X3 = one_one_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9659_real__root__lt__0__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ ( root @ N @ X3 ) @ zero_zero_real )
% 9.10/5.33          = ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9660_real__root__gt__0__iff,axiom,
% 9.10/5.33      ! [N: nat,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 9.10/5.33          = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9661_real__root__le__0__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ zero_zero_real )
% 9.10/5.33          = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9662_real__root__ge__0__iff,axiom,
% 9.10/5.33      ! [N: nat,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 9.10/5.33          = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9663_real__root__gt__1__iff,axiom,
% 9.10/5.33      ! [N: nat,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 9.10/5.33          = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9664_real__root__lt__1__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ ( root @ N @ X3 ) @ one_one_real )
% 9.10/5.33          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9665_real__root__le__1__iff,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ ( root @ N @ X3 ) @ one_one_real )
% 9.10/5.33          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9666_real__root__ge__1__iff,axiom,
% 9.10/5.33      ! [N: nat,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 9.10/5.33          = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9667_real__root__pow__pos2,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
% 9.10/5.33            = X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9668_real__root__inverse,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( root @ N @ ( inverse_inverse_real @ X3 ) )
% 9.10/5.33        = ( inverse_inverse_real @ ( root @ N @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9669_real__root__minus,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( root @ N @ ( uminus_uminus_real @ X3 ) )
% 9.10/5.33        = ( uminus_uminus_real @ ( root @ N @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9670_real__root__commute,axiom,
% 9.10/5.33      ! [M: nat,N: nat,X3: real] :
% 9.10/5.33        ( ( root @ M @ ( root @ N @ X3 ) )
% 9.10/5.33        = ( root @ N @ ( root @ M @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9671_real__root__mult,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( root @ N @ ( times_times_real @ X3 @ Y3 ) )
% 9.10/5.33        = ( times_times_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9672_real__root__mult__exp,axiom,
% 9.10/5.33      ! [M: nat,N: nat,X3: real] :
% 9.10/5.33        ( ( root @ ( times_times_nat @ M @ N ) @ X3 )
% 9.10/5.33        = ( root @ M @ ( root @ N @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9673_real__root__divide,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( root @ N @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.10/5.33        = ( divide_divide_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9674_real__root__pos__pos__le,axiom,
% 9.10/5.33      ! [X3: real,N: nat] :
% 9.10/5.33        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9675_real__root__less__mono,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.33         => ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9676_real__root__le__mono,axiom,
% 9.10/5.33      ! [N: nat,X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.10/5.33         => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9677_real__root__power,axiom,
% 9.10/5.33      ! [N: nat,X3: real,K: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( root @ N @ ( power_power_real @ X3 @ K ) )
% 9.10/5.33          = ( power_power_real @ ( root @ N @ X3 ) @ K ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9678_real__root__abs,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( root @ N @ ( abs_abs_real @ X3 ) )
% 9.10/5.33          = ( abs_abs_real @ ( root @ N @ X3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9679_sgn__root,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( sgn_sgn_real @ ( root @ N @ X3 ) )
% 9.10/5.33          = ( sgn_sgn_real @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9680_real__root__gt__zero,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( ord_less_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9681_real__root__strict__decreasing,axiom,
% 9.10/5.33      ! [N: nat,N4: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_nat @ N @ N4 )
% 9.10/5.33         => ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.33           => ( ord_less_real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9682_sqrt__def,axiom,
% 9.10/5.33      ( sqrt
% 9.10/5.33      = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9683_root__abs__power,axiom,
% 9.10/5.33      ! [N: nat,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 9.10/5.33          = ( abs_abs_real @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9684_real__root__pos__pos,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9685_real__root__strict__increasing,axiom,
% 9.10/5.33      ! [N: nat,N4: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_nat @ N @ N4 )
% 9.10/5.33         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33           => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33             => ( ord_less_real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9686_real__root__decreasing,axiom,
% 9.10/5.33      ! [N: nat,N4: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_nat @ N @ N4 )
% 9.10/5.33         => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.10/5.33           => ( ord_less_eq_real @ ( root @ N4 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9687_real__root__pow__pos,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
% 9.10/5.33            = X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9688_real__root__pos__unique,axiom,
% 9.10/5.33      ! [N: nat,Y3: real,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.10/5.33         => ( ( ( power_power_real @ Y3 @ N )
% 9.10/5.33              = X3 )
% 9.10/5.33           => ( ( root @ N @ X3 )
% 9.10/5.33              = Y3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9689_real__root__power__cancel,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( ( root @ N @ ( power_power_real @ X3 @ N ) )
% 9.10/5.33            = X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9690_odd__real__root__power__cancel,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33       => ( ( root @ N @ ( power_power_real @ X3 @ N ) )
% 9.10/5.33          = X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9691_odd__real__root__unique,axiom,
% 9.10/5.33      ! [N: nat,Y3: real,X3: real] :
% 9.10/5.33        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33       => ( ( ( power_power_real @ Y3 @ N )
% 9.10/5.33            = X3 )
% 9.10/5.33         => ( ( root @ N @ X3 )
% 9.10/5.33            = Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9692_odd__real__root__pow,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33       => ( ( power_power_real @ ( root @ N @ X3 ) @ N )
% 9.10/5.33          = X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9693_real__root__increasing,axiom,
% 9.10/5.33      ! [N: nat,N4: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_eq_nat @ N @ N4 )
% 9.10/5.33         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33           => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.33             => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N4 @ X3 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9694_sgn__power__root,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X3 ) ) @ N ) )
% 9.10/5.33          = X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9695_root__sgn__power,axiom,
% 9.10/5.33      ! [N: nat,Y3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 9.10/5.33          = Y3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9696_ln__root,axiom,
% 9.10/5.33      ! [N: nat,B2: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.33         => ( ( ln_ln_real @ ( root @ N @ B2 ) )
% 9.10/5.33            = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9697_log__root,axiom,
% 9.10/5.33      ! [N: nat,A: real,B2: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.10/5.33         => ( ( log @ B2 @ ( root @ N @ A ) )
% 9.10/5.33            = ( divide_divide_real @ ( log @ B2 @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9698_log__base__root,axiom,
% 9.10/5.33      ! [N: nat,B2: real,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ B2 )
% 9.10/5.33         => ( ( log @ ( root @ N @ B2 ) @ X3 )
% 9.10/5.33            = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B2 @ X3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9699_split__root,axiom,
% 9.10/5.33      ! [P: real > $o,N: nat,X3: real] :
% 9.10/5.33        ( ( P @ ( root @ N @ X3 ) )
% 9.10/5.33        = ( ( ( N = zero_zero_nat )
% 9.10/5.33           => ( P @ zero_zero_real ) )
% 9.10/5.33          & ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33           => ! [Y: real] :
% 9.10/5.33                ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 9.10/5.33                  = X3 )
% 9.10/5.33               => ( P @ Y ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9700_bij__betw__nth__root__unity,axiom,
% 9.10/5.33      ! [C: complex,N: nat] :
% 9.10/5.33        ( ( C != zero_zero_complex )
% 9.10/5.33       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 9.10/5.33            @ ( collect_complex
% 9.10/5.33              @ ^ [Z2: complex] :
% 9.10/5.33                  ( ( power_power_complex @ Z2 @ N )
% 9.10/5.33                  = one_one_complex ) )
% 9.10/5.33            @ ( collect_complex
% 9.10/5.33              @ ^ [Z2: complex] :
% 9.10/5.33                  ( ( power_power_complex @ Z2 @ N )
% 9.10/5.33                  = C ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9701_xor__minus__numerals_I2_J,axiom,
% 9.10/5.33      ! [K: int,N: num] :
% 9.10/5.33        ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9702_xor__minus__numerals_I1_J,axiom,
% 9.10/5.33      ! [N: num,K: int] :
% 9.10/5.33        ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 9.10/5.33        = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9703_sub__BitM__One__eq,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 9.10/5.33        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9704_Suc__funpow,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( compow_nat_nat @ N @ suc )
% 9.10/5.33        = ( plus_plus_nat @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9705_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.10/5.33      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 9.10/5.33      @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 9.10/5.33      @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9706_times__int_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( abs_Integ
% 9.10/5.33          @ ( produc27273713700761075at_nat
% 9.10/5.33            @ ^ [X: nat,Y: nat] :
% 9.10/5.33                ( produc2626176000494625587at_nat
% 9.10/5.33                @ ^ [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 ) ) ) )
% 9.10/5.33            @ Xa2
% 9.10/5.33            @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9707_eq__Abs__Integ,axiom,
% 9.10/5.33      ! [Z: int] :
% 9.10/5.33        ~ ! [X5: nat,Y4: nat] :
% 9.10/5.33            ( Z
% 9.10/5.33           != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9708_nat_Oabs__eq,axiom,
% 9.10/5.33      ! [X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( nat2 @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9709_zero__int__def,axiom,
% 9.10/5.33      ( zero_zero_int
% 9.10/5.33      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9710_int__def,axiom,
% 9.10/5.33      ( semiri1314217659103216013at_int
% 9.10/5.33      = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9711_uminus__int_Oabs__eq,axiom,
% 9.10/5.33      ! [X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( abs_Integ
% 9.10/5.33          @ ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
% 9.10/5.33            @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9712_one__int__def,axiom,
% 9.10/5.33      ( one_one_int
% 9.10/5.33      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9713_less__int_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( produc8739625826339149834_nat_o
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc6081775807080527818_nat_o
% 9.10/5.33              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 9.10/5.33          @ Xa2
% 9.10/5.33          @ X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9714_less__eq__int_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( produc8739625826339149834_nat_o
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc6081775807080527818_nat_o
% 9.10/5.33              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 9.10/5.33          @ Xa2
% 9.10/5.33          @ X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9715_plus__int_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( abs_Integ
% 9.10/5.33          @ ( produc27273713700761075at_nat
% 9.10/5.33            @ ^ [X: nat,Y: nat] :
% 9.10/5.33                ( produc2626176000494625587at_nat
% 9.10/5.33                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 9.10/5.33            @ Xa2
% 9.10/5.33            @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9716_minus__int_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.10/5.33        ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.10/5.33        = ( abs_Integ
% 9.10/5.33          @ ( produc27273713700761075at_nat
% 9.10/5.33            @ ^ [X: nat,Y: nat] :
% 9.10/5.33                ( produc2626176000494625587at_nat
% 9.10/5.33                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
% 9.10/5.33            @ Xa2
% 9.10/5.33            @ X3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9717_less__eq__int_Orep__eq,axiom,
% 9.10/5.33      ( ord_less_eq_int
% 9.10/5.33      = ( ^ [X: int,Xa4: int] :
% 9.10/5.33            ( produc8739625826339149834_nat_o
% 9.10/5.33            @ ^ [Y: nat,Z2: nat] :
% 9.10/5.33                ( produc6081775807080527818_nat_o
% 9.10/5.33                @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 9.10/5.33            @ ( rep_Integ @ X )
% 9.10/5.33            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9718_less__int_Orep__eq,axiom,
% 9.10/5.33      ( ord_less_int
% 9.10/5.33      = ( ^ [X: int,Xa4: int] :
% 9.10/5.33            ( produc8739625826339149834_nat_o
% 9.10/5.33            @ ^ [Y: nat,Z2: nat] :
% 9.10/5.33                ( produc6081775807080527818_nat_o
% 9.10/5.33                @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 9.10/5.33            @ ( rep_Integ @ X )
% 9.10/5.33            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9719_nat_Orep__eq,axiom,
% 9.10/5.33      ( nat2
% 9.10/5.33      = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9720_prod__encode__def,axiom,
% 9.10/5.33      ( nat_prod_encode
% 9.10/5.33      = ( produc6842872674320459806at_nat
% 9.10/5.33        @ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9721_uminus__int__def,axiom,
% 9.10/5.33      ( uminus_uminus_int
% 9.10/5.33      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 9.10/5.33        @ ( produc2626176000494625587at_nat
% 9.10/5.33          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9722_le__prod__encode__1,axiom,
% 9.10/5.33      ! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9723_le__prod__encode__2,axiom,
% 9.10/5.33      ! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9724_prod__encode__prod__decode__aux,axiom,
% 9.10/5.33      ! [K: nat,M: nat] :
% 9.10/5.33        ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 9.10/5.33        = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9725_times__int__def,axiom,
% 9.10/5.33      ( times_times_int
% 9.10/5.33      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.10/5.33        @ ( produc27273713700761075at_nat
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc2626176000494625587at_nat
% 9.10/5.33              @ ^ [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 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9726_minus__int__def,axiom,
% 9.10/5.33      ( minus_minus_int
% 9.10/5.33      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.10/5.33        @ ( produc27273713700761075at_nat
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc2626176000494625587at_nat
% 9.10/5.33              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9727_plus__int__def,axiom,
% 9.10/5.33      ( plus_plus_int
% 9.10/5.33      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.10/5.33        @ ( produc27273713700761075at_nat
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc2626176000494625587at_nat
% 9.10/5.33              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9728_num__of__nat_Osimps_I2_J,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( ( num_of_nat @ ( suc @ N ) )
% 9.10/5.33            = ( inc @ ( num_of_nat @ N ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( ( num_of_nat @ ( suc @ N ) )
% 9.10/5.33            = one ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9729_pow_Osimps_I3_J,axiom,
% 9.10/5.33      ! [X3: num,Y3: num] :
% 9.10/5.33        ( ( pow @ X3 @ ( bit1 @ Y3 ) )
% 9.10/5.33        = ( times_times_num @ ( sqr @ ( pow @ X3 @ Y3 ) ) @ X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9730_num__of__nat__numeral__eq,axiom,
% 9.10/5.33      ! [Q2: num] :
% 9.10/5.33        ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 9.10/5.33        = Q2 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9731_sqr_Osimps_I2_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( sqr @ ( bit0 @ N ) )
% 9.10/5.33        = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9732_sqr_Osimps_I1_J,axiom,
% 9.10/5.33      ( ( sqr @ one )
% 9.10/5.33      = one ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9733_sqr__conv__mult,axiom,
% 9.10/5.33      ( sqr
% 9.10/5.33      = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9734_num__of__nat_Osimps_I1_J,axiom,
% 9.10/5.33      ( ( num_of_nat @ zero_zero_nat )
% 9.10/5.33      = one ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9735_numeral__num__of__nat,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 9.10/5.33          = N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9736_num__of__nat__One,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ N @ one_one_nat )
% 9.10/5.33       => ( ( num_of_nat @ N )
% 9.10/5.33          = one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9737_pow_Osimps_I2_J,axiom,
% 9.10/5.33      ! [X3: num,Y3: num] :
% 9.10/5.33        ( ( pow @ X3 @ ( bit0 @ Y3 ) )
% 9.10/5.33        = ( sqr @ ( pow @ X3 @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9738_num__of__nat__double,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 9.10/5.33          = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9739_num__of__nat__plus__distrib,axiom,
% 9.10/5.33      ! [M: nat,N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.10/5.33       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 9.10/5.33            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9740_sqr_Osimps_I3_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( sqr @ ( bit1 @ N ) )
% 9.10/5.33        = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9741_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.33        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.10/5.33       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.10/5.33                ( X3
% 9.10/5.33                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.33           => ( Xa2 = one_one_nat ) )
% 9.10/5.33         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.33               => ( ( Deg2 = Xa2 )
% 9.10/5.33                  & ! [X5: vEBT_VEBT] :
% 9.10/5.33                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                     => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.33                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                  & ( case_o184042715313410164at_nat
% 9.10/5.33                    @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 9.10/5.33                      & ! [X: vEBT_VEBT] :
% 9.10/5.33                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                         => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                    @ ( produc6081775807080527818_nat_o
% 9.10/5.33                      @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                          & ! [I4: nat] :
% 9.10/5.33                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                             => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 9.10/5.33                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.10/5.33                          & ( ( Mi3 = Ma3 )
% 9.10/5.33                           => ! [X: vEBT_VEBT] :
% 9.10/5.33                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                          & ( ( Mi3 != Ma3 )
% 9.10/5.33                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.10/5.33                              & ! [X: nat] :
% 9.10/5.33                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.10/5.33                                   => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33                    @ Mima ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9742_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.33        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.10/5.33       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.10/5.33                ( X3
% 9.10/5.33                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.33           => ( Xa2 != one_one_nat ) )
% 9.10/5.33         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.33               => ~ ( ( Deg2 = Xa2 )
% 9.10/5.33                    & ! [X6: vEBT_VEBT] :
% 9.10/5.33                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                       => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.33                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                    & ( case_o184042715313410164at_nat
% 9.10/5.33                      @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 9.10/5.33                        & ! [X: vEBT_VEBT] :
% 9.10/5.33                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                           => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                      @ ( produc6081775807080527818_nat_o
% 9.10/5.33                        @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                            & ! [I4: nat] :
% 9.10/5.33                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                               => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 9.10/5.33                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.10/5.33                            & ( ( Mi3 = Ma3 )
% 9.10/5.33                             => ! [X: vEBT_VEBT] :
% 9.10/5.33                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                            & ( ( Mi3 != Ma3 )
% 9.10/5.33                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.10/5.33                                & ! [X: nat] :
% 9.10/5.33                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.10/5.33                                     => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33                      @ Mima ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9743_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.33        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.10/5.33                ( X3
% 9.10/5.33                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.33           => ( Y3
% 9.10/5.33              = ( Xa2 != one_one_nat ) ) )
% 9.10/5.33         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.33               => ( Y3
% 9.10/5.33                  = ( ~ ( ( Deg2 = Xa2 )
% 9.10/5.33                        & ! [X: vEBT_VEBT] :
% 9.10/5.33                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.33                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                        & ( case_o184042715313410164at_nat
% 9.10/5.33                          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 9.10/5.33                            & ! [X: vEBT_VEBT] :
% 9.10/5.33                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                          @ ( produc6081775807080527818_nat_o
% 9.10/5.33                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                & ! [I4: nat] :
% 9.10/5.33                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 9.10/5.33                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.10/5.33                                & ( ( Mi3 = Ma3 )
% 9.10/5.33                                 => ! [X: vEBT_VEBT] :
% 9.10/5.33                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                                & ( ( Mi3 != Ma3 )
% 9.10/5.33                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.10/5.33                                    & ! [X: nat] :
% 9.10/5.33                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.10/5.33                                         => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33                          @ Mima ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9744_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 9.10/5.33      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 9.10/5.33        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
% 9.10/5.33        = ( ( Deg = Deg4 )
% 9.10/5.33          & ! [X: vEBT_VEBT] :
% 9.10/5.33              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.10/5.33             => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33          & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.10/5.33            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33          & ( case_o184042715313410164at_nat
% 9.10/5.33            @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
% 9.10/5.33              & ! [X: vEBT_VEBT] :
% 9.10/5.33                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.10/5.33                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33            @ ( produc6081775807080527818_nat_o
% 9.10/5.33              @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                  ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                  & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.10/5.33                  & ! [I4: nat] :
% 9.10/5.33                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
% 9.10/5.33                        = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 9.10/5.33                  & ( ( Mi3 = Ma3 )
% 9.10/5.33                   => ! [X: vEBT_VEBT] :
% 9.10/5.33                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.10/5.33                       => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                  & ( ( Mi3 != Ma3 )
% 9.10/5.33                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 9.10/5.33                      & ! [X: nat] :
% 9.10/5.33                          ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.10/5.33                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 9.10/5.33                           => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33            @ Mima2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9745_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.33        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.10/5.33       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.33         => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.33               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.10/5.33                 => ( Xa2 = one_one_nat ) ) )
% 9.10/5.33           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.33                  ( ( X3
% 9.10/5.33                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.33                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 9.10/5.33                   => ( ( Deg2 = Xa2 )
% 9.10/5.33                      & ! [X5: vEBT_VEBT] :
% 9.10/5.33                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                         => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.33                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                      & ( case_o184042715313410164at_nat
% 9.10/5.33                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 9.10/5.33                          & ! [X: vEBT_VEBT] :
% 9.10/5.33                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                        @ ( produc6081775807080527818_nat_o
% 9.10/5.33                          @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                              & ! [I4: nat] :
% 9.10/5.33                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 9.10/5.33                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.10/5.33                              & ( ( Mi3 = Ma3 )
% 9.10/5.33                               => ! [X: vEBT_VEBT] :
% 9.10/5.33                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                              & ( ( Mi3 != Ma3 )
% 9.10/5.33                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.10/5.33                                  & ! [X: nat] :
% 9.10/5.33                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.10/5.33                                       => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33                        @ Mima ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9746_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.10/5.33        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.10/5.33       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.33         => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.33               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.10/5.33                 => ( Xa2 != one_one_nat ) ) )
% 9.10/5.33           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.33                  ( ( X3
% 9.10/5.33                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.33                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 9.10/5.33                   => ~ ( ( Deg2 = Xa2 )
% 9.10/5.33                        & ! [X6: vEBT_VEBT] :
% 9.10/5.33                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                           => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.33                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                        & ( case_o184042715313410164at_nat
% 9.10/5.33                          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 9.10/5.33                            & ! [X: vEBT_VEBT] :
% 9.10/5.33                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                          @ ( produc6081775807080527818_nat_o
% 9.10/5.33                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                & ! [I4: nat] :
% 9.10/5.33                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 9.10/5.33                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.10/5.33                                & ( ( Mi3 = Ma3 )
% 9.10/5.33                                 => ! [X: vEBT_VEBT] :
% 9.10/5.33                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                                & ( ( Mi3 != Ma3 )
% 9.10/5.33                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.10/5.33                                    & ! [X: nat] :
% 9.10/5.33                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.10/5.33                                         => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33                          @ Mima ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9747_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.10/5.33        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.10/5.33         => ( ! [Uu2: $o,Uv2: $o] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.10/5.33               => ( ( Y3
% 9.10/5.33                    = ( Xa2 = one_one_nat ) )
% 9.10/5.33                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.10/5.33           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.10/5.33                  ( ( X3
% 9.10/5.33                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.10/5.33                 => ( ( Y3
% 9.10/5.33                      = ( ( Deg2 = Xa2 )
% 9.10/5.33                        & ! [X: vEBT_VEBT] :
% 9.10/5.33                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.10/5.33                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                        & ( case_o184042715313410164at_nat
% 9.10/5.33                          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
% 9.10/5.33                            & ! [X: vEBT_VEBT] :
% 9.10/5.33                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                          @ ( produc6081775807080527818_nat_o
% 9.10/5.33                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.10/5.33                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.10/5.33                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                & ! [I4: nat] :
% 9.10/5.33                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X7 ) )
% 9.10/5.33                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.10/5.33                                & ( ( Mi3 = Ma3 )
% 9.10/5.33                                 => ! [X: vEBT_VEBT] :
% 9.10/5.33                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.10/5.33                                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X @ X7 ) ) )
% 9.10/5.33                                & ( ( Mi3 != Ma3 )
% 9.10/5.33                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.10/5.33                                    & ! [X: nat] :
% 9.10/5.33                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.10/5.33                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.10/5.33                                         => ( ( ord_less_nat @ Mi3 @ X )
% 9.10/5.33                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.10/5.33                          @ Mima ) ) )
% 9.10/5.33                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9748_Sup__int__def,axiom,
% 9.10/5.33      ( complete_Sup_Sup_int
% 9.10/5.33      = ( ^ [X7: set_int] :
% 9.10/5.33            ( the_int
% 9.10/5.33            @ ^ [X: int] :
% 9.10/5.33                ( ( member_int @ X @ X7 )
% 9.10/5.33                & ! [Y: int] :
% 9.10/5.33                    ( ( member_int @ Y @ X7 )
% 9.10/5.33                   => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9749_take__bit__numeral__minus__numeral__int,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int
% 9.10/5.33          @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
% 9.10/5.33          @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9750_and__minus__numerals_I3_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9751_take__bit__num__simps_I1_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 9.10/5.33        = none_num ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9752_take__bit__num__simps_I2_J,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 9.10/5.33        = ( some_num @ one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9753_take__bit__num__simps_I5_J,axiom,
% 9.10/5.33      ! [R: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 9.10/5.33        = ( some_num @ one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9754_take__bit__num__simps_I3_J,axiom,
% 9.10/5.33      ! [N: nat,M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 9.10/5.33        = ( case_o6005452278849405969um_num @ none_num
% 9.10/5.33          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.10/5.33          @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9755_take__bit__num__simps_I4_J,axiom,
% 9.10/5.33      ! [N: nat,M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 9.10/5.33        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9756_take__bit__num__simps_I6_J,axiom,
% 9.10/5.33      ! [R: num,M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 9.10/5.33        = ( case_o6005452278849405969um_num @ none_num
% 9.10/5.33          @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.10/5.33          @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9757_take__bit__num__simps_I7_J,axiom,
% 9.10/5.33      ! [R: num,M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 9.10/5.33        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9758_and__minus__numerals_I4_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9759_and__minus__numerals_I8_J,axiom,
% 9.10/5.33      ! [N: num,M: num] :
% 9.10/5.33        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9760_and__minus__numerals_I7_J,axiom,
% 9.10/5.33      ! [N: num,M: num] :
% 9.10/5.33        ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9761_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 9.10/5.33      ! [N: nat,M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 9.10/5.33        = ( case_nat_option_num @ none_num
% 9.10/5.33          @ ^ [N2: nat] :
% 9.10/5.33              ( case_o6005452278849405969um_num @ none_num
% 9.10/5.33              @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.10/5.33              @ ( bit_take_bit_num @ N2 @ M ) )
% 9.10/5.33          @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9762_and__not__num_Osimps_I1_J,axiom,
% 9.10/5.33      ( ( bit_and_not_num @ one @ one )
% 9.10/5.33      = none_num ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9763_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( bit_take_bit_num @ N @ one )
% 9.10/5.33        = ( case_nat_option_num @ none_num
% 9.10/5.33          @ ^ [N2: nat] : ( some_num @ one )
% 9.10/5.33          @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9764_and__not__num_Osimps_I4_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 9.10/5.33        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9765_and__not__num_Osimps_I2_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 9.10/5.33        = ( some_num @ one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9766_GreatestI__ex__nat,axiom,
% 9.10/5.33      ! [P: nat > $o,B2: nat] :
% 9.10/5.33        ( ? [X_12: nat] : ( P @ X_12 )
% 9.10/5.33       => ( ! [Y4: nat] :
% 9.10/5.33              ( ( P @ Y4 )
% 9.10/5.33             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 9.10/5.33         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9767_Greatest__le__nat,axiom,
% 9.10/5.33      ! [P: nat > $o,K: nat,B2: nat] :
% 9.10/5.33        ( ( P @ K )
% 9.10/5.33       => ( ! [Y4: nat] :
% 9.10/5.33              ( ( P @ Y4 )
% 9.10/5.33             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 9.10/5.33         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9768_GreatestI__nat,axiom,
% 9.10/5.33      ! [P: nat > $o,K: nat,B2: nat] :
% 9.10/5.33        ( ( P @ K )
% 9.10/5.33       => ( ! [Y4: nat] :
% 9.10/5.33              ( ( P @ Y4 )
% 9.10/5.33             => ( ord_less_eq_nat @ Y4 @ B2 ) )
% 9.10/5.33         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9769_and__not__num_Osimps_I3_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 9.10/5.33        = none_num ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9770_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 9.10/5.33      ! [N: nat,M: num] :
% 9.10/5.33        ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 9.10/5.33        = ( case_nat_option_num @ none_num
% 9.10/5.33          @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 9.10/5.33          @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9771_and__not__num_Osimps_I7_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 9.10/5.33        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9772_and__not__num__eq__Some__iff,axiom,
% 9.10/5.33      ! [M: num,N: num,Q2: num] :
% 9.10/5.33        ( ( ( bit_and_not_num @ M @ N )
% 9.10/5.33          = ( some_num @ Q2 ) )
% 9.10/5.33        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33          = ( numeral_numeral_int @ Q2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9773_and__not__num_Osimps_I8_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.33        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.10/5.33          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.10/5.33          @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9774_and__not__num__eq__None__iff,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( ( bit_and_not_num @ M @ N )
% 9.10/5.33          = none_num )
% 9.10/5.33        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33          = zero_zero_int ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9775_int__numeral__not__and__num,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9776_int__numeral__and__not__num,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33        = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9777_take__bit__num__def,axiom,
% 9.10/5.33      ( bit_take_bit_num
% 9.10/5.33      = ( ^ [N2: nat,M3: num] :
% 9.10/5.33            ( if_option_num
% 9.10/5.33            @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
% 9.10/5.33              = zero_zero_nat )
% 9.10/5.33            @ none_num
% 9.10/5.33            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9778_Bit__Operations_Otake__bit__num__code,axiom,
% 9.10/5.33      ( bit_take_bit_num
% 9.10/5.33      = ( ^ [N2: nat,M3: num] :
% 9.10/5.33            ( produc478579273971653890on_num
% 9.10/5.33            @ ^ [A5: nat,X: num] :
% 9.10/5.33                ( case_nat_option_num @ none_num
% 9.10/5.33                @ ^ [O: nat] :
% 9.10/5.33                    ( case_num_option_num @ ( some_num @ one )
% 9.10/5.33                    @ ^ [P5: num] :
% 9.10/5.33                        ( case_o6005452278849405969um_num @ none_num
% 9.10/5.33                        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 9.10/5.33                        @ ( bit_take_bit_num @ O @ P5 ) )
% 9.10/5.33                    @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
% 9.10/5.33                    @ X )
% 9.10/5.33                @ A5 )
% 9.10/5.33            @ ( product_Pair_nat_num @ N2 @ M3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9779_integer__of__num__triv_I2_J,axiom,
% 9.10/5.33      ( ( code_integer_of_num @ ( bit0 @ one ) )
% 9.10/5.33      = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9780_integer__of__num__def,axiom,
% 9.10/5.33      code_integer_of_num = numera6620942414471956472nteger ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9781_integer__of__num__triv_I1_J,axiom,
% 9.10/5.33      ( ( code_integer_of_num @ one )
% 9.10/5.33      = one_one_Code_integer ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9782_integer__of__num_I2_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( code_integer_of_num @ ( bit0 @ N ) )
% 9.10/5.33        = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9783_and__not__num_Oelims,axiom,
% 9.10/5.33      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.10/5.33        ( ( ( bit_and_not_num @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( ( X3 = one )
% 9.10/5.33           => ( ( Xa2 = one )
% 9.10/5.33             => ( Y3 != none_num ) ) )
% 9.10/5.33         => ( ( ( X3 = one )
% 9.10/5.33             => ( ? [N3: num] :
% 9.10/5.33                    ( Xa2
% 9.10/5.33                    = ( bit0 @ N3 ) )
% 9.10/5.33               => ( Y3
% 9.10/5.33                 != ( some_num @ one ) ) ) )
% 9.10/5.33           => ( ( ( X3 = one )
% 9.10/5.33               => ( ? [N3: num] :
% 9.10/5.33                      ( Xa2
% 9.10/5.33                      = ( bit1 @ N3 ) )
% 9.10/5.33                 => ( Y3 != none_num ) ) )
% 9.10/5.33             => ( ! [M4: num] :
% 9.10/5.33                    ( ( X3
% 9.10/5.33                      = ( bit0 @ M4 ) )
% 9.10/5.33                   => ( ( Xa2 = one )
% 9.10/5.33                     => ( Y3
% 9.10/5.33                       != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 9.10/5.33               => ( ! [M4: num] :
% 9.10/5.33                      ( ( X3
% 9.10/5.33                        = ( bit0 @ M4 ) )
% 9.10/5.33                     => ! [N3: num] :
% 9.10/5.33                          ( ( Xa2
% 9.10/5.33                            = ( bit0 @ N3 ) )
% 9.10/5.33                         => ( Y3
% 9.10/5.33                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                 => ( ! [M4: num] :
% 9.10/5.33                        ( ( X3
% 9.10/5.33                          = ( bit0 @ M4 ) )
% 9.10/5.33                       => ! [N3: num] :
% 9.10/5.33                            ( ( Xa2
% 9.10/5.33                              = ( bit1 @ N3 ) )
% 9.10/5.33                           => ( Y3
% 9.10/5.33                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                   => ( ! [M4: num] :
% 9.10/5.33                          ( ( X3
% 9.10/5.33                            = ( bit1 @ M4 ) )
% 9.10/5.33                         => ( ( Xa2 = one )
% 9.10/5.33                           => ( Y3
% 9.10/5.33                             != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 9.10/5.33                     => ( ! [M4: num] :
% 9.10/5.33                            ( ( X3
% 9.10/5.33                              = ( bit1 @ M4 ) )
% 9.10/5.33                           => ! [N3: num] :
% 9.10/5.33                                ( ( Xa2
% 9.10/5.33                                  = ( bit0 @ N3 ) )
% 9.10/5.33                               => ( Y3
% 9.10/5.33                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.10/5.33                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.10/5.33                                    @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                       => ~ ! [M4: num] :
% 9.10/5.33                              ( ( X3
% 9.10/5.33                                = ( bit1 @ M4 ) )
% 9.10/5.33                             => ! [N3: num] :
% 9.10/5.33                                  ( ( Xa2
% 9.10/5.33                                    = ( bit1 @ N3 ) )
% 9.10/5.33                                 => ( Y3
% 9.10/5.33                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9784_xor__num_Osimps_I6_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.33        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9785_xor__num_Osimps_I8_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.33        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9786_xor__num_Osimps_I5_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9787_xor__num_Osimps_I9_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9788_and__not__num_Osimps_I5_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9789_and__not__num_Osimps_I6_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9790_and__not__num_Osimps_I9_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9791_xor__num_Osimps_I1_J,axiom,
% 9.10/5.33      ( ( bit_un2480387367778600638or_num @ one @ one )
% 9.10/5.33      = none_num ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9792_xor__num_Oelims,axiom,
% 9.10/5.33      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.10/5.33        ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( ( X3 = one )
% 9.10/5.33           => ( ( Xa2 = one )
% 9.10/5.33             => ( Y3 != none_num ) ) )
% 9.10/5.33         => ( ( ( X3 = one )
% 9.10/5.33             => ! [N3: num] :
% 9.10/5.33                  ( ( Xa2
% 9.10/5.33                    = ( bit0 @ N3 ) )
% 9.10/5.33                 => ( Y3
% 9.10/5.33                   != ( some_num @ ( bit1 @ N3 ) ) ) ) )
% 9.10/5.33           => ( ( ( X3 = one )
% 9.10/5.33               => ! [N3: num] :
% 9.10/5.33                    ( ( Xa2
% 9.10/5.33                      = ( bit1 @ N3 ) )
% 9.10/5.33                   => ( Y3
% 9.10/5.33                     != ( some_num @ ( bit0 @ N3 ) ) ) ) )
% 9.10/5.33             => ( ! [M4: num] :
% 9.10/5.33                    ( ( X3
% 9.10/5.33                      = ( bit0 @ M4 ) )
% 9.10/5.33                   => ( ( Xa2 = one )
% 9.10/5.33                     => ( Y3
% 9.10/5.33                       != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 9.10/5.33               => ( ! [M4: num] :
% 9.10/5.33                      ( ( X3
% 9.10/5.33                        = ( bit0 @ M4 ) )
% 9.10/5.33                     => ! [N3: num] :
% 9.10/5.33                          ( ( Xa2
% 9.10/5.33                            = ( bit0 @ N3 ) )
% 9.10/5.33                         => ( Y3
% 9.10/5.33                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                 => ( ! [M4: num] :
% 9.10/5.33                        ( ( X3
% 9.10/5.33                          = ( bit0 @ M4 ) )
% 9.10/5.33                       => ! [N3: num] :
% 9.10/5.33                            ( ( Xa2
% 9.10/5.33                              = ( bit1 @ N3 ) )
% 9.10/5.33                           => ( Y3
% 9.10/5.33                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
% 9.10/5.33                   => ( ! [M4: num] :
% 9.10/5.33                          ( ( X3
% 9.10/5.33                            = ( bit1 @ M4 ) )
% 9.10/5.33                         => ( ( Xa2 = one )
% 9.10/5.33                           => ( Y3
% 9.10/5.33                             != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 9.10/5.33                     => ( ! [M4: num] :
% 9.10/5.33                            ( ( X3
% 9.10/5.33                              = ( bit1 @ M4 ) )
% 9.10/5.33                           => ! [N3: num] :
% 9.10/5.33                                ( ( Xa2
% 9.10/5.33                                  = ( bit0 @ N3 ) )
% 9.10/5.33                               => ( Y3
% 9.10/5.33                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) )
% 9.10/5.33                       => ~ ! [M4: num] :
% 9.10/5.33                              ( ( X3
% 9.10/5.33                                = ( bit1 @ M4 ) )
% 9.10/5.33                             => ! [N3: num] :
% 9.10/5.33                                  ( ( Xa2
% 9.10/5.33                                    = ( bit1 @ N3 ) )
% 9.10/5.33                                 => ( Y3
% 9.10/5.33                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9793_xor__num_Osimps_I7_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 9.10/5.33        = ( some_num @ ( bit0 @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9794_xor__num_Osimps_I4_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 9.10/5.33        = ( some_num @ ( bit1 @ M ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9795_xor__num_Osimps_I3_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 9.10/5.33        = ( some_num @ ( bit0 @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9796_xor__num_Osimps_I2_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 9.10/5.33        = ( some_num @ ( bit1 @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9797_and__not__num_Opelims,axiom,
% 9.10/5.33      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.10/5.33        ( ( ( bit_and_not_num @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 9.10/5.33         => ( ( ( X3 = one )
% 9.10/5.33             => ( ( Xa2 = one )
% 9.10/5.33               => ( ( Y3 = none_num )
% 9.10/5.33                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.10/5.33           => ( ( ( X3 = one )
% 9.10/5.33               => ! [N3: num] :
% 9.10/5.33                    ( ( Xa2
% 9.10/5.33                      = ( bit0 @ N3 ) )
% 9.10/5.33                   => ( ( Y3
% 9.10/5.33                        = ( some_num @ one ) )
% 9.10/5.33                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33             => ( ( ( X3 = one )
% 9.10/5.33                 => ! [N3: num] :
% 9.10/5.33                      ( ( Xa2
% 9.10/5.33                        = ( bit1 @ N3 ) )
% 9.10/5.33                     => ( ( Y3 = none_num )
% 9.10/5.33                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 9.10/5.33               => ( ! [M4: num] :
% 9.10/5.33                      ( ( X3
% 9.10/5.33                        = ( bit0 @ M4 ) )
% 9.10/5.33                     => ( ( Xa2 = one )
% 9.10/5.33                       => ( ( Y3
% 9.10/5.33                            = ( some_num @ ( bit0 @ M4 ) ) )
% 9.10/5.33                         => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 9.10/5.33                 => ( ! [M4: num] :
% 9.10/5.33                        ( ( X3
% 9.10/5.33                          = ( bit0 @ M4 ) )
% 9.10/5.33                       => ! [N3: num] :
% 9.10/5.33                            ( ( Xa2
% 9.10/5.33                              = ( bit0 @ N3 ) )
% 9.10/5.33                           => ( ( Y3
% 9.10/5.33                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 9.10/5.33                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33                   => ( ! [M4: num] :
% 9.10/5.33                          ( ( X3
% 9.10/5.33                            = ( bit0 @ M4 ) )
% 9.10/5.33                         => ! [N3: num] :
% 9.10/5.33                              ( ( Xa2
% 9.10/5.33                                = ( bit1 @ N3 ) )
% 9.10/5.33                             => ( ( Y3
% 9.10/5.33                                  = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 9.10/5.33                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 9.10/5.33                     => ( ! [M4: num] :
% 9.10/5.33                            ( ( X3
% 9.10/5.33                              = ( bit1 @ M4 ) )
% 9.10/5.33                           => ( ( Xa2 = one )
% 9.10/5.33                             => ( ( Y3
% 9.10/5.33                                  = ( some_num @ ( bit0 @ M4 ) ) )
% 9.10/5.33                               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 9.10/5.33                       => ( ! [M4: num] :
% 9.10/5.33                              ( ( X3
% 9.10/5.33                                = ( bit1 @ M4 ) )
% 9.10/5.33                             => ! [N3: num] :
% 9.10/5.33                                  ( ( Xa2
% 9.10/5.33                                    = ( bit0 @ N3 ) )
% 9.10/5.33                                 => ( ( Y3
% 9.10/5.33                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.10/5.33                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.10/5.33                                        @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 9.10/5.33                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33                         => ~ ! [M4: num] :
% 9.10/5.33                                ( ( X3
% 9.10/5.33                                  = ( bit1 @ M4 ) )
% 9.10/5.33                               => ! [N3: num] :
% 9.10/5.33                                    ( ( Xa2
% 9.10/5.33                                      = ( bit1 @ N3 ) )
% 9.10/5.33                                   => ( ( Y3
% 9.10/5.33                                        = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) )
% 9.10/5.33                                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9798_xor__num_Opelims,axiom,
% 9.10/5.33      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.10/5.33        ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 9.10/5.33         => ( ( ( X3 = one )
% 9.10/5.33             => ( ( Xa2 = one )
% 9.10/5.33               => ( ( Y3 = none_num )
% 9.10/5.33                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.10/5.33           => ( ( ( X3 = one )
% 9.10/5.33               => ! [N3: num] :
% 9.10/5.33                    ( ( Xa2
% 9.10/5.33                      = ( bit0 @ N3 ) )
% 9.10/5.33                   => ( ( Y3
% 9.10/5.33                        = ( some_num @ ( bit1 @ N3 ) ) )
% 9.10/5.33                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33             => ( ( ( X3 = one )
% 9.10/5.33                 => ! [N3: num] :
% 9.10/5.33                      ( ( Xa2
% 9.10/5.33                        = ( bit1 @ N3 ) )
% 9.10/5.33                     => ( ( Y3
% 9.10/5.33                          = ( some_num @ ( bit0 @ N3 ) ) )
% 9.10/5.33                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 9.10/5.33               => ( ! [M4: num] :
% 9.10/5.33                      ( ( X3
% 9.10/5.33                        = ( bit0 @ M4 ) )
% 9.10/5.33                     => ( ( Xa2 = one )
% 9.10/5.33                       => ( ( Y3
% 9.10/5.33                            = ( some_num @ ( bit1 @ M4 ) ) )
% 9.10/5.33                         => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 9.10/5.33                 => ( ! [M4: num] :
% 9.10/5.33                        ( ( X3
% 9.10/5.33                          = ( bit0 @ M4 ) )
% 9.10/5.33                       => ! [N3: num] :
% 9.10/5.33                            ( ( Xa2
% 9.10/5.33                              = ( bit0 @ N3 ) )
% 9.10/5.33                           => ( ( Y3
% 9.10/5.33                                = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
% 9.10/5.33                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33                   => ( ! [M4: num] :
% 9.10/5.33                          ( ( X3
% 9.10/5.33                            = ( bit0 @ M4 ) )
% 9.10/5.33                         => ! [N3: num] :
% 9.10/5.33                              ( ( Xa2
% 9.10/5.33                                = ( bit1 @ N3 ) )
% 9.10/5.33                             => ( ( Y3
% 9.10/5.33                                  = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
% 9.10/5.33                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 9.10/5.33                     => ( ! [M4: num] :
% 9.10/5.33                            ( ( X3
% 9.10/5.33                              = ( bit1 @ M4 ) )
% 9.10/5.33                           => ( ( Xa2 = one )
% 9.10/5.33                             => ( ( Y3
% 9.10/5.33                                  = ( some_num @ ( bit0 @ M4 ) ) )
% 9.10/5.33                               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 9.10/5.33                       => ( ! [M4: num] :
% 9.10/5.33                              ( ( X3
% 9.10/5.33                                = ( bit1 @ M4 ) )
% 9.10/5.33                             => ! [N3: num] :
% 9.10/5.33                                  ( ( Xa2
% 9.10/5.33                                    = ( bit0 @ N3 ) )
% 9.10/5.33                                 => ( ( Y3
% 9.10/5.33                                      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) ) )
% 9.10/5.33                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33                         => ~ ! [M4: num] :
% 9.10/5.33                                ( ( X3
% 9.10/5.33                                  = ( bit1 @ M4 ) )
% 9.10/5.33                               => ! [N3: num] :
% 9.10/5.33                                    ( ( Xa2
% 9.10/5.33                                      = ( bit1 @ N3 ) )
% 9.10/5.33                                   => ( ( Y3
% 9.10/5.33                                        = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N3 ) ) )
% 9.10/5.33                                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9799_and__num_Oelims,axiom,
% 9.10/5.33      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.10/5.33        ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( ( X3 = one )
% 9.10/5.33           => ( ( Xa2 = one )
% 9.10/5.33             => ( Y3
% 9.10/5.33               != ( some_num @ one ) ) ) )
% 9.10/5.33         => ( ( ( X3 = one )
% 9.10/5.33             => ( ? [N3: num] :
% 9.10/5.33                    ( Xa2
% 9.10/5.33                    = ( bit0 @ N3 ) )
% 9.10/5.33               => ( Y3 != none_num ) ) )
% 9.10/5.33           => ( ( ( X3 = one )
% 9.10/5.33               => ( ? [N3: num] :
% 9.10/5.33                      ( Xa2
% 9.10/5.33                      = ( bit1 @ N3 ) )
% 9.10/5.33                 => ( Y3
% 9.10/5.33                   != ( some_num @ one ) ) ) )
% 9.10/5.33             => ( ( ? [M4: num] :
% 9.10/5.33                      ( X3
% 9.10/5.33                      = ( bit0 @ M4 ) )
% 9.10/5.33                 => ( ( Xa2 = one )
% 9.10/5.33                   => ( Y3 != none_num ) ) )
% 9.10/5.33               => ( ! [M4: num] :
% 9.10/5.33                      ( ( X3
% 9.10/5.33                        = ( bit0 @ M4 ) )
% 9.10/5.33                     => ! [N3: num] :
% 9.10/5.33                          ( ( Xa2
% 9.10/5.33                            = ( bit0 @ N3 ) )
% 9.10/5.33                         => ( Y3
% 9.10/5.33                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                 => ( ! [M4: num] :
% 9.10/5.33                        ( ( X3
% 9.10/5.33                          = ( bit0 @ M4 ) )
% 9.10/5.33                       => ! [N3: num] :
% 9.10/5.33                            ( ( Xa2
% 9.10/5.33                              = ( bit1 @ N3 ) )
% 9.10/5.33                           => ( Y3
% 9.10/5.33                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                   => ( ( ? [M4: num] :
% 9.10/5.33                            ( X3
% 9.10/5.33                            = ( bit1 @ M4 ) )
% 9.10/5.33                       => ( ( Xa2 = one )
% 9.10/5.33                         => ( Y3
% 9.10/5.33                           != ( some_num @ one ) ) ) )
% 9.10/5.33                     => ( ! [M4: num] :
% 9.10/5.33                            ( ( X3
% 9.10/5.33                              = ( bit1 @ M4 ) )
% 9.10/5.33                           => ! [N3: num] :
% 9.10/5.33                                ( ( Xa2
% 9.10/5.33                                  = ( bit0 @ N3 ) )
% 9.10/5.33                               => ( Y3
% 9.10/5.33                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) )
% 9.10/5.33                       => ~ ! [M4: num] :
% 9.10/5.33                              ( ( X3
% 9.10/5.33                                = ( bit1 @ M4 ) )
% 9.10/5.33                             => ! [N3: num] :
% 9.10/5.33                                  ( ( Xa2
% 9.10/5.33                                    = ( bit1 @ N3 ) )
% 9.10/5.33                                 => ( Y3
% 9.10/5.33                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.10/5.33                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.10/5.33                                      @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9800_xor__num__rel__dict,axiom,
% 9.10/5.33      bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9801_xor__num__dict,axiom,
% 9.10/5.33      bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9802_and__num_Osimps_I1_J,axiom,
% 9.10/5.33      ( ( bit_un7362597486090784418nd_num @ one @ one )
% 9.10/5.33      = ( some_num @ one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9803_and__num_Osimps_I5_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9804_and__num_Osimps_I3_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 9.10/5.33        = ( some_num @ one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9805_and__num_Osimps_I7_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 9.10/5.33        = ( some_num @ one ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9806_and__num_Osimps_I4_J,axiom,
% 9.10/5.33      ! [M: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 9.10/5.33        = none_num ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9807_and__num_Osimps_I2_J,axiom,
% 9.10/5.33      ! [N: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 9.10/5.33        = none_num ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9808_and__num_Osimps_I6_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9809_and__num_Osimps_I8_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.10/5.33        = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9810_and__num_Osimps_I9_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.10/5.33        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.10/5.33          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.10/5.33          @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9811_and__num_Opelims,axiom,
% 9.10/5.33      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.10/5.33        ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
% 9.10/5.33         => ( ( ( X3 = one )
% 9.10/5.33             => ( ( Xa2 = one )
% 9.10/5.33               => ( ( Y3
% 9.10/5.33                    = ( some_num @ one ) )
% 9.10/5.33                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 9.10/5.33           => ( ( ( X3 = one )
% 9.10/5.33               => ! [N3: num] :
% 9.10/5.33                    ( ( Xa2
% 9.10/5.33                      = ( bit0 @ N3 ) )
% 9.10/5.33                   => ( ( Y3 = none_num )
% 9.10/5.33                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33             => ( ( ( X3 = one )
% 9.10/5.33                 => ! [N3: num] :
% 9.10/5.33                      ( ( Xa2
% 9.10/5.33                        = ( bit1 @ N3 ) )
% 9.10/5.33                     => ( ( Y3
% 9.10/5.33                          = ( some_num @ one ) )
% 9.10/5.33                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
% 9.10/5.33               => ( ! [M4: num] :
% 9.10/5.33                      ( ( X3
% 9.10/5.33                        = ( bit0 @ M4 ) )
% 9.10/5.33                     => ( ( Xa2 = one )
% 9.10/5.33                       => ( ( Y3 = none_num )
% 9.10/5.33                         => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 9.10/5.33                 => ( ! [M4: num] :
% 9.10/5.33                        ( ( X3
% 9.10/5.33                          = ( bit0 @ M4 ) )
% 9.10/5.33                       => ! [N3: num] :
% 9.10/5.33                            ( ( Xa2
% 9.10/5.33                              = ( bit0 @ N3 ) )
% 9.10/5.33                           => ( ( Y3
% 9.10/5.33                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 9.10/5.33                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33                   => ( ! [M4: num] :
% 9.10/5.33                          ( ( X3
% 9.10/5.33                            = ( bit0 @ M4 ) )
% 9.10/5.33                         => ! [N3: num] :
% 9.10/5.33                              ( ( Xa2
% 9.10/5.33                                = ( bit1 @ N3 ) )
% 9.10/5.33                             => ( ( Y3
% 9.10/5.33                                  = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 9.10/5.33                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 9.10/5.33                     => ( ! [M4: num] :
% 9.10/5.33                            ( ( X3
% 9.10/5.33                              = ( bit1 @ M4 ) )
% 9.10/5.33                           => ( ( Xa2 = one )
% 9.10/5.33                             => ( ( Y3
% 9.10/5.33                                  = ( some_num @ one ) )
% 9.10/5.33                               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 9.10/5.33                       => ( ! [M4: num] :
% 9.10/5.33                              ( ( X3
% 9.10/5.33                                = ( bit1 @ M4 ) )
% 9.10/5.33                             => ! [N3: num] :
% 9.10/5.33                                  ( ( Xa2
% 9.10/5.33                                    = ( bit0 @ N3 ) )
% 9.10/5.33                                 => ( ( Y3
% 9.10/5.33                                      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 9.10/5.33                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 9.10/5.33                         => ~ ! [M4: num] :
% 9.10/5.33                                ( ( X3
% 9.10/5.33                                  = ( bit1 @ M4 ) )
% 9.10/5.33                               => ! [N3: num] :
% 9.10/5.33                                    ( ( Xa2
% 9.10/5.33                                      = ( bit1 @ N3 ) )
% 9.10/5.33                                   => ( ( Y3
% 9.10/5.33                                        = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.10/5.33                                          @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.10/5.33                                          @ ( bit_un7362597486090784418nd_num @ M4 @ N3 ) ) )
% 9.10/5.33                                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9812_and__num__dict,axiom,
% 9.10/5.33      bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9813_and__num__rel__dict,axiom,
% 9.10/5.33      bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9814_Rats__eq__int__div__nat,axiom,
% 9.10/5.33      ( field_5140801741446780682s_real
% 9.10/5.33      = ( collect_real
% 9.10/5.33        @ ^ [Uu3: real] :
% 9.10/5.33          ? [I4: int,N2: nat] :
% 9.10/5.33            ( ( Uu3
% 9.10/5.33              = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.10/5.33            & ( N2 != zero_zero_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9815_rat__floor__lemma,axiom,
% 9.10/5.33      ! [A: int,B2: int] :
% 9.10/5.33        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B2 ) ) @ ( fract @ A @ B2 ) )
% 9.10/5.33        & ( ord_less_rat @ ( fract @ A @ B2 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B2 ) @ one_one_int ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9816_Rats__abs__iff,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( member_real @ ( abs_abs_real @ X3 ) @ field_5140801741446780682s_real )
% 9.10/5.33        = ( member_real @ X3 @ field_5140801741446780682s_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9817_mult__rat,axiom,
% 9.10/5.33      ! [A: int,B2: int,C: int,D: int] :
% 9.10/5.33        ( ( times_times_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 9.10/5.33        = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9818_divide__rat,axiom,
% 9.10/5.33      ! [A: int,B2: int,C: int,D: int] :
% 9.10/5.33        ( ( divide_divide_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 9.10/5.33        = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ C ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9819_less__rat,axiom,
% 9.10/5.33      ! [B2: int,D: int,A: int,C: int] :
% 9.10/5.33        ( ( B2 != zero_zero_int )
% 9.10/5.33       => ( ( D != zero_zero_int )
% 9.10/5.33         => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 9.10/5.33            = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9820_add__rat,axiom,
% 9.10/5.33      ! [B2: int,D: int,A: int,C: int] :
% 9.10/5.33        ( ( B2 != zero_zero_int )
% 9.10/5.33       => ( ( D != zero_zero_int )
% 9.10/5.33         => ( ( plus_plus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 9.10/5.33            = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9821_le__rat,axiom,
% 9.10/5.33      ! [B2: int,D: int,A: int,C: int] :
% 9.10/5.33        ( ( B2 != zero_zero_int )
% 9.10/5.33       => ( ( D != zero_zero_int )
% 9.10/5.33         => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 9.10/5.33            = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B2 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B2 ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9822_diff__rat,axiom,
% 9.10/5.33      ! [B2: int,D: int,A: int,C: int] :
% 9.10/5.33        ( ( B2 != zero_zero_int )
% 9.10/5.33       => ( ( D != zero_zero_int )
% 9.10/5.33         => ( ( minus_minus_rat @ ( fract @ A @ B2 ) @ ( fract @ C @ D ) )
% 9.10/5.33            = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B2 ) ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9823_sgn__rat,axiom,
% 9.10/5.33      ! [A: int,B2: int] :
% 9.10/5.33        ( ( sgn_sgn_rat @ ( fract @ A @ B2 ) )
% 9.10/5.33        = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9824_Rats__no__top__le,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33      ? [X5: real] :
% 9.10/5.33        ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 9.10/5.33        & ( ord_less_eq_real @ X3 @ X5 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9825_Rats__no__bot__less,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33      ? [X5: real] :
% 9.10/5.33        ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 9.10/5.33        & ( ord_less_real @ X5 @ X3 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9826_Rats__dense__in__real,axiom,
% 9.10/5.33      ! [X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.33       => ? [X5: real] :
% 9.10/5.33            ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 9.10/5.33            & ( ord_less_real @ X3 @ X5 )
% 9.10/5.33            & ( ord_less_real @ X5 @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9827_Rat__induct__pos,axiom,
% 9.10/5.33      ! [P: rat > $o,Q2: rat] :
% 9.10/5.33        ( ! [A3: int,B3: int] :
% 9.10/5.33            ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.10/5.33           => ( P @ ( fract @ A3 @ B3 ) ) )
% 9.10/5.33       => ( P @ Q2 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9828_eq__rat_I1_J,axiom,
% 9.10/5.33      ! [B2: int,D: int,A: int,C: int] :
% 9.10/5.33        ( ( B2 != zero_zero_int )
% 9.10/5.33       => ( ( D != zero_zero_int )
% 9.10/5.33         => ( ( ( fract @ A @ B2 )
% 9.10/5.33              = ( fract @ C @ D ) )
% 9.10/5.33            = ( ( times_times_int @ A @ D )
% 9.10/5.33              = ( times_times_int @ C @ B2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9829_mult__rat__cancel,axiom,
% 9.10/5.33      ! [C: int,A: int,B2: int] :
% 9.10/5.33        ( ( C != zero_zero_int )
% 9.10/5.33       => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
% 9.10/5.33          = ( fract @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9830_quotient__of__eq,axiom,
% 9.10/5.33      ! [A: int,B2: int,P4: int,Q2: int] :
% 9.10/5.33        ( ( ( quotient_of @ ( fract @ A @ B2 ) )
% 9.10/5.33          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.33       => ( ( fract @ P4 @ Q2 )
% 9.10/5.33          = ( fract @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9831_normalize__eq,axiom,
% 9.10/5.33      ! [A: int,B2: int,P4: int,Q2: int] :
% 9.10/5.33        ( ( ( normalize @ ( product_Pair_int_int @ A @ B2 ) )
% 9.10/5.33          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.33       => ( ( fract @ P4 @ Q2 )
% 9.10/5.33          = ( fract @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9832_rat__number__expand_I3_J,axiom,
% 9.10/5.33      ( numeral_numeral_rat
% 9.10/5.33      = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9833_rat__number__collapse_I3_J,axiom,
% 9.10/5.33      ! [W: num] :
% 9.10/5.33        ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 9.10/5.33        = ( numeral_numeral_rat @ W ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9834_quotient__of__Fract,axiom,
% 9.10/5.33      ! [A: int,B2: int] :
% 9.10/5.33        ( ( quotient_of @ ( fract @ A @ B2 ) )
% 9.10/5.33        = ( normalize @ ( product_Pair_int_int @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9835_zero__less__Fract__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 9.10/5.33          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9836_Fract__less__zero__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 9.10/5.33          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9837_one__less__Fract__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 9.10/5.33          = ( ord_less_int @ B2 @ A ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9838_Fract__less__one__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 9.10/5.33          = ( ord_less_int @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9839_Rats__eq__int__div__int,axiom,
% 9.10/5.33      ( field_5140801741446780682s_real
% 9.10/5.33      = ( collect_real
% 9.10/5.33        @ ^ [Uu3: real] :
% 9.10/5.33          ? [I4: int,J3: int] :
% 9.10/5.33            ( ( Uu3
% 9.10/5.33              = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 9.10/5.33            & ( J3 != zero_zero_int ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9840_Fract__le__zero__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ zero_zero_rat )
% 9.10/5.33          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9841_zero__le__Fract__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B2 ) )
% 9.10/5.33          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9842_Fract__le__one__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_eq_rat @ ( fract @ A @ B2 ) @ one_one_rat )
% 9.10/5.33          = ( ord_less_eq_int @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9843_one__le__Fract__iff,axiom,
% 9.10/5.33      ! [B2: int,A: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ B2 )
% 9.10/5.33       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B2 ) )
% 9.10/5.33          = ( ord_less_eq_int @ B2 @ A ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9844_rat__number__expand_I5_J,axiom,
% 9.10/5.33      ! [K: num] :
% 9.10/5.33        ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 9.10/5.33        = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9845_rat__number__collapse_I4_J,axiom,
% 9.10/5.33      ! [W: num] :
% 9.10/5.33        ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 9.10/5.33        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9846_image__minus__const__atLeastLessThan__nat,axiom,
% 9.10/5.33      ! [C: nat,Y3: nat,X3: nat] :
% 9.10/5.33        ( ( ( ord_less_nat @ C @ Y3 )
% 9.10/5.33         => ( ( image_nat_nat
% 9.10/5.33              @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.10/5.33              @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 9.10/5.33            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_nat @ C @ Y3 )
% 9.10/5.33         => ( ( ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.33             => ( ( image_nat_nat
% 9.10/5.33                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.10/5.33                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 9.10/5.33                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 9.10/5.33            & ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.10/5.33             => ( ( image_nat_nat
% 9.10/5.33                  @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.10/5.33                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 9.10/5.33                = bot_bot_set_nat ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9847_of__nat__eq__id,axiom,
% 9.10/5.33      semiri1316708129612266289at_nat = id_nat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9848_bij__betw__Suc,axiom,
% 9.10/5.33      ! [M7: set_nat,N4: set_nat] :
% 9.10/5.33        ( ( bij_betw_nat_nat @ suc @ M7 @ N4 )
% 9.10/5.33        = ( ( image_nat_nat @ suc @ M7 )
% 9.10/5.33          = N4 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9849_less__int__def,axiom,
% 9.10/5.33      ( ord_less_int
% 9.10/5.33      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.10/5.33        @ ( produc8739625826339149834_nat_o
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc6081775807080527818_nat_o
% 9.10/5.33              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9850_less__eq__int__def,axiom,
% 9.10/5.33      ( ord_less_eq_int
% 9.10/5.33      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.10/5.33        @ ( produc8739625826339149834_nat_o
% 9.10/5.33          @ ^ [X: nat,Y: nat] :
% 9.10/5.33              ( produc6081775807080527818_nat_o
% 9.10/5.33              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9851_nat__def,axiom,
% 9.10/5.33      ( nat2
% 9.10/5.33      = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9852_positive__rat,axiom,
% 9.10/5.33      ! [A: int,B2: int] :
% 9.10/5.33        ( ( positive @ ( fract @ A @ B2 ) )
% 9.10/5.33        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9853_Inf__real__def,axiom,
% 9.10/5.33      ( comple4887499456419720421f_real
% 9.10/5.33      = ( ^ [X7: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X7 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9854_Rat_Opositive__mult,axiom,
% 9.10/5.33      ! [X3: rat,Y3: rat] :
% 9.10/5.33        ( ( positive @ X3 )
% 9.10/5.33       => ( ( positive @ Y3 )
% 9.10/5.33         => ( positive @ ( times_times_rat @ X3 @ Y3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9855_finite__int__iff__bounded__le,axiom,
% 9.10/5.33      ( finite_finite_int
% 9.10/5.33      = ( ^ [S4: set_int] :
% 9.10/5.33          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9856_finite__int__iff__bounded,axiom,
% 9.10/5.33      ( finite_finite_int
% 9.10/5.33      = ( ^ [S4: set_int] :
% 9.10/5.33          ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S4 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9857_suminf__eq__SUP__real,axiom,
% 9.10/5.33      ! [X9: nat > real] :
% 9.10/5.33        ( ( summable_real @ X9 )
% 9.10/5.33       => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X9 @ I3 ) )
% 9.10/5.33         => ( ( suminf_real @ X9 )
% 9.10/5.33            = ( comple1385675409528146559p_real
% 9.10/5.33              @ ( image_nat_real
% 9.10/5.33                @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X9 @ ( set_ord_lessThan_nat @ I4 ) )
% 9.10/5.33                @ top_top_set_nat ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9858_less__rat__def,axiom,
% 9.10/5.33      ( ord_less_rat
% 9.10/5.33      = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9859_image__atLeastZeroLessThan__int,axiom,
% 9.10/5.33      ! [U: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.10/5.33       => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 9.10/5.33          = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9860_range__mod,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( image_nat_nat
% 9.10/5.33            @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 9.10/5.33            @ top_top_set_nat )
% 9.10/5.33          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9861_Rat_Opositive_Orep__eq,axiom,
% 9.10/5.33      ( positive
% 9.10/5.33      = ( ^ [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 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9862_card__UNIV__bool,axiom,
% 9.10/5.33      ( ( finite_card_o @ top_top_set_o )
% 9.10/5.33      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9863_range__mult,axiom,
% 9.10/5.33      ! [A: real] :
% 9.10/5.33        ( ( ( A = zero_zero_real )
% 9.10/5.33         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.10/5.33            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 9.10/5.33        & ( ( A != zero_zero_real )
% 9.10/5.33         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.10/5.33            = top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9864_Rat_Opositive__def,axiom,
% 9.10/5.33      ( positive
% 9.10/5.33      = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 9.10/5.33        @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9865_root__def,axiom,
% 9.10/5.33      ( root
% 9.10/5.33      = ( ^ [N2: nat,X: real] :
% 9.10/5.33            ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 9.10/5.33            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 9.10/5.33              @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 9.10/5.33              @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9866_card__UNIV__char,axiom,
% 9.10/5.33      ( ( finite_card_char @ top_top_set_char )
% 9.10/5.33      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9867_UNIV__char__of__nat,axiom,
% 9.10/5.33      ( top_top_set_char
% 9.10/5.33      = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9868_nat__of__char__less__256,axiom,
% 9.10/5.33      ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9869_range__nat__of__char,axiom,
% 9.10/5.33      ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 9.10/5.33      = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9870_integer__of__char__code,axiom,
% 9.10/5.33      ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 9.10/5.33        ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 9.10/5.33        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9871_of__rat__dense,axiom,
% 9.10/5.33      ! [X3: real,Y3: real] :
% 9.10/5.33        ( ( ord_less_real @ X3 @ Y3 )
% 9.10/5.33       => ? [Q3: rat] :
% 9.10/5.33            ( ( ord_less_real @ X3 @ ( field_7254667332652039916t_real @ Q3 ) )
% 9.10/5.33            & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9872_plus__rat__def,axiom,
% 9.10/5.33      ( plus_plus_rat
% 9.10/5.33      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 9.10/5.33        @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9873_inverse__rat__def,axiom,
% 9.10/5.33      ( inverse_inverse_rat
% 9.10/5.33      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 9.10/5.33        @ ^ [X: product_prod_int_int] :
% 9.10/5.33            ( if_Pro3027730157355071871nt_int
% 9.10/5.33            @ ( ( product_fst_int_int @ X )
% 9.10/5.33              = zero_zero_int )
% 9.10/5.33            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.10/5.33            @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9874_String_Ochar__of__ascii__of,axiom,
% 9.10/5.33      ! [C: char] :
% 9.10/5.33        ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 9.10/5.33        = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9875_one__rat__def,axiom,
% 9.10/5.33      ( one_one_rat
% 9.10/5.33      = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9876_Fract_Oabs__eq,axiom,
% 9.10/5.33      ( fract
% 9.10/5.33      = ( ^ [Xa4: int,X: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa4 @ X ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9877_zero__rat__def,axiom,
% 9.10/5.33      ( zero_zero_rat
% 9.10/5.33      = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9878_uminus__rat__def,axiom,
% 9.10/5.33      ( uminus_uminus_rat
% 9.10/5.33      = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
% 9.10/5.33        @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9879_times__rat__def,axiom,
% 9.10/5.33      ( times_times_rat
% 9.10/5.33      = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
% 9.10/5.33        @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9880_upto__aux__rec,axiom,
% 9.10/5.33      ( upto_aux
% 9.10/5.33      = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9881_upto_Opelims,axiom,
% 9.10/5.33      ! [X3: int,Xa2: int,Y3: list_int] :
% 9.10/5.33        ( ( ( upto @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 9.10/5.33         => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 9.10/5.33                 => ( Y3
% 9.10/5.33                    = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 9.10/5.33                & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 9.10/5.33                 => ( Y3 = nil_int ) ) )
% 9.10/5.33             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9882_upto__empty,axiom,
% 9.10/5.33      ! [J: int,I: int] :
% 9.10/5.33        ( ( ord_less_int @ J @ I )
% 9.10/5.33       => ( ( upto @ I @ J )
% 9.10/5.33          = nil_int ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9883_upto__Nil2,axiom,
% 9.10/5.33      ! [I: int,J: int] :
% 9.10/5.33        ( ( nil_int
% 9.10/5.33          = ( upto @ I @ J ) )
% 9.10/5.33        = ( ord_less_int @ J @ I ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9884_upto__Nil,axiom,
% 9.10/5.33      ! [I: int,J: int] :
% 9.10/5.33        ( ( ( upto @ I @ J )
% 9.10/5.33          = nil_int )
% 9.10/5.33        = ( ord_less_int @ J @ I ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9885_nth__upto,axiom,
% 9.10/5.33      ! [I: int,K: nat,J: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 9.10/5.33       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 9.10/5.33          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9886_upto__rec__numeral_I1_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33            = nil_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9887_upto__rec__numeral_I4_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33            = nil_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9888_upto__rec__numeral_I3_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33            = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.10/5.33            = nil_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9889_upto__rec__numeral_I2_J,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.10/5.33            = nil_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9890_atLeastAtMost__upto,axiom,
% 9.10/5.33      ( set_or1266510415728281911st_int
% 9.10/5.33      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ J3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9891_upto__split2,axiom,
% 9.10/5.33      ! [I: int,J: int,K: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ I @ J )
% 9.10/5.33       => ( ( ord_less_eq_int @ J @ K )
% 9.10/5.33         => ( ( upto @ I @ K )
% 9.10/5.33            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9892_upto__split1,axiom,
% 9.10/5.33      ! [I: int,J: int,K: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ I @ J )
% 9.10/5.33       => ( ( ord_less_eq_int @ J @ K )
% 9.10/5.33         => ( ( upto @ I @ K )
% 9.10/5.33            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9893_atLeastLessThan__upto,axiom,
% 9.10/5.33      ( set_or4662586982721622107an_int
% 9.10/5.33      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9894_upto__rec1,axiom,
% 9.10/5.33      ! [I: int,J: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ I @ J )
% 9.10/5.33       => ( ( upto @ I @ J )
% 9.10/5.33          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9895_upto_Osimps,axiom,
% 9.10/5.33      ( upto
% 9.10/5.33      = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9896_upto_Oelims,axiom,
% 9.10/5.33      ! [X3: int,Xa2: int,Y3: list_int] :
% 9.10/5.33        ( ( ( upto @ X3 @ Xa2 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 9.10/5.33           => ( Y3
% 9.10/5.33              = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 9.10/5.33          & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 9.10/5.33           => ( Y3 = nil_int ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9897_upto__rec2,axiom,
% 9.10/5.33      ! [I: int,J: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ I @ J )
% 9.10/5.33       => ( ( upto @ I @ J )
% 9.10/5.33          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9898_upto__split3,axiom,
% 9.10/5.33      ! [I: int,J: int,K: int] :
% 9.10/5.33        ( ( ord_less_eq_int @ I @ J )
% 9.10/5.33       => ( ( ord_less_eq_int @ J @ K )
% 9.10/5.33         => ( ( upto @ I @ K )
% 9.10/5.33            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9899_upto_Opsimps,axiom,
% 9.10/5.33      ! [I: int,J: int] :
% 9.10/5.33        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 9.10/5.33       => ( ( ( ord_less_eq_int @ I @ J )
% 9.10/5.33           => ( ( upto @ I @ J )
% 9.10/5.33              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 9.10/5.33          & ( ~ ( ord_less_eq_int @ I @ J )
% 9.10/5.33           => ( ( upto @ I @ J )
% 9.10/5.33              = nil_int ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9900_plus__rat_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
% 9.10/5.33        ( ( ratrel @ Xa2 @ Xa2 )
% 9.10/5.33       => ( ( ratrel @ X3 @ X3 )
% 9.10/5.33         => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
% 9.10/5.33            = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9901_ratrel__iff,axiom,
% 9.10/5.33      ( ratrel
% 9.10/5.33      = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 9.10/5.33            ( ( ( product_snd_int_int @ X )
% 9.10/5.33             != zero_zero_int )
% 9.10/5.33            & ( ( product_snd_int_int @ Y )
% 9.10/5.33             != zero_zero_int )
% 9.10/5.33            & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 9.10/5.33              = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9902_one__rat_Orsp,axiom,
% 9.10/5.33      ratrel @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9903_zero__rat_Orsp,axiom,
% 9.10/5.33      ratrel @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9904_ratrel__def,axiom,
% 9.10/5.33      ( ratrel
% 9.10/5.33      = ( ^ [X: product_prod_int_int,Y: product_prod_int_int] :
% 9.10/5.33            ( ( ( product_snd_int_int @ X )
% 9.10/5.33             != zero_zero_int )
% 9.10/5.33            & ( ( product_snd_int_int @ Y )
% 9.10/5.33             != zero_zero_int )
% 9.10/5.33            & ( ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) )
% 9.10/5.33              = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9905_uminus__rat_Oabs__eq,axiom,
% 9.10/5.33      ! [X3: product_prod_int_int] :
% 9.10/5.33        ( ( ratrel @ X3 @ X3 )
% 9.10/5.33       => ( ( uminus_uminus_rat @ ( abs_Rat @ X3 ) )
% 9.10/5.33          = ( abs_Rat @ ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X3 ) ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9906_times__rat_Oabs__eq,axiom,
% 9.10/5.33      ! [Xa2: product_prod_int_int,X3: product_prod_int_int] :
% 9.10/5.33        ( ( ratrel @ Xa2 @ Xa2 )
% 9.10/5.33       => ( ( ratrel @ X3 @ X3 )
% 9.10/5.33         => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X3 ) )
% 9.10/5.33            = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9907_Rat_Opositive_Oabs__eq,axiom,
% 9.10/5.33      ! [X3: product_prod_int_int] :
% 9.10/5.33        ( ( ratrel @ X3 @ X3 )
% 9.10/5.33       => ( ( positive @ ( abs_Rat @ X3 ) )
% 9.10/5.33          = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9908_inverse__rat_Oabs__eq,axiom,
% 9.10/5.33      ! [X3: product_prod_int_int] :
% 9.10/5.33        ( ( ratrel @ X3 @ X3 )
% 9.10/5.33       => ( ( inverse_inverse_rat @ ( abs_Rat @ X3 ) )
% 9.10/5.33          = ( abs_Rat
% 9.10/5.33            @ ( if_Pro3027730157355071871nt_int
% 9.10/5.33              @ ( ( product_fst_int_int @ X3 )
% 9.10/5.33                = zero_zero_int )
% 9.10/5.33              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.10/5.33              @ ( product_Pair_int_int @ ( product_snd_int_int @ X3 ) @ ( product_fst_int_int @ X3 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9909_DERIV__even__real__root,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33         => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.33           => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9910_DERIV__real__root__generic,axiom,
% 9.10/5.33      ! [N: nat,X3: real,D4: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( X3 != zero_zero_real )
% 9.10/5.33         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33               => ( D4
% 9.10/5.33                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 9.10/5.33           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33               => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.33                 => ( D4
% 9.10/5.33                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 9.10/5.33             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33                 => ( D4
% 9.10/5.33                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 9.10/5.33               => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9911_DERIV__neg__imp__decreasing,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33               => ? [Y5: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 9.10/5.33         => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9912_DERIV__pos__imp__increasing,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33               => ? [Y5: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 9.10/5.33         => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9913_deriv__nonneg__imp__mono,axiom,
% 9.10/5.33      ! [A: real,B2: real,G: real > real,G2: real > real] :
% 9.10/5.33        ( ! [X5: real] :
% 9.10/5.33            ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 9.10/5.33           => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 9.10/5.33             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X5 ) ) )
% 9.10/5.33         => ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.33           => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9914_DERIV__nonneg__imp__nondecreasing,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33               => ? [Y5: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 9.10/5.33         => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9915_DERIV__nonpos__imp__nonincreasing,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33               => ? [Y5: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 9.10/5.33         => ( ord_less_eq_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9916_DERIV__const__ratio__const,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,K: real] :
% 9.10/5.33        ( ( A != B2 )
% 9.10/5.33       => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33         => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 9.10/5.33            = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ K ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9917_has__real__derivative__pos__inc__left,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
% 9.10/5.33                   => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9918_has__real__derivative__neg__dec__left,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 9.10/5.33       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S2 )
% 9.10/5.33                   => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9919_has__real__derivative__pos__inc__right,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
% 9.10/5.33                   => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9920_has__real__derivative__neg__dec__right,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,S2: set_real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S2 ) )
% 9.10/5.33       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S2 )
% 9.10/5.33                   => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9921_DERIV__pos__inc__right,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9922_DERIV__neg__dec__right,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9923_DERIV__neg__dec__left,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9924_DERIV__pos__inc__left,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.10/5.33         => ? [D3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.10/5.33              & ! [H4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.10/5.33                 => ( ( ord_less_real @ H4 @ D3 )
% 9.10/5.33                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9925_MVT2,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,F4: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33               => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33         => ? [Z4: real] :
% 9.10/5.33              ( ( ord_less_real @ A @ Z4 )
% 9.10/5.33              & ( ord_less_real @ Z4 @ B2 )
% 9.10/5.33              & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 9.10/5.33                = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ ( F4 @ Z4 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9926_DERIV__local__const,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,D: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.10/5.33         => ( ! [Y4: real] :
% 9.10/5.33                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
% 9.10/5.33               => ( ( F @ X3 )
% 9.10/5.33                  = ( F @ Y4 ) ) )
% 9.10/5.33           => ( L = zero_zero_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9927_DERIV__ln,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9928_DERIV__const__average,axiom,
% 9.10/5.33      ! [A: real,B2: real,V: real > real,K: real] :
% 9.10/5.33        ( ( A != B2 )
% 9.10/5.33       => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33         => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.10/5.33            = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9929_DERIV__local__max,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,D: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.10/5.33         => ( ! [Y4: real] :
% 9.10/5.33                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
% 9.10/5.33               => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X3 ) ) )
% 9.10/5.33           => ( L = zero_zero_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9930_DERIV__local__min,axiom,
% 9.10/5.33      ! [F: real > real,L: real,X3: real,D: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.10/5.33         => ( ! [Y4: real] :
% 9.10/5.33                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) @ D )
% 9.10/5.33               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 9.10/5.33           => ( L = zero_zero_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9931_DERIV__ln__divide,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9932_DERIV__pow,axiom,
% 9.10/5.33      ! [N: nat,X3: real,S: set_real] :
% 9.10/5.33        ( has_fi5821293074295781190e_real
% 9.10/5.33        @ ^ [X: real] : ( power_power_real @ X @ N )
% 9.10/5.33        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 9.10/5.33        @ ( topolo2177554685111907308n_real @ X3 @ S ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9933_DERIV__fun__pow,axiom,
% 9.10/5.33      ! [G: real > real,M: real,X3: real,N: nat] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( has_fi5821293074295781190e_real
% 9.10/5.33          @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N )
% 9.10/5.33          @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 9.10/5.33          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9934_has__real__derivative__powr,axiom,
% 9.10/5.33      ! [Z: real,R: real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ Z )
% 9.10/5.33       => ( has_fi5821293074295781190e_real
% 9.10/5.33          @ ^ [Z2: real] : ( powr_real @ Z2 @ R )
% 9.10/5.33          @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
% 9.10/5.33          @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9935_DERIV__log,axiom,
% 9.10/5.33      ! [X3: real,B2: real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9936_DERIV__fun__powr,axiom,
% 9.10/5.33      ! [G: real > real,M: real,X3: real,R: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 9.10/5.33         => ( has_fi5821293074295781190e_real
% 9.10/5.33            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R )
% 9.10/5.33            @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 9.10/5.33            @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9937_DERIV__powr,axiom,
% 9.10/5.33      ! [G: real > real,M: real,X3: real,F: real > real,R: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 9.10/5.33         => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.10/5.33           => ( has_fi5821293074295781190e_real
% 9.10/5.33              @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 9.10/5.33              @ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
% 9.10/5.33              @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9938_DERIV__real__sqrt,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9939_DERIV__arctan,axiom,
% 9.10/5.33      ! [X3: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9940_arsinh__real__has__field__derivative,axiom,
% 9.10/5.33      ! [X3: real,A4: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A4 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9941_DERIV__real__sqrt__generic,axiom,
% 9.10/5.33      ! [X3: real,D4: real] :
% 9.10/5.33        ( ( X3 != zero_zero_real )
% 9.10/5.33       => ( ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33           => ( D4
% 9.10/5.33              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33         => ( ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.10/5.33             => ( D4
% 9.10/5.33                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.10/5.33           => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9942_arcosh__real__has__field__derivative,axiom,
% 9.10/5.33      ! [X3: real,A4: set_real] :
% 9.10/5.33        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.33       => ( 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 @ A4 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9943_artanh__real__has__field__derivative,axiom,
% 9.10/5.33      ! [X3: real,A4: set_real] :
% 9.10/5.33        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.33       => ( 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 @ A4 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9944_DERIV__real__root,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.10/5.33         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9945_DERIV__arccos,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33         => ( 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 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9946_DERIV__arcsin,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33         => ( 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 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9947_Maclaurin__all__le,axiom,
% 9.10/5.33      ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
% 9.10/5.33        ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33          = F )
% 9.10/5.33       => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33         => ? [T4: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.33              & ( ( F @ X3 )
% 9.10/5.33                = ( plus_plus_real
% 9.10/5.33                  @ ( groups6591440286371151544t_real
% 9.10/5.33                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.33                    @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9948_Maclaurin__all__le__objl,axiom,
% 9.10/5.33      ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
% 9.10/5.33        ( ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33            = F )
% 9.10/5.33          & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.10/5.33       => ? [T4: real] :
% 9.10/5.33            ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.33            & ( ( F @ X3 )
% 9.10/5.33              = ( plus_plus_real
% 9.10/5.33                @ ( groups6591440286371151544t_real
% 9.10/5.33                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.33                  @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9949_DERIV__odd__real__root,axiom,
% 9.10/5.33      ! [N: nat,X3: real] :
% 9.10/5.33        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33       => ( ( X3 != zero_zero_real )
% 9.10/5.33         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X3 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9950_Maclaurin,axiom,
% 9.10/5.33      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.10/5.33       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33              = F )
% 9.10/5.33           => ( ! [M4: nat,T4: real] :
% 9.10/5.33                  ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                    & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.33                    & ( ord_less_eq_real @ T4 @ H2 ) )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33             => ? [T4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ T4 )
% 9.10/5.33                  & ( ord_less_real @ T4 @ H2 )
% 9.10/5.33                  & ( ( F @ H2 )
% 9.10/5.33                    = ( plus_plus_real
% 9.10/5.33                      @ ( groups6591440286371151544t_real
% 9.10/5.33                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 9.10/5.33                        @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9951_Maclaurin2,axiom,
% 9.10/5.33      ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.10/5.33       => ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33            = F )
% 9.10/5.33         => ( ! [M4: nat,T4: real] :
% 9.10/5.33                ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                  & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.33                  & ( ord_less_eq_real @ T4 @ H2 ) )
% 9.10/5.33               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33           => ? [T4: real] :
% 9.10/5.33                ( ( ord_less_real @ zero_zero_real @ T4 )
% 9.10/5.33                & ( ord_less_eq_real @ T4 @ H2 )
% 9.10/5.33                & ( ( F @ H2 )
% 9.10/5.33                  = ( plus_plus_real
% 9.10/5.33                    @ ( groups6591440286371151544t_real
% 9.10/5.33                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 9.10/5.33                      @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9952_Maclaurin__minus,axiom,
% 9.10/5.33      ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ H2 @ zero_zero_real )
% 9.10/5.33       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33              = F )
% 9.10/5.33           => ( ! [M4: nat,T4: real] :
% 9.10/5.33                  ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                    & ( ord_less_eq_real @ H2 @ T4 )
% 9.10/5.33                    & ( ord_less_eq_real @ T4 @ zero_zero_real ) )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33             => ? [T4: real] :
% 9.10/5.33                  ( ( ord_less_real @ H2 @ T4 )
% 9.10/5.33                  & ( ord_less_real @ T4 @ zero_zero_real )
% 9.10/5.33                  & ( ( F @ H2 )
% 9.10/5.33                    = ( plus_plus_real
% 9.10/5.33                      @ ( groups6591440286371151544t_real
% 9.10/5.33                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 9.10/5.33                        @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9953_Maclaurin__all__lt,axiom,
% 9.10/5.33      ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
% 9.10/5.33        ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33          = F )
% 9.10/5.33       => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33         => ( ( X3 != zero_zero_real )
% 9.10/5.33           => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33             => ? [T4: real] :
% 9.10/5.33                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T4 ) )
% 9.10/5.33                  & ( ord_less_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.33                  & ( ( F @ X3 )
% 9.10/5.33                    = ( plus_plus_real
% 9.10/5.33                      @ ( groups6591440286371151544t_real
% 9.10/5.33                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.33                        @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9954_Maclaurin__bi__le,axiom,
% 9.10/5.33      ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
% 9.10/5.33        ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33          = F )
% 9.10/5.33       => ( ! [M4: nat,T4: real] :
% 9.10/5.33              ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                & ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) ) )
% 9.10/5.33             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33         => ? [T4: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ ( abs_abs_real @ T4 ) @ ( abs_abs_real @ X3 ) )
% 9.10/5.33              & ( ( F @ X3 )
% 9.10/5.33                = ( plus_plus_real
% 9.10/5.33                  @ ( groups6591440286371151544t_real
% 9.10/5.33                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X3 @ M3 ) )
% 9.10/5.33                    @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9955_Taylor,axiom,
% 9.10/5.33      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real,X3: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33            = F )
% 9.10/5.33         => ( ! [M4: nat,T4: real] :
% 9.10/5.33                ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                  & ( ord_less_eq_real @ A @ T4 )
% 9.10/5.33                  & ( ord_less_eq_real @ T4 @ B2 ) )
% 9.10/5.33               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33           => ( ( ord_less_eq_real @ A @ C )
% 9.10/5.33             => ( ( ord_less_eq_real @ C @ B2 )
% 9.10/5.33               => ( ( ord_less_eq_real @ A @ X3 )
% 9.10/5.33                 => ( ( ord_less_eq_real @ X3 @ B2 )
% 9.10/5.33                   => ( ( X3 != C )
% 9.10/5.33                     => ? [T4: real] :
% 9.10/5.33                          ( ( ( ord_less_real @ X3 @ C )
% 9.10/5.33                           => ( ( ord_less_real @ X3 @ T4 )
% 9.10/5.33                              & ( ord_less_real @ T4 @ C ) ) )
% 9.10/5.33                          & ( ~ ( ord_less_real @ X3 @ C )
% 9.10/5.33                           => ( ( ord_less_real @ C @ T4 )
% 9.10/5.33                              & ( ord_less_real @ T4 @ X3 ) ) )
% 9.10/5.33                          & ( ( F @ X3 )
% 9.10/5.33                            = ( plus_plus_real
% 9.10/5.33                              @ ( groups6591440286371151544t_real
% 9.10/5.33                                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ M3 ) )
% 9.10/5.33                                @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9956_Taylor__up,axiom,
% 9.10/5.33      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33            = F )
% 9.10/5.33         => ( ! [M4: nat,T4: real] :
% 9.10/5.33                ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                  & ( ord_less_eq_real @ A @ T4 )
% 9.10/5.33                  & ( ord_less_eq_real @ T4 @ B2 ) )
% 9.10/5.33               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33           => ( ( ord_less_eq_real @ A @ C )
% 9.10/5.33             => ( ( ord_less_real @ C @ B2 )
% 9.10/5.33               => ? [T4: real] :
% 9.10/5.33                    ( ( ord_less_real @ C @ T4 )
% 9.10/5.33                    & ( ord_less_real @ T4 @ B2 )
% 9.10/5.33                    & ( ( F @ B2 )
% 9.10/5.33                      = ( plus_plus_real
% 9.10/5.33                        @ ( groups6591440286371151544t_real
% 9.10/5.33                          @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ M3 ) )
% 9.10/5.33                          @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9957_Taylor__down,axiom,
% 9.10/5.33      ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B2: real,C: real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( ( Diff @ zero_zero_nat )
% 9.10/5.33            = F )
% 9.10/5.33         => ( ! [M4: nat,T4: real] :
% 9.10/5.33                ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33                  & ( ord_less_eq_real @ A @ T4 )
% 9.10/5.33                  & ( ord_less_eq_real @ T4 @ B2 ) )
% 9.10/5.33               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33           => ( ( ord_less_real @ A @ C )
% 9.10/5.33             => ( ( ord_less_eq_real @ C @ B2 )
% 9.10/5.33               => ? [T4: real] :
% 9.10/5.33                    ( ( ord_less_real @ A @ T4 )
% 9.10/5.33                    & ( ord_less_real @ T4 @ C )
% 9.10/5.33                    & ( ( F @ A )
% 9.10/5.33                      = ( plus_plus_real
% 9.10/5.33                        @ ( groups6591440286371151544t_real
% 9.10/5.33                          @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 9.10/5.33                          @ ( set_ord_lessThan_nat @ N ) )
% 9.10/5.33                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T4 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9958_Maclaurin__lemma2,axiom,
% 9.10/5.33      ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
% 9.10/5.33        ( ! [M4: nat,T4: real] :
% 9.10/5.33            ( ( ( ord_less_nat @ M4 @ N )
% 9.10/5.33              & ( ord_less_eq_real @ zero_zero_real @ T4 )
% 9.10/5.33              & ( ord_less_eq_real @ T4 @ H2 ) )
% 9.10/5.33           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) )
% 9.10/5.33       => ( ( N
% 9.10/5.33            = ( suc @ K ) )
% 9.10/5.33         => ! [M2: nat,T5: real] :
% 9.10/5.33              ( ( ( ord_less_nat @ M2 @ N )
% 9.10/5.33                & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 9.10/5.33                & ( ord_less_eq_real @ T5 @ H2 ) )
% 9.10/5.33             => ( has_fi5821293074295781190e_real
% 9.10/5.33                @ ^ [U2: real] :
% 9.10/5.33                    ( minus_minus_real @ ( Diff @ M2 @ U2 )
% 9.10/5.33                    @ ( plus_plus_real
% 9.10/5.33                      @ ( groups6591440286371151544t_real
% 9.10/5.33                        @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
% 9.10/5.33                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 9.10/5.33                      @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 9.10/5.33                @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T5 )
% 9.10/5.33                  @ ( plus_plus_real
% 9.10/5.33                    @ ( groups6591440286371151544t_real
% 9.10/5.33                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T5 @ P5 ) )
% 9.10/5.33                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 9.10/5.33                    @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T5 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 9.10/5.33                @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9959_DERIV__arctan__series,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.33       => ( has_fi5821293074295781190e_real
% 9.10/5.33          @ ^ [X10: real] :
% 9.10/5.33              ( suminf_real
% 9.10/5.33              @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 9.10/5.33          @ ( suminf_real
% 9.10/5.33            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9960_DERIV__power__series_H,axiom,
% 9.10/5.33      ! [R2: real,F: nat > real,X0: real] :
% 9.10/5.33        ( ! [X5: real] :
% 9.10/5.33            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 9.10/5.33           => ( summable_real
% 9.10/5.33              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 9.10/5.33       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 9.10/5.33         => ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.10/5.33           => ( has_fi5821293074295781190e_real
% 9.10/5.33              @ ^ [X: real] :
% 9.10/5.33                  ( suminf_real
% 9.10/5.33                  @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
% 9.10/5.33              @ ( suminf_real
% 9.10/5.33                @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 9.10/5.33              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9961_DERIV__isconst3,axiom,
% 9.10/5.33      ! [A: real,B2: real,X3: real,Y3: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33         => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33           => ( ! [X5: real] :
% 9.10/5.33                  ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.10/5.33             => ( ( F @ X3 )
% 9.10/5.33                = ( F @ Y3 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9962_DERIV__series_H,axiom,
% 9.10/5.33      ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B2: real,L5: nat > real] :
% 9.10/5.33        ( ! [N3: nat] :
% 9.10/5.33            ( has_fi5821293074295781190e_real
% 9.10/5.33            @ ^ [X: real] : ( F @ X @ N3 )
% 9.10/5.33            @ ( F4 @ X0 @ N3 )
% 9.10/5.33            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33             => ( summable_real @ ( F @ X5 ) ) )
% 9.10/5.33         => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33           => ( ( summable_real @ ( F4 @ X0 ) )
% 9.10/5.33             => ( ( summable_real @ L5 )
% 9.10/5.33               => ( ! [N3: nat,X5: real,Y4: real] :
% 9.10/5.33                      ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33                     => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33                       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y4 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real
% 9.10/5.33                    @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 9.10/5.33                    @ ( suminf_real @ ( F4 @ X0 ) )
% 9.10/5.33                    @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9963_card__greaterThanLessThan,axiom,
% 9.10/5.33      ! [L: nat,U: nat] :
% 9.10/5.33        ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 9.10/5.33        = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9964_isCont__Lb__Ub,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33                & ( ord_less_eq_real @ X5 @ B2 ) )
% 9.10/5.33             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 9.10/5.33         => ? [L6: real,M8: real] :
% 9.10/5.33              ( ! [X6: real] :
% 9.10/5.33                  ( ( ( ord_less_eq_real @ A @ X6 )
% 9.10/5.33                    & ( ord_less_eq_real @ X6 @ B2 ) )
% 9.10/5.33                 => ( ( ord_less_eq_real @ L6 @ ( F @ X6 ) )
% 9.10/5.33                    & ( ord_less_eq_real @ ( F @ X6 ) @ M8 ) ) )
% 9.10/5.33              & ! [Y5: real] :
% 9.10/5.33                  ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 9.10/5.33                    & ( ord_less_eq_real @ Y5 @ M8 ) )
% 9.10/5.33                 => ? [X5: real] :
% 9.10/5.33                      ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33                      & ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33                      & ( ( F @ X5 )
% 9.10/5.33                        = Y5 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9965_LIM__fun__less__zero,axiom,
% 9.10/5.33      ! [F: real > real,L: real,C: real] :
% 9.10/5.33        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.10/5.33         => ? [R3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.10/5.33              & ! [X6: real] :
% 9.10/5.33                  ( ( ( X6 != C )
% 9.10/5.33                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
% 9.10/5.33                 => ( ord_less_real @ ( F @ X6 ) @ zero_zero_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9966_LIM__fun__not__zero,axiom,
% 9.10/5.33      ! [F: real > real,L: real,C: real] :
% 9.10/5.33        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.10/5.33       => ( ( L != zero_zero_real )
% 9.10/5.33         => ? [R3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.10/5.33              & ! [X6: real] :
% 9.10/5.33                  ( ( ( X6 != C )
% 9.10/5.33                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
% 9.10/5.33                 => ( ( F @ X6 )
% 9.10/5.33                   != zero_zero_real ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9967_LIM__fun__gt__zero,axiom,
% 9.10/5.33      ! [F: real > real,L: real,C: real] :
% 9.10/5.33        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.10/5.33       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.10/5.33         => ? [R3: real] :
% 9.10/5.33              ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.10/5.33              & ! [X6: real] :
% 9.10/5.33                  ( ( ( X6 != C )
% 9.10/5.33                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X6 ) ) @ R3 ) )
% 9.10/5.33                 => ( ord_less_real @ zero_zero_real @ ( F @ X6 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9968_isCont__real__sqrt,axiom,
% 9.10/5.33      ! [X3: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ sqrt ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9969_isCont__real__root,axiom,
% 9.10/5.33      ! [X3: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ ( root @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9970_isCont__inverse__function2,axiom,
% 9.10/5.33      ! [A: real,X3: real,B2: real,G: real > real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ B2 )
% 9.10/5.33         => ( ! [Z4: real] :
% 9.10/5.33                ( ( ord_less_eq_real @ A @ Z4 )
% 9.10/5.33               => ( ( ord_less_eq_real @ Z4 @ B2 )
% 9.10/5.33                 => ( ( G @ ( F @ Z4 ) )
% 9.10/5.33                    = Z4 ) ) )
% 9.10/5.33           => ( ! [Z4: real] :
% 9.10/5.33                  ( ( ord_less_eq_real @ A @ Z4 )
% 9.10/5.33                 => ( ( ord_less_eq_real @ Z4 @ B2 )
% 9.10/5.33                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 9.10/5.33             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9971_isCont__arcosh,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.33       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcosh_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9972_LIM__cos__div__sin,axiom,
% 9.10/5.33      ( filterlim_real_real
% 9.10/5.33      @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
% 9.10/5.33      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33      @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9973_DERIV__inverse__function,axiom,
% 9.10/5.33      ! [F: real > real,D4: real,G: real > real,X3: real,A: real,B2: real] :
% 9.10/5.33        ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
% 9.10/5.33       => ( ( D4 != zero_zero_real )
% 9.10/5.33         => ( ( ord_less_real @ A @ X3 )
% 9.10/5.33           => ( ( ord_less_real @ X3 @ B2 )
% 9.10/5.33             => ( ! [Y4: real] :
% 9.10/5.33                    ( ( ord_less_real @ A @ Y4 )
% 9.10/5.33                   => ( ( ord_less_real @ Y4 @ B2 )
% 9.10/5.33                     => ( ( F @ ( G @ Y4 ) )
% 9.10/5.33                        = Y4 ) ) )
% 9.10/5.33               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9974_isCont__arccos,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arccos ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9975_isCont__arcsin,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcsin ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9976_LIM__less__bound,axiom,
% 9.10/5.33      ! [B2: real,X3: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ B2 @ X3 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B2 @ X3 ) )
% 9.10/5.33             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.10/5.33         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
% 9.10/5.33           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9977_isCont__artanh,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9978_greaterThanLessThan__upto,axiom,
% 9.10/5.33      ( set_or5832277885323065728an_int
% 9.10/5.33      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9979_isCont__inverse__function,axiom,
% 9.10/5.33      ! [D: real,X3: real,G: real > real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ D )
% 9.10/5.33       => ( ! [Z4: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X3 ) ) @ D )
% 9.10/5.33             => ( ( G @ ( F @ Z4 ) )
% 9.10/5.33                = Z4 ) )
% 9.10/5.33         => ( ! [Z4: real] :
% 9.10/5.33                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X3 ) ) @ D )
% 9.10/5.33               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 9.10/5.33           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9980_GMVT_H,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [Z4: real] :
% 9.10/5.33              ( ( ord_less_eq_real @ A @ Z4 )
% 9.10/5.33             => ( ( ord_less_eq_real @ Z4 @ B2 )
% 9.10/5.33               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 9.10/5.33         => ( ! [Z4: real] :
% 9.10/5.33                ( ( ord_less_eq_real @ A @ Z4 )
% 9.10/5.33               => ( ( ord_less_eq_real @ Z4 @ B2 )
% 9.10/5.33                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
% 9.10/5.33           => ( ! [Z4: real] :
% 9.10/5.33                  ( ( ord_less_real @ A @ Z4 )
% 9.10/5.33                 => ( ( ord_less_real @ Z4 @ B2 )
% 9.10/5.33                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 9.10/5.33             => ( ! [Z4: real] :
% 9.10/5.33                    ( ( ord_less_real @ A @ Z4 )
% 9.10/5.33                   => ( ( ord_less_real @ Z4 @ B2 )
% 9.10/5.33                     => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 9.10/5.33               => ? [C2: real] :
% 9.10/5.33                    ( ( ord_less_real @ A @ C2 )
% 9.10/5.33                    & ( ord_less_real @ C2 @ B2 )
% 9.10/5.33                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
% 9.10/5.33                      = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ ( F4 @ C2 ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9981_summable__Leibniz_I3_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ( topolo6980174941875973593q_real @ A )
% 9.10/5.33         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 9.10/5.33           => ! [N9: nat] :
% 9.10/5.33                ( member_real
% 9.10/5.33                @ ( suminf_real
% 9.10/5.33                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 9.10/5.33                @ ( set_or1222579329274155063t_real
% 9.10/5.33                  @ ( groups6591440286371151544t_real
% 9.10/5.33                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
% 9.10/5.33                  @ ( groups6591440286371151544t_real
% 9.10/5.33                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9982_summable__Leibniz_I2_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ( topolo6980174941875973593q_real @ A )
% 9.10/5.33         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 9.10/5.33           => ! [N9: nat] :
% 9.10/5.33                ( member_real
% 9.10/5.33                @ ( suminf_real
% 9.10/5.33                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 9.10/5.33                @ ( set_or1222579329274155063t_real
% 9.10/5.33                  @ ( groups6591440286371151544t_real
% 9.10/5.33                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 9.10/5.33                  @ ( groups6591440286371151544t_real
% 9.10/5.33                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9983_mult__nat__left__at__top,axiom,
% 9.10/5.33      ! [C: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.10/5.33       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9984_mult__nat__right__at__top,axiom,
% 9.10/5.33      ! [C: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.10/5.33       => ( filterlim_nat_nat
% 9.10/5.33          @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 9.10/5.33          @ at_top_nat
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9985_monoseq__convergent,axiom,
% 9.10/5.33      ! [X9: nat > real,B5: real] :
% 9.10/5.33        ( ( topolo6980174941875973593q_real @ X9 )
% 9.10/5.33       => ( ! [I3: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X9 @ I3 ) ) @ B5 )
% 9.10/5.33         => ~ ! [L6: real] :
% 9.10/5.33                ~ ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9986_LIMSEQ__root,axiom,
% 9.10/5.33      ( filterlim_nat_real
% 9.10/5.33      @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.10/5.33      @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.10/5.33      @ at_top_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9987_nested__sequence__unique,axiom,
% 9.10/5.33      ! [F: nat > real,G: nat > real] :
% 9.10/5.33        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
% 9.10/5.33         => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.10/5.33           => ( ( filterlim_nat_real
% 9.10/5.33                @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 9.10/5.33                @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33                @ at_top_nat )
% 9.10/5.33             => ? [L4: real] :
% 9.10/5.33                  ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
% 9.10/5.33                  & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 9.10/5.33                  & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
% 9.10/5.33                  & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9988_LIMSEQ__inverse__zero,axiom,
% 9.10/5.33      ! [X9: nat > real] :
% 9.10/5.33        ( ! [R3: real] :
% 9.10/5.33          ? [N7: nat] :
% 9.10/5.33          ! [N3: nat] :
% 9.10/5.33            ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.10/5.33           => ( ord_less_real @ R3 @ ( X9 @ N3 ) ) )
% 9.10/5.33       => ( filterlim_nat_real
% 9.10/5.33          @ ^ [N2: nat] : ( inverse_inverse_real @ ( X9 @ N2 ) )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9989_LIMSEQ__root__const,axiom,
% 9.10/5.33      ! [C: real] :
% 9.10/5.33        ( ( ord_less_real @ zero_zero_real @ C )
% 9.10/5.33       => ( filterlim_nat_real
% 9.10/5.33          @ ^ [N2: nat] : ( root @ N2 @ C )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9990_increasing__LIMSEQ,axiom,
% 9.10/5.33      ! [F: nat > real,L: real] :
% 9.10/5.33        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 9.10/5.33         => ( ! [E2: real] :
% 9.10/5.33                ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.10/5.33               => ? [N9: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
% 9.10/5.33           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9991_LIMSEQ__realpow__zero,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.10/5.33         => ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9992_LIMSEQ__divide__realpow__zero,axiom,
% 9.10/5.33      ! [X3: real,A: real] :
% 9.10/5.33        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.33       => ( filterlim_nat_real
% 9.10/5.33          @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N2 ) )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9993_LIMSEQ__abs__realpow__zero,axiom,
% 9.10/5.33      ! [C: real] :
% 9.10/5.33        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.10/5.33       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9994_LIMSEQ__abs__realpow__zero2,axiom,
% 9.10/5.33      ! [C: real] :
% 9.10/5.33        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.10/5.33       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9995_LIMSEQ__inverse__realpow__zero,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_real @ one_one_real @ X3 )
% 9.10/5.33       => ( filterlim_nat_real
% 9.10/5.33          @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N2 ) )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9996_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 9.10/5.33      ! [R: real] :
% 9.10/5.33        ( filterlim_nat_real
% 9.10/5.33        @ ^ [N2: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
% 9.10/5.33        @ ( topolo2815343760600316023s_real @ R )
% 9.10/5.33        @ at_top_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9997_summable__Leibniz_I1_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ( topolo6980174941875973593q_real @ A )
% 9.10/5.33         => ( summable_real
% 9.10/5.33            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9998_summable,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.33         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.10/5.33           => ( summable_real
% 9.10/5.33              @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_9999_cos__diff__limit__1,axiom,
% 9.10/5.33      ! [Theta: nat > real,Theta2: real] :
% 9.10/5.33        ( ( filterlim_nat_real
% 9.10/5.33          @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.10/5.33          @ at_top_nat )
% 9.10/5.33       => ~ ! [K3: nat > int] :
% 9.10/5.33              ~ ( filterlim_nat_real
% 9.10/5.33                @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.10/5.33                @ ( topolo2815343760600316023s_real @ Theta2 )
% 9.10/5.33                @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10000_cos__limit__1,axiom,
% 9.10/5.33      ! [Theta: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real
% 9.10/5.33          @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.10/5.33          @ at_top_nat )
% 9.10/5.33       => ? [K3: nat > int] :
% 9.10/5.33            ( filterlim_nat_real
% 9.10/5.33            @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 9.10/5.33            @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33            @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10001_summable__Leibniz_I4_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ( topolo6980174941875973593q_real @ A )
% 9.10/5.33         => ( filterlim_nat_real
% 9.10/5.33            @ ^ [N2: nat] :
% 9.10/5.33                ( groups6591440286371151544t_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.10/5.33            @ ( topolo2815343760600316023s_real
% 9.10/5.33              @ ( suminf_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 9.10/5.33            @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10002_zeroseq__arctan__series,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.10/5.33       => ( filterlim_nat_real
% 9.10/5.33          @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 9.10/5.33          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10003_summable__Leibniz_H_I2_J,axiom,
% 9.10/5.33      ! [A: nat > real,N: nat] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.33         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.10/5.33           => ( ord_less_eq_real
% 9.10/5.33              @ ( groups6591440286371151544t_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.10/5.33              @ ( suminf_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10004_summable__Leibniz_H_I3_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.33         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.10/5.33           => ( filterlim_nat_real
% 9.10/5.33              @ ^ [N2: nat] :
% 9.10/5.33                  ( groups6591440286371151544t_real
% 9.10/5.33                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.10/5.33              @ ( topolo2815343760600316023s_real
% 9.10/5.33                @ ( suminf_real
% 9.10/5.33                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 9.10/5.33              @ at_top_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10005_sums__alternating__upper__lower,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.33         => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33           => ? [L4: real] :
% 9.10/5.33                ( ! [N9: nat] :
% 9.10/5.33                    ( ord_less_eq_real
% 9.10/5.33                    @ ( groups6591440286371151544t_real
% 9.10/5.33                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 9.10/5.33                    @ L4 )
% 9.10/5.33                & ( filterlim_nat_real
% 9.10/5.33                  @ ^ [N2: nat] :
% 9.10/5.33                      ( groups6591440286371151544t_real
% 9.10/5.33                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.10/5.33                  @ ( topolo2815343760600316023s_real @ L4 )
% 9.10/5.33                  @ at_top_nat )
% 9.10/5.33                & ! [N9: nat] :
% 9.10/5.33                    ( ord_less_eq_real @ L4
% 9.10/5.33                    @ ( groups6591440286371151544t_real
% 9.10/5.33                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
% 9.10/5.33                & ( filterlim_nat_real
% 9.10/5.33                  @ ^ [N2: nat] :
% 9.10/5.33                      ( groups6591440286371151544t_real
% 9.10/5.33                      @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 9.10/5.33                  @ ( topolo2815343760600316023s_real @ L4 )
% 9.10/5.33                  @ at_top_nat ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10006_summable__Leibniz_I5_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ( topolo6980174941875973593q_real @ A )
% 9.10/5.33         => ( filterlim_nat_real
% 9.10/5.33            @ ^ [N2: nat] :
% 9.10/5.33                ( groups6591440286371151544t_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 9.10/5.33            @ ( topolo2815343760600316023s_real
% 9.10/5.33              @ ( suminf_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 9.10/5.33            @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10007_summable__Leibniz_H_I5_J,axiom,
% 9.10/5.33      ! [A: nat > real] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.33         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.10/5.33           => ( filterlim_nat_real
% 9.10/5.33              @ ^ [N2: nat] :
% 9.10/5.33                  ( groups6591440286371151544t_real
% 9.10/5.33                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 9.10/5.33              @ ( topolo2815343760600316023s_real
% 9.10/5.33                @ ( suminf_real
% 9.10/5.33                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 9.10/5.33              @ at_top_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10008_summable__Leibniz_H_I4_J,axiom,
% 9.10/5.33      ! [A: nat > real,N: nat] :
% 9.10/5.33        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.10/5.33       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.10/5.33         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.10/5.33           => ( ord_less_eq_real
% 9.10/5.33              @ ( suminf_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 9.10/5.33              @ ( groups6591440286371151544t_real
% 9.10/5.33                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 9.10/5.33                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10009_eventually__sequentially__seg,axiom,
% 9.10/5.33      ! [P: nat > $o,K: nat] :
% 9.10/5.33        ( ( eventually_nat
% 9.10/5.33          @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
% 9.10/5.33          @ at_top_nat )
% 9.10/5.33        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10010_sequentially__offset,axiom,
% 9.10/5.33      ! [P: nat > $o,K: nat] :
% 9.10/5.33        ( ( eventually_nat @ P @ at_top_nat )
% 9.10/5.33       => ( eventually_nat
% 9.10/5.33          @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 9.10/5.33          @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10011_eventually__sequentially,axiom,
% 9.10/5.33      ! [P: nat > $o] :
% 9.10/5.33        ( ( eventually_nat @ P @ at_top_nat )
% 9.10/5.33        = ( ? [N6: nat] :
% 9.10/5.33            ! [N2: nat] :
% 9.10/5.33              ( ( ord_less_eq_nat @ N6 @ N2 )
% 9.10/5.33             => ( P @ N2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10012_eventually__sequentiallyI,axiom,
% 9.10/5.33      ! [C: nat,P: nat > $o] :
% 9.10/5.33        ( ! [X5: nat] :
% 9.10/5.33            ( ( ord_less_eq_nat @ C @ X5 )
% 9.10/5.33           => ( P @ X5 ) )
% 9.10/5.33       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10013_le__sequentially,axiom,
% 9.10/5.33      ! [F5: filter_nat] :
% 9.10/5.33        ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 9.10/5.33        = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10014_real__bounded__linear,axiom,
% 9.10/5.33      ( real_V5970128139526366754l_real
% 9.10/5.33      = ( ^ [F3: real > real] :
% 9.10/5.33          ? [C3: real] :
% 9.10/5.33            ( F3
% 9.10/5.33            = ( ^ [X: real] : ( times_times_real @ X @ C3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10015_card__greaterThanAtMost,axiom,
% 9.10/5.33      ! [L: nat,U: nat] :
% 9.10/5.33        ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 9.10/5.33        = ( minus_minus_nat @ U @ L ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10016_sqrt__at__top,axiom,
% 9.10/5.33      filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10017_eventually__at__left__real,axiom,
% 9.10/5.33      ! [B2: real,A: real] :
% 9.10/5.33        ( ( ord_less_real @ B2 @ A )
% 9.10/5.33       => ( eventually_real
% 9.10/5.33          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B2 @ A ) )
% 9.10/5.33          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10018_filterlim__tan__at__left,axiom,
% 9.10/5.33      filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10019_DERIV__neg__imp__decreasing__at__top,axiom,
% 9.10/5.33      ! [B2: real,F: real > real,Flim: real] :
% 9.10/5.33        ( ! [X5: real] :
% 9.10/5.33            ( ( ord_less_eq_real @ B2 @ X5 )
% 9.10/5.33           => ? [Y5: real] :
% 9.10/5.33                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 9.10/5.33       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 9.10/5.33         => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10020_tendsto__arctan__at__top,axiom,
% 9.10/5.33      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10021_at__top__le__at__infinity,axiom,
% 9.10/5.33      ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10022_greaterThanAtMost__upto,axiom,
% 9.10/5.33      ( set_or6656581121297822940st_int
% 9.10/5.33      = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10023_filterlim__pow__at__bot__even,axiom,
% 9.10/5.33      ! [N: nat,F: real > real,F5: filter_real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 9.10/5.33         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33           => ( filterlim_real_real
% 9.10/5.33              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 9.10/5.33              @ at_top_real
% 9.10/5.33              @ F5 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10024_at__bot__le__at__infinity,axiom,
% 9.10/5.33      ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10025_DERIV__pos__imp__increasing__at__bot,axiom,
% 9.10/5.33      ! [B2: real,F: real > real,Flim: real] :
% 9.10/5.33        ( ! [X5: real] :
% 9.10/5.33            ( ( ord_less_eq_real @ X5 @ B2 )
% 9.10/5.33           => ? [Y5: real] :
% 9.10/5.33                ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 9.10/5.33       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 9.10/5.33         => ( ord_less_real @ Flim @ ( F @ B2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10026_filterlim__pow__at__bot__odd,axiom,
% 9.10/5.33      ! [N: nat,F: real > real,F5: filter_real] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 9.10/5.33         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.10/5.33           => ( filterlim_real_real
% 9.10/5.33              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 9.10/5.33              @ at_bot_real
% 9.10/5.33              @ F5 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10027_tendsto__arctan__at__bot,axiom,
% 9.10/5.33      filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10028_Bseq__eq__bounded,axiom,
% 9.10/5.33      ! [F: nat > real,A: real,B2: real] :
% 9.10/5.33        ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 9.10/5.33       => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10029_Bseq__realpow,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.10/5.33         => ( bfun_nat_real @ ( power_power_real @ X3 ) @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10030_tendsto__exp__limit__at__right,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( filterlim_real_real
% 9.10/5.33        @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X3 @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 9.10/5.33        @ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
% 9.10/5.33        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10031_filterlim__tan__at__right,axiom,
% 9.10/5.33      filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10032_eventually__at__right__real,axiom,
% 9.10/5.33      ! [A: real,B2: real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( eventually_real
% 9.10/5.33          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B2 ) )
% 9.10/5.33          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10033_decseq__bounded,axiom,
% 9.10/5.33      ! [X9: nat > real,B5: real] :
% 9.10/5.33        ( ( order_9091379641038594480t_real @ X9 )
% 9.10/5.33       => ( ! [I3: nat] : ( ord_less_eq_real @ B5 @ ( X9 @ I3 ) )
% 9.10/5.33         => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10034_decseq__convergent,axiom,
% 9.10/5.33      ! [X9: nat > real,B5: real] :
% 9.10/5.33        ( ( order_9091379641038594480t_real @ X9 )
% 9.10/5.33       => ( ! [I3: nat] : ( ord_less_eq_real @ B5 @ ( X9 @ I3 ) )
% 9.10/5.33         => ~ ! [L6: real] :
% 9.10/5.33                ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 9.10/5.33               => ~ ! [I2: nat] : ( ord_less_eq_real @ L6 @ ( X9 @ I2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10035_GMVT,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,G: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33                & ( ord_less_eq_real @ X5 @ B2 ) )
% 9.10/5.33             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 9.10/5.33         => ( ! [X5: real] :
% 9.10/5.33                ( ( ( ord_less_real @ A @ X5 )
% 9.10/5.33                  & ( ord_less_real @ X5 @ B2 ) )
% 9.10/5.33               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.10/5.33           => ( ! [X5: real] :
% 9.10/5.33                  ( ( ( ord_less_eq_real @ A @ X5 )
% 9.10/5.33                    & ( ord_less_eq_real @ X5 @ B2 ) )
% 9.10/5.33                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
% 9.10/5.33             => ( ! [X5: real] :
% 9.10/5.33                    ( ( ( ord_less_real @ A @ X5 )
% 9.10/5.33                      & ( ord_less_real @ X5 @ B2 ) )
% 9.10/5.33                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.10/5.33               => ? [G_c: real,F_c: real,C2: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 9.10/5.33                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_real @ A @ C2 )
% 9.10/5.33                    & ( ord_less_real @ C2 @ B2 )
% 9.10/5.33                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) ) @ G_c )
% 9.10/5.33                      = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10036_MVT,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33         => ( ! [X5: real] :
% 9.10/5.33                ( ( ord_less_real @ A @ X5 )
% 9.10/5.33               => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33           => ? [L4: real,Z4: real] :
% 9.10/5.33                ( ( ord_less_real @ A @ Z4 )
% 9.10/5.33                & ( ord_less_real @ Z4 @ B2 )
% 9.10/5.33                & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 9.10/5.33                & ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 9.10/5.33                  = ( times_times_real @ ( minus_minus_real @ B2 @ A ) @ L4 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10037_continuous__on__arcosh_H,axiom,
% 9.10/5.33      ! [A4: set_real,F: real > real] :
% 9.10/5.33        ( ( topolo5044208981011980120l_real @ A4 @ F )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( member_real @ X5 @ A4 )
% 9.10/5.33             => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 9.10/5.33         => ( topolo5044208981011980120l_real @ A4
% 9.10/5.33            @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10038_continuous__image__closed__interval,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_eq_real @ A @ B2 )
% 9.10/5.33       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33         => ? [C2: real,D3: real] :
% 9.10/5.33              ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B2 ) )
% 9.10/5.33                = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
% 9.10/5.33              & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10039_continuous__on__arcosh,axiom,
% 9.10/5.33      ! [A4: set_real] :
% 9.10/5.33        ( ( ord_less_eq_set_real @ A4 @ ( set_ord_atLeast_real @ one_one_real ) )
% 9.10/5.33       => ( topolo5044208981011980120l_real @ A4 @ arcosh_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10040_Rolle__deriv,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( ( F @ A )
% 9.10/5.33            = ( F @ B2 ) )
% 9.10/5.33         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33           => ( ! [X5: real] :
% 9.10/5.33                  ( ( ord_less_real @ A @ X5 )
% 9.10/5.33                 => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33                   => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33             => ? [Z4: real] :
% 9.10/5.33                  ( ( ord_less_real @ A @ Z4 )
% 9.10/5.33                  & ( ord_less_real @ Z4 @ B2 )
% 9.10/5.33                  & ( ( F4 @ Z4 )
% 9.10/5.33                    = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10041_mvt,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,F4: real > real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33         => ( ! [X5: real] :
% 9.10/5.33                ( ( ord_less_real @ A @ X5 )
% 9.10/5.33               => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33                 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33           => ~ ! [Xi: real] :
% 9.10/5.33                  ( ( ord_less_real @ A @ Xi )
% 9.10/5.33                 => ( ( ord_less_real @ Xi @ B2 )
% 9.10/5.33                   => ( ( minus_minus_real @ ( F @ B2 ) @ ( F @ A ) )
% 9.10/5.33                     != ( F4 @ Xi @ ( minus_minus_real @ B2 @ A ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10042_DERIV__pos__imp__increasing__open,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33               => ? [Y5: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 9.10/5.33         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33           => ( ord_less_real @ ( F @ A ) @ ( F @ B2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10043_DERIV__neg__imp__decreasing__open,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ! [X5: real] :
% 9.10/5.33              ( ( ord_less_real @ A @ X5 )
% 9.10/5.33             => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33               => ? [Y5: real] :
% 9.10/5.33                    ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33                    & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 9.10/5.33         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33           => ( ord_less_real @ ( F @ B2 ) @ ( F @ A ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10044_DERIV__isconst__end,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33         => ( ! [X5: real] :
% 9.10/5.33                ( ( ord_less_real @ A @ X5 )
% 9.10/5.33               => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33           => ( ( F @ B2 )
% 9.10/5.33              = ( F @ A ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10045_continuous__on__artanh,axiom,
% 9.10/5.33      ! [A4: set_real] :
% 9.10/5.33        ( ( ord_less_eq_set_real @ A4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 9.10/5.33       => ( topolo5044208981011980120l_real @ A4 @ artanh_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10046_DERIV__isconst2,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real,X3: real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33         => ( ! [X5: real] :
% 9.10/5.33                ( ( ord_less_real @ A @ X5 )
% 9.10/5.33               => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33           => ( ( ord_less_eq_real @ A @ X3 )
% 9.10/5.33             => ( ( ord_less_eq_real @ X3 @ B2 )
% 9.10/5.33               => ( ( F @ X3 )
% 9.10/5.33                  = ( F @ A ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10047_Rolle,axiom,
% 9.10/5.33      ! [A: real,B2: real,F: real > real] :
% 9.10/5.33        ( ( ord_less_real @ A @ B2 )
% 9.10/5.33       => ( ( ( F @ A )
% 9.10/5.33            = ( F @ B2 ) )
% 9.10/5.33         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B2 ) @ F )
% 9.10/5.33           => ( ! [X5: real] :
% 9.10/5.33                  ( ( ord_less_real @ A @ X5 )
% 9.10/5.33                 => ( ( ord_less_real @ X5 @ B2 )
% 9.10/5.33                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.10/5.33             => ? [Z4: real] :
% 9.10/5.33                  ( ( ord_less_real @ A @ Z4 )
% 9.10/5.33                  & ( ord_less_real @ Z4 @ B2 )
% 9.10/5.33                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10048_mono__Suc,axiom,
% 9.10/5.33      order_mono_nat_nat @ suc ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10049_mono__times__nat,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10050_incseq__bounded,axiom,
% 9.10/5.33      ! [X9: nat > real,B5: real] :
% 9.10/5.33        ( ( order_mono_nat_real @ X9 )
% 9.10/5.33       => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B5 )
% 9.10/5.33         => ( bfun_nat_real @ X9 @ at_top_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10051_incseq__convergent,axiom,
% 9.10/5.33      ! [X9: nat > real,B5: real] :
% 9.10/5.33        ( ( order_mono_nat_real @ X9 )
% 9.10/5.33       => ( ! [I3: nat] : ( ord_less_eq_real @ ( X9 @ I3 ) @ B5 )
% 9.10/5.33         => ~ ! [L6: real] :
% 9.10/5.33                ( ( filterlim_nat_real @ X9 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 9.10/5.33               => ~ ! [I2: nat] : ( ord_less_eq_real @ ( X9 @ I2 ) @ L6 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10052_mono__ge2__power__minus__self,axiom,
% 9.10/5.33      ! [K: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.10/5.33       => ( order_mono_nat_nat
% 9.10/5.33          @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10053_nonneg__incseq__Bseq__subseq__iff,axiom,
% 9.10/5.33      ! [F: nat > real,G: nat > nat] :
% 9.10/5.33        ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.33       => ( ( order_mono_nat_real @ F )
% 9.10/5.33         => ( ( order_5726023648592871131at_nat @ G )
% 9.10/5.33           => ( ( bfun_nat_real
% 9.10/5.33                @ ^ [X: nat] : ( F @ ( G @ X ) )
% 9.10/5.33                @ at_top_nat )
% 9.10/5.33              = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10054_strict__mono__imp__increasing,axiom,
% 9.10/5.33      ! [F: nat > nat,N: nat] :
% 9.10/5.33        ( ( order_5726023648592871131at_nat @ F )
% 9.10/5.33       => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10055_inj__sgn__power,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( inj_on_real_real
% 9.10/5.33          @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 9.10/5.33          @ top_top_set_real ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10056_list__encode_Oelims,axiom,
% 9.10/5.33      ! [X3: list_nat,Y3: nat] :
% 9.10/5.33        ( ( ( nat_list_encode @ X3 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( ( X3 = nil_nat )
% 9.10/5.33           => ( Y3 != zero_zero_nat ) )
% 9.10/5.33         => ~ ! [X5: nat,Xs3: list_nat] :
% 9.10/5.33                ( ( X3
% 9.10/5.33                  = ( cons_nat @ X5 @ Xs3 ) )
% 9.10/5.33               => ( Y3
% 9.10/5.33                 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10057_log__inj,axiom,
% 9.10/5.33      ! [B2: real] :
% 9.10/5.33        ( ( ord_less_real @ one_one_real @ B2 )
% 9.10/5.33       => ( inj_on_real_real @ ( log @ B2 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10058_list__encode_Osimps_I2_J,axiom,
% 9.10/5.33      ! [X3: nat,Xs2: list_nat] :
% 9.10/5.33        ( ( nat_list_encode @ ( cons_nat @ X3 @ Xs2 ) )
% 9.10/5.33        = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10059_pos__deriv__imp__strict__mono,axiom,
% 9.10/5.33      ! [F: real > real,F4: real > real] :
% 9.10/5.33        ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.10/5.33       => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X5 ) )
% 9.10/5.33         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10060_list__encode_Opelims,axiom,
% 9.10/5.33      ! [X3: list_nat,Y3: nat] :
% 9.10/5.33        ( ( ( nat_list_encode @ X3 )
% 9.10/5.33          = Y3 )
% 9.10/5.33       => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
% 9.10/5.33         => ( ( ( X3 = nil_nat )
% 9.10/5.33             => ( ( Y3 = zero_zero_nat )
% 9.10/5.33               => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 9.10/5.33           => ~ ! [X5: nat,Xs3: list_nat] :
% 9.10/5.33                  ( ( X3
% 9.10/5.33                    = ( cons_nat @ X5 @ Xs3 ) )
% 9.10/5.33                 => ( ( Y3
% 9.10/5.33                      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 9.10/5.33                   => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10061_inj__on__diff__nat,axiom,
% 9.10/5.33      ! [N4: set_nat,K: nat] :
% 9.10/5.33        ( ! [N3: nat] :
% 9.10/5.33            ( ( member_nat @ N3 @ N4 )
% 9.10/5.33           => ( ord_less_eq_nat @ K @ N3 ) )
% 9.10/5.33       => ( inj_on_nat_nat
% 9.10/5.33          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 9.10/5.33          @ N4 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10062_inj__Suc,axiom,
% 9.10/5.33      ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10063_summable__reindex,axiom,
% 9.10/5.33      ! [F: nat > real,G: nat > nat] :
% 9.10/5.33        ( ( summable_real @ F )
% 9.10/5.33       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 9.10/5.33         => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.33           => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10064_suminf__reindex__mono,axiom,
% 9.10/5.33      ! [F: nat > real,G: nat > nat] :
% 9.10/5.33        ( ( summable_real @ F )
% 9.10/5.33       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 9.10/5.33         => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.33           => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10065_inj__on__char__of__nat,axiom,
% 9.10/5.33      inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10066_suminf__reindex,axiom,
% 9.10/5.33      ! [F: nat > real,G: nat > nat] :
% 9.10/5.33        ( ( summable_real @ F )
% 9.10/5.33       => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 9.10/5.33         => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.10/5.33           => ( ! [X5: nat] :
% 9.10/5.33                  ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 9.10/5.33                 => ( ( F @ X5 )
% 9.10/5.33                    = zero_zero_real ) )
% 9.10/5.33             => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 9.10/5.33                = ( suminf_real @ F ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10067_powr__real__of__int_H,axiom,
% 9.10/5.33      ! [X3: real,N: int] :
% 9.10/5.33        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.10/5.33       => ( ( ( X3 != zero_zero_real )
% 9.10/5.33            | ( ord_less_int @ zero_zero_int @ N ) )
% 9.10/5.33         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N ) )
% 9.10/5.33            = ( power_int_real @ X3 @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10068_uniformity__real__def,axiom,
% 9.10/5.33      ( topolo1511823702728130853y_real
% 9.10/5.33      = ( comple2936214249959783750l_real
% 9.10/5.33        @ ( image_2178119161166701260l_real
% 9.10/5.33          @ ^ [E3: real] :
% 9.10/5.33              ( princi6114159922880469582l_real
% 9.10/5.33              @ ( collec3799799289383736868l_real
% 9.10/5.33                @ ( produc5414030515140494994real_o
% 9.10/5.33                  @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 9.10/5.33          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10069_uniformity__complex__def,axiom,
% 9.10/5.33      ( topolo896644834953643431omplex
% 9.10/5.33      = ( comple8358262395181532106omplex
% 9.10/5.33        @ ( image_5971271580939081552omplex
% 9.10/5.33          @ ^ [E3: real] :
% 9.10/5.33              ( princi3496590319149328850omplex
% 9.10/5.33              @ ( collec8663557070575231912omplex
% 9.10/5.33                @ ( produc6771430404735790350plex_o
% 9.10/5.33                  @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 9.10/5.33          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10070_eventually__prod__sequentially,axiom,
% 9.10/5.33      ! [P: product_prod_nat_nat > $o] :
% 9.10/5.33        ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 9.10/5.33        = ( ? [N6: nat] :
% 9.10/5.33            ! [M3: nat] :
% 9.10/5.33              ( ( ord_less_eq_nat @ N6 @ M3 )
% 9.10/5.33             => ! [N2: nat] :
% 9.10/5.33                  ( ( ord_less_eq_nat @ N6 @ N2 )
% 9.10/5.33                 => ( P @ ( product_Pair_nat_nat @ N2 @ M3 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10071_min__Suc__Suc,axiom,
% 9.10/5.33      ! [M: nat,N: nat] :
% 9.10/5.33        ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.10/5.33        = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10072_min__0L,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_min_nat @ zero_zero_nat @ N )
% 9.10/5.33        = zero_zero_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10073_min__0R,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_min_nat @ N @ zero_zero_nat )
% 9.10/5.33        = zero_zero_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10074_min__numeral__Suc,axiom,
% 9.10/5.33      ! [K: num,N: nat] :
% 9.10/5.33        ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.10/5.33        = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10075_min__Suc__numeral,axiom,
% 9.10/5.33      ! [N: nat,K: num] :
% 9.10/5.33        ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.10/5.33        = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10076_inf__nat__def,axiom,
% 9.10/5.33      inf_inf_nat = ord_min_nat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10077_min__diff,axiom,
% 9.10/5.33      ! [M: nat,I: nat,N: nat] :
% 9.10/5.33        ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 9.10/5.33        = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10078_nat__mult__min__right,axiom,
% 9.10/5.33      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.33        ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 9.10/5.33        = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10079_nat__mult__min__left,axiom,
% 9.10/5.33      ! [M: nat,N: nat,Q2: nat] :
% 9.10/5.33        ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 9.10/5.33        = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10080_concat__bit__assoc__sym,axiom,
% 9.10/5.33      ! [M: nat,N: nat,K: int,L: int,R: int] :
% 9.10/5.33        ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R )
% 9.10/5.33        = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10081_take__bit__concat__bit__eq,axiom,
% 9.10/5.33      ! [M: nat,N: nat,K: int,L: int] :
% 9.10/5.33        ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
% 9.10/5.33        = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10082_min__Suc1,axiom,
% 9.10/5.33      ! [N: nat,M: nat] :
% 9.10/5.33        ( ( ord_min_nat @ ( suc @ N ) @ M )
% 9.10/5.33        = ( case_nat_nat @ zero_zero_nat
% 9.10/5.33          @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
% 9.10/5.33          @ M ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10083_min__Suc2,axiom,
% 9.10/5.33      ! [M: nat,N: nat] :
% 9.10/5.33        ( ( ord_min_nat @ M @ ( suc @ N ) )
% 9.10/5.33        = ( case_nat_nat @ zero_zero_nat
% 9.10/5.33          @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
% 9.10/5.33          @ M ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10084_min__enat__simps_I3_J,axiom,
% 9.10/5.33      ! [Q2: extended_enat] :
% 9.10/5.33        ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 9.10/5.33        = zero_z5237406670263579293d_enat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10085_min__enat__simps_I2_J,axiom,
% 9.10/5.33      ! [Q2: extended_enat] :
% 9.10/5.33        ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 9.10/5.33        = zero_z5237406670263579293d_enat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10086_inf__enat__def,axiom,
% 9.10/5.33      inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10087_hd__upt,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ord_less_nat @ I @ J )
% 9.10/5.33       => ( ( hd_nat @ ( upt @ I @ J ) )
% 9.10/5.33          = I ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10088_drop__upt,axiom,
% 9.10/5.33      ! [M: nat,I: nat,J: nat] :
% 9.10/5.33        ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 9.10/5.33        = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10089_length__upt,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 9.10/5.33        = ( minus_minus_nat @ J @ I ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10090_nth__upt,axiom,
% 9.10/5.33      ! [I: nat,K: nat,J: nat] :
% 9.10/5.33        ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 9.10/5.33       => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 9.10/5.33          = ( plus_plus_nat @ I @ K ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10091_take__upt,axiom,
% 9.10/5.33      ! [I: nat,M: nat,N: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 9.10/5.33       => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 9.10/5.33          = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10092_upt__conv__Nil,axiom,
% 9.10/5.33      ! [J: nat,I: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ J @ I )
% 9.10/5.33       => ( ( upt @ I @ J )
% 9.10/5.33          = nil_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10093_upt__eq__Nil__conv,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ( upt @ I @ J )
% 9.10/5.33          = nil_nat )
% 9.10/5.33        = ( ( J = zero_zero_nat )
% 9.10/5.33          | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10094_upt__rec__numeral,axiom,
% 9.10/5.33      ! [M: num,N: num] :
% 9.10/5.33        ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.33         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.33            = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.33         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.10/5.33            = nil_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10095_map__add__upt,axiom,
% 9.10/5.33      ! [N: nat,M: nat] :
% 9.10/5.33        ( ( map_nat_nat
% 9.10/5.33          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 9.10/5.33          @ ( upt @ zero_zero_nat @ M ) )
% 9.10/5.33        = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10096_upt__add__eq__append,axiom,
% 9.10/5.33      ! [I: nat,J: nat,K: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.33       => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 9.10/5.33          = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10097_greaterThanAtMost__upt,axiom,
% 9.10/5.33      ( set_or6659071591806873216st_nat
% 9.10/5.33      = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10098_greaterThanLessThan__upt,axiom,
% 9.10/5.33      ( set_or5834768355832116004an_nat
% 9.10/5.33      = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10099_atLeastAtMost__upt,axiom,
% 9.10/5.33      ( set_or1269000886237332187st_nat
% 9.10/5.33      = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M3 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10100_atLeastLessThan__upt,axiom,
% 9.10/5.33      ( set_or4665077453230672383an_nat
% 9.10/5.33      = ( ^ [I4: nat,J3: nat] : ( set_nat2 @ ( upt @ I4 @ J3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10101_atLeast__upt,axiom,
% 9.10/5.33      ( set_ord_lessThan_nat
% 9.10/5.33      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10102_atMost__upto,axiom,
% 9.10/5.33      ( set_ord_atMost_nat
% 9.10/5.33      = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10103_upt__conv__Cons,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ord_less_nat @ I @ J )
% 9.10/5.33       => ( ( upt @ I @ J )
% 9.10/5.33          = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10104_map__decr__upt,axiom,
% 9.10/5.33      ! [M: nat,N: nat] :
% 9.10/5.33        ( ( map_nat_nat
% 9.10/5.33          @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.10/5.33          @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 9.10/5.33        = ( upt @ M @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10105_upt__eq__Cons__conv,axiom,
% 9.10/5.33      ! [I: nat,J: nat,X3: nat,Xs2: list_nat] :
% 9.10/5.33        ( ( ( upt @ I @ J )
% 9.10/5.33          = ( cons_nat @ X3 @ Xs2 ) )
% 9.10/5.33        = ( ( ord_less_nat @ I @ J )
% 9.10/5.33          & ( I = X3 )
% 9.10/5.33          & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 9.10/5.33            = Xs2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10106_upt__rec,axiom,
% 9.10/5.33      ( upt
% 9.10/5.33      = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10107_upt__Suc__append,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.33       => ( ( upt @ I @ ( suc @ J ) )
% 9.10/5.33          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10108_upt__Suc,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.33         => ( ( upt @ I @ ( suc @ J ) )
% 9.10/5.33            = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 9.10/5.33        & ( ~ ( ord_less_eq_nat @ I @ J )
% 9.10/5.33         => ( ( upt @ I @ ( suc @ J ) )
% 9.10/5.33            = nil_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10109_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 9.10/5.33      ! [N: nat,J: nat,I: nat] :
% 9.10/5.33        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 9.10/5.33       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 9.10/5.33          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10110_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 9.10/5.33      ! [N: nat,J: nat,I: nat] :
% 9.10/5.33        ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 9.10/5.33       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 9.10/5.33          = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10111_sorted__list__of__set__greaterThanAtMost,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 9.10/5.33       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 9.10/5.33          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10112_sorted__list__of__set__greaterThanLessThan,axiom,
% 9.10/5.33      ! [I: nat,J: nat] :
% 9.10/5.33        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 9.10/5.33       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 9.10/5.33          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10113_sum__list__upt,axiom,
% 9.10/5.33      ! [M: nat,N: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ M @ N )
% 9.10/5.33       => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 9.10/5.33          = ( groups3542108847815614940at_nat
% 9.10/5.33            @ ^ [X: nat] : X
% 9.10/5.33            @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10114_card__length__sum__list__rec,axiom,
% 9.10/5.33      ! [M: nat,N4: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.10/5.33       => ( ( finite_card_list_nat
% 9.10/5.33            @ ( collect_list_nat
% 9.10/5.33              @ ^ [L2: list_nat] :
% 9.10/5.33                  ( ( ( size_size_list_nat @ L2 )
% 9.10/5.33                    = M )
% 9.10/5.33                  & ( ( groups4561878855575611511st_nat @ L2 )
% 9.10/5.33                    = N4 ) ) ) )
% 9.10/5.33          = ( plus_plus_nat
% 9.10/5.33            @ ( finite_card_list_nat
% 9.10/5.33              @ ( collect_list_nat
% 9.10/5.33                @ ^ [L2: list_nat] :
% 9.10/5.33                    ( ( ( size_size_list_nat @ L2 )
% 9.10/5.33                      = ( minus_minus_nat @ M @ one_one_nat ) )
% 9.10/5.33                    & ( ( groups4561878855575611511st_nat @ L2 )
% 9.10/5.33                      = N4 ) ) ) )
% 9.10/5.33            @ ( finite_card_list_nat
% 9.10/5.33              @ ( collect_list_nat
% 9.10/5.33                @ ^ [L2: list_nat] :
% 9.10/5.33                    ( ( ( size_size_list_nat @ L2 )
% 9.10/5.33                      = M )
% 9.10/5.33                    & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 9.10/5.33                      = N4 ) ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10115_card__length__sum__list,axiom,
% 9.10/5.33      ! [M: nat,N4: nat] :
% 9.10/5.33        ( ( finite_card_list_nat
% 9.10/5.33          @ ( collect_list_nat
% 9.10/5.33            @ ^ [L2: list_nat] :
% 9.10/5.33                ( ( ( size_size_list_nat @ L2 )
% 9.10/5.33                  = M )
% 9.10/5.33                & ( ( groups4561878855575611511st_nat @ L2 )
% 9.10/5.33                  = N4 ) ) ) )
% 9.10/5.33        = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ one_one_nat ) @ N4 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10116_sorted__upt,axiom,
% 9.10/5.33      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10117_sorted__wrt__upt,axiom,
% 9.10/5.33      ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10118_sorted__wrt__less__idx,axiom,
% 9.10/5.33      ! [Ns: list_nat,I: nat] :
% 9.10/5.33        ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 9.10/5.33       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 9.10/5.33         => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10119_sorted__wrt__upto,axiom,
% 9.10/5.33      ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10120_sorted__upto,axiom,
% 9.10/5.33      ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10121_VEBT_Osize_I3_J,axiom,
% 9.10/5.33      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.10/5.33        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.10/5.33        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10122_VEBT_Osize__gen_I1_J,axiom,
% 9.10/5.33      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.10/5.33        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.10/5.33        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10123_card__le__Suc__Max,axiom,
% 9.10/5.33      ! [S2: set_nat] :
% 9.10/5.33        ( ( finite_finite_nat @ S2 )
% 9.10/5.33       => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10124_divide__nat__def,axiom,
% 9.10/5.33      ( divide_divide_nat
% 9.10/5.33      = ( ^ [M3: nat,N2: nat] :
% 9.10/5.33            ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 9.10/5.33            @ ( lattic8265883725875713057ax_nat
% 9.10/5.33              @ ( collect_nat
% 9.10/5.33                @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M3 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10125_gcd__is__Max__divisors__nat,axiom,
% 9.10/5.33      ! [N: nat,M: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( ( gcd_gcd_nat @ M @ N )
% 9.10/5.33          = ( lattic8265883725875713057ax_nat
% 9.10/5.33            @ ( collect_nat
% 9.10/5.33              @ ^ [D2: nat] :
% 9.10/5.33                  ( ( dvd_dvd_nat @ D2 @ M )
% 9.10/5.33                  & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10126_atLeastLessThan__add__Un,axiom,
% 9.10/5.33      ! [I: nat,J: nat,K: nat] :
% 9.10/5.33        ( ( ord_less_eq_nat @ I @ J )
% 9.10/5.33       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 9.10/5.33          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10127_sup__nat__def,axiom,
% 9.10/5.33      sup_sup_nat = ord_max_nat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10128_sup__enat__def,axiom,
% 9.10/5.33      sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10129_quotient__of__def,axiom,
% 9.10/5.33      ( quotient_of
% 9.10/5.33      = ( ^ [X: rat] :
% 9.10/5.33            ( the_Pr4378521158711661632nt_int
% 9.10/5.33            @ ^ [Pair: product_prod_int_int] :
% 9.10/5.33                ( ( X
% 9.10/5.33                  = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 9.10/5.33                & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 9.10/5.33                & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10130_coprime__abs__left__iff,axiom,
% 9.10/5.33      ! [K: int,L: int] :
% 9.10/5.33        ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L )
% 9.10/5.33        = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10131_coprime__abs__right__iff,axiom,
% 9.10/5.33      ! [K: int,L: int] :
% 9.10/5.33        ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L ) )
% 9.10/5.33        = ( algebr932160517623751201me_int @ K @ L ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10132_normalize__stable,axiom,
% 9.10/5.33      ! [Q2: int,P4: int] :
% 9.10/5.33        ( ( ord_less_int @ zero_zero_int @ Q2 )
% 9.10/5.33       => ( ( algebr932160517623751201me_int @ P4 @ Q2 )
% 9.10/5.33         => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.33            = ( product_Pair_int_int @ P4 @ Q2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10133_Rat__cases,axiom,
% 9.10/5.33      ! [Q2: rat] :
% 9.10/5.33        ~ ! [A3: int,B3: int] :
% 9.10/5.33            ( ( Q2
% 9.10/5.33              = ( fract @ A3 @ B3 ) )
% 9.10/5.33           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.10/5.33             => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10134_Rat__induct,axiom,
% 9.10/5.33      ! [P: rat > $o,Q2: rat] :
% 9.10/5.33        ( ! [A3: int,B3: int] :
% 9.10/5.33            ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.10/5.33           => ( ( algebr932160517623751201me_int @ A3 @ B3 )
% 9.10/5.33             => ( P @ ( fract @ A3 @ B3 ) ) ) )
% 9.10/5.33       => ( P @ Q2 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10135_quotient__of__coprime,axiom,
% 9.10/5.33      ! [R: rat,P4: int,Q2: int] :
% 9.10/5.33        ( ( ( quotient_of @ R )
% 9.10/5.33          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.33       => ( algebr932160517623751201me_int @ P4 @ Q2 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10136_normalize__coprime,axiom,
% 9.10/5.33      ! [R: product_prod_int_int,P4: int,Q2: int] :
% 9.10/5.33        ( ( ( normalize @ R )
% 9.10/5.33          = ( product_Pair_int_int @ P4 @ Q2 ) )
% 9.10/5.33       => ( algebr932160517623751201me_int @ P4 @ Q2 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10137_coprime__crossproduct__int,axiom,
% 9.10/5.33      ! [A: int,D: int,B2: int,C: int] :
% 9.10/5.33        ( ( algebr932160517623751201me_int @ A @ D )
% 9.10/5.33       => ( ( algebr932160517623751201me_int @ B2 @ C )
% 9.10/5.33         => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 9.10/5.33              = ( times_times_int @ ( abs_abs_int @ B2 ) @ ( abs_abs_int @ D ) ) )
% 9.10/5.33            = ( ( ( abs_abs_int @ A )
% 9.10/5.33                = ( abs_abs_int @ B2 ) )
% 9.10/5.33              & ( ( abs_abs_int @ C )
% 9.10/5.33                = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10138_coprime__common__divisor__int,axiom,
% 9.10/5.33      ! [A: int,B2: int,X3: int] :
% 9.10/5.33        ( ( algebr932160517623751201me_int @ A @ B2 )
% 9.10/5.33       => ( ( dvd_dvd_int @ X3 @ A )
% 9.10/5.33         => ( ( dvd_dvd_int @ X3 @ B2 )
% 9.10/5.33           => ( ( abs_abs_int @ X3 )
% 9.10/5.33              = one_one_int ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10139_Rat__cases__nonzero,axiom,
% 9.10/5.33      ! [Q2: rat] :
% 9.10/5.33        ( ! [A3: int,B3: int] :
% 9.10/5.33            ( ( Q2
% 9.10/5.33              = ( fract @ A3 @ B3 ) )
% 9.10/5.33           => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.10/5.33             => ( ( A3 != zero_zero_int )
% 9.10/5.33               => ~ ( algebr932160517623751201me_int @ A3 @ B3 ) ) ) )
% 9.10/5.33       => ( Q2 = zero_zero_rat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10140_quotient__of__unique,axiom,
% 9.10/5.33      ! [R: rat] :
% 9.10/5.33      ? [X5: product_prod_int_int] :
% 9.10/5.33        ( ( R
% 9.10/5.33          = ( fract @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) )
% 9.10/5.33        & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X5 ) )
% 9.10/5.33        & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) )
% 9.10/5.33        & ! [Y5: product_prod_int_int] :
% 9.10/5.33            ( ( ( R
% 9.10/5.33                = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 9.10/5.33              & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 9.10/5.33              & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 9.10/5.33           => ( Y5 = X5 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10141_coprime__int__iff,axiom,
% 9.10/5.33      ! [M: nat,N: nat] :
% 9.10/5.33        ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.10/5.33        = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10142_coprime__nat__abs__right__iff,axiom,
% 9.10/5.33      ! [N: nat,K: int] :
% 9.10/5.33        ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 9.10/5.33        = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10143_coprime__nat__abs__left__iff,axiom,
% 9.10/5.33      ! [K: int,N: nat] :
% 9.10/5.33        ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 9.10/5.33        = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10144_coprime__Suc__0__left,axiom,
% 9.10/5.33      ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10145_coprime__Suc__0__right,axiom,
% 9.10/5.33      ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10146_coprime__crossproduct__nat,axiom,
% 9.10/5.33      ! [A: nat,D: nat,B2: nat,C: nat] :
% 9.10/5.33        ( ( algebr934650988132801477me_nat @ A @ D )
% 9.10/5.33       => ( ( algebr934650988132801477me_nat @ B2 @ C )
% 9.10/5.33         => ( ( ( times_times_nat @ A @ C )
% 9.10/5.33              = ( times_times_nat @ B2 @ D ) )
% 9.10/5.33            = ( ( A = B2 )
% 9.10/5.33              & ( C = D ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10147_coprime__common__divisor__nat,axiom,
% 9.10/5.33      ! [A: nat,B2: nat,X3: nat] :
% 9.10/5.33        ( ( algebr934650988132801477me_nat @ A @ B2 )
% 9.10/5.33       => ( ( dvd_dvd_nat @ X3 @ A )
% 9.10/5.33         => ( ( dvd_dvd_nat @ X3 @ B2 )
% 9.10/5.33           => ( X3 = one_one_nat ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10148_Gcd__int__set__eq__fold,axiom,
% 9.10/5.33      ! [Xs2: list_int] :
% 9.10/5.33        ( ( gcd_Gcd_int @ ( set_int2 @ Xs2 ) )
% 9.10/5.33        = ( fold_int_int @ gcd_gcd_int @ Xs2 @ zero_zero_int ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10149_Gcd__nat__set__eq__fold,axiom,
% 9.10/5.33      ! [Xs2: list_nat] :
% 9.10/5.33        ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
% 9.10/5.33        = ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10150_Gcd__int__greater__eq__0,axiom,
% 9.10/5.33      ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10151_coprime__diff__one__right__nat,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10152_coprime__diff__one__left__nat,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.10/5.33       => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10153_Rats__abs__nat__div__natE,axiom,
% 9.10/5.33      ! [X3: real] :
% 9.10/5.33        ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 9.10/5.33       => ~ ! [M4: nat,N3: nat] :
% 9.10/5.33              ( ( N3 != zero_zero_nat )
% 9.10/5.33             => ( ( ( abs_abs_real @ X3 )
% 9.10/5.33                  = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 9.10/5.33               => ~ ( algebr934650988132801477me_nat @ M4 @ N3 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10154_Rat_Opositive_Orsp,axiom,
% 9.10/5.33      ( bNF_re8699439704749558557nt_o_o @ ratrel
% 9.10/5.33      @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 )
% 9.10/5.33      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 9.10/5.33      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10155_less__eq__integer_Orsp,axiom,
% 9.10/5.33      ( bNF_re3403563459893282935_int_o
% 9.10/5.33      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re5089333283451836215nt_o_o
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ord_less_eq_int
% 9.10/5.33      @ ord_less_eq_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10156_less__eq__natural_Orsp,axiom,
% 9.10/5.33      ( bNF_re578469030762574527_nat_o
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re4705727531993890431at_o_o
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ord_less_eq_nat
% 9.10/5.33      @ ord_less_eq_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10157_plus__rat_Orsp,axiom,
% 9.10/5.33      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 9.10/5.33      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 9.10/5.33      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10158_times__rat_Orsp,axiom,
% 9.10/5.33      ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 9.10/5.33      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 9.10/5.33      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10159_Fract_Orsp,axiom,
% 9.10/5.33      ( bNF_re157797125943740599nt_int
% 9.10/5.33      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re6250860962936578807nt_int
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33        @ ratrel )
% 9.10/5.33      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
% 9.10/5.33      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10160_plus__natural_Orsp,axiom,
% 9.10/5.33      ( bNF_re1345281282404953727at_nat
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re5653821019739307937at_nat
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.33      @ plus_plus_nat
% 9.10/5.33      @ plus_plus_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10161_minus__natural_Orsp,axiom,
% 9.10/5.33      ( bNF_re1345281282404953727at_nat
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re5653821019739307937at_nat
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.33      @ minus_minus_nat
% 9.10/5.33      @ minus_minus_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10162_times__natural_Orsp,axiom,
% 9.10/5.33      ( bNF_re1345281282404953727at_nat
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re5653821019739307937at_nat
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.33      @ times_times_nat
% 9.10/5.33      @ times_times_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10163_times__integer_Orsp,axiom,
% 9.10/5.33      ( bNF_re711492959462206631nt_int
% 9.10/5.33      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re4712519889275205905nt_int
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.10/5.33      @ times_times_int
% 9.10/5.33      @ times_times_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10164_divide__natural_Orsp,axiom,
% 9.10/5.33      ( bNF_re1345281282404953727at_nat
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re5653821019739307937at_nat
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 ) )
% 9.10/5.33      @ divide_divide_nat
% 9.10/5.33      @ divide_divide_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10165_sub_Orsp,axiom,
% 9.10/5.33      ( bNF_re8402795839162346335um_int
% 9.10/5.33      @ ^ [Y6: num,Z5: num] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re1822329894187522285nt_int
% 9.10/5.33        @ ^ [Y6: num,Z5: num] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) )
% 9.10/5.33      @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10166_less__natural_Orsp,axiom,
% 9.10/5.33      ( bNF_re578469030762574527_nat_o
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re4705727531993890431at_o_o
% 9.10/5.33        @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ord_less_nat
% 9.10/5.33      @ ord_less_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10167_less__integer_Orsp,axiom,
% 9.10/5.33      ( bNF_re3403563459893282935_int_o
% 9.10/5.33      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re5089333283451836215nt_o_o
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ord_less_int
% 9.10/5.33      @ ord_less_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10168_uminus__rat_Orsp,axiom,
% 9.10/5.33      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 9.10/5.33      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) )
% 9.10/5.33      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10169_inverse__rat_Orsp,axiom,
% 9.10/5.33      ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
% 9.10/5.33      @ ^ [X: product_prod_int_int] :
% 9.10/5.33          ( if_Pro3027730157355071871nt_int
% 9.10/5.33          @ ( ( product_fst_int_int @ X )
% 9.10/5.33            = zero_zero_int )
% 9.10/5.33          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.10/5.33          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) )
% 9.10/5.33      @ ^ [X: product_prod_int_int] :
% 9.10/5.33          ( if_Pro3027730157355071871nt_int
% 9.10/5.33          @ ( ( product_fst_int_int @ X )
% 9.10/5.33            = zero_zero_int )
% 9.10/5.33          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.10/5.33          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10170_plus__rat_Otransfer,axiom,
% 9.10/5.33      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 9.10/5.33      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 9.10/5.33      @ plus_plus_rat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10171_one__rat_Otransfer,axiom,
% 9.10/5.33      pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10172_zero__rat_Otransfer,axiom,
% 9.10/5.33      pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10173_Fract_Otransfer,axiom,
% 9.10/5.33      ( bNF_re3461391660133120880nt_rat
% 9.10/5.33      @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33      @ ( bNF_re2214769303045360666nt_rat
% 9.10/5.33        @ ^ [Y6: int,Z5: int] : ( Y6 = Z5 )
% 9.10/5.33        @ pcr_rat )
% 9.10/5.33      @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( B4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A5 @ B4 ) )
% 9.10/5.33      @ fract ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10174_uminus__rat_Otransfer,axiom,
% 9.10/5.33      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 9.10/5.33      @ ^ [X: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X ) ) @ ( product_snd_int_int @ X ) )
% 9.10/5.33      @ uminus_uminus_rat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10175_times__rat_Otransfer,axiom,
% 9.10/5.33      ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 9.10/5.33      @ ^ [X: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X ) @ ( product_snd_int_int @ Y ) ) )
% 9.10/5.33      @ times_times_rat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10176_Rat_Opositive_Otransfer,axiom,
% 9.10/5.33      ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 9.10/5.33      @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 )
% 9.10/5.33      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 9.10/5.33      @ positive ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10177_inverse__rat_Otransfer,axiom,
% 9.10/5.33      ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
% 9.10/5.33      @ ^ [X: product_prod_int_int] :
% 9.10/5.33          ( if_Pro3027730157355071871nt_int
% 9.10/5.33          @ ( ( product_fst_int_int @ X )
% 9.10/5.33            = zero_zero_int )
% 9.10/5.33          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.10/5.33          @ ( product_Pair_int_int @ ( product_snd_int_int @ X ) @ ( product_fst_int_int @ X ) ) )
% 9.10/5.33      @ inverse_inverse_rat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10178_times__int_Otransfer,axiom,
% 9.10/5.33      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [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 ) ) ) ) )
% 9.10/5.33      @ times_times_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10179_zero__int_Otransfer,axiom,
% 9.10/5.33      pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10180_int__transfer,axiom,
% 9.10/5.33      ( bNF_re6830278522597306478at_int
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ pcr_int
% 9.10/5.33      @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
% 9.10/5.33      @ semiri1314217659103216013at_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10181_uminus__int_Otransfer,axiom,
% 9.10/5.33      ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
% 9.10/5.33      @ ( produc2626176000494625587at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
% 9.10/5.33      @ uminus_uminus_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10182_nat_Otransfer,axiom,
% 9.10/5.33      ( bNF_re4555766996558763186at_nat @ pcr_int
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 9.10/5.33      @ nat2 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10183_one__int_Otransfer,axiom,
% 9.10/5.33      pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10184_less__int_Otransfer,axiom,
% 9.10/5.33      ( bNF_re717283939379294677_int_o @ pcr_int
% 9.10/5.33      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.10/5.33      @ ord_less_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10185_less__eq__int_Otransfer,axiom,
% 9.10/5.33      ( bNF_re717283939379294677_int_o @ pcr_int
% 9.10/5.33      @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.10/5.33      @ ord_less_eq_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10186_plus__int_Otransfer,axiom,
% 9.10/5.33      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 9.10/5.33      @ plus_plus_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10187_minus__int_Otransfer,axiom,
% 9.10/5.33      ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
% 9.10/5.33      @ minus_minus_int ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10188_times__int_Orsp,axiom,
% 9.10/5.33      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [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 ) ) ) ) )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [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 ) ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10189_intrel__iff,axiom,
% 9.10/5.33      ! [X3: nat,Y3: nat,U: nat,V: nat] :
% 9.10/5.33        ( ( intrel @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 9.10/5.33        = ( ( plus_plus_nat @ X3 @ V )
% 9.10/5.33          = ( plus_plus_nat @ U @ Y3 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10190_nat_Orsp,axiom,
% 9.10/5.33      ( bNF_re8246922863344978751at_nat @ intrel
% 9.10/5.33      @ ^ [Y6: nat,Z5: nat] : ( Y6 = Z5 )
% 9.10/5.33      @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 9.10/5.33      @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10191_uminus__int_Orsp,axiom,
% 9.10/5.33      ( bNF_re2241393799969408733at_nat @ intrel @ intrel
% 9.10/5.33      @ ( produc2626176000494625587at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) )
% 9.10/5.33      @ ( produc2626176000494625587at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10192_one__int_Orsp,axiom,
% 9.10/5.33      intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10193_zero__int_Orsp,axiom,
% 9.10/5.33      intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10194_intrel__def,axiom,
% 9.10/5.33      ( intrel
% 9.10/5.33      = ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] :
% 9.10/5.33                ( ( plus_plus_nat @ X @ V4 )
% 9.10/5.33                = ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10195_less__int_Orsp,axiom,
% 9.10/5.33      ( bNF_re4202695980764964119_nat_o @ intrel
% 9.10/5.33      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.10/5.33      @ ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10196_less__eq__int_Orsp,axiom,
% 9.10/5.33      ( bNF_re4202695980764964119_nat_o @ intrel
% 9.10/5.33      @ ( bNF_re3666534408544137501at_o_o @ intrel
% 9.10/5.33        @ ^ [Y6: $o,Z5: $o] : ( Y6 = Z5 ) )
% 9.10/5.33      @ ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.10/5.33      @ ( produc8739625826339149834_nat_o
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc6081775807080527818_nat_o
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10197_minus__int_Orsp,axiom,
% 9.10/5.33      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10198_plus__int_Orsp,axiom,
% 9.10/5.33      ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 9.10/5.33      @ ( produc27273713700761075at_nat
% 9.10/5.33        @ ^ [X: nat,Y: nat] :
% 9.10/5.33            ( produc2626176000494625587at_nat
% 9.10/5.33            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10199_vanishes__mult__bounded,axiom,
% 9.10/5.33      ! [X9: nat > rat,Y7: nat > rat] :
% 9.10/5.33        ( ? [A8: rat] :
% 9.10/5.33            ( ( ord_less_rat @ zero_zero_rat @ A8 )
% 9.10/5.33            & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N3 ) ) @ A8 ) )
% 9.10/5.33       => ( ( vanishes @ Y7 )
% 9.10/5.33         => ( vanishes
% 9.10/5.33            @ ^ [N2: nat] : ( times_times_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10200_vanishes__const,axiom,
% 9.10/5.33      ! [C: rat] :
% 9.10/5.33        ( ( vanishes
% 9.10/5.33          @ ^ [N2: nat] : C )
% 9.10/5.33        = ( C = zero_zero_rat ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10201_vanishes__minus,axiom,
% 9.10/5.33      ! [X9: nat > rat] :
% 9.10/5.33        ( ( vanishes @ X9 )
% 9.10/5.33       => ( vanishes
% 9.10/5.33          @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X9 @ N2 ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10202_vanishes__add,axiom,
% 9.10/5.33      ! [X9: nat > rat,Y7: nat > rat] :
% 9.10/5.33        ( ( vanishes @ X9 )
% 9.10/5.33       => ( ( vanishes @ Y7 )
% 9.10/5.33         => ( vanishes
% 9.10/5.33            @ ^ [N2: nat] : ( plus_plus_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10203_vanishes__diff,axiom,
% 9.10/5.33      ! [X9: nat > rat,Y7: nat > rat] :
% 9.10/5.33        ( ( vanishes @ X9 )
% 9.10/5.33       => ( ( vanishes @ Y7 )
% 9.10/5.33         => ( vanishes
% 9.10/5.33            @ ^ [N2: nat] : ( minus_minus_rat @ ( X9 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10204_vanishes__def,axiom,
% 9.10/5.33      ( vanishes
% 9.10/5.33      = ( ^ [X7: nat > rat] :
% 9.10/5.33          ! [R5: rat] :
% 9.10/5.33            ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.10/5.33           => ? [K2: nat] :
% 9.10/5.33              ! [N2: nat] :
% 9.10/5.33                ( ( ord_less_eq_nat @ K2 @ N2 )
% 9.10/5.33               => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10205_vanishesI,axiom,
% 9.10/5.33      ! [X9: nat > rat] :
% 9.10/5.33        ( ! [R3: rat] :
% 9.10/5.33            ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 9.10/5.33           => ? [K4: nat] :
% 9.10/5.33              ! [N3: nat] :
% 9.10/5.33                ( ( ord_less_eq_nat @ K4 @ N3 )
% 9.10/5.33               => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N3 ) ) @ R3 ) ) )
% 9.10/5.33       => ( vanishes @ X9 ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10206_vanishesD,axiom,
% 9.10/5.33      ! [X9: nat > rat,R: rat] :
% 9.10/5.33        ( ( vanishes @ X9 )
% 9.10/5.33       => ( ( ord_less_rat @ zero_zero_rat @ R )
% 9.10/5.33         => ? [K3: nat] :
% 9.10/5.33            ! [N9: nat] :
% 9.10/5.33              ( ( ord_less_eq_nat @ K3 @ N9 )
% 9.10/5.33             => ( ord_less_rat @ ( abs_abs_rat @ ( X9 @ N9 ) ) @ R ) ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10207_Field__natLeq__on,axiom,
% 9.10/5.33      ! [N: nat] :
% 9.10/5.33        ( ( field_nat
% 9.10/5.33          @ ( collec3392354462482085612at_nat
% 9.10/5.33            @ ( produc6081775807080527818_nat_o
% 9.10/5.33              @ ^ [X: nat,Y: nat] :
% 9.10/5.33                  ( ( ord_less_nat @ X @ N )
% 9.10/5.33                  & ( ord_less_nat @ Y @ N )
% 9.10/5.33                  & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 9.10/5.33        = ( collect_nat
% 9.10/5.33          @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(fact_10208_natLess__def,axiom,
% 9.10/5.33      ( bNF_Ca8459412986667044542atLess
% 9.10/5.33      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 9.10/5.33      ! [X3: int,Y3: int] :
% 9.10/5.33        ( ( if_int @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 9.10/5.33      ! [X3: int,Y3: int] :
% 9.10/5.33        ( ( if_int @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 9.10/5.33      ! [X3: nat,Y3: nat] :
% 9.10/5.33        ( ( if_nat @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 9.10/5.33      ! [X3: nat,Y3: nat] :
% 9.10/5.33        ( ( if_nat @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 9.10/5.33      ! [X3: num,Y3: num] :
% 9.10/5.33        ( ( if_num @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 9.10/5.33      ! [X3: num,Y3: num] :
% 9.10/5.33        ( ( if_num @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 9.10/5.33      ! [X3: rat,Y3: rat] :
% 9.10/5.33        ( ( if_rat @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 9.10/5.33      ! [X3: rat,Y3: rat] :
% 9.10/5.33        ( ( if_rat @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 9.10/5.33      ! [X3: real,Y3: real] :
% 9.10/5.33        ( ( if_real @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 9.10/5.33      ! [X3: real,Y3: real] :
% 9.10/5.33        ( ( if_real @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 9.10/5.33      ! [P: real > $o] :
% 9.10/5.33        ( ( P @ ( fChoice_real @ P ) )
% 9.10/5.33        = ( ? [X7: real] : ( P @ X7 ) ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.10/5.33      ! [X3: complex,Y3: complex] :
% 9.10/5.33        ( ( if_complex @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.10/5.33      ! [X3: complex,Y3: complex] :
% 9.10/5.33        ( ( if_complex @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.10/5.33      ! [X3: extended_enat,Y3: extended_enat] :
% 9.10/5.33        ( ( if_Extended_enat @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.10/5.33      ! [X3: extended_enat,Y3: extended_enat] :
% 9.10/5.33        ( ( if_Extended_enat @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.10/5.33      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.33        ( ( if_Code_integer @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.10/5.33      ! [X3: code_integer,Y3: code_integer] :
% 9.10/5.33        ( ( if_Code_integer @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.10/5.33      ! [X3: set_int,Y3: set_int] :
% 9.10/5.33        ( ( if_set_int @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.10/5.33      ! [X3: set_int,Y3: set_int] :
% 9.10/5.33        ( ( if_set_int @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
% 9.10/5.33        ( ( if_VEBT_VEBT @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 9.10/5.33      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT] :
% 9.10/5.33        ( ( if_VEBT_VEBT @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.10/5.33      ! [X3: list_int,Y3: list_int] :
% 9.10/5.33        ( ( if_list_int @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.10/5.33      ! [X3: list_int,Y3: list_int] :
% 9.10/5.33        ( ( if_list_int @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.10/5.33      ! [X3: list_nat,Y3: list_nat] :
% 9.10/5.33        ( ( if_list_nat @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.10/5.33      ! [X3: list_nat,Y3: list_nat] :
% 9.10/5.33        ( ( if_list_nat @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 9.10/5.33      ! [X3: option_nat,Y3: option_nat] :
% 9.10/5.33        ( ( if_option_nat @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 9.10/5.33      ! [X3: option_nat,Y3: option_nat] :
% 9.10/5.33        ( ( if_option_nat @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.10/5.33      ! [X3: option_num,Y3: option_num] :
% 9.10/5.33        ( ( if_option_num @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.10/5.33      ! [X3: option_num,Y3: option_num] :
% 9.10/5.33        ( ( if_option_num @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.10/5.33      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 9.10/5.33        ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.10/5.33      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 9.10/5.33        ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.10/5.33      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.10/5.33        ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.10/5.33      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.10/5.33        ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.10/5.33      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 9.10/5.33        ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.10/5.33      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 9.10/5.33        ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.10/5.33      ! [P: $o] :
% 9.10/5.33        ( ( P = $true )
% 9.10/5.33        | ( P = $false ) ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.10/5.33      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 9.10/5.33        ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y3 )
% 9.10/5.33        = Y3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.10/5.33      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 9.10/5.33        ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y3 )
% 9.10/5.33        = X3 ) ).
% 9.10/5.33  
% 9.10/5.33  thf(conj_0,hypothesis,
% 9.10/5.33      ! [Minilow: nat] :
% 9.10/5.33        ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.10/5.33          = ( some_nat @ Minilow ) )
% 9.10/5.33       => thesis ) ).
% 9.10/5.33  
% 9.10/5.33  thf(conj_1,conjecture,
% 9.10/5.33      thesis ).
% 9.10/5.33  ------------------------
% 9.11/5.86  % (22700)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/18Mi)
% 9.11/5.87  % (22694)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/183Mi)
% 9.11/5.87  % (22695)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_22586 for (2994ds/4Mi)
% 9.11/5.87  % (22697)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/2Mi)
% 9.11/5.87  % (22696)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_22586 for (2994ds/27Mi)
% 9.11/5.87  % (22698)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_22586 for (2994ds/2Mi)
% 9.11/5.87  % (22699)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_22586 for (2994ds/275Mi)
% 9.11/5.87  % (22697)Instruction limit reached!
% 9.11/5.87  % (22697)------------------------------
% 9.11/5.87  % (22697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.87  % (22697)Termination reason: Unknown
% 9.11/5.87  % (22697)Termination phase: shuffling
% 9.11/5.87  
% 9.11/5.87  % (22697)Memory used [KB]: 13176
% 9.11/5.87  % (22697)Time elapsed: 0.003 s
% 9.11/5.87  % (22697)Instructions burned: 2 (million)
% 9.11/5.87  % (22697)------------------------------
% 9.11/5.87  % (22697)------------------------------
% 9.11/5.87  % (22698)Instruction limit reached!
% 9.11/5.87  % (22698)------------------------------
% 9.11/5.87  % (22698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.87  % (22698)Termination reason: Unknown
% 9.11/5.87  % (22698)Termination phase: shuffling
% 9.11/5.87  
% 9.11/5.87  % (22698)Memory used [KB]: 13176
% 9.11/5.87  % (22698)Time elapsed: 0.003 s
% 9.11/5.87  % (22698)Instructions burned: 2 (million)
% 9.11/5.87  % (22698)------------------------------
% 9.11/5.87  % (22698)------------------------------
% 9.11/5.87  % (22695)Instruction limit reached!
% 9.11/5.87  % (22695)------------------------------
% 9.11/5.87  % (22695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.87  % (22695)Termination reason: Unknown
% 9.11/5.87  % (22695)Termination phase: shuffling
% 9.11/5.87  
% 9.11/5.87  % (22695)Memory used [KB]: 13304
% 9.11/5.87  % (22695)Time elapsed: 0.007 s
% 9.11/5.87  % (22695)Instructions burned: 4 (million)
% 9.11/5.87  % (22695)------------------------------
% 9.11/5.87  % (22695)------------------------------
% 9.11/5.88  % (22700)Instruction limit reached!
% 9.11/5.88  % (22700)------------------------------
% 9.11/5.88  % (22700)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.88  % (22700)Termination reason: Unknown
% 9.11/5.88  % (22700)Termination phase: shuffling
% 9.11/5.88  
% 9.11/5.88  % (22700)Memory used [KB]: 14072
% 9.11/5.88  % (22700)Time elapsed: 0.016 s
% 9.11/5.88  % (22700)Instructions burned: 20 (million)
% 9.11/5.88  % (22700)------------------------------
% 9.11/5.88  % (22700)------------------------------
% 9.11/5.89  % (22704)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_22586 for (2994ds/3Mi)
% 9.11/5.89  % (22704)Instruction limit reached!
% 9.11/5.89  % (22704)------------------------------
% 9.11/5.89  % (22704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.89  % (22704)Termination reason: Unknown
% 9.11/5.89  % (22704)Termination phase: shuffling
% 9.11/5.89  
% 9.11/5.89  % (22704)Memory used [KB]: 13304
% 9.11/5.89  % (22704)Time elapsed: 0.004 s
% 9.11/5.89  % (22704)Instructions burned: 3 (million)
% 9.11/5.89  % (22704)------------------------------
% 9.11/5.89  % (22704)------------------------------
% 9.11/5.89  % (22702)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_22586 for (2994ds/37Mi)
% 9.11/5.89  % (22701)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/3Mi)
% 9.11/5.89  % (22701)Instruction limit reached!
% 9.11/5.89  % (22701)------------------------------
% 9.11/5.89  % (22701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.89  % (22701)Termination reason: Unknown
% 9.11/5.89  % (22701)Termination phase: shuffling
% 9.11/5.89  
% 9.11/5.89  % (22701)Memory used [KB]: 13176
% 9.11/5.89  % (22701)Time elapsed: 0.005 s
% 9.11/5.89  % (22701)Instructions burned: 3 (million)
% 9.11/5.89  % (22701)------------------------------
% 9.11/5.89  % (22701)------------------------------
% 9.11/5.89  % (22703)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_22586 for (2994ds/15Mi)
% 9.11/5.90  % (22696)Instruction limit reached!
% 9.11/5.90  % (22696)------------------------------
% 9.11/5.90  % (22696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.90  % (22696)Termination reason: Unknown
% 9.11/5.90  % (22696)Termination phase: shuffling
% 9.11/5.90  
% 9.11/5.90  % (22696)Memory used [KB]: 14072
% 9.11/5.90  % (22696)Time elapsed: 0.030 s
% 9.11/5.90  % (22696)Instructions burned: 28 (million)
% 9.11/5.90  % (22696)------------------------------
% 9.11/5.90  % (22696)------------------------------
% 9.11/5.90  % (22705)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_22586 for (2994ds/1041Mi)
% 9.11/5.91  % (22706)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/7Mi)
% 9.11/5.91  % (22703)Instruction limit reached!
% 9.11/5.91  % (22703)------------------------------
% 9.11/5.91  % (22703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.91  % (22703)Termination reason: Unknown
% 9.11/5.91  % (22703)Termination phase: shuffling
% 9.11/5.91  
% 9.11/5.91  % (22703)Memory used [KB]: 14072
% 9.11/5.91  % (22703)Time elapsed: 0.022 s
% 9.11/5.91  % (22703)Instructions burned: 15 (million)
% 9.11/5.91  % (22703)------------------------------
% 9.11/5.91  % (22703)------------------------------
% 9.11/5.92  % (22707)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/16Mi)
% 9.11/5.92  % (22706)Instruction limit reached!
% 9.11/5.92  % (22706)------------------------------
% 9.11/5.92  % (22706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.92  % (22706)Termination reason: Unknown
% 9.11/5.92  % (22706)Termination phase: shuffling
% 9.11/5.92  
% 9.11/5.92  % (22706)Memory used [KB]: 13560
% 9.11/5.92  % (22706)Time elapsed: 0.011 s
% 9.11/5.92  % (22706)Instructions burned: 7 (million)
% 9.11/5.92  % (22706)------------------------------
% 9.11/5.92  % (22706)------------------------------
% 9.11/5.92  % (22702)Instruction limit reached!
% 9.11/5.92  % (22702)------------------------------
% 9.11/5.92  % (22702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.92  % (22702)Termination reason: Unknown
% 9.11/5.92  % (22702)Termination phase: shuffling
% 9.11/5.92  
% 9.11/5.92  % (22702)Memory used [KB]: 14200
% 9.11/5.92  % (22702)Time elapsed: 0.034 s
% 9.11/5.92  % (22702)Instructions burned: 38 (million)
% 9.11/5.92  % (22702)------------------------------
% 9.11/5.92  % (22702)------------------------------
% 9.11/5.94  % (22708)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_22586 for (2994ds/3Mi)
% 9.11/5.94  % (22708)Instruction limit reached!
% 9.11/5.94  % (22708)------------------------------
% 9.11/5.94  % (22708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.94  % (22708)Termination reason: Unknown
% 9.11/5.94  % (22708)Termination phase: shuffling
% 9.11/5.94  
% 9.11/5.94  % (22708)Memory used [KB]: 13304
% 9.11/5.94  % (22708)Time elapsed: 0.006 s
% 9.11/5.94  % (22708)Instructions burned: 3 (million)
% 9.11/5.94  % (22708)------------------------------
% 9.11/5.94  % (22708)------------------------------
% 9.11/5.94  % (22707)Instruction limit reached!
% 9.11/5.94  % (22707)------------------------------
% 9.11/5.94  % (22707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.94  % (22707)Termination reason: Unknown
% 9.11/5.94  % (22707)Termination phase: shuffling
% 9.11/5.94  
% 9.11/5.94  % (22707)Memory used [KB]: 14072
% 9.11/5.94  % (22707)Time elapsed: 0.023 s
% 9.11/5.94  % (22707)Instructions burned: 16 (million)
% 9.11/5.94  % (22707)------------------------------
% 9.11/5.94  % (22707)------------------------------
% 9.11/5.94  % (22709)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_22586 for (2994ds/3Mi)
% 9.11/5.94  % (22710)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/7Mi)
% 9.11/5.94  % (22709)Instruction limit reached!
% 9.11/5.94  % (22709)------------------------------
% 9.11/5.94  % (22709)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.94  % (22709)Termination reason: Unknown
% 9.11/5.94  % (22709)Termination phase: shuffling
% 9.11/5.94  
% 9.11/5.94  % (22709)Memory used [KB]: 13176
% 9.11/5.94  % (22709)Time elapsed: 0.005 s
% 9.11/5.94  % (22709)Instructions burned: 3 (million)
% 9.11/5.94  % (22709)------------------------------
% 9.11/5.94  % (22709)------------------------------
% 9.11/5.95  % (22710)Instruction limit reached!
% 9.11/5.95  % (22710)------------------------------
% 9.11/5.95  % (22710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.95  % (22710)Termination reason: Unknown
% 9.11/5.95  % (22710)Termination phase: shuffling
% 9.11/5.95  
% 9.11/5.95  % (22710)Memory used [KB]: 13560
% 9.11/5.95  % (22710)Time elapsed: 0.011 s
% 9.11/5.95  % (22710)Instructions burned: 7 (million)
% 9.11/5.95  % (22710)------------------------------
% 9.11/5.95  % (22710)------------------------------
% 9.11/5.96  % (22711)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/3Mi)
% 9.11/5.96  % (22712)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/4Mi)
% 9.11/5.96  % (22711)Instruction limit reached!
% 9.11/5.96  % (22711)------------------------------
% 9.11/5.96  % (22711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.96  % (22711)Termination reason: Unknown
% 9.11/5.96  % (22711)Termination phase: shuffling
% 9.11/5.96  
% 9.11/5.96  % (22711)Memory used [KB]: 13304
% 9.11/5.96  % (22711)Time elapsed: 0.005 s
% 9.11/5.96  % (22711)Instructions burned: 3 (million)
% 9.11/5.96  % (22711)------------------------------
% 9.11/5.96  % (22711)------------------------------
% 9.11/5.96  % (22712)Instruction limit reached!
% 9.11/5.96  % (22712)------------------------------
% 9.11/5.96  % (22712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.96  % (22712)Termination reason: Unknown
% 9.11/5.96  % (22712)Termination phase: shuffling
% 9.11/5.96  
% 9.11/5.96  % (22712)Memory used [KB]: 13304
% 9.11/5.96  % (22712)Time elapsed: 0.006 s
% 9.11/5.96  % (22712)Instructions burned: 4 (million)
% 9.11/5.96  % (22712)------------------------------
% 9.11/5.96  % (22712)------------------------------
% 9.11/5.96  % (22713)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22586 for (2994ds/18Mi)
% 9.11/5.97  % (22714)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22586 for (2993ds/710Mi)
% 9.11/5.98  % (22715)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22586 for (2993ds/6Mi)
% 9.11/5.98  % (22716)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_22586 for (2993ds/902Mi)
% 9.11/5.98  % (22713)Instruction limit reached!
% 9.11/5.98  % (22713)------------------------------
% 9.11/5.98  % (22713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.98  % (22713)Termination reason: Unknown
% 9.11/5.98  % (22713)Termination phase: shuffling
% 9.11/5.98  
% 9.11/5.98  % (22713)Memory used [KB]: 14072
% 9.11/5.98  % (22713)Time elapsed: 0.020 s
% 9.11/5.98  % (22713)Instructions burned: 19 (million)
% 9.11/5.98  % (22713)------------------------------
% 9.11/5.98  % (22713)------------------------------
% 9.11/5.99  % (22694)Instruction limit reached!
% 9.11/5.99  % (22694)------------------------------
% 9.11/5.99  % (22694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.99  % (22694)Termination reason: Unknown
% 9.11/5.99  % (22694)Termination phase: shuffling
% 9.11/5.99  
% 9.11/5.99  % (22694)Memory used [KB]: 16375
% 9.11/5.99  % (22694)Time elapsed: 0.118 s
% 9.11/5.99  % (22694)Instructions burned: 184 (million)
% 9.11/5.99  % (22694)------------------------------
% 9.11/5.99  % (22694)------------------------------
% 9.11/5.99  % (22715)Instruction limit reached!
% 9.11/5.99  % (22715)------------------------------
% 9.11/5.99  % (22715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/5.99  % (22715)Termination reason: Unknown
% 9.11/5.99  % (22715)Termination phase: shuffling
% 9.11/5.99  
% 9.11/5.99  % (22715)Memory used [KB]: 13432
% 9.11/5.99  % (22715)Time elapsed: 0.008 s
% 9.11/5.99  % (22715)Instructions burned: 6 (million)
% 9.11/5.99  % (22715)------------------------------
% 9.11/5.99  % (22715)------------------------------
% 9.11/5.99  % (22717)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_22586 for (2993ds/21Mi)
% 9.11/6.00  % (22719)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_22586 for (2993ds/6Mi)
% 9.11/6.00  % (22719)Instruction limit reached!
% 9.11/6.00  % (22719)------------------------------
% 9.11/6.00  % (22719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/6.00  % (22719)Termination reason: Unknown
% 9.11/6.00  % (22719)Termination phase: shuffling
% 9.11/6.00  
% 9.11/6.00  % (22719)Memory used [KB]: 13432
% 9.11/6.00  % (22719)Time elapsed: 0.006 s
% 9.11/6.00  % (22719)Instructions burned: 6 (million)
% 9.11/6.00  % (22719)------------------------------
% 9.11/6.00  % (22719)------------------------------
% 9.11/6.00  % (22718)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22586 for (2993ds/5Mi)
% 9.11/6.01  % (22718)Instruction limit reached!
% 9.11/6.01  % (22718)------------------------------
% 9.11/6.01  % (22718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/6.01  % (22718)Termination reason: Unknown
% 9.11/6.01  % (22718)Termination phase: shuffling
% 9.11/6.01  
% 9.11/6.01  % (22718)Memory used [KB]: 13304
% 9.11/6.01  % (22718)Time elapsed: 0.005 s
% 9.11/6.01  % (22718)Instructions burned: 5 (million)
% 9.11/6.01  % (22718)------------------------------
% 9.11/6.01  % (22718)------------------------------
% 9.11/6.01  % (22717)Instruction limit reached!
% 9.11/6.01  % (22717)------------------------------
% 9.11/6.01  % (22717)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.11/6.01  % (22717)Termination reason: Unknown
% 9.11/6.01  % (22717)Termination phase: shuffling
% 9.11/6.01  
% 9.11/6.01  % (22717)Memory used [KB]: 14072
% 9.11/6.01  % (22717)Time elapsed: 0.018 s
% 9.11/6.01  % (22717)Instructions burned: 23 (million)
% 9.11/6.01  % (22717)------------------------------
% 9.11/6.01  % (22717)------------------------------
% 9.11/6.01  % (22720)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22586 for (2993ds/377Mi)
% 10.65/6.02  % (22699)Instruction limit reached!
% 10.65/6.02  % (22699)------------------------------
% 10.65/6.02  % (22699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.65/6.02  % (22699)Termination reason: Unknown
% 10.65/6.02  % (22699)Termination phase: shuffling
% 10.65/6.02  
% 10.65/6.02  % (22699)Memory used [KB]: 17654
% 10.65/6.02  % (22699)Time elapsed: 0.155 s
% 10.65/6.02  % (22699)Instructions burned: 275 (million)
% 10.65/6.02  % (22699)------------------------------
% 10.65/6.02  % (22699)------------------------------
% 10.65/6.02  % (22722)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_22586 for (2993ds/19Mi)
% 10.65/6.03  % (22721)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_22586 for (2993ds/779Mi)
% 10.65/6.04  % (22723)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22586 for (2993ds/879Mi)
% 10.65/6.04  % (22722)Instruction limit reached!
% 10.65/6.04  % (22722)------------------------------
% 10.65/6.04  % (22722)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.65/6.04  % (22722)Termination reason: Unknown
% 10.65/6.04  % (22722)Termination phase: shuffling
% 10.65/6.04  
% 10.65/6.04  % (22722)Memory used [KB]: 14072
% 10.65/6.04  % (22722)Time elapsed: 0.017 s
% 10.65/6.04  % (22722)Instructions burned: 19 (million)
% 10.65/6.04  % (22722)------------------------------
% 10.65/6.04  % (22722)------------------------------
% 10.65/6.05  % (22724)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22586 for (2992ds/17Mi)
% 10.65/6.07  % (22724)Instruction limit reached!
% 10.65/6.07  % (22724)------------------------------
% 10.65/6.07  % (22724)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.65/6.07  % (22724)Termination reason: Unknown
% 10.65/6.07  % (22724)Termination phase: shuffling
% 10.65/6.07  
% 10.65/6.07  % (22724)Memory used [KB]: 14072
% 10.65/6.07  % (22724)Time elapsed: 0.017 s
% 10.65/6.07  % (22724)Instructions burned: 19 (million)
% 10.65/6.07  % (22724)------------------------------
% 10.65/6.07  % (22724)------------------------------
% 10.65/6.08  % (22725)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22586 for (2992ds/3Mi)
% 10.65/6.08  % (22725)Instruction limit reached!
% 10.65/6.08  % (22725)------------------------------
% 10.65/6.08  % (22725)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.65/6.08  % (22725)Termination reason: Unknown
% 10.65/6.08  % (22725)Termination phase: shuffling
% 10.65/6.08  
% 10.65/6.08  % (22725)Memory used [KB]: 13304
% 10.65/6.08  % (22725)Time elapsed: 0.004 s
% 10.65/6.08  % (22725)Instructions burned: 3 (million)
% 10.65/6.08  % (22725)------------------------------
% 10.65/6.08  % (22725)------------------------------
% 10.65/6.09  % (22726)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22586 for (2992ds/30Mi)
% 10.65/6.11  % (22726)Instruction limit reached!
% 10.65/6.11  % (22726)------------------------------
% 10.65/6.11  % (22726)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 10.65/6.11  % (22726)Termination reason: Unknown
% 10.65/6.11  % (22726)Termination phase: shuffling
% 10.65/6.11  
% 10.65/6.11  % (22726)Memory used [KB]: 14072
% 10.65/6.11  % (22726)Time elapsed: 0.020 s
% 10.65/6.11  % (22726)Instructions burned: 31 (million)
% 10.65/6.11  % (22726)------------------------------
% 10.65/6.11  % (22726)------------------------------
% 11.11/6.12  % (22727)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22586 for (2992ds/127Mi)
% 11.11/6.15  % (22720)Instruction limit reached!
% 11.11/6.15  % (22720)------------------------------
% 11.11/6.15  % (22720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.11/6.15  % (22720)Termination reason: Unknown
% 11.11/6.15  % (22720)Termination phase: shuffling
% 11.11/6.15  
% 11.11/6.15  % (22720)Memory used [KB]: 18933
% 11.11/6.15  % (22720)Time elapsed: 0.135 s
% 11.11/6.15  % (22720)Instructions burned: 378 (million)
% 11.11/6.15  % (22720)------------------------------
% 11.11/6.15  % (22720)------------------------------
% 11.11/6.16  % (22728)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22586 for (2991ds/100Mi)
% 11.11/6.18  % (22727)Instruction limit reached!
% 11.11/6.18  % (22727)------------------------------
% 11.11/6.18  % (22727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.11/6.18  % (22727)Termination reason: Unknown
% 11.11/6.18  % (22727)Termination phase: shuffling
% 11.11/6.18  
% 11.11/6.18  % (22727)Memory used [KB]: 15607
% 11.11/6.18  % (22727)Time elapsed: 0.055 s
% 11.11/6.18  % (22727)Instructions burned: 129 (million)
% 11.11/6.18  % (22727)------------------------------
% 11.11/6.18  % (22727)------------------------------
% 11.11/6.19  % (22729)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_22586 for (2991ds/3Mi)
% 11.11/6.19  % (22729)Instruction limit reached!
% 11.11/6.19  % (22729)------------------------------
% 11.11/6.19  % (22729)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.11/6.19  % (22729)Termination reason: Unknown
% 11.11/6.19  % (22729)Termination phase: shuffling
% 11.11/6.19  
% 11.11/6.19  % (22729)Memory used [KB]: 13304
% 11.11/6.19  % (22729)Time elapsed: 0.004 s
% 11.11/6.19  % (22729)Instructions burned: 3 (million)
% 11.11/6.19  % (22729)------------------------------
% 11.11/6.19  % (22729)------------------------------
% 11.11/6.19  % (22728)Instruction limit reached!
% 11.11/6.19  % (22728)------------------------------
% 11.11/6.19  % (22728)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.11/6.19  % (22728)Termination reason: Unknown
% 11.11/6.19  % (22728)Termination phase: shuffling
% 11.11/6.19  
% 11.11/6.19  % (22728)Memory used [KB]: 15223
% 11.11/6.19  % (22728)Time elapsed: 0.038 s
% 11.11/6.19  % (22728)Instructions burned: 102 (million)
% 11.11/6.19  % (22728)------------------------------
% 11.11/6.19  % (22728)------------------------------
% 11.62/6.20  % (22730)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_22586 for (2991ds/20Mi)
% 11.62/6.20  % (22731)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_22586 for (2991ds/86Mi)
% 11.62/6.21  % (22705)Instruction limit reached!
% 11.62/6.21  % (22705)------------------------------
% 11.62/6.21  % (22705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.21  % (22705)Termination reason: Unknown
% 11.62/6.21  % (22705)Termination phase: shuffling
% 11.62/6.21  
% 11.62/6.21  % (22705)Memory used [KB]: 34285
% 11.62/6.21  % (22705)Time elapsed: 0.307 s
% 11.62/6.21  % (22705)Instructions burned: 1042 (million)
% 11.62/6.21  % (22705)------------------------------
% 11.62/6.21  % (22705)------------------------------
% 11.62/6.22  % (22730)Instruction limit reached!
% 11.62/6.22  % (22730)------------------------------
% 11.62/6.22  % (22730)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.22  % (22730)Termination reason: Unknown
% 11.62/6.22  % (22730)Termination phase: shuffling
% 11.62/6.22  
% 11.62/6.22  % (22730)Memory used [KB]: 14072
% 11.62/6.22  % (22730)Time elapsed: 0.017 s
% 11.62/6.22  % (22730)Instructions burned: 21 (million)
% 11.62/6.22  % (22730)------------------------------
% 11.62/6.22  % (22730)------------------------------
% 11.62/6.22  % (22732)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_22586 for (2991ds/107Mi)
% 11.62/6.23  % (22733)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_22586 for (2991ds/39Mi)
% 11.62/6.24  % (22731)Instruction limit reached!
% 11.62/6.24  % (22731)------------------------------
% 11.62/6.24  % (22731)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.24  % (22731)Termination reason: Unknown
% 11.62/6.24  % (22731)Termination phase: shuffling
% 11.62/6.24  
% 11.62/6.24  % (22731)Memory used [KB]: 14967
% 11.62/6.24  % (22731)Time elapsed: 0.037 s
% 11.62/6.24  % (22731)Instructions burned: 88 (million)
% 11.62/6.24  % (22731)------------------------------
% 11.62/6.24  % (22731)------------------------------
% 11.62/6.25  % (22714)Instruction limit reached!
% 11.62/6.25  % (22714)------------------------------
% 11.62/6.25  % (22714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.25  % (22714)Termination reason: Unknown
% 11.62/6.25  % (22714)Termination phase: shuffling
% 11.62/6.25  
% 11.62/6.25  % (22714)Memory used [KB]: 25458
% 11.62/6.25  % (22714)Time elapsed: 0.276 s
% 11.62/6.25  % (22714)Instructions burned: 710 (million)
% 11.62/6.25  % (22714)------------------------------
% 11.62/6.25  % (22714)------------------------------
% 11.62/6.25  % (22734)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_22586 for (2991ds/448Mi)
% 11.62/6.25  % (22733)Instruction limit reached!
% 11.62/6.25  % (22733)------------------------------
% 11.62/6.25  % (22733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.25  % (22733)Termination reason: Unknown
% 11.62/6.25  % (22733)Termination phase: shuffling
% 11.62/6.25  
% 11.62/6.25  % (22733)Memory used [KB]: 14328
% 11.62/6.25  % (22733)Time elapsed: 0.023 s
% 11.62/6.25  % (22733)Instructions burned: 42 (million)
% 11.62/6.25  % (22733)------------------------------
% 11.62/6.25  % (22733)------------------------------
% 11.62/6.26  % (22735)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_22586 for (2990ds/46Mi)
% 11.62/6.26  % (22732)Instruction limit reached!
% 11.62/6.26  % (22732)------------------------------
% 11.62/6.26  % (22732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.26  % (22732)Termination reason: Unknown
% 11.62/6.26  % (22732)Termination phase: shuffling
% 11.62/6.26  
% 11.62/6.26  % (22732)Memory used [KB]: 15223
% 11.62/6.26  % (22732)Time elapsed: 0.045 s
% 11.62/6.26  % (22732)Instructions burned: 108 (million)
% 11.62/6.26  % (22732)------------------------------
% 11.62/6.26  % (22732)------------------------------
% 11.62/6.27  % (22736)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_22586 for (2990ds/98Mi)
% 11.62/6.27  % (22737)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_22586 for (2990ds/507Mi)
% 11.62/6.28  % (22735)Instruction limit reached!
% 11.62/6.28  % (22735)------------------------------
% 11.62/6.28  % (22735)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.28  % (22735)Termination reason: Unknown
% 11.62/6.28  % (22735)Termination phase: shuffling
% 11.62/6.28  
% 11.62/6.28  % (22735)Memory used [KB]: 14328
% 11.62/6.28  % (22735)Time elapsed: 0.025 s
% 11.62/6.28  % (22735)Instructions burned: 47 (million)
% 11.62/6.28  % (22735)------------------------------
% 11.62/6.28  % (22735)------------------------------
% 11.62/6.29  % (22721)Instruction limit reached!
% 11.62/6.29  % (22721)------------------------------
% 11.62/6.29  % (22721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.62/6.29  % (22721)Termination reason: Unknown
% 11.62/6.29  % (22721)Termination phase: shuffling
% 11.62/6.29  
% 11.62/6.29  % (22721)Memory used [KB]: 26609
% 11.62/6.29  % (22721)Time elapsed: 0.264 s
% 11.62/6.29  % (22721)Instructions burned: 780 (million)
% 11.62/6.29  % (22721)------------------------------
% 11.62/6.29  % (22721)------------------------------
% 11.62/6.29  % (22738)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_22586 for (2990ds/149Mi)
% 12.28/6.30  % (22739)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_22586 for (2990ds/39Mi)
% 12.28/6.30  % (22736)Instruction limit reached!
% 12.28/6.30  % (22736)------------------------------
% 12.28/6.30  % (22736)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.30  % (22736)Termination reason: Unknown
% 12.28/6.30  % (22736)Termination phase: shuffling
% 12.28/6.30  
% 12.28/6.30  % (22736)Memory used [KB]: 15095
% 12.28/6.30  % (22736)Time elapsed: 0.040 s
% 12.28/6.30  % (22736)Instructions burned: 98 (million)
% 12.28/6.30  % (22736)------------------------------
% 12.28/6.30  % (22736)------------------------------
% 12.28/6.32  % (22740)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_22586 for (2990ds/6Mi)
% 12.28/6.32  % (22740)Instruction limit reached!
% 12.28/6.32  % (22740)------------------------------
% 12.28/6.32  % (22740)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.32  % (22740)Termination reason: Unknown
% 12.28/6.32  % (22740)Termination phase: shuffling
% 12.28/6.32  
% 12.28/6.32  % (22740)Memory used [KB]: 13432
% 12.28/6.32  % (22740)Time elapsed: 0.007 s
% 12.28/6.32  % (22740)Instructions burned: 6 (million)
% 12.28/6.32  % (22740)------------------------------
% 12.28/6.32  % (22740)------------------------------
% 12.28/6.32  % (22716)Instruction limit reached!
% 12.28/6.32  % (22716)------------------------------
% 12.28/6.32  % (22716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.32  % (22716)Termination reason: Unknown
% 12.28/6.32  % (22716)Termination phase: Property scanning
% 12.28/6.32  
% 12.28/6.32  % (22716)Memory used [KB]: 27121
% 12.28/6.32  % (22716)Time elapsed: 0.342 s
% 12.28/6.32  % (22716)Instructions burned: 905 (million)
% 12.28/6.32  % (22716)------------------------------
% 12.28/6.32  % (22716)------------------------------
% 12.28/6.32  % (22739)Instruction limit reached!
% 12.28/6.32  % (22739)------------------------------
% 12.28/6.32  % (22739)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.32  % (22739)Termination reason: Unknown
% 12.28/6.32  % (22739)Termination phase: shuffling
% 12.28/6.32  
% 12.28/6.32  % (22739)Memory used [KB]: 14328
% 12.28/6.32  % (22739)Time elapsed: 0.024 s
% 12.28/6.32  % (22739)Instructions burned: 40 (million)
% 12.28/6.32  % (22739)------------------------------
% 12.28/6.32  % (22739)------------------------------
% 12.28/6.33  % (22741)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_22586 for (2990ds/1087Mi)
% 12.28/6.33  % (22742)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_22586 for (2990ds/47Mi)
% 12.28/6.33  % (22743)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_22586 for (2990ds/44Mi)
% 12.28/6.35  % (22738)Instruction limit reached!
% 12.28/6.35  % (22738)------------------------------
% 12.28/6.35  % (22738)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.35  % (22738)Termination reason: Unknown
% 12.28/6.35  % (22738)Termination phase: shuffling
% 12.28/6.35  
% 12.28/6.35  % (22738)Memory used [KB]: 15863
% 12.28/6.35  % (22738)Time elapsed: 0.058 s
% 12.28/6.35  % (22738)Instructions burned: 151 (million)
% 12.28/6.35  % (22738)------------------------------
% 12.28/6.35  % (22738)------------------------------
% 12.28/6.36  % (22743)Instruction limit reached!
% 12.28/6.36  % (22743)------------------------------
% 12.28/6.36  % (22743)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.36  % (22743)Termination reason: Unknown
% 12.28/6.36  % (22743)Termination phase: shuffling
% 12.28/6.36  
% 12.28/6.36  % (22743)Memory used [KB]: 14328
% 12.28/6.36  % (22743)Time elapsed: 0.022 s
% 12.28/6.36  % (22743)Instructions burned: 47 (million)
% 12.28/6.36  % (22743)------------------------------
% 12.28/6.36  % (22743)------------------------------
% 12.28/6.36  % (22742)Instruction limit reached!
% 12.28/6.36  % (22742)------------------------------
% 12.28/6.36  % (22742)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.36  % (22742)Termination reason: Unknown
% 12.28/6.36  % (22742)Termination phase: shuffling
% 12.28/6.36  
% 12.28/6.36  % (22742)Memory used [KB]: 14456
% 12.28/6.36  % (22742)Time elapsed: 0.026 s
% 12.28/6.36  % (22742)Instructions burned: 49 (million)
% 12.28/6.36  % (22742)------------------------------
% 12.28/6.36  % (22742)------------------------------
% 12.28/6.36  % (22723)Instruction limit reached!
% 12.28/6.36  % (22723)------------------------------
% 12.28/6.36  % (22723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.36  % (22723)Termination reason: Unknown
% 12.28/6.36  % (22723)Termination phase: Property scanning
% 12.28/6.36  
% 12.28/6.36  % (22723)Memory used [KB]: 26993
% 12.28/6.36  % (22723)Time elapsed: 0.326 s
% 12.28/6.36  % (22723)Instructions burned: 880 (million)
% 12.28/6.36  % (22723)------------------------------
% 12.28/6.36  % (22723)------------------------------
% 12.28/6.36  % (22744)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_22586 for (2990ds/432Mi)
% 12.28/6.37  % (22745)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_22586 for (2989ds/62Mi)
% 12.28/6.37  % (22746)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_22586 for (2989ds/31Mi)
% 12.28/6.37  % (22747)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_22586 for (2989ds/96Mi)
% 12.28/6.39  % (22734)Instruction limit reached!
% 12.28/6.39  % (22734)------------------------------
% 12.28/6.39  % (22734)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.39  % (22734)Termination reason: Unknown
% 12.28/6.39  % (22734)Termination phase: shuffling
% 12.28/6.39  
% 12.28/6.39  % (22734)Memory used [KB]: 22131
% 12.28/6.39  % (22734)Time elapsed: 0.137 s
% 12.28/6.39  % (22734)Instructions burned: 449 (million)
% 12.28/6.39  % (22734)------------------------------
% 12.28/6.39  % (22734)------------------------------
% 12.28/6.39  % (22746)Instruction limit reached!
% 12.28/6.39  % (22746)------------------------------
% 12.28/6.39  % (22746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.39  % (22746)Termination reason: Unknown
% 12.28/6.39  % (22746)Termination phase: shuffling
% 12.28/6.39  
% 12.28/6.39  % (22746)Memory used [KB]: 14072
% 12.28/6.39  % (22746)Time elapsed: 0.021 s
% 12.28/6.39  % (22746)Instructions burned: 33 (million)
% 12.28/6.39  % (22746)------------------------------
% 12.28/6.39  % (22746)------------------------------
% 12.28/6.39  % (22745)Instruction limit reached!
% 12.28/6.39  % (22745)------------------------------
% 12.28/6.39  % (22745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.28/6.39  % (22745)Termination reason: Unknown
% 12.28/6.39  % (22745)Termination phase: shuffling
% 12.28/6.39  
% 12.28/6.39  % (22745)Memory used [KB]: 14583
% 12.28/6.39  % (22745)Time elapsed: 0.027 s
% 12.28/6.39  % (22745)Instructions burned: 62 (million)
% 12.28/6.39  % (22745)------------------------------
% 12.28/6.39  % (22745)------------------------------
% 13.10/6.40  % (22748)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_22586 for (2989ds/453Mi)
% 13.10/6.40  % (22749)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_22586 for (2989ds/71Mi)
% 13.10/6.40  % (22750)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_22586 for (2989ds/83Mi)
% 13.10/6.41  % (22747)Instruction limit reached!
% 13.10/6.41  % (22747)------------------------------
% 13.10/6.41  % (22747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.41  % (22747)Termination reason: Unknown
% 13.10/6.41  % (22747)Termination phase: shuffling
% 13.10/6.41  
% 13.10/6.41  % (22747)Memory used [KB]: 15095
% 13.10/6.41  % (22747)Time elapsed: 0.041 s
% 13.10/6.41  % (22747)Instructions burned: 99 (million)
% 13.10/6.41  % (22747)------------------------------
% 13.10/6.41  % (22747)------------------------------
% 13.10/6.42  % (22751)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_22586 for (2989ds/11Mi)
% 13.10/6.43  % (22751)Instruction limit reached!
% 13.10/6.43  % (22751)------------------------------
% 13.10/6.43  % (22751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.43  % (22751)Termination reason: Unknown
% 13.10/6.43  % (22751)Termination phase: shuffling
% 13.10/6.43  
% 13.10/6.43  % (22751)Memory used [KB]: 14072
% 13.10/6.43  % (22751)Time elapsed: 0.013 s
% 13.10/6.43  % (22751)Instructions burned: 12 (million)
% 13.10/6.43  % (22751)------------------------------
% 13.10/6.43  % (22751)------------------------------
% 13.10/6.44  % (22749)Instruction limit reached!
% 13.10/6.44  % (22749)------------------------------
% 13.10/6.44  % (22749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.44  % (22749)Termination reason: Unknown
% 13.10/6.44  % (22749)Termination phase: shuffling
% 13.10/6.44  
% 13.10/6.44  % (22749)Memory used [KB]: 14711
% 13.10/6.44  % (22749)Time elapsed: 0.034 s
% 13.10/6.44  % (22749)Instructions burned: 72 (million)
% 13.10/6.44  % (22749)------------------------------
% 13.10/6.44  % (22749)------------------------------
% 13.10/6.44  % (22750)Instruction limit reached!
% 13.10/6.44  % (22750)------------------------------
% 13.10/6.44  % (22750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.44  % (22750)Termination reason: Unknown
% 13.10/6.44  % (22750)Termination phase: shuffling
% 13.10/6.44  
% 13.10/6.44  % (22750)Memory used [KB]: 14967
% 13.10/6.44  % (22750)Time elapsed: 0.037 s
% 13.10/6.44  % (22750)Instructions burned: 86 (million)
% 13.10/6.44  % (22750)------------------------------
% 13.10/6.44  % (22750)------------------------------
% 13.10/6.45  % (22752)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_22586 for (2988ds/41Mi)
% 13.10/6.45  % (22753)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22586 for (2988ds/6Mi)
% 13.10/6.45  % (22737)Instruction limit reached!
% 13.10/6.45  % (22737)------------------------------
% 13.10/6.45  % (22737)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.45  % (22737)Termination reason: Unknown
% 13.10/6.45  % (22737)Termination phase: shuffling
% 13.10/6.45  
% 13.10/6.45  % (22737)Memory used [KB]: 20340
% 13.10/6.45  % (22737)Time elapsed: 0.176 s
% 13.10/6.45  % (22737)Instructions burned: 508 (million)
% 13.10/6.45  % (22737)------------------------------
% 13.10/6.45  % (22737)------------------------------
% 13.10/6.45  % (22754)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_22586 for (2988ds/150Mi)
% 13.10/6.45  % (22753)Instruction limit reached!
% 13.10/6.45  % (22753)------------------------------
% 13.10/6.45  % (22753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.45  % (22753)Termination reason: Unknown
% 13.10/6.45  % (22753)Termination phase: shuffling
% 13.10/6.45  
% 13.10/6.45  % (22753)Memory used [KB]: 13432
% 13.10/6.45  % (22753)Time elapsed: 0.006 s
% 13.10/6.45  % (22753)Instructions burned: 6 (million)
% 13.10/6.45  % (22753)------------------------------
% 13.10/6.45  % (22753)------------------------------
% 13.10/6.46  % (22755)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_22586 for (2988ds/343Mi)
% 13.10/6.46  % (22756)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_22586 for (2988ds/109Mi)
% 13.10/6.47  % (22752)Instruction limit reached!
% 13.10/6.47  % (22752)------------------------------
% 13.10/6.47  % (22752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.47  % (22752)Termination reason: Unknown
% 13.10/6.47  % (22752)Termination phase: shuffling
% 13.10/6.47  
% 13.10/6.47  % (22752)Memory used [KB]: 14328
% 13.10/6.47  % (22752)Time elapsed: 0.022 s
% 13.10/6.47  % (22752)Instructions burned: 44 (million)
% 13.10/6.47  % (22752)------------------------------
% 13.10/6.47  % (22752)------------------------------
% 13.10/6.48  % (22757)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_22586 for (2988ds/253Mi)
% 13.10/6.49  % (22744)Instruction limit reached!
% 13.10/6.49  % (22744)------------------------------
% 13.10/6.49  % (22744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.10/6.49  % (22744)Termination reason: Unknown
% 13.10/6.49  % (22744)Termination phase: shuffling
% 13.10/6.49  
% 13.10/6.49  % (22744)Memory used [KB]: 21875
% 13.10/6.49  % (22744)Time elapsed: 0.132 s
% 13.10/6.49  % (22744)Instructions burned: 432 (million)
% 13.10/6.49  % (22744)------------------------------
% 13.10/6.49  % (22744)------------------------------
% 13.65/6.50  % (22758)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_22586 for (2988ds/140Mi)
% 13.65/6.51  % (22754)Instruction limit reached!
% 13.65/6.51  % (22754)------------------------------
% 13.65/6.51  % (22754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.65/6.51  % (22754)Termination reason: Unknown
% 13.65/6.51  % (22754)Termination phase: shuffling
% 13.65/6.51  
% 13.65/6.51  % (22754)Memory used [KB]: 15735
% 13.65/6.51  % (22754)Time elapsed: 0.058 s
% 13.65/6.51  % (22754)Instructions burned: 150 (million)
% 13.65/6.51  % (22754)------------------------------
% 13.65/6.51  % (22754)------------------------------
% 13.65/6.51  % (22756)Instruction limit reached!
% 13.65/6.51  % (22756)------------------------------
% 13.65/6.51  % (22756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.65/6.51  % (22756)Termination reason: Unknown
% 13.65/6.51  % (22756)Termination phase: shuffling
% 13.65/6.51  
% 13.65/6.51  % (22756)Memory used [KB]: 15351
% 13.65/6.51  % (22756)Time elapsed: 0.049 s
% 13.65/6.51  % (22756)Instructions burned: 111 (million)
% 13.65/6.51  % (22756)------------------------------
% 13.65/6.51  % (22756)------------------------------
% 13.65/6.52  % (22759)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_22586 for (2988ds/614Mi)
% 13.65/6.52  % (22760)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_22586 for (2988ds/202Mi)
% 13.65/6.54  % (22748)Instruction limit reached!
% 13.65/6.54  % (22748)------------------------------
% 13.65/6.54  % (22748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.65/6.54  % (22748)Termination reason: Unknown
% 13.65/6.54  % (22748)Termination phase: shuffling
% 13.65/6.54  
% 13.65/6.54  % (22748)Memory used [KB]: 22131
% 13.65/6.54  % (22748)Time elapsed: 0.138 s
% 13.65/6.54  % (22748)Instructions burned: 453 (million)
% 13.65/6.54  % (22748)------------------------------
% 13.65/6.54  % (22748)------------------------------
% 13.65/6.55  % (22761)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_22586 for (2988ds/339Mi)
% 13.65/6.56  % (22758)Instruction limit reached!
% 13.65/6.56  % (22758)------------------------------
% 13.65/6.56  % (22758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.65/6.56  % (22758)Termination reason: Unknown
% 13.65/6.56  % (22758)Termination phase: shuffling
% 13.65/6.56  
% 13.65/6.56  % (22758)Memory used [KB]: 15607
% 13.65/6.56  % (22758)Time elapsed: 0.055 s
% 13.65/6.56  % (22758)Instructions burned: 141 (million)
% 13.65/6.56  % (22758)------------------------------
% 13.65/6.56  % (22758)------------------------------
% 13.65/6.57  % (22757)Instruction limit reached!
% 13.65/6.57  % (22757)------------------------------
% 13.65/6.57  % (22757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.65/6.57  % (22757)Termination reason: Unknown
% 13.65/6.57  % (22757)Termination phase: shuffling
% 13.65/6.57  
% 13.65/6.57  % (22757)Memory used [KB]: 17398
% 13.65/6.57  % (22757)Time elapsed: 0.094 s
% 13.65/6.57  % (22757)Instructions burned: 254 (million)
% 13.65/6.57  % (22757)------------------------------
% 13.65/6.57  % (22757)------------------------------
% 13.65/6.57  % (22762)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_22586 for (2987ds/283Mi)
% 13.65/6.58  % (22755)Instruction limit reached!
% 13.65/6.58  % (22755)------------------------------
% 13.65/6.58  % (22755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.65/6.58  % (22755)Termination reason: Unknown
% 13.65/6.58  % (22755)Termination phase: shuffling
% 13.65/6.58  
% 13.65/6.58  % (22755)Memory used [KB]: 18421
% 13.65/6.58  % (22755)Time elapsed: 0.122 s
% 13.65/6.58  % (22755)Instructions burned: 343 (million)
% 13.65/6.58  % (22755)------------------------------
% 13.65/6.58  % (22755)------------------------------
% 13.65/6.58  % (22763)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_22586 for (2987ds/134Mi)
% 13.65/6.59  % (22764)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_22586 for (2987ds/298Mi)
% 14.22/6.60  % (22760)Instruction limit reached!
% 14.22/6.60  % (22760)------------------------------
% 14.22/6.60  % (22760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.60  % (22760)Termination reason: Unknown
% 14.22/6.60  % (22760)Termination phase: shuffling
% 14.22/6.60  
% 14.22/6.60  % (22760)Memory used [KB]: 16630
% 14.22/6.60  % (22760)Time elapsed: 0.077 s
% 14.22/6.60  % (22760)Instructions burned: 202 (million)
% 14.22/6.60  % (22760)------------------------------
% 14.22/6.60  % (22760)------------------------------
% 14.22/6.61  % (22765)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_22586 for (2987ds/192Mi)
% 14.22/6.63  % (22763)Instruction limit reached!
% 14.22/6.63  % (22763)------------------------------
% 14.22/6.63  % (22763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.63  % (22763)Termination reason: Unknown
% 14.22/6.63  % (22763)Termination phase: shuffling
% 14.22/6.63  
% 14.22/6.63  % (22763)Memory used [KB]: 15607
% 14.22/6.63  % (22763)Time elapsed: 0.053 s
% 14.22/6.63  % (22763)Instructions burned: 134 (million)
% 14.22/6.63  % (22763)------------------------------
% 14.22/6.63  % (22763)------------------------------
% 14.22/6.64  % (22766)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_22586 for (2987ds/258Mi)
% 14.22/6.66  % (22762)Instruction limit reached!
% 14.22/6.66  % (22762)------------------------------
% 14.22/6.66  % (22762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.66  % (22762)Termination reason: Unknown
% 14.22/6.66  % (22762)Termination phase: shuffling
% 14.22/6.66  
% 14.22/6.66  % (22762)Memory used [KB]: 19829
% 14.22/6.66  % (22762)Time elapsed: 0.090 s
% 14.22/6.66  % (22762)Instructions burned: 285 (million)
% 14.22/6.66  % (22762)------------------------------
% 14.22/6.66  % (22762)------------------------------
% 14.22/6.67  % (22761)Instruction limit reached!
% 14.22/6.67  % (22761)------------------------------
% 14.22/6.67  % (22761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.67  % (22761)Termination reason: Unknown
% 14.22/6.67  % (22761)Termination phase: shuffling
% 14.22/6.67  
% 14.22/6.67  % (22761)Memory used [KB]: 18421
% 14.22/6.67  % (22761)Time elapsed: 0.121 s
% 14.22/6.67  % (22761)Instructions burned: 340 (million)
% 14.22/6.67  % (22761)------------------------------
% 14.22/6.67  % (22761)------------------------------
% 14.22/6.67  % (22767)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_22586 for (2986ds/90Mi)
% 14.22/6.68  % (22768)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_22586 for (2986ds/231Mi)
% 14.22/6.68  % (22765)Instruction limit reached!
% 14.22/6.68  % (22765)------------------------------
% 14.22/6.68  % (22765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.68  % (22765)Termination reason: Unknown
% 14.22/6.68  % (22765)Termination phase: shuffling
% 14.22/6.68  
% 14.22/6.68  % (22765)Memory used [KB]: 16375
% 14.22/6.68  % (22765)Time elapsed: 0.072 s
% 14.22/6.68  % (22765)Instructions burned: 192 (million)
% 14.22/6.68  % (22765)------------------------------
% 14.22/6.68  % (22765)------------------------------
% 14.22/6.68  % (22764)Instruction limit reached!
% 14.22/6.68  % (22764)------------------------------
% 14.22/6.68  % (22764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.68  % (22764)Termination reason: Unknown
% 14.22/6.68  % (22764)Termination phase: shuffling
% 14.22/6.68  
% 14.22/6.68  % (22764)Memory used [KB]: 20084
% 14.22/6.68  % (22764)Time elapsed: 0.094 s
% 14.22/6.68  % (22764)Instructions burned: 300 (million)
% 14.22/6.68  % (22764)------------------------------
% 14.22/6.68  % (22764)------------------------------
% 14.22/6.69  % (22741)Instruction limit reached!
% 14.22/6.69  % (22741)------------------------------
% 14.22/6.69  % (22741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.22/6.69  % (22741)Termination reason: Unknown
% 14.22/6.69  % (22741)Termination phase: Property scanning
% 14.22/6.69  
% 14.22/6.69  % (22741)Memory used [KB]: 27121
% 14.22/6.69  % (22741)Time elapsed: 0.360 s
% 14.22/6.69  % (22741)Instructions burned: 1088 (million)
% 14.22/6.69  % (22741)------------------------------
% 14.22/6.69  % (22741)------------------------------
% 14.22/6.69  % (22769)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_22586 for (2986ds/458Mi)
% 14.22/6.70  % (22770)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_22586 for (2986ds/31Mi)
% 15.17/6.70  % (22771)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_22586 for (2986ds/297Mi)
% 15.17/6.71  % (22767)Instruction limit reached!
% 15.17/6.71  % (22767)------------------------------
% 15.17/6.71  % (22767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.71  % (22767)Termination reason: Unknown
% 15.17/6.71  % (22767)Termination phase: shuffling
% 15.17/6.71  
% 15.17/6.71  % (22767)Memory used [KB]: 14967
% 15.17/6.71  % (22767)Time elapsed: 0.039 s
% 15.17/6.71  % (22767)Instructions burned: 90 (million)
% 15.17/6.71  % (22767)------------------------------
% 15.17/6.71  % (22767)------------------------------
% 15.17/6.71  % (22770)Instruction limit reached!
% 15.17/6.71  % (22770)------------------------------
% 15.17/6.71  % (22770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.71  % (22770)Termination reason: Unknown
% 15.17/6.71  % (22770)Termination phase: shuffling
% 15.17/6.71  
% 15.17/6.71  % (22770)Memory used [KB]: 14072
% 15.17/6.71  % (22770)Time elapsed: 0.019 s
% 15.17/6.71  % (22770)Instructions burned: 31 (million)
% 15.17/6.71  % (22770)------------------------------
% 15.17/6.71  % (22770)------------------------------
% 15.17/6.72  % (22772)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_22586 for (2986ds/987Mi)
% 15.17/6.72  % (22773)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_22586 for (2986ds/182Mi)
% 15.17/6.73  % (22759)Instruction limit reached!
% 15.17/6.73  % (22759)------------------------------
% 15.17/6.73  % (22759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.73  % (22759)Termination reason: Unknown
% 15.17/6.73  % (22759)Termination phase: shuffling
% 15.17/6.73  
% 15.17/6.73  % (22759)Memory used [KB]: 24050
% 15.17/6.73  % (22759)Time elapsed: 0.216 s
% 15.17/6.73  % (22759)Instructions burned: 616 (million)
% 15.17/6.73  % (22759)------------------------------
% 15.17/6.73  % (22759)------------------------------
% 15.17/6.74  % (22766)Instruction limit reached!
% 15.17/6.74  % (22766)------------------------------
% 15.17/6.74  % (22766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.74  % (22766)Termination reason: Unknown
% 15.17/6.74  % (22766)Termination phase: shuffling
% 15.17/6.74  
% 15.17/6.74  % (22766)Memory used [KB]: 17398
% 15.17/6.74  % (22766)Time elapsed: 0.095 s
% 15.17/6.74  % (22766)Instructions burned: 260 (million)
% 15.17/6.74  % (22766)------------------------------
% 15.17/6.74  % (22766)------------------------------
% 15.17/6.74  % (22774)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_22586 for (2986ds/182Mi)
% 15.17/6.75  % (22775)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_22586 for (2986ds/154Mi)
% 15.17/6.75  % (22768)Instruction limit reached!
% 15.17/6.75  % (22768)------------------------------
% 15.17/6.75  % (22768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.75  % (22768)Termination reason: Unknown
% 15.17/6.75  % (22768)Termination phase: shuffling
% 15.17/6.75  
% 15.17/6.75  % (22768)Memory used [KB]: 17014
% 15.17/6.75  % (22768)Time elapsed: 0.074 s
% 15.17/6.75  % (22768)Instructions burned: 231 (million)
% 15.17/6.75  % (22768)------------------------------
% 15.17/6.75  % (22768)------------------------------
% 15.17/6.76  % (22776)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_22586 for (2986ds/79Mi)
% 15.17/6.79  % (22773)Instruction limit reached!
% 15.17/6.79  % (22773)------------------------------
% 15.17/6.79  % (22773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.79  % (22773)Termination reason: Unknown
% 15.17/6.79  % (22773)Termination phase: shuffling
% 15.17/6.79  
% 15.17/6.79  % (22773)Memory used [KB]: 16247
% 15.17/6.79  % (22773)Time elapsed: 0.069 s
% 15.17/6.79  % (22773)Instructions burned: 183 (million)
% 15.17/6.79  % (22773)------------------------------
% 15.17/6.79  % (22773)------------------------------
% 15.17/6.80  % (22771)Instruction limit reached!
% 15.17/6.80  % (22771)------------------------------
% 15.17/6.80  % (22771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.17/6.80  % (22771)Termination reason: Unknown
% 15.17/6.80  % (22771)Termination phase: shuffling
% 15.17/6.80  
% 15.17/6.80  % (22771)Memory used [KB]: 19957
% 15.17/6.80  % (22771)Time elapsed: 0.094 s
% 15.17/6.80  % (22771)Instructions burned: 299 (million)
% 15.73/6.80  % (22771)------------------------------
% 15.73/6.80  % (22771)------------------------------
% 15.73/6.80  % (22776)Instruction limit reached!
% 15.73/6.80  % (22776)------------------------------
% 15.73/6.80  % (22776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.80  % (22776)Termination reason: Unknown
% 15.73/6.80  % (22776)Termination phase: shuffling
% 15.73/6.80  
% 15.73/6.80  % (22776)Memory used [KB]: 14839
% 15.73/6.80  % (22776)Time elapsed: 0.036 s
% 15.73/6.80  % (22776)Instructions burned: 80 (million)
% 15.73/6.80  % (22776)------------------------------
% 15.73/6.80  % (22776)------------------------------
% 15.73/6.80  % (22775)Instruction limit reached!
% 15.73/6.80  % (22775)------------------------------
% 15.73/6.80  % (22775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.80  % (22775)Termination reason: Unknown
% 15.73/6.80  % (22775)Termination phase: shuffling
% 15.73/6.80  
% 15.73/6.80  % (22775)Memory used [KB]: 15863
% 15.73/6.80  % (22775)Time elapsed: 0.052 s
% 15.73/6.80  % (22775)Instructions burned: 154 (million)
% 15.73/6.80  % (22775)------------------------------
% 15.73/6.80  % (22775)------------------------------
% 15.73/6.80  % (22777)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_22586 for (2985ds/216Mi)
% 15.73/6.81  % (22778)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_22586 for (2985ds/114Mi)
% 15.73/6.81  % (22779)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_22586 for (2985ds/922Mi)
% 15.73/6.81  % (22780)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_22586 for (2985ds/182Mi)
% 15.73/6.81  % (22774)Instruction limit reached!
% 15.73/6.81  % (22774)------------------------------
% 15.73/6.81  % (22774)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.81  % (22774)Termination reason: Unknown
% 15.73/6.81  % (22774)Termination phase: shuffling
% 15.73/6.81  
% 15.73/6.81  % (22774)Memory used [KB]: 16375
% 15.73/6.81  % (22774)Time elapsed: 0.071 s
% 15.73/6.81  % (22774)Instructions burned: 183 (million)
% 15.73/6.81  % (22774)------------------------------
% 15.73/6.81  % (22774)------------------------------
% 15.73/6.83  % (22781)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_22586 for (2985ds/164Mi)
% 15.73/6.85  % (22769)Instruction limit reached!
% 15.73/6.85  % (22769)------------------------------
% 15.73/6.85  % (22769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.85  % (22769)Termination reason: Unknown
% 15.73/6.85  % (22769)Termination phase: shuffling
% 15.73/6.85  
% 15.73/6.85  % (22769)Memory used [KB]: 19957
% 15.73/6.85  % (22769)Time elapsed: 0.161 s
% 15.73/6.85  % (22769)Instructions burned: 458 (million)
% 15.73/6.85  % (22769)------------------------------
% 15.73/6.85  % (22769)------------------------------
% 15.73/6.85  % (22778)Instruction limit reached!
% 15.73/6.85  % (22778)------------------------------
% 15.73/6.85  % (22778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.85  % (22778)Termination reason: Unknown
% 15.73/6.85  % (22778)Termination phase: shuffling
% 15.73/6.85  
% 15.73/6.85  % (22778)Memory used [KB]: 15351
% 15.73/6.85  % (22778)Time elapsed: 0.046 s
% 15.73/6.85  % (22778)Instructions burned: 114 (million)
% 15.73/6.85  % (22778)------------------------------
% 15.73/6.85  % (22778)------------------------------
% 15.73/6.86  % (22783)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_22586 for (2984ds/368Mi)
% 15.73/6.86  % (22782)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_22586 for (2984ds/50Mi)
% 15.73/6.87  % (22780)Instruction limit reached!
% 15.73/6.87  % (22780)------------------------------
% 15.73/6.87  % (22780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.87  % (22780)Termination reason: Unknown
% 15.73/6.87  % (22780)Termination phase: shuffling
% 15.73/6.87  
% 15.73/6.87  % (22780)Memory used [KB]: 16375
% 15.73/6.87  % (22780)Time elapsed: 0.060 s
% 15.73/6.87  % (22780)Instructions burned: 185 (million)
% 15.73/6.87  % (22780)------------------------------
% 15.73/6.87  % (22780)------------------------------
% 15.73/6.87  % (22777)Instruction limit reached!
% 15.73/6.87  % (22777)------------------------------
% 15.73/6.87  % (22777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.87  % (22777)Termination reason: Unknown
% 15.73/6.87  % (22777)Termination phase: shuffling
% 15.73/6.87  
% 15.73/6.87  % (22777)Memory used [KB]: 16886
% 15.73/6.87  % (22777)Time elapsed: 0.070 s
% 15.73/6.87  % (22777)Instructions burned: 219 (million)
% 15.73/6.87  % (22777)------------------------------
% 15.73/6.87  % (22777)------------------------------
% 15.73/6.88  % (22781)Instruction limit reached!
% 15.73/6.88  % (22781)------------------------------
% 15.73/6.88  % (22781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.88  % (22781)Termination reason: Unknown
% 15.73/6.88  % (22781)Termination phase: shuffling
% 15.73/6.88  
% 15.73/6.88  % (22781)Memory used [KB]: 16119
% 15.73/6.88  % (22784)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_22586 for (2984ds/1021Mi)
% 15.73/6.88  % (22781)Time elapsed: 0.056 s
% 15.73/6.88  % (22781)Instructions burned: 166 (million)
% 15.73/6.88  % (22781)------------------------------
% 15.73/6.88  % (22781)------------------------------
% 15.73/6.88  % (22785)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_22586 for (2984ds/267Mi)
% 15.73/6.89  % (22782)Instruction limit reached!
% 15.73/6.89  % (22782)------------------------------
% 15.73/6.89  % (22782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.73/6.89  % (22782)Termination reason: Unknown
% 15.73/6.89  % (22782)Termination phase: shuffling
% 15.73/6.89  
% 15.73/6.89  % (22782)Memory used [KB]: 14456
% 15.73/6.89  % (22782)Time elapsed: 0.024 s
% 15.73/6.89  % (22782)Instructions burned: 51 (million)
% 15.73/6.89  % (22782)------------------------------
% 15.73/6.89  % (22782)------------------------------
% 15.73/6.89  % (22786)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_22586 for (2984ds/204Mi)
% 16.51/6.90  % (22787)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_22586 for (2984ds/26Mi)
% 16.51/6.91  % (22787)Instruction limit reached!
% 16.51/6.91  % (22787)------------------------------
% 16.51/6.91  % (22787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/6.91  % (22787)Termination reason: Unknown
% 16.51/6.91  % (22787)Termination phase: shuffling
% 16.51/6.91  
% 16.51/6.91  % (22787)Memory used [KB]: 14200
% 16.51/6.91  % (22787)Time elapsed: 0.018 s
% 16.51/6.91  % (22787)Instructions burned: 28 (million)
% 16.51/6.91  % (22787)------------------------------
% 16.51/6.91  % (22787)------------------------------
% 16.51/6.93  % (22788)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_22586 for (2984ds/53Mi)
% 16.51/6.95  % (22788)Instruction limit reached!
% 16.51/6.95  % (22788)------------------------------
% 16.51/6.95  % (22788)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/6.95  % (22788)Termination reason: Unknown
% 16.51/6.95  % (22788)Termination phase: shuffling
% 16.51/6.95  
% 16.51/6.95  % (22788)Memory used [KB]: 14456
% 16.51/6.95  % (22788)Time elapsed: 0.025 s
% 16.51/6.95  % (22788)Instructions burned: 53 (million)
% 16.51/6.95  % (22788)------------------------------
% 16.51/6.95  % (22788)------------------------------
% 16.51/6.96  % (22789)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_22586 for (2983ds/4436Mi)
% 16.51/6.97  % (22786)Instruction limit reached!
% 16.51/6.97  % (22786)------------------------------
% 16.51/6.97  % (22786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/6.97  % (22786)Termination reason: Unknown
% 16.51/6.97  % (22786)Termination phase: shuffling
% 16.51/6.97  
% 16.51/6.97  % (22786)Memory used [KB]: 16630
% 16.51/6.97  % (22786)Time elapsed: 0.076 s
% 16.51/6.97  % (22786)Instructions burned: 204 (million)
% 16.51/6.97  % (22786)------------------------------
% 16.51/6.97  % (22786)------------------------------
% 16.51/6.97  % (22785)Instruction limit reached!
% 16.51/6.97  % (22785)------------------------------
% 16.51/6.97  % (22785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/6.97  % (22785)Termination reason: Unknown
% 16.51/6.97  % (22785)Termination phase: shuffling
% 16.51/6.97  
% 16.51/6.97  % (22785)Memory used [KB]: 19573
% 16.51/6.97  % (22785)Time elapsed: 0.085 s
% 16.51/6.97  % (22785)Instructions burned: 269 (million)
% 16.51/6.97  % (22785)------------------------------
% 16.51/6.97  % (22785)------------------------------
% 16.51/6.98  % (22790)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_22586 for (2983ds/1539Mi)
% 16.51/6.98  % (22791)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_22586 for (2983ds/408Mi)
% 16.51/6.99  % (22783)Instruction limit reached!
% 16.51/6.99  % (22783)------------------------------
% 16.51/6.99  % (22783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/6.99  % (22783)Termination reason: Unknown
% 16.51/6.99  % (22783)Termination phase: shuffling
% 16.51/6.99  
% 16.51/6.99  % (22783)Memory used [KB]: 18805
% 16.51/6.99  % (22783)Time elapsed: 0.131 s
% 16.51/6.99  % (22783)Instructions burned: 369 (million)
% 16.51/6.99  % (22783)------------------------------
% 16.51/6.99  % (22783)------------------------------
% 16.51/7.01  % (22792)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_22586 for (2983ds/5205Mi)
% 16.51/7.01  % (22772)Instruction limit reached!
% 16.51/7.01  % (22772)------------------------------
% 16.51/7.01  % (22772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.51/7.01  % (22772)Termination reason: Unknown
% 16.51/7.01  % (22772)Termination phase: shuffling
% 16.51/7.01  
% 16.51/7.01  % (22772)Memory used [KB]: 33517
% 16.51/7.01  % (22772)Time elapsed: 0.292 s
% 16.51/7.01  % (22772)Instructions burned: 987 (million)
% 16.51/7.01  % (22772)------------------------------
% 16.51/7.01  % (22772)------------------------------
% 16.51/7.02  % (22793)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_22586 for (2983ds/1913Mi)
% 17.69/7.12  % (22791)Instruction limit reached!
% 17.69/7.12  % (22791)------------------------------
% 17.69/7.12  % (22791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.69/7.12  % (22791)Termination reason: Unknown
% 17.69/7.12  % (22791)Termination phase: shuffling
% 17.69/7.12  
% 17.69/7.12  % (22791)Memory used [KB]: 19445
% 17.69/7.12  % (22791)Time elapsed: 0.145 s
% 17.69/7.12  % (22791)Instructions burned: 409 (million)
% 17.69/7.12  % (22791)------------------------------
% 17.69/7.12  % (22791)------------------------------
% 18.18/7.13  % (22779)Instruction limit reached!
% 18.18/7.13  % (22779)------------------------------
% 18.18/7.13  % (22779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.18/7.13  % (22779)Termination reason: Unknown
% 18.18/7.13  % (22779)Termination phase: Property scanning
% 18.18/7.13  
% 18.18/7.13  % (22779)Memory used [KB]: 27121
% 18.18/7.13  % (22779)Time elapsed: 0.324 s
% 18.18/7.13  % (22779)Instructions burned: 922 (million)
% 18.18/7.13  % (22779)------------------------------
% 18.18/7.13  % (22779)------------------------------
% 18.18/7.13  % (22794)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_22586 for (2982ds/4350Mi)
% 18.18/7.14  % (22795)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_22586 for (2982ds/68Mi)
% 18.18/7.17  % (22795)Instruction limit reached!
% 18.18/7.17  % (22795)------------------------------
% 18.18/7.17  % (22795)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.18/7.17  % (22795)Termination reason: Unknown
% 18.18/7.17  % (22795)Termination phase: shuffling
% 18.18/7.17  
% 18.18/7.17  % (22795)Memory used [KB]: 14711
% 18.18/7.17  % (22795)Time elapsed: 0.032 s
% 18.18/7.17  % (22795)Instructions burned: 70 (million)
% 18.18/7.17  % (22795)------------------------------
% 18.18/7.17  % (22795)------------------------------
% 18.18/7.19  % (22796)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_22586 for (2981ds/358Mi)
% 18.71/7.22  % (22784)Instruction limit reached!
% 18.71/7.22  % (22784)------------------------------
% 18.71/7.22  % (22784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.71/7.22  % (22784)Termination reason: Unknown
% 18.71/7.22  % (22784)Termination phase: Property scanning
% 18.71/7.22  
% 18.71/7.22  % (22784)Memory used [KB]: 27121
% 18.71/7.22  % (22784)Time elapsed: 0.342 s
% 18.71/7.22  % (22784)Instructions burned: 1023 (million)
% 18.71/7.22  % (22784)------------------------------
% 18.71/7.22  % (22784)------------------------------
% 18.71/7.23  % (22797)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_22586 for (2981ds/31Mi)
% 18.71/7.25  % (22797)Instruction limit reached!
% 18.71/7.25  % (22797)------------------------------
% 18.71/7.25  % (22797)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.71/7.25  % (22797)Termination reason: Unknown
% 18.71/7.25  % (22797)Termination phase: shuffling
% 18.71/7.25  
% 18.71/7.25  % (22797)Memory used [KB]: 14072
% 18.71/7.25  % (22797)Time elapsed: 0.019 s
% 18.71/7.25  % (22797)Instructions burned: 31 (million)
% 18.71/7.25  % (22797)------------------------------
% 18.71/7.25  % (22797)------------------------------
% 18.71/7.26  % (22798)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_22586 for (2980ds/279Mi)
% 19.05/7.31  % (22796)Instruction limit reached!
% 19.05/7.31  % (22796)------------------------------
% 19.05/7.31  % (22796)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.05/7.31  % (22796)Termination reason: Unknown
% 19.05/7.31  % (22796)Termination phase: shuffling
% 19.05/7.31  
% 19.05/7.31  % (22796)Memory used [KB]: 18677
% 19.05/7.31  % (22796)Time elapsed: 0.126 s
% 19.05/7.31  % (22796)Instructions burned: 358 (million)
% 19.05/7.31  % (22796)------------------------------
% 19.05/7.31  % (22796)------------------------------
% 19.05/7.32  % (22799)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_22586 for (2980ds/258Mi)
% 19.05/7.35  % (22798)Instruction limit reached!
% 19.05/7.35  % (22798)------------------------------
% 19.05/7.35  % (22798)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.05/7.35  % (22798)Termination reason: Unknown
% 19.05/7.35  % (22798)Termination phase: shuffling
% 19.05/7.35  
% 19.05/7.35  % (22798)Memory used [KB]: 19829
% 19.05/7.35  % (22798)Time elapsed: 0.088 s
% 19.05/7.35  % (22798)Instructions burned: 279 (million)
% 19.05/7.35  % (22798)------------------------------
% 19.05/7.35  % (22798)------------------------------
% 19.05/7.36  % (22800)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_22586 for (2979ds/26Mi)
% 19.05/7.38  % (22800)Instruction limit reached!
% 19.05/7.38  % (22800)------------------------------
% 19.05/7.38  % (22800)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.05/7.38  % (22800)Termination reason: Unknown
% 19.05/7.38  % (22800)Termination phase: shuffling
% 19.05/7.38  
% 19.05/7.38  % (22800)Memory used [KB]: 14200
% 19.05/7.38  % (22800)Time elapsed: 0.017 s
% 19.05/7.38  % (22800)Instructions burned: 26 (million)
% 19.05/7.38  % (22800)------------------------------
% 19.05/7.38  % (22800)------------------------------
% 19.05/7.39  % (22801)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_22586 for (2979ds/995Mi)
% 19.63/7.41  % (22799)Instruction limit reached!
% 19.63/7.41  % (22799)------------------------------
% 19.63/7.41  % (22799)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.63/7.41  % (22799)Termination reason: Unknown
% 19.63/7.41  % (22799)Termination phase: shuffling
% 19.63/7.41  
% 19.63/7.41  % (22799)Memory used [KB]: 17270
% 19.63/7.41  % (22799)Time elapsed: 0.093 s
% 19.63/7.41  % (22799)Instructions burned: 258 (million)
% 19.63/7.41  % (22799)------------------------------
% 19.63/7.41  % (22799)------------------------------
% 19.63/7.42  % (22802)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_22586 for (2979ds/373Mi)
% 19.63/7.46  % (22790)Instruction limit reached!
% 19.63/7.46  % (22790)------------------------------
% 19.63/7.46  % (22790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.63/7.46  % (22790)Termination reason: Unknown
% 19.63/7.46  % (22790)Termination phase: SInE selection
% 19.63/7.46  
% 19.63/7.46  % (22790)Memory used [KB]: 27121
% 19.63/7.46  % (22790)Time elapsed: 0.480 s
% 19.63/7.46  % (22790)Instructions burned: 1539 (million)
% 19.63/7.46  % (22790)------------------------------
% 19.63/7.46  % (22790)------------------------------
% 19.63/7.47  % (22803)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_22586 for (2978ds/253Mi)
% 20.00/7.55  % (22803)Instruction limit reached!
% 20.00/7.55  % (22803)------------------------------
% 20.00/7.55  % (22803)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/7.55  % (22803)Termination reason: Unknown
% 20.00/7.55  % (22803)Termination phase: shuffling
% 20.00/7.55  
% 20.00/7.55  % (22803)Memory used [KB]: 19445
% 20.00/7.55  % (22803)Time elapsed: 0.081 s
% 20.00/7.55  % (22803)Instructions burned: 255 (million)
% 20.00/7.55  % (22803)------------------------------
% 20.00/7.55  % (22803)------------------------------
% 20.00/7.55  % (22802)Instruction limit reached!
% 20.00/7.55  % (22802)------------------------------
% 20.00/7.55  % (22802)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/7.55  % (22802)Termination reason: Unknown
% 20.00/7.55  % (22802)Termination phase: shuffling
% 20.00/7.55  
% 20.00/7.55  % (22802)Memory used [KB]: 18933
% 20.00/7.55  % (22802)Time elapsed: 0.132 s
% 20.00/7.55  % (22802)Instructions burned: 374 (million)
% 20.00/7.55  % (22802)------------------------------
% 20.00/7.55  % (22802)------------------------------
% 20.00/7.56  % (22804)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_22586 for (2977ds/832Mi)
% 20.00/7.57  % (22805)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_22586 for (2977ds/668Mi)
% 20.42/7.62  % (22793)Instruction limit reached!
% 20.42/7.62  % (22793)------------------------------
% 20.42/7.62  % (22793)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.42/7.62  % (22793)Termination reason: Unknown
% 20.42/7.62  % (22793)Termination phase: SInE selection
% 20.42/7.62  
% 20.42/7.62  % (22793)Memory used [KB]: 27888
% 20.42/7.62  % (22793)Time elapsed: 0.598 s
% 20.42/7.62  % (22793)Instructions burned: 1913 (million)
% 20.42/7.62  % (22793)------------------------------
% 20.42/7.62  % (22793)------------------------------
% 20.42/7.63  % (22806)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_22586 for (2977ds/40Mi)
% 20.42/7.65  % (22806)Instruction limit reached!
% 20.42/7.65  % (22806)------------------------------
% 20.42/7.65  % (22806)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.42/7.65  % (22806)Termination reason: Unknown
% 20.42/7.65  % (22806)Termination phase: shuffling
% 20.42/7.65  
% 20.42/7.65  % (22806)Memory used [KB]: 14328
% 20.42/7.65  % (22806)Time elapsed: 0.022 s
% 20.42/7.65  % (22806)Instructions burned: 42 (million)
% 20.42/7.65  % (22806)------------------------------
% 20.42/7.65  % (22806)------------------------------
% 20.42/7.66  % (22807)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_22586 for (2976ds/294Mi)
% 20.42/7.68  % (22801)Instruction limit reached!
% 20.42/7.68  % (22801)------------------------------
% 20.42/7.68  % (22801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.42/7.68  % (22801)Termination reason: Unknown
% 20.42/7.68  % (22801)Termination phase: shuffling
% 20.42/7.68  
% 20.42/7.68  % (22801)Memory used [KB]: 33517
% 20.42/7.68  % (22801)Time elapsed: 0.293 s
% 20.42/7.68  % (22801)Instructions burned: 998 (million)
% 20.42/7.68  % (22801)------------------------------
% 20.42/7.68  % (22801)------------------------------
% 20.42/7.69  % (22808)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_22586 for (2976ds/187Mi)
% 21.57/7.75  % (22807)Instruction limit reached!
% 21.57/7.75  % (22807)------------------------------
% 21.57/7.75  % (22807)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/7.75  % (22807)Termination reason: Unknown
% 21.57/7.75  % (22807)Termination phase: shuffling
% 21.57/7.75  
% 21.57/7.75  % (22807)Memory used [KB]: 19957
% 21.57/7.75  % (22807)Time elapsed: 0.091 s
% 21.57/7.75  % (22807)Instructions burned: 294 (million)
% 21.57/7.75  % (22807)------------------------------
% 21.57/7.75  % (22807)------------------------------
% 21.57/7.76  % (22805)Instruction limit reached!
% 21.57/7.76  % (22805)------------------------------
% 21.57/7.76  % (22805)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/7.76  % (22805)Termination reason: Unknown
% 21.57/7.76  % (22805)Termination phase: shuffling
% 21.57/7.76  
% 21.57/7.76  % (22805)Memory used [KB]: 25202
% 21.57/7.76  % (22805)Time elapsed: 0.196 s
% 21.57/7.76  % (22805)Instructions burned: 669 (million)
% 21.57/7.76  % (22805)------------------------------
% 21.57/7.76  % (22805)------------------------------
% 21.57/7.76  % (22808)Instruction limit reached!
% 21.57/7.76  % (22808)------------------------------
% 21.57/7.76  % (22808)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.57/7.76  % (22808)Termination reason: Unknown
% 21.57/7.76  % (22808)Termination phase: shuffling
% 21.57/7.76  
% 21.57/7.76  % (22808)Memory used [KB]: 16375
% 21.57/7.76  % (22808)Time elapsed: 0.070 s
% 21.57/7.76  % (22808)Instructions burned: 187 (million)
% 21.57/7.76  % (22808)------------------------------
% 21.57/7.76  % (22808)------------------------------
% 21.57/7.76  % (22809)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_22586 for (2975ds/371Mi)
% 21.57/7.77  % (22811)lrs+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:tnu=1:i=70:si=on:rtra=on_0 on DTF2THF_22586 for (2975ds/70Mi)
% 21.57/7.77  % (22810)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_22586 for (2975ds/1827Mi)
% 22.17/7.80  % (22811)Instruction limit reached!
% 22.17/7.80  % (22811)------------------------------
% 22.17/7.80  % (22811)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.17/7.80  % (22811)Termination reason: Unknown
% 22.17/7.80  % (22811)Termination phase: shuffling
% 22.17/7.80  
% 22.17/7.80  % (22811)Memory used [KB]: 14711
% 22.17/7.80  % (22811)Time elapsed: 0.029 s
% 22.17/7.80  % (22811)Instructions burned: 72 (million)
% 22.17/7.80  % (22811)------------------------------
% 22.17/7.80  % (22811)------------------------------
% 22.17/7.81  % (22812)lrs+10_1:20_bd=off:cnfonf=lazy_not_be_gen:fe=axiom:hud=10:ins=1:nm=20:prag=on:sos=on:i=316:si=on:rtra=on_0 on DTF2THF_22586 for (2975ds/316Mi)
% 22.17/7.85  % (22804)Instruction limit reached!
% 22.17/7.85  % (22804)------------------------------
% 22.17/7.85  % (22804)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.17/7.85  % (22804)Termination reason: Unknown
% 22.17/7.85  % (22804)Termination phase: shuffling
% 22.17/7.85  
% 22.17/7.85  % (22804)Memory used [KB]: 27121
% 22.17/7.85  % (22804)Time elapsed: 0.291 s
% 22.17/7.85  % (22804)Instructions burned: 833 (million)
% 22.17/7.85  % (22804)------------------------------
% 22.17/7.85  % (22804)------------------------------
% 22.17/7.86  % (22813)ott+1010_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=3548551,65536:s2a=on:i=932:si=on:rtra=on_0 on DTF2THF_22586 for (2974ds/932Mi)
% 22.17/7.89  % (22809)Instruction limit reached!
% 22.17/7.89  % (22809)------------------------------
% 22.17/7.89  % (22809)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.17/7.89  % (22809)Termination reason: Unknown
% 22.17/7.89  % (22809)Termination phase: shuffling
% 22.17/7.89  
% 22.17/7.89  % (22809)Memory used [KB]: 18805
% 22.17/7.89  % (22809)Time elapsed: 0.130 s
% 22.17/7.89  % (22809)Instructions burned: 372 (million)
% 22.17/7.89  % (22809)------------------------------
% 22.17/7.89  % (22809)------------------------------
% 22.85/7.90  % (22814)lrs+1010_1:128_au=on:bd=preordered:cnfonf=lazy_pi_sigma_gen:plsq=on:plsqc=1:plsqr=32,1:sos=all:sp=occurrence:i=4714:si=on:rtra=on_0 on DTF2THF_22586 for (2974ds/4714Mi)
% 22.85/7.91  % (22812)Instruction limit reached!
% 22.85/7.91  % (22812)------------------------------
% 22.85/7.91  % (22812)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.85/7.91  % (22812)Termination reason: Unknown
% 22.85/7.91  % (22812)Termination phase: shuffling
% 22.85/7.91  
% 22.85/7.91  % (22812)Memory used [KB]: 20340
% 22.85/7.91  % (22812)Time elapsed: 0.098 s
% 22.85/7.91  % (22812)Instructions burned: 319 (million)
% 22.85/7.91  % (22812)------------------------------
% 22.85/7.91  % (22812)------------------------------
% 22.85/7.92  % (22815)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=8250:si=on:rtra=on_0 on DTF2THF_22586 for (2974ds/8250Mi)
% 25.45/8.15  % (22813)Instruction limit reached!
% 25.45/8.15  % (22813)------------------------------
% 25.45/8.15  % (22813)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.45/8.15  % (22813)Termination reason: Unknown
% 25.45/8.15  % (22813)Termination phase: shuffling
% 25.45/8.15  
% 25.45/8.15  % (22813)Memory used [KB]: 32750
% 25.45/8.15  % (22813)Time elapsed: 0.285 s
% 25.45/8.15  % (22813)Instructions burned: 932 (million)
% 25.45/8.15  % (22813)------------------------------
% 25.45/8.15  % (22813)------------------------------
% 25.45/8.16  % (22816)lrs+1002_1:1_apa=on:cnfonf=off:fd=off:flr=on:hud=1:lcm=reverse:prag=on:sd=1:sos=on:ss=axioms:i=770:si=on:rtra=on_0 on DTF2THF_22586 for (2972ds/770Mi)
% 26.50/8.33  % (22789)Instruction limit reached!
% 26.50/8.33  % (22789)------------------------------
% 26.50/8.33  % (22789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.50/8.33  % (22789)Termination reason: Unknown
% 26.50/8.33  % (22789)Termination phase: Saturation
% 26.50/8.33  
% 26.50/8.33  % (22789)Memory used [KB]: 35052
% 26.50/8.33  % (22789)Time elapsed: 1.369 s
% 26.50/8.33  % (22789)Instructions burned: 4438 (million)
% 26.50/8.33  % (22789)------------------------------
% 26.50/8.33  % (22789)------------------------------
% 26.50/8.34  % (22817)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_22586 for (2970ds/794Mi)
% 26.50/8.35  % (22810)Instruction limit reached!
% 26.50/8.35  % (22810)------------------------------
% 26.50/8.35  % (22810)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.50/8.35  % (22810)Termination reason: Unknown
% 26.50/8.35  % (22810)Termination phase: Preprocessing 3
% 26.50/8.35  
% 26.50/8.35  % (22810)Memory used [KB]: 34285
% 26.50/8.35  % (22810)Time elapsed: 0.579 s
% 26.50/8.35  % (22810)Instructions burned: 1829 (million)
% 26.50/8.35  % (22810)------------------------------
% 26.50/8.35  % (22810)------------------------------
% 26.50/8.36  % (22818)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_22586 for (2969ds/464Mi)
% 27.02/8.42  % (22792)Instruction limit reached!
% 27.02/8.42  % (22792)------------------------------
% 27.02/8.42  % (22792)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.02/8.42  % (22792)Termination reason: Unknown
% 27.02/8.42  % (22792)Termination phase: Property scanning
% 27.02/8.42  
% 27.02/8.42  % (22792)Memory used [KB]: 60126
% 27.02/8.42  % (22792)Time elapsed: 1.417 s
% 27.02/8.42  % (22792)Instructions burned: 5205 (million)
% 27.02/8.42  % (22792)------------------------------
% 27.02/8.42  % (22792)------------------------------
% 27.02/8.42  % (22816)Instruction limit reached!
% 27.02/8.42  % (22816)------------------------------
% 27.02/8.42  % (22816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.02/8.42  % (22816)Termination reason: Unknown
% 27.02/8.42  % (22816)Termination phase: shuffling
% 27.02/8.42  
% 27.02/8.42  % (22816)Memory used [KB]: 25713
% 27.02/8.42  % (22816)Time elapsed: 0.264 s
% 27.02/8.42  % (22816)Instructions burned: 771 (million)
% 27.02/8.42  % (22816)------------------------------
% 27.02/8.42  % (22816)------------------------------
% 27.02/8.43  % (22819)ott+21_1:3_avsq=on:bd=off:cnfonf=lazy_gen:fe=off:hud=10:ixr=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=1376:si=on:rtra=on_0 on DTF2THF_22586 for (2969ds/1376Mi)
% 27.02/8.43  % (22794)Refutation not found, non-redundant clauses discarded% (22794)------------------------------
% 27.02/8.43  % (22794)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.02/8.43  % (22794)Termination reason: Refutation not found, non-redundant clauses discarded
% 27.02/8.43  
% 27.02/8.43  % (22794)Memory used [KB]: 34541
% 27.02/8.43  % (22794)Time elapsed: 1.301 s
% 27.02/8.43  % (22794)Instructions burned: 4161 (million)
% 27.02/8.43  % (22794)------------------------------
% 27.02/8.43  % (22794)------------------------------
% 27.02/8.43  % (22820)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_22586 for (2969ds/906Mi)
% 27.02/8.45  % (22821)lrs+1002_1:16_bd=off:e2e=on:fde=unused:kws=precedence:sac=on:sos=on:ss=axioms:i=434:si=on:rtra=on_0 on DTF2THF_22586 for (2968ds/434Mi)
% 27.58/8.52  % (22818)Instruction limit reached!
% 27.58/8.52  % (22818)------------------------------
% 27.58/8.52  % (22818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.58/8.52  % (22818)Termination reason: Unknown
% 27.58/8.52  % (22818)Termination phase: shuffling
% 27.58/8.52  
% 27.58/8.52  % (22818)Memory used [KB]: 20084
% 27.58/8.52  % (22818)Time elapsed: 0.161 s
% 27.58/8.52  % (22818)Instructions burned: 466 (million)
% 27.58/8.52  % (22818)------------------------------
% 27.58/8.52  % (22818)------------------------------
% 27.58/8.54  % (22822)lrs+10_5:4_cnfonf=conj_eager:fe=axiom:hflw=5:hfsq=on:hfsqc=10:hfsql=off:hud=10:plsq=on:sgt=5:spb=goal_then_units:ss=axioms:i=865:si=on:rtra=on_0 on DTF2THF_22586 for (2968ds/865Mi)
% 27.95/8.60  % (22821)Instruction limit reached!
% 27.95/8.60  % (22821)------------------------------
% 27.95/8.60  % (22821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.95/8.60  % (22821)Termination reason: Unknown
% 27.95/8.60  % (22821)Termination phase: shuffling
% 27.95/8.60  
% 27.95/8.60  % (22821)Memory used [KB]: 19829
% 27.95/8.60  % (22821)Time elapsed: 0.153 s
% 27.95/8.60  % (22821)Instructions burned: 434 (million)
% 27.95/8.60  % (22821)------------------------------
% 27.95/8.60  % (22821)------------------------------
% 27.95/8.61  % (22817)Instruction limit reached!
% 27.95/8.61  % (22817)------------------------------
% 27.95/8.61  % (22817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.95/8.61  % (22817)Termination reason: Unknown
% 27.95/8.61  % (22817)Termination phase: shuffling
% 27.95/8.61  
% 27.95/8.61  % (22817)Memory used [KB]: 26353
% 27.95/8.61  % (22817)Time elapsed: 0.272 s
% 27.95/8.61  % (22817)Instructions burned: 795 (million)
% 27.95/8.61  % (22817)------------------------------
% 27.95/8.61  % (22817)------------------------------
% 27.95/8.61  % (22823)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=593:si=on:rtra=on_0 on DTF2THF_22586 for (2967ds/593Mi)
% 27.95/8.62  % (22824)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_22586 for (2967ds/443Mi)
% 28.61/8.74  % (22820)Instruction limit reached!
% 28.61/8.74  % (22820)------------------------------
% 28.61/8.74  % (22820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.61/8.74  % (22820)Termination reason: Unknown
% 28.61/8.74  % (22820)Termination phase: Property scanning
% 28.61/8.74  
% 28.61/8.74  % (22820)Memory used [KB]: 27121
% 28.61/8.74  % (22820)Time elapsed: 0.304 s
% 28.61/8.74  % (22820)Instructions burned: 906 (million)
% 28.61/8.74  % (22820)------------------------------
% 28.61/8.74  % (22820)------------------------------
% 28.61/8.75  % (22825)dis+10_1:1_atotf=0.1:cbe=off:sd=3:ss=axioms:st=2.0:i=594:si=on:rtra=on_0 on DTF2THF_22586 for (2965ds/594Mi)
% 28.61/8.77  % (22824)Instruction limit reached!
% 28.61/8.77  % (22824)------------------------------
% 28.61/8.77  % (22824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.61/8.77  % (22824)Termination reason: Unknown
% 28.61/8.77  % (22824)Termination phase: shuffling
% 28.61/8.77  
% 28.61/8.77  % (22824)Memory used [KB]: 19829
% 28.61/8.77  % (22824)Time elapsed: 0.153 s
% 28.61/8.77  % (22824)Instructions burned: 444 (million)
% 28.61/8.77  % (22824)------------------------------
% 28.61/8.77  % (22824)------------------------------
% 28.61/8.79  % (22822)Instruction limit reached!
% 28.61/8.79  % (22822)------------------------------
% 28.61/8.79  % (22822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.61/8.79  % (22822)Termination reason: Unknown
% 28.61/8.79  % (22822)Termination phase: shuffling
% 28.61/8.79  
% 28.61/8.79  % (22822)Memory used [KB]: 31854
% 28.61/8.79  % (22826)ott+10_1:1024_anc=none:bd=off:e2e=on:fde=none:fsr=off:hud=10:ixr=off:nm=4:nwc=5.0:spb=goal:i=327:si=on:rtra=on_0 on DTF2THF_22586 for (2965ds/327Mi)
% 28.61/8.79  % (22822)Time elapsed: 0.253 s
% 28.61/8.79  % (22822)Instructions burned: 865 (million)
% 28.61/8.79  % (22822)------------------------------
% 28.61/8.79  % (22822)------------------------------
% 28.61/8.79  % (22823)Instruction limit reached!
% 28.61/8.79  % (22823)------------------------------
% 28.61/8.79  % (22823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.61/8.79  % (22823)Termination reason: Unknown
% 28.61/8.79  % (22823)Termination phase: shuffling
% 28.61/8.79  
% 28.61/8.79  % (22823)Memory used [KB]: 24050
% 28.61/8.79  % (22823)Time elapsed: 0.176 s
% 28.61/8.79  % (22823)Instructions burned: 595 (million)
% 28.61/8.79  % (22823)------------------------------
% 28.61/8.79  % (22823)------------------------------
% 29.74/8.80  % (22827)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=5411:si=on:rtra=on_0 on DTF2THF_22586 for (2965ds/5411Mi)
% 29.74/8.80  % (22829)lrs+10_1:20_au=on:c=on:cbe=off:cnfonf=lazy_gen:ins=1:plsq=on:plsqc=1:plsqr=46,31:sac=on:i=1762:si=on:rtra=on_0 on DTF2THF_22586 for (2965ds/1762Mi)
% 29.74/8.83  % (22819)Instruction limit reached!
% 29.74/8.83  % (22819)------------------------------
% 29.74/8.83  % (22819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.74/8.83  % (22819)Termination reason: Unknown
% 29.74/8.83  % (22819)Termination phase: shuffling
% 29.74/8.83  
% 29.74/8.83  % (22819)Memory used [KB]: 38634
% 29.74/8.83  % (22819)Time elapsed: 0.401 s
% 29.74/8.83  % (22819)Instructions burned: 1378 (million)
% 29.74/8.83  % (22819)------------------------------
% 29.74/8.83  % (22819)------------------------------
% 29.74/8.85  % (22830)dis+1003_1:64_cha=on:cnfonf=lazy_gen:hud=9:nwc=8.0:prag=on:i=96:si=on:rtra=on_0 on DTF2THF_22586 for (2965ds/96Mi)
% 29.74/8.88  % (22830)Instruction limit reached!
% 29.74/8.88  % (22830)------------------------------
% 29.74/8.88  % (22830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.74/8.88  % (22830)Termination reason: Unknown
% 29.74/8.88  % (22830)Termination phase: shuffling
% 29.74/8.88  
% 29.74/8.88  % (22830)Memory used [KB]: 15095
% 29.74/8.88  % (22830)Time elapsed: 0.036 s
% 29.74/8.88  % (22830)Instructions burned: 97 (million)
% 29.74/8.88  % (22830)------------------------------
% 29.74/8.88  % (22830)------------------------------
% 29.74/8.89  % (22831)lrs+21_1:1_hud=5:kws=inv_frequency:nm=16:nwc=2.0:prag=on:sos=on:ss=axioms:i=503:si=on:rtra=on_0 on DTF2THF_22586 for (2964ds/503Mi)
% 30.40/8.90  % (22826)Instruction limit reached!
% 30.40/8.90  % (22826)------------------------------
% 30.40/8.90  % (22826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.40/8.90  % (22826)Termination reason: Unknown
% 30.40/8.90  % (22826)Termination phase: shuffling
% 30.40/8.90  
% 30.40/8.90  % (22826)Memory used [KB]: 18293
% 30.40/8.90  % (22826)Time elapsed: 0.118 s
% 30.40/8.90  % (22826)Instructions burned: 329 (million)
% 30.40/8.90  % (22826)------------------------------
% 30.40/8.90  % (22826)------------------------------
% 30.40/8.91  % (22832)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_22586 for (2964ds/90Mi)
% 30.40/8.95  % (22832)Instruction limit reached!
% 30.40/8.95  % (22832)------------------------------
% 30.40/8.95  % (22832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.40/8.95  % (22832)Termination reason: Unknown
% 30.40/8.95  % (22832)Termination phase: shuffling
% 30.40/8.95  
% 30.40/8.95  % (22832)Memory used [KB]: 15095
% 30.40/8.95  % (22832)Time elapsed: 0.039 s
% 30.40/8.95  % (22832)Instructions burned: 92 (million)
% 30.40/8.95  % (22832)------------------------------
% 30.40/8.95  % (22832)------------------------------
% 30.40/8.95  % (22825)Instruction limit reached!
% 30.40/8.95  % (22825)------------------------------
% 30.40/8.95  % (22825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.40/8.95  % (22825)Termination reason: Unknown
% 30.40/8.95  % (22825)Termination phase: shuffling
% 30.40/8.95  
% 30.40/8.95  % (22825)Memory used [KB]: 23666
% 30.40/8.95  % (22825)Time elapsed: 0.205 s
% 30.40/8.95  % (22825)Instructions burned: 594 (million)
% 30.40/8.95  % (22825)------------------------------
% 30.40/8.95  % (22825)------------------------------
% 30.40/8.96  % (22833)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_22586 for (2963ds/593Mi)
% 30.40/8.97  % (22834)dis-1002_1:1_alpa=false:apa=on:au=on:bd=off:cnfonf=off:e2e=on:er=filter:fde=unused:plsq=on:plsqr=9699663,1048576:s2a=on:sos=on:sp=unary_first:i=1512:si=on:rtra=on_0 on DTF2THF_22586 for (2963ds/1512Mi)
% 31.74/9.07  % (22831)Instruction limit reached!
% 31.74/9.07  % (22831)------------------------------
% 31.74/9.07  % (22831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.74/9.07  % (22831)Termination reason: Unknown
% 31.74/9.07  % (22831)Termination phase: shuffling
% 31.74/9.07  
% 31.74/9.07  % (22831)Memory used [KB]: 20596
% 31.74/9.07  % (22831)Time elapsed: 0.175 s
% 31.74/9.07  % (22831)Instructions burned: 504 (million)
% 31.74/9.07  % (22831)------------------------------
% 31.74/9.07  % (22831)------------------------------
% 31.74/9.08  % (22835)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_22586 for (2962ds/569Mi)
% 32.40/9.15  % (22814)Instruction limit reached!
% 32.40/9.15  % (22814)------------------------------
% 32.40/9.15  % (22814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.40/9.15  % (22814)Termination reason: Unknown
% 32.40/9.15  % (22814)Termination phase: Function definition elimination
% 32.40/9.15  
% 32.40/9.15  % (22814)Memory used [KB]: 46694
% 32.40/9.15  % (22814)Time elapsed: 1.249 s
% 32.40/9.15  % (22814)Instructions burned: 4717 (million)
% 32.40/9.15  % (22814)------------------------------
% 32.40/9.15  % (22814)------------------------------
% 32.40/9.17  % (22836)ott+1002_1:1_bd=off:cnfonf=off:e2e=on:fd=off:fe=axiom:hud=5:prag=on:sos=on:i=501:si=on:rtra=on_0 on DTF2THF_22586 for (2961ds/501Mi)
% 32.40/9.17  % (22833)Instruction limit reached!
% 32.40/9.17  % (22833)------------------------------
% 32.40/9.17  % (22833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.40/9.17  % (22833)Termination reason: Unknown
% 32.40/9.17  % (22833)Termination phase: shuffling
% 32.40/9.17  
% 32.40/9.17  % (22833)Memory used [KB]: 23794
% 32.40/9.17  % (22833)Time elapsed: 0.207 s
% 32.40/9.17  % (22833)Instructions burned: 595 (million)
% 32.40/9.17  % (22833)------------------------------
% 32.40/9.17  % (22833)------------------------------
% 32.40/9.18  % (22837)lrs+1010_1:3_cnfonf=conj_eager:fsr=off:prag=on:sd=4:sgt=10:ss=axioms:i=562:si=on:rtra=on_0 on DTF2THF_22586 for (2961ds/562Mi)
% 32.78/9.25  % (22835)Instruction limit reached!
% 32.78/9.25  % (22835)------------------------------
% 32.78/9.25  % (22835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.78/9.25  % (22835)Termination reason: Unknown
% 32.78/9.25  % (22835)Termination phase: shuffling
% 32.78/9.25  
% 32.78/9.25  % (22835)Memory used [KB]: 23794
% 32.78/9.25  % (22835)Time elapsed: 0.169 s
% 32.78/9.25  % (22835)Instructions burned: 570 (million)
% 32.78/9.25  % (22835)------------------------------
% 32.78/9.25  % (22835)------------------------------
% 32.78/9.26  % (22838)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_22586 for (2961ds/406Mi)
% 33.37/9.34  % (22836)Instruction limit reached!
% 33.37/9.34  % (22836)------------------------------
% 33.37/9.34  % (22836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.37/9.34  % (22836)Termination reason: Unknown
% 33.37/9.34  % (22836)Termination phase: shuffling
% 33.37/9.34  
% 33.37/9.34  % (22836)Memory used [KB]: 20468
% 33.37/9.34  % (22836)Time elapsed: 0.174 s
% 33.37/9.34  % (22836)Instructions burned: 502 (million)
% 33.37/9.34  % (22836)------------------------------
% 33.37/9.34  % (22836)------------------------------
% 33.37/9.34  % (22829)Instruction limit reached!
% 33.37/9.34  % (22829)------------------------------
% 33.37/9.34  % (22829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.37/9.34  % (22829)Termination reason: Unknown
% 33.37/9.34  % (22829)Termination phase: Property scanning
% 33.37/9.34  
% 33.37/9.34  % (22829)Memory used [KB]: 42088
% 33.37/9.34  % (22829)Time elapsed: 0.542 s
% 33.37/9.34  % (22829)Instructions burned: 1764 (million)
% 33.37/9.34  % (22829)------------------------------
% 33.37/9.34  % (22829)------------------------------
% 33.37/9.35  % (22837)Instruction limit reached!
% 33.37/9.35  % (22837)------------------------------
% 33.37/9.35  % (22837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.37/9.35  % (22837)Termination reason: Unknown
% 33.37/9.35  % (22837)Termination phase: shuffling
% 33.37/9.35  
% 33.37/9.35  % (22837)Memory used [KB]: 23666
% 33.37/9.35  % (22837)Time elapsed: 0.166 s
% 33.37/9.35  % (22837)Instructions burned: 564 (million)
% 33.37/9.35  % (22837)------------------------------
% 33.37/9.35  % (22837)------------------------------
% 33.37/9.35  % (22839)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_22586 for (2959ds/582Mi)
% 33.37/9.35  % (22840)lrs+2_1:1_au=on:bd=off:bsr=unit_only:sd=2:sgt=32:sos=all:ss=axioms:i=456:si=on:rtra=on_0 on DTF2THF_22586 for (2959ds/456Mi)
% 33.37/9.36  % (22841)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_22586 for (2959ds/393Mi)
% 33.37/9.40  % (22838)Instruction limit reached!
% 33.37/9.40  % (22838)------------------------------
% 33.37/9.40  % (22838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.37/9.40  % (22838)Termination reason: Unknown
% 33.37/9.40  % (22838)Termination phase: shuffling
% 33.37/9.40  
% 33.37/9.40  % (22838)Memory used [KB]: 19061
% 33.37/9.40  % (22838)Time elapsed: 0.141 s
% 33.37/9.40  % (22838)Instructions burned: 406 (million)
% 33.37/9.40  % (22838)------------------------------
% 33.37/9.40  % (22838)------------------------------
% 34.09/9.41  % (22842)lrs+1010_1:6_au=on:cnfonf=conj_eager:nwc=10.0:pe=on:prag=on:s2a=on:ss=axioms:st=1.5:i=1335:si=on:rtra=on_0 on DTF2THF_22586 for (2959ds/1335Mi)
% 34.09/9.44  % (22834)Instruction limit reached!
% 34.09/9.44  % (22834)------------------------------
% 34.09/9.44  % (22834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.09/9.44  % (22834)Termination reason: Unknown
% 34.09/9.44  % (22834)Termination phase: Property scanning
% 34.09/9.44  
% 34.09/9.44  % (22834)Memory used [KB]: 29551
% 34.09/9.44  % (22834)Time elapsed: 0.479 s
% 34.09/9.44  % (22834)Instructions burned: 1512 (million)
% 34.09/9.44  % (22834)------------------------------
% 34.09/9.44  % (22834)------------------------------
% 34.09/9.46  % (22843)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=3373:si=on:rtra=on_0 on DTF2THF_22586 for (2958ds/3373Mi)
% 34.09/9.47  % (22841)Instruction limit reached!
% 34.09/9.47  % (22841)------------------------------
% 34.09/9.47  % (22841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.09/9.47  % (22841)Termination reason: Unknown
% 34.09/9.47  % (22841)Termination phase: shuffling
% 34.09/9.47  
% 34.09/9.47  % (22841)Memory used [KB]: 21364
% 34.09/9.47  % (22841)Time elapsed: 0.118 s
% 34.09/9.47  % (22841)Instructions burned: 394 (million)
% 34.09/9.47  % (22841)------------------------------
% 34.09/9.47  % (22841)------------------------------
% 34.09/9.49  % (22844)dis+10_1:1024_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=32,1:spb=goal_then_units:i=474:si=on:rtra=on_0 on DTF2THF_22586 for (2958ds/474Mi)
% 34.63/9.51  % (22840)Instruction limit reached!
% 34.63/9.51  % (22840)------------------------------
% 34.63/9.51  % (22840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.63/9.51  % (22840)Termination reason: Unknown
% 34.63/9.51  % (22840)Termination phase: shuffling
% 34.63/9.51  
% 34.63/9.51  % (22840)Memory used [KB]: 19957
% 34.63/9.51  % (22840)Time elapsed: 0.157 s
% 34.63/9.51  % (22840)Instructions burned: 456 (million)
% 34.63/9.51  % (22840)------------------------------
% 34.63/9.51  % (22840)------------------------------
% 34.63/9.52  % (22845)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_22586 for (2958ds/157Mi)
% 34.63/9.55  % (22839)Instruction limit reached!
% 34.63/9.55  % (22839)------------------------------
% 34.63/9.55  % (22839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.63/9.55  % (22839)Termination reason: Unknown
% 34.63/9.55  % (22839)Termination phase: shuffling
% 34.63/9.55  
% 34.63/9.55  % (22839)Memory used [KB]: 21875
% 34.63/9.55  % (22839)Time elapsed: 0.205 s
% 34.63/9.55  % (22839)Instructions burned: 584 (million)
% 34.63/9.55  % (22839)------------------------------
% 34.63/9.55  % (22839)------------------------------
% 34.63/9.57  % (22846)dis+1010_1:1_amm=off:cs=on:kws=precedence:nwc=10.0:piset=all:s2a=on:sac=on:sp=unary_frequency:spb=goal:ss=axioms:st=3.0:i=590:si=on:rtra=on_0 on DTF2THF_22586 for (2957ds/590Mi)
% 34.63/9.57  % (22845)Instruction limit reached!
% 34.63/9.57  % (22845)------------------------------
% 34.63/9.57  % (22845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.63/9.57  % (22845)Termination reason: Unknown
% 34.63/9.57  % (22845)Termination phase: shuffling
% 34.63/9.57  
% 34.63/9.57  % (22845)Memory used [KB]: 15991
% 34.63/9.57  % (22845)Time elapsed: 0.053 s
% 34.63/9.57  % (22845)Instructions burned: 159 (million)
% 34.63/9.57  % (22845)------------------------------
% 34.63/9.57  % (22845)------------------------------
% 34.63/9.59  % (22847)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_22586 for (2957ds/104Mi)
% 35.29/9.63  % (22844)Instruction limit reached!
% 35.29/9.63  % (22844)------------------------------
% 35.29/9.63  % (22844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.29/9.63  % (22844)Termination reason: Unknown
% 35.29/9.63  % (22844)Termination phase: shuffling
% 35.29/9.63  
% 35.29/9.63  % (22844)Memory used [KB]: 22515
% 35.29/9.63  % (22844)Time elapsed: 0.141 s
% 35.29/9.63  % (22844)Instructions burned: 474 (million)
% 35.29/9.63  % (22844)------------------------------
% 35.29/9.63  % (22844)------------------------------
% 35.29/9.63  % (22847)Instruction limit reached!
% 35.29/9.63  % (22847)------------------------------
% 35.29/9.63  % (22847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.29/9.63  % (22847)Termination reason: Unknown
% 35.29/9.63  % (22847)Termination phase: shuffling
% 35.29/9.63  
% 35.29/9.63  % (22847)Memory used [KB]: 15223
% 35.29/9.63  % (22847)Time elapsed: 0.043 s
% 35.29/9.63  % (22847)Instructions burned: 106 (million)
% 35.29/9.63  % (22847)------------------------------
% 35.29/9.63  % (22847)------------------------------
% 35.29/9.64  % (22848)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_22586 for (2957ds/414Mi)
% 35.29/9.64  % (22849)dis+10_1:1_av=off:bsd=on:bsr=unit_only:fd=off:hud=8:nm=16:prag=on:i=715:si=on:rtra=on_0 on DTF2THF_22586 for (2957ds/715Mi)
% 35.69/9.76  % (22848)Instruction limit reached!
% 35.69/9.76  % (22848)------------------------------
% 35.69/9.76  % (22848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.69/9.76  % (22848)Termination reason: Unknown
% 35.69/9.76  % (22848)Termination phase: shuffling
% 35.69/9.76  
% 35.69/9.76  % (22848)Memory used [KB]: 21620
% 35.69/9.76  % (22848)Time elapsed: 0.124 s
% 35.69/9.76  % (22848)Instructions burned: 415 (million)
% 35.69/9.76  % (22848)------------------------------
% 35.69/9.76  % (22848)------------------------------
% 35.69/9.77  % (22846)Instruction limit reached!
% 35.69/9.77  % (22846)------------------------------
% 35.69/9.77  % (22846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.69/9.77  % (22846)Termination reason: Unknown
% 35.69/9.77  % (22846)Termination phase: shuffling
% 35.69/9.77  
% 35.69/9.77  % (22846)Memory used [KB]: 23666
% 35.69/9.77  % (22846)Time elapsed: 0.203 s
% 35.69/9.77  % (22846)Instructions burned: 591 (million)
% 35.69/9.77  % (22846)------------------------------
% 35.69/9.77  % (22846)------------------------------
% 35.69/9.77  % (22850)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=961:si=on:rtra=on_0 on DTF2THF_22586 for (2955ds/961Mi)
% 35.69/9.78  % (22851)lrs-1002_1:128_abs=on:bd=preordered:cbe=off:e2e=on:fde=unused:fe=off:tgt=full:i=1213:si=on:rtra=on_0 on DTF2THF_22586 for (2955ds/1213Mi)
% 35.69/9.79  % (22842)Instruction limit reached!
% 35.69/9.79  % (22842)------------------------------
% 35.69/9.79  % (22842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.69/9.79  % (22842)Termination reason: Unknown
% 35.69/9.79  % (22842)Termination phase: shuffling
% 35.69/9.79  
% 35.69/9.79  % (22842)Memory used [KB]: 38123
% 35.69/9.79  % (22842)Time elapsed: 0.387 s
% 35.69/9.79  % (22842)Instructions burned: 1335 (million)
% 35.69/9.79  % (22842)------------------------------
% 35.69/9.79  % (22842)------------------------------
% 36.24/9.81  % (22852)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_22586 for (2955ds/1979Mi)
% 36.24/9.88  % (22849)Instruction limit reached!
% 36.24/9.88  % (22849)------------------------------
% 36.24/9.88  % (22849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.24/9.88  % (22849)Termination reason: Unknown
% 36.24/9.88  % (22849)Termination phase: shuffling
% 36.24/9.88  
% 36.24/9.88  % (22849)Memory used [KB]: 25330
% 36.24/9.88  % (22849)Time elapsed: 0.245 s
% 36.24/9.88  % (22849)Instructions burned: 717 (million)
% 36.24/9.88  % (22849)------------------------------
% 36.24/9.88  % (22849)------------------------------
% 36.24/9.89  % (22853)dis+1002_1:1_amm=off:au=on:bsd=on:cbe=off:cnfonf=lazy_gen:sp=unary_frequency:i=481:si=on:rtra=on_0 on DTF2THF_22586 for (2954ds/481Mi)
% 38.75/10.04  % (22853)Instruction limit reached!
% 38.75/10.04  % (22853)------------------------------
% 38.75/10.04  % (22853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.75/10.04  % (22853)Termination reason: Unknown
% 38.75/10.04  % (22853)Termination phase: shuffling
% 38.75/10.04  
% 38.75/10.04  % (22853)Memory used [KB]: 22515
% 38.75/10.04  % (22853)Time elapsed: 0.144 s
% 38.75/10.04  % (22853)Instructions burned: 483 (million)
% 38.75/10.04  % (22853)------------------------------
% 38.75/10.04  % (22853)------------------------------
% 38.75/10.06  % (22854)dis+10_8:1_apa=on:cnfonf=off:e2e=on:plsq=on:plsqc=1:plsqr=32,1:ss=axioms:i=783:si=on:rtra=on_0 on DTF2THF_22586 for (2953ds/783Mi)
% 38.75/10.09  % (22850)Instruction limit reached!
% 38.75/10.09  % (22850)------------------------------
% 38.75/10.09  % (22850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.75/10.09  % (22850)Termination reason: Unknown
% 38.75/10.09  % (22850)Termination phase: Property scanning
% 38.75/10.09  
% 38.75/10.09  % (22850)Memory used [KB]: 26993
% 38.75/10.09  % (22850)Time elapsed: 0.322 s
% 38.75/10.09  % (22850)Instructions burned: 963 (million)
% 38.75/10.09  % (22850)------------------------------
% 38.75/10.09  % (22850)------------------------------
% 38.75/10.11  % (22855)lrs+1010_1:1_avsq=on:avsqr=1,16:kws=inv_arity:nwc=5.0:sd=1:ss=axioms:i=432:si=on:rtra=on_0 on DTF2THF_22586 for (2952ds/432Mi)
% 39.35/10.17  % (22851)Instruction limit reached!
% 39.35/10.17  % (22851)------------------------------
% 39.35/10.17  % (22851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.35/10.17  % (22851)Termination reason: Unknown
% 39.35/10.17  % (22851)Termination phase: Property scanning
% 39.35/10.17  
% 39.35/10.17  % (22851)Memory used [KB]: 27121
% 39.35/10.17  % (22851)Time elapsed: 0.395 s
% 39.35/10.17  % (22851)Instructions burned: 1215 (million)
% 39.35/10.17  % (22851)------------------------------
% 39.35/10.17  % (22851)------------------------------
% 39.35/10.19  % (22856)lrs+1010_1:1_atotf=0.5:au=on:cbe=off:cnfonf=lazy_not_be_gen:fe=off:nwc=5.0:spb=units:i=1670:si=on:rtra=on_0 on DTF2THF_22586 for (2951ds/1670Mi)
% 39.71/10.25  % (22855)Instruction limit reached!
% 39.71/10.25  % (22855)------------------------------
% 39.71/10.25  % (22855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.71/10.25  % (22855)Termination reason: Unknown
% 39.71/10.25  % (22855)Termination phase: shuffling
% 39.71/10.25  
% 39.71/10.25  % (22855)Memory used [KB]: 19573
% 39.71/10.25  % (22855)Time elapsed: 0.150 s
% 39.71/10.25  % (22855)Instructions burned: 433 (million)
% 39.71/10.25  % (22855)------------------------------
% 39.71/10.25  % (22855)------------------------------
% 39.71/10.27  % (22857)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_22586 for (2950ds/90Mi)
% 40.28/10.30  % (22857)Instruction limit reached!
% 40.28/10.30  % (22857)------------------------------
% 40.28/10.30  % (22857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.28/10.30  % (22857)Termination reason: Unknown
% 40.28/10.30  % (22857)Termination phase: shuffling
% 40.28/10.30  
% 40.28/10.30  % (22857)Memory used [KB]: 15095
% 40.28/10.30  % (22857)Time elapsed: 0.040 s
% 40.28/10.30  % (22857)Instructions burned: 92 (million)
% 40.28/10.30  % (22857)------------------------------
% 40.28/10.30  % (22857)------------------------------
% 40.28/10.32  % (22858)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_22586 for (2950ds/346Mi)
% 40.28/10.33  % (22854)Instruction limit reached!
% 40.28/10.33  % (22854)------------------------------
% 40.28/10.33  % (22854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.28/10.33  % (22854)Termination reason: Unknown
% 40.28/10.33  % (22854)Termination phase: shuffling
% 40.28/10.33  
% 40.28/10.33  % (22854)Memory used [KB]: 25969
% 40.28/10.33  % (22854)Time elapsed: 0.276 s
% 40.28/10.33  % (22854)Instructions burned: 784 (million)
% 40.28/10.33  % (22854)------------------------------
% 40.28/10.33  % (22854)------------------------------
% 40.28/10.34  % (22859)dis+1002_1:1_cnfonf=lazy_not_gen:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=401:si=on:rtra=on_0 on DTF2THF_22586 for (2950ds/401Mi)
% 41.02/10.42  % (22858)Instruction limit reached!
% 41.02/10.42  % (22858)------------------------------
% 41.02/10.42  % (22858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.02/10.42  % (22858)Termination reason: Unknown
% 41.02/10.42  % (22858)Termination phase: shuffling
% 41.02/10.42  
% 41.02/10.42  % (22858)Memory used [KB]: 20724
% 41.02/10.42  % (22858)Time elapsed: 0.106 s
% 41.02/10.42  % (22858)Instructions burned: 348 (million)
% 41.02/10.42  % (22858)------------------------------
% 41.02/10.42  % (22858)------------------------------
% 41.02/10.43  % (22860)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_22586 for (2949ds/6454Mi)
% 41.02/10.45  % (22852)Instruction limit reached!
% 41.02/10.45  % (22852)------------------------------
% 41.02/10.45  % (22852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.02/10.45  % (22852)Termination reason: Unknown
% 41.02/10.45  % (22852)Termination phase: Preprocessing 3
% 41.02/10.45  
% 41.02/10.45  % (22852)Memory used [KB]: 36076
% 41.02/10.45  % (22852)Time elapsed: 0.641 s
% 41.02/10.45  % (22852)Instructions burned: 1979 (million)
% 41.02/10.45  % (22852)------------------------------
% 41.02/10.45  % (22852)------------------------------
% 41.02/10.45  % (22843)Instruction limit reached!
% 41.02/10.45  % (22843)------------------------------
% 41.02/10.45  % (22843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.02/10.45  % (22843)Termination reason: Unknown
% 41.02/10.45  % (22843)Termination phase: Function definition elimination
% 41.02/10.45  
% 41.02/10.45  % (22843)Memory used [KB]: 45926
% 41.02/10.45  % (22843)Time elapsed: 0.999 s
% 41.02/10.45  % (22843)Instructions burned: 3373 (million)
% 41.02/10.45  % (22843)------------------------------
% 41.02/10.45  % (22843)------------------------------
% 41.02/10.46  % (22861)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:kws=arity:prag=on:i=283:si=on:rtra=on_0 on DTF2THF_22586 for (2948ds/283Mi)
% 41.02/10.46  % (22859)Instruction limit reached!
% 41.02/10.46  % (22859)------------------------------
% 41.02/10.46  % (22859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.02/10.46  % (22859)Termination reason: Unknown
% 41.02/10.46  % (22859)Termination phase: shuffling
% 41.02/10.46  
% 41.02/10.46  % (22859)Memory used [KB]: 21492
% 41.02/10.46  % (22859)Time elapsed: 0.122 s
% 41.02/10.46  % (22859)Instructions burned: 402 (million)
% 41.02/10.46  % (22859)------------------------------
% 41.02/10.46  % (22859)------------------------------
% 41.02/10.47  % (22862)ott+10_1:1_aac=none:apa=on:au=on:cnfonf=off:ntd=on:nwc=12.0:sos=on:i=865:si=on:rtra=on_0 on DTF2THF_22586 for (2948ds/865Mi)
% 41.02/10.47  % (22863)lrs+1002_3:4_au=on:av=off:cnfonf=off:fd=off:fsr=off:ins=2:sd=1:sos=on:ss=axioms:i=4528:si=on:rtra=on_0 on DTF2THF_22586 for (2948ds/4528Mi)
% 41.02/10.48  % (22815)Instruction limit reached!
% 41.02/10.48  % (22815)------------------------------
% 41.02/10.48  % (22815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.02/10.48  % (22815)Termination reason: Unknown
% 41.02/10.48  % (22815)Termination phase: Saturation
% 41.02/10.48  
% 41.02/10.48  % (22815)Memory used [KB]: 58719
% 41.02/10.48  % (22815)Time elapsed: 2.566 s
% 41.02/10.48  % (22815)Instructions burned: 8251 (million)
% 41.02/10.48  % (22815)------------------------------
% 41.02/10.48  % (22815)------------------------------
% 41.60/10.50  % (22864)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=1001:si=on:rtra=on_0 on DTF2THF_22586 for (2948ds/1001Mi)
% 41.60/10.56  % (22827)Instruction limit reached!
% 41.60/10.56  % (22827)------------------------------
% 41.60/10.56  % (22827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.60/10.56  % (22827)Termination reason: Unknown
% 41.60/10.56  % (22827)Termination phase: Saturation
% 41.60/10.56  
% 41.60/10.56  % (22827)Memory used [KB]: 36332
% 41.60/10.56  % (22827)Time elapsed: 1.758 s
% 41.60/10.56  % (22827)Instructions burned: 5413 (million)
% 41.60/10.56  % (22827)------------------------------
% 41.60/10.56  % (22827)------------------------------
% 41.60/10.56  % (22861)Instruction limit reached!
% 41.60/10.56  % (22861)------------------------------
% 41.60/10.56  % (22861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.60/10.56  % (22861)Termination reason: Unknown
% 41.60/10.56  % (22861)Termination phase: shuffling
% 41.60/10.56  
% 41.60/10.56  % (22861)Memory used [KB]: 17654
% 41.60/10.56  % (22861)Time elapsed: 0.102 s
% 41.60/10.56  % (22861)Instructions burned: 284 (million)
% 41.60/10.56  % (22861)------------------------------
% 41.60/10.56  % (22861)------------------------------
% 41.60/10.57  % (22865)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_22586 for (2947ds/102Mi)
% 41.60/10.57  % (22866)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_22586 for (2947ds/952Mi)
% 41.78/10.60  % (22865)Instruction limit reached!
% 41.78/10.60  % (22865)------------------------------
% 41.78/10.60  % (22865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.78/10.60  % (22865)Termination reason: Unknown
% 41.78/10.61  % (22865)Termination phase: shuffling
% 41.78/10.61  
% 41.78/10.61  % (22865)Memory used [KB]: 15223
% 41.78/10.61  % (22865)Time elapsed: 0.038 s
% 41.78/10.61  % (22865)Instructions burned: 103 (million)
% 41.78/10.61  % (22865)------------------------------
% 41.78/10.61  % (22865)------------------------------
% 41.78/10.62  % (22867)lrs+1010_1:1024_add=large:c=on:cbe=off:cnfonf=lazy_gen:fde=none:nm=30:piset=and:sac=on:uhcvi=on:i=578:si=on:rtra=on_0 on DTF2THF_22586 for (2947ds/578Mi)
% 41.78/10.67  % (22856)Instruction limit reached!
% 41.78/10.67  % (22856)------------------------------
% 41.78/10.67  % (22856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.78/10.67  % (22856)Termination reason: Unknown
% 41.78/10.67  % (22856)Termination phase: Property scanning
% 41.78/10.67  
% 41.78/10.67  % (22856)Memory used [KB]: 42088
% 41.78/10.67  % (22856)Time elapsed: 0.485 s
% 41.78/10.67  % (22856)Instructions burned: 1673 (million)
% 41.78/10.67  % (22856)------------------------------
% 41.78/10.67  % (22856)------------------------------
% 41.78/10.68  % (22868)dis+1010_1:4_avsq=on:avsqr=16,1:bet=on:cbe=off:cnfonf=lazy_not_be_gen:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=643:si=on:rtra=on_0 on DTF2THF_22586 for (2946ds/643Mi)
% 42.33/10.76  % (22862)Instruction limit reached!
% 42.33/10.76  % (22862)------------------------------
% 42.33/10.76  % (22862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.33/10.76  % (22862)Termination reason: Unknown
% 42.33/10.76  % (22862)Termination phase: shuffling
% 42.33/10.76  
% 42.33/10.76  % (22862)Memory used [KB]: 26865
% 42.33/10.76  % (22862)Time elapsed: 0.292 s
% 42.33/10.76  % (22862)Instructions burned: 865 (million)
% 42.33/10.76  % (22862)------------------------------
% 42.33/10.76  % (22862)------------------------------
% 42.33/10.77  % (22869)lrs+1010_1:1_cnfonf=off:ntd=on:nwc=3.0:s2a=on:sp=const_frequency:i=865:si=on:rtra=on_0 on DTF2THF_22586 for (2945ds/865Mi)
% 42.33/10.78  % (22867)Instruction limit reached!
% 42.33/10.78  % (22867)------------------------------
% 42.33/10.78  % (22867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.33/10.78  % (22867)Termination reason: Unknown
% 42.33/10.78  % (22867)Termination phase: shuffling
% 42.33/10.78  
% 42.33/10.78  % (22867)Memory used [KB]: 23922
% 42.33/10.78  % (22867)Time elapsed: 0.170 s
% 42.33/10.78  % (22867)Instructions burned: 580 (million)
% 42.33/10.78  % (22867)------------------------------
% 42.33/10.78  % (22867)------------------------------
% 42.82/10.80  % (22870)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_22586 for (2945ds/1174Mi)
% 42.82/10.83  % (22864)Instruction limit reached!
% 42.82/10.83  % (22864)------------------------------
% 42.82/10.83  % (22864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.82/10.83  % (22864)Termination reason: Unknown
% 42.82/10.83  % (22864)Termination phase: Property scanning
% 42.82/10.83  
% 42.82/10.83  % (22864)Memory used [KB]: 26993
% 42.82/10.83  % (22864)Time elapsed: 0.329 s
% 42.82/10.83  % (22864)Instructions burned: 1002 (million)
% 42.82/10.83  % (22864)------------------------------
% 42.82/10.83  % (22864)------------------------------
% 42.82/10.84  % (22871)ott+1010_1:32_apa=on:atotf=0.3:au=on:c=on:cnfonf=off:fe=off:fsr=off:nm=6:piset=or:sac=on:sos=on:i=644:si=on:rtra=on_0 on DTF2THF_22586 for (2945ds/644Mi)
% 42.82/10.87  % (22868)Instruction limit reached!
% 42.82/10.87  % (22868)------------------------------
% 42.82/10.87  % (22868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.82/10.87  % (22868)Termination reason: Unknown
% 42.82/10.87  % (22868)Termination phase: shuffling
% 42.82/10.87  
% 42.82/10.87  % (22868)Memory used [KB]: 24818
% 42.82/10.87  % (22868)Time elapsed: 0.188 s
% 42.82/10.87  % (22868)Instructions burned: 643 (million)
% 42.82/10.87  % (22868)------------------------------
% 42.82/10.87  % (22868)------------------------------
% 42.82/10.88  % (22872)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_22586 for (2944ds/1201Mi)
% 42.82/10.89  % (22866)Instruction limit reached!
% 42.82/10.89  % (22866)------------------------------
% 42.82/10.89  % (22866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.82/10.89  % (22866)Termination reason: Unknown
% 42.82/10.89  % (22866)Termination phase: Property scanning
% 42.82/10.89  
% 42.82/10.89  % (22866)Memory used [KB]: 26993
% 42.82/10.89  % (22866)Time elapsed: 0.317 s
% 42.82/10.89  % (22866)Instructions burned: 953 (million)
% 42.82/10.89  % (22866)------------------------------
% 42.82/10.89  % (22866)------------------------------
% 43.69/10.90  % (22873)lrs+1666_1:64_cnfonf=lazy_simp:hud=5:plsq=on:plsql=on:plsqr=152565,524288:tnu=2:i=1046:si=on:rtra=on_0 on DTF2THF_22586 for (2944ds/1046Mi)
% 45.67/11.06  % (22871)Instruction limit reached!
% 45.67/11.06  % (22871)------------------------------
% 45.67/11.06  % (22871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.67/11.06  % (22871)Termination reason: Unknown
% 45.67/11.06  % (22871)Termination phase: shuffling
% 45.67/11.06  
% 45.67/11.06  % (22871)Memory used [KB]: 24178
% 45.67/11.06  % (22871)Time elapsed: 0.219 s
% 45.67/11.06  % (22871)Instructions burned: 644 (million)
% 45.67/11.06  % (22871)------------------------------
% 45.67/11.06  % (22871)------------------------------
% 45.67/11.06  % (22869)Instruction limit reached!
% 45.67/11.06  % (22869)------------------------------
% 45.67/11.06  % (22869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.67/11.06  % (22869)Termination reason: Unknown
% 45.67/11.06  % (22869)Termination phase: shuffling
% 45.67/11.06  
% 45.67/11.06  % (22869)Memory used [KB]: 26865
% 45.67/11.06  % (22869)Time elapsed: 0.293 s
% 45.67/11.06  % (22869)Instructions burned: 865 (million)
% 45.67/11.06  % (22869)------------------------------
% 45.67/11.06  % (22869)------------------------------
% 45.67/11.07  % (22874)lrs+1010_1:1_c=on:cnfonf=lazy_simp:fsr=off:hud=5:kws=arity:nm=32:plsq=on:plsqc=1:plsqr=642,29:prag=on:i=222:si=on:rtra=on_0 on DTF2THF_22586 for (2942ds/222Mi)
% 45.67/11.07  % (22875)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_22586 for (2942ds/68Mi)
% 45.67/11.10  % (22875)Instruction limit reached!
% 45.67/11.10  % (22875)------------------------------
% 45.67/11.10  % (22875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.67/11.10  % (22875)Termination reason: Unknown
% 45.67/11.10  % (22875)Termination phase: shuffling
% 45.67/11.10  
% 45.67/11.10  % (22875)Memory used [KB]: 14711
% 45.67/11.10  % (22875)Time elapsed: 0.032 s
% 45.67/11.10  % (22875)Instructions burned: 68 (million)
% 45.67/11.10  % (22875)------------------------------
% 45.67/11.10  % (22875)------------------------------
% 46.33/11.12  % (22876)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=454:si=on:rtra=on_0 on DTF2THF_22586 for (2942ds/454Mi)
% 46.33/11.14  % (22874)Instruction limit reached!
% 46.33/11.14  % (22874)------------------------------
% 46.33/11.14  % (22874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.33/11.14  % (22874)Termination reason: Unknown
% 46.33/11.14  % (22874)Termination phase: shuffling
% 46.33/11.14  
% 46.33/11.14  % (22874)Memory used [KB]: 17014
% 46.33/11.14  % (22874)Time elapsed: 0.075 s
% 46.33/11.14  % (22874)Instructions burned: 224 (million)
% 46.33/11.14  % (22874)------------------------------
% 46.33/11.14  % (22874)------------------------------
% 46.33/11.15  % (22877)dis+1002_1:1_cnfonf=lazy_simp:hud=5:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=360:si=on:rtra=on_0 on DTF2THF_22586 for (2941ds/360Mi)
% 46.33/11.18  % (22870)Instruction limit reached!
% 46.33/11.18  % (22870)------------------------------
% 46.33/11.18  % (22870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.33/11.18  % (22870)Termination reason: Unknown
% 46.33/11.18  % (22870)Termination phase: Property scanning
% 46.33/11.18  
% 46.33/11.18  % (22870)Memory used [KB]: 26993
% 46.33/11.18  % (22870)Time elapsed: 0.380 s
% 46.33/11.18  % (22870)Instructions burned: 1175 (million)
% 46.33/11.18  % (22870)------------------------------
% 46.33/11.18  % (22870)------------------------------
% 46.33/11.19  % (22878)lrs+1002_1:16_amm=sco:cnfonf=lazy_pi_sigma_gen:fe=off:nm=60:piset=all_but_not_eq:plsq=on:plsqc=1:plsqr=2,1:rawr=on:sp=occurrence:ss=included:st=-1.0:i=157:si=on:rtra=on_0 on DTF2THF_22586 for (2941ds/157Mi)
% 46.84/11.21  % (22873)Instruction limit reached!
% 46.84/11.21  % (22873)------------------------------
% 46.84/11.21  % (22873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.84/11.21  % (22873)Termination reason: Unknown
% 46.84/11.21  % (22873)Termination phase: shuffling
% 46.84/11.21  
% 46.84/11.21  % (22873)Memory used [KB]: 34285
% 46.84/11.21  % (22873)Time elapsed: 0.311 s
% 46.84/11.21  % (22873)Instructions burned: 1049 (million)
% 46.84/11.21  % (22873)------------------------------
% 46.84/11.21  % (22873)------------------------------
% 46.84/11.22  % (22879)ott+1004_1:24_anc=none:cbe=off:cnfonf=lazy_not_gen:fe=off:piset=not:prag=on:rp=on:i=5061:si=on:rtra=on_0 on DTF2THF_22586 for (2941ds/5061Mi)
% 46.84/11.24  % (22878)Instruction limit reached!
% 46.84/11.24  % (22878)------------------------------
% 46.84/11.24  % (22878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.84/11.24  % (22878)Termination reason: Unknown
% 46.84/11.24  % (22878)Termination phase: shuffling
% 46.84/11.24  
% 46.84/11.24  % (22878)Memory used [KB]: 15991
% 46.84/11.24  % (22878)Time elapsed: 0.052 s
% 46.84/11.24  % (22878)Instructions burned: 157 (million)
% 46.84/11.24  % (22878)------------------------------
% 46.84/11.24  % (22878)------------------------------
% 46.84/11.25  % (22880)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_22586 for (2941ds/2401Mi)
% 46.84/11.25  % (22876)Instruction limit reached!
% 46.84/11.25  % (22876)------------------------------
% 46.84/11.25  % (22876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.84/11.25  % (22876)Termination reason: Unknown
% 46.84/11.25  % (22876)Termination phase: shuffling
% 46.84/11.25  
% 46.84/11.25  % (22876)Memory used [KB]: 22259
% 46.84/11.25  % (22876)Time elapsed: 0.138 s
% 46.84/11.25  % (22876)Instructions burned: 457 (million)
% 46.84/11.25  % (22876)------------------------------
% 46.84/11.25  % (22876)------------------------------
% 46.84/11.26  % (22877)Instruction limit reached!
% 46.84/11.26  % (22877)------------------------------
% 46.84/11.26  % (22877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.84/11.26  % (22877)Termination reason: Unknown
% 46.84/11.26  % (22877)Termination phase: shuffling
% 46.84/11.26  
% 46.84/11.26  % (22877)Memory used [KB]: 20980
% 46.84/11.26  % (22877)Time elapsed: 0.111 s
% 46.84/11.26  % (22877)Instructions burned: 363 (million)
% 46.84/11.26  % (22877)------------------------------
% 46.84/11.26  % (22877)------------------------------
% 46.84/11.27  % (22881)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_22586 for (2940ds/5798Mi)
% 46.84/11.27  % (22872)Instruction limit reached!
% 46.84/11.27  % (22872)------------------------------
% 46.84/11.27  % (22872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.84/11.27  % (22872)Termination reason: Unknown
% 46.84/11.27  % (22872)Termination phase: Property scanning
% 46.84/11.27  
% 46.84/11.27  % (22872)Memory used [KB]: 26993
% 46.84/11.27  % (22872)Time elapsed: 0.394 s
% 46.84/11.27  % (22872)Instructions burned: 1203 (million)
% 46.84/11.27  % (22872)------------------------------
% 46.84/11.27  % (22872)------------------------------
% 46.84/11.28  % (22882)dis+10_1:1_au=on:cnfonf=lazy_gen:ixr=off:plsq=on:plsqc=1:plsqr=32,1:i=346:si=on:rtra=on_0 on DTF2THF_22586 for (2940ds/346Mi)
% 46.84/11.29  % (22883)lrs+1002_1:1_av=off:bd=off:bet=on:e2e=on:fde=unused:fe=off:plsq=on:prag=on:sos=on:i=1321:si=on:rtra=on_0 on DTF2THF_22586 for (2940ds/1321Mi)
% 47.47/11.38  % (22882)Instruction limit reached!
% 47.47/11.38  % (22882)------------------------------
% 47.47/11.38  % (22882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.47/11.38  % (22882)Termination reason: Unknown
% 47.47/11.38  % (22882)Termination phase: shuffling
% 47.47/11.38  
% 47.47/11.38  % (22882)Memory used [KB]: 20724
% 47.47/11.38  % (22882)Time elapsed: 0.106 s
% 47.47/11.38  % (22882)Instructions burned: 347 (million)
% 47.47/11.38  % (22882)------------------------------
% 47.47/11.38  % (22882)------------------------------
% 47.47/11.39  % (22884)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_22586 for (2939ds/1312Mi)
% 48.41/11.70  % (22883)Instruction limit reached!
% 48.41/11.70  % (22883)------------------------------
% 48.41/11.70  % (22883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.41/11.70  % (22883)Termination reason: Unknown
% 48.41/11.70  % (22883)Termination phase: Property scanning
% 48.41/11.70  
% 48.41/11.70  % (22883)Memory used [KB]: 27121
% 48.41/11.70  % (22883)Time elapsed: 0.417 s
% 48.41/11.70  % (22883)Instructions burned: 1323 (million)
% 48.41/11.70  % (22883)------------------------------
% 48.41/11.70  % (22883)------------------------------
% 48.41/11.71  % (22885)ott+2_8:1_add=large:c=on:cnfonf=off:e2e=on:er=filter:hud=5:plsq=on:plsqr=1,1:ss=axioms:i=672:si=on:rtra=on_0 on DTF2THF_22586 for (2936ds/672Mi)
% 48.41/11.77  % (22884)Instruction limit reached!
% 48.41/11.77  % (22884)------------------------------
% 48.41/11.77  % (22884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.41/11.77  % (22884)Termination reason: Unknown
% 48.41/11.77  % (22884)Termination phase: shuffling
% 48.41/11.77  
% 48.41/11.77  % (22884)Memory used [KB]: 37739
% 48.41/11.77  % (22884)Time elapsed: 0.379 s
% 48.41/11.77  % (22884)Instructions burned: 1314 (million)
% 48.41/11.77  % (22884)------------------------------
% 48.41/11.77  % (22884)------------------------------
% 48.41/11.78  % (22886)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=1897:si=on:rtra=on_0 on DTF2THF_22586 for (2935ds/1897Mi)
% 49.42/11.89  % (22863)Instruction limit reached!
% 49.42/11.89  % (22863)------------------------------
% 49.42/11.89  % (22863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.42/11.89  % (22863)Termination reason: Unknown
% 49.42/11.89  % (22863)Termination phase: Saturation
% 49.42/11.89  
% 49.42/11.89  % (22863)Memory used [KB]: 33133
% 49.42/11.89  % (22863)Time elapsed: 1.416 s
% 49.42/11.89  % (22863)Instructions burned: 4528 (million)
% 49.42/11.89  % (22863)------------------------------
% 49.42/11.89  % (22863)------------------------------
% 49.63/11.90  % (22887)lrs+10_1:40_av=off:fde=unused:ntd=on:plsq=on:plsqc=2:plsqr=411915,1048576:sp=unary_frequency:tgt=full:i=1202:si=on:rtra=on_0 on DTF2THF_22586 for (2934ds/1202Mi)
% 49.63/11.94  % (22885)Instruction limit reached!
% 49.63/11.94  % (22885)------------------------------
% 49.63/11.94  % (22885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.63/11.94  % (22885)Termination reason: Unknown
% 49.63/11.94  % (22885)Termination phase: shuffling
% 49.63/11.94  
% 49.63/11.94  % (22885)Memory used [KB]: 24562
% 49.63/11.94  % (22885)Time elapsed: 0.232 s
% 49.63/11.94  % (22885)Instructions burned: 673 (million)
% 49.63/11.94  % (22885)------------------------------
% 49.63/11.94  % (22885)------------------------------
% 49.63/11.96  % (22888)dis+21_1:1_au=on:cnfonf=lazy_pi_sigma_gen:hfsq=on:hfsqc=40:hfsqr=8,1:ntd=on:sos=on:sp=occurrence:ss=axioms:i=1493:si=on:rtra=on_0 on DTF2THF_22586 for (2934ds/1493Mi)
% 49.63/11.98  % (22880)Instruction limit reached!
% 49.63/11.98  % (22880)------------------------------
% 49.63/11.98  % (22880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.63/11.98  % (22880)Termination reason: Unknown
% 49.63/11.98  % (22880)Termination phase: Property scanning
% 49.63/11.98  
% 49.63/11.98  % (22880)Memory used [KB]: 37739
% 49.63/11.98  % (22880)Time elapsed: 0.733 s
% 49.63/11.98  % (22880)Instructions burned: 2403 (million)
% 49.63/11.98  % (22880)------------------------------
% 49.63/11.98  % (22880)------------------------------
% 49.63/11.99  % (22889)dis+1010_3:1_av=off:fde=unused:hud=10:nwc=10.0:s2a=on:ss=axioms:i=7679:si=on:rtra=on_0 on DTF2THF_22586 for (2933ds/7679Mi)
% 54.68/12.31  % (22887)Instruction limit reached!
% 54.68/12.31  % (22887)------------------------------
% 54.68/12.31  % (22887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.68/12.31  % (22887)Termination reason: Unknown
% 54.68/12.31  % (22887)Termination phase: Property scanning
% 54.68/12.31  
% 54.68/12.31  % (22887)Memory used [KB]: 26993
% 54.68/12.31  % (22887)Time elapsed: 0.408 s
% 54.68/12.31  % (22887)Instructions burned: 1203 (million)
% 54.68/12.31  % (22887)------------------------------
% 54.68/12.31  % (22887)------------------------------
% 54.68/12.32  % (22890)lrs+2_1:1024_anc=all:cnfonf=off:cs=on:e2e=on:nm=20:s2at=1.5:slsq=on:slsqr=1,2:spb=goal:i=340:si=on:rtra=on_0 on DTF2THF_22586 for (2930ds/340Mi)
% 54.68/12.38  % (22886)Instruction limit reached!
% 54.68/12.38  % (22886)------------------------------
% 54.68/12.38  % (22886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.68/12.38  % (22886)Termination reason: Unknown
% 54.68/12.38  % (22886)Termination phase: Property scanning
% 54.68/12.38  
% 54.68/12.38  % (22886)Memory used [KB]: 32238
% 54.68/12.38  % (22886)Time elapsed: 0.597 s
% 54.68/12.38  % (22886)Instructions burned: 1897 (million)
% 54.68/12.38  % (22886)------------------------------
% 54.68/12.38  % (22886)------------------------------
% 54.68/12.39  % (22888)Instruction limit reached!
% 54.68/12.39  % (22888)------------------------------
% 54.68/12.39  % (22888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.68/12.39  % (22888)Termination reason: Unknown
% 54.68/12.39  % (22888)Termination phase: shuffling
% 54.68/12.39  
% 54.68/12.39  % (22888)Memory used [KB]: 40169
% 54.68/12.39  % (22888)Time elapsed: 0.431 s
% 54.68/12.39  % (22888)Instructions burned: 1494 (million)
% 54.68/12.39  % (22888)------------------------------
% 54.68/12.39  % (22888)------------------------------
% 54.68/12.39  % (22891)dis+1002_1:1_acc=model:au=on:nwc=5.0:sd=5:ss=axioms:st=2.5:i=3672:si=on:rtra=on_0 on DTF2THF_22586 for (2929ds/3672Mi)
% 55.29/12.40  % (22892)dis-1002_3:1_acc=model:bd=off:bsd=on:bsdmm=1:cnfonf=lazy_not_be_gen:fd=off:fe=off:ntd=on:nwc=10.0:piset=equals:plsq=on:plsqc=1:plsqr=32,1:sp=const_max:uhcvi=on:i=1535:si=on:rtra=on_0 on DTF2THF_22586 for (2929ds/1535Mi)
% 55.29/12.44  % (22890)Instruction limit reached!
% 55.29/12.44  % (22890)------------------------------
% 55.29/12.44  % (22890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.29/12.44  % (22890)Termination reason: Unknown
% 55.29/12.44  % (22890)Termination phase: shuffling
% 55.29/12.44  
% 55.29/12.44  % (22890)Memory used [KB]: 18421
% 55.29/12.44  % (22890)Time elapsed: 0.121 s
% 55.29/12.44  % (22890)Instructions burned: 342 (million)
% 55.29/12.44  % (22890)------------------------------
% 55.29/12.44  % (22890)------------------------------
% 55.29/12.45  % (22860)Instruction limit reached!
% 55.29/12.45  % (22860)------------------------------
% 55.29/12.45  % (22860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.29/12.45  % (22860)Termination reason: Unknown
% 55.29/12.45  % (22860)Termination phase: Saturation
% 55.29/12.45  
% 55.29/12.45  % (22860)Memory used [KB]: 36204
% 55.29/12.45  % (22860)Time elapsed: 2.014 s
% 55.29/12.45  % (22860)Instructions burned: 6455 (million)
% 55.29/12.45  % (22860)------------------------------
% 55.29/12.45  % (22860)------------------------------
% 55.29/12.45  % (22893)dis+10_16:1_hud=1:plsq=on:plsqc=1:plsqr=32,1:prag=on:sd=1:sgt=64:ss=axioms:i=4637:si=on:rtra=on_0 on DTF2THF_22586 for (2928ds/4637Mi)
% 55.29/12.46  % (22894)lrs+10_1:1_fde=unused:prag=on:sd=2:sgt=32:ss=axioms:i=3174:si=on:rtra=on_0 on DTF2THF_22586 for (2928ds/3174Mi)
% 55.78/12.58  % (22879)Instruction limit reached!
% 55.78/12.58  % (22879)------------------------------
% 55.78/12.58  % (22879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.78/12.58  % (22879)Termination reason: Unknown
% 55.78/12.58  % (22879)Termination phase: Property scanning
% 55.78/12.58  
% 55.78/12.58  % (22879)Memory used [KB]: 60126
% 55.78/12.58  % (22879)Time elapsed: 1.364 s
% 55.78/12.58  % (22879)Instructions burned: 5063 (million)
% 55.78/12.58  % (22879)------------------------------
% 55.78/12.58  % (22879)------------------------------
% 56.09/12.60  % (22895)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_22586 for (2927ds/307Mi)
% 56.09/12.69  % (22895)Instruction limit reached!
% 56.09/12.69  % (22895)------------------------------
% 56.09/12.69  % (22895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.09/12.69  % (22895)Termination reason: Unknown
% 56.09/12.69  % (22895)Termination phase: shuffling
% 56.09/12.69  
% 56.09/12.69  % (22895)Memory used [KB]: 20084
% 56.09/12.69  % (22895)Time elapsed: 0.095 s
% 56.09/12.69  % (22895)Instructions burned: 307 (million)
% 56.09/12.69  % (22895)------------------------------
% 56.09/12.69  % (22895)------------------------------
% 56.40/12.70  % (22896)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=3619:si=on:rtra=on_0 on DTF2THF_22586 for (2926ds/3619Mi)
% 56.60/12.83  % (22881)Instruction limit reached!
% 56.60/12.83  % (22881)------------------------------
% 56.60/12.83  % (22881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.60/12.83  % (22881)Termination reason: Unknown
% 56.60/12.83  % (22881)Termination phase: Property scanning
% 56.60/12.83  
% 56.60/12.83  % (22881)Memory used [KB]: 52067
% 56.60/12.83  % (22881)Time elapsed: 1.570 s
% 56.60/12.83  % (22881)Instructions burned: 5798 (million)
% 56.60/12.83  % (22881)------------------------------
% 56.60/12.83  % (22881)------------------------------
% 56.60/12.84  % (22892)Instruction limit reached!
% 56.60/12.84  % (22892)------------------------------
% 56.60/12.84  % (22892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.60/12.84  % (22892)Termination reason: Unknown
% 56.60/12.84  % (22892)Termination phase: shuffling
% 56.60/12.84  
% 56.60/12.84  % (22892)Memory used [KB]: 40809
% 56.60/12.84  % (22892)Time elapsed: 0.441 s
% 56.60/12.84  % (22892)Instructions burned: 1538 (million)
% 56.60/12.84  % (22892)------------------------------
% 56.60/12.84  % (22892)------------------------------
% 56.60/12.85  % (22897)lrs+1002_1:1_au=on:cbe=off:fde=none:ins=1:sp=const_max:i=2325:si=on:rtra=on_0 on DTF2THF_22586 for (2925ds/2325Mi)
% 56.60/12.85  % (22898)dis+1002_1:1_au=on:sac=on:sd=5:slsq=on:slsqr=1,4:ss=axioms:st=2.0:i=2341:si=on:rtra=on_0 on DTF2THF_22586 for (2924ds/2341Mi)
% 59.76/13.05  % (22889)First to succeed.
% 59.76/13.06  % (22889)Refutation found. Thanks to Tanya!
% 59.76/13.06  % SZS status Theorem for DTF2THF_22586
% 59.76/13.06  % SZS output start Proof for DTF2THF_22586
% 59.76/13.06  thf(type_def_5, type, real: $tType).
% 59.76/13.06  thf(type_def_6, type, int: $tType).
% 59.76/13.06  thf(type_def_8, type, rat: $tType).
% 59.76/13.06  thf(type_def_9, type, set_Pr1261947904930325089at_nat: $tType).
% 59.76/13.06  thf(type_def_10, type, product_prod_int_int: $tType).
% 59.76/13.06  thf(type_def_11, type, nat: $tType).
% 59.76/13.06  thf(type_def_12, type, product_prod_nat_nat: $tType).
% 59.76/13.06  thf(type_def_13, type, num: $tType).
% 59.76/13.06  thf(type_def_14, type, complex: $tType).
% 59.76/13.06  thf(type_def_15, type, option_num: $tType).
% 59.76/13.06  thf(type_def_16, type, product_prod_num_num: $tType).
% 59.76/13.06  thf(type_def_17, type, code_integer: $tType).
% 59.76/13.06  thf(type_def_18, type, produc6271795597528267376eger_o: $tType).
% 59.76/13.06  thf(type_def_19, type, produc8923325533196201883nteger: $tType).
% 59.76/13.06  thf(type_def_20, type, set_fi4554929511873752355omplex: $tType).
% 59.76/13.06  thf(type_def_21, type, filter6041513312241820739omplex: $tType).
% 59.76/13.06  thf(type_def_22, type, set_fi7789364187291644575l_real: $tType).
% 59.76/13.06  thf(type_def_23, type, filter2146258269922977983l_real: $tType).
% 59.76/13.06  thf(type_def_24, type, set_real: $tType).
% 59.76/13.06  thf(type_def_25, type, set_int: $tType).
% 59.76/13.06  thf(type_def_26, type, filter_real: $tType).
% 59.76/13.06  thf(type_def_27, type, filter_nat: $tType).
% 59.76/13.06  thf(type_def_28, type, filter1242075044329608583at_nat: $tType).
% 59.76/13.06  thf(type_def_29, type, set_Pr5085853215250843933omplex: $tType).
% 59.76/13.06  thf(type_def_30, type, set_Pr6218003697084177305l_real: $tType).
% 59.76/13.06  thf(type_def_31, type, set_o: $tType).
% 59.76/13.06  thf(type_def_32, type, set_complex: $tType).
% 59.76/13.06  thf(type_def_33, type, set_list_nat: $tType).
% 59.76/13.06  thf(type_def_34, type, set_nat: $tType).
% 59.76/13.06  thf(type_def_35, type, set_char: $tType).
% 59.76/13.06  thf(type_def_36, type, set_list_o: $tType).
% 59.76/13.06  thf(type_def_37, type, set_list_complex: $tType).
% 59.76/13.06  thf(type_def_38, type, set_list_int: $tType).
% 59.76/13.06  thf(type_def_39, type, set_list_VEBT_VEBT: $tType).
% 59.76/13.06  thf(type_def_40, type, set_num: $tType).
% 59.76/13.06  thf(type_def_41, type, set_rat: $tType).
% 59.76/13.06  thf(type_def_42, type, set_set_complex: $tType).
% 59.76/13.06  thf(type_def_43, type, set_set_int: $tType).
% 59.76/13.06  thf(type_def_44, type, set_set_nat: $tType).
% 59.76/13.06  thf(type_def_45, type, set_VEBT_VEBT: $tType).
% 59.76/13.06  thf(type_def_46, type, char: $tType).
% 59.76/13.06  thf(type_def_47, type, extended_enat: $tType).
% 59.76/13.06  thf(type_def_48, type, list_o: $tType).
% 59.76/13.06  thf(type_def_49, type, list_nat: $tType).
% 59.76/13.06  thf(type_def_50, type, list_int: $tType).
% 59.76/13.06  thf(type_def_51, type, option_nat: $tType).
% 59.76/13.06  thf(type_def_52, type, vEBT_VEBT: $tType).
% 59.76/13.06  thf(type_def_53, type, set_Pr958786334691620121nt_int: $tType).
% 59.76/13.06  thf(type_def_54, type, list_complex: $tType).
% 59.76/13.06  thf(type_def_55, type, list_list_nat: $tType).
% 59.76/13.06  thf(type_def_56, type, list_real: $tType).
% 59.76/13.06  thf(type_def_57, type, list_VEBT_VEBT: $tType).
% 59.76/13.06  thf(type_def_58, type, list_P3795440434834930179_o_int: $tType).
% 59.76/13.06  thf(type_def_59, type, product_prod_o_int: $tType).
% 59.76/13.06  thf(type_def_60, type, list_P7495141550334521929T_VEBT: $tType).
% 59.76/13.06  thf(type_def_61, type, produc2504756804600209347T_VEBT: $tType).
% 59.76/13.06  thf(type_def_62, type, list_P5087981734274514673_int_o: $tType).
% 59.76/13.06  thf(type_def_63, type, product_prod_int_o: $tType).
% 59.76/13.06  thf(type_def_64, type, list_P5707943133018811711nt_int: $tType).
% 59.76/13.06  thf(type_def_65, type, list_P8198026277950538467nt_nat: $tType).
% 59.76/13.06  thf(type_def_66, type, product_prod_int_nat: $tType).
% 59.76/13.06  thf(type_def_67, type, list_P7524865323317820941T_VEBT: $tType).
% 59.76/13.06  thf(type_def_68, type, produc1531783533982839933T_VEBT: $tType).
% 59.76/13.06  thf(type_def_69, type, list_P3126845725202233233VEBT_o: $tType).
% 59.76/13.06  thf(type_def_70, type, produc334124729049499915VEBT_o: $tType).
% 59.76/13.06  thf(type_def_71, type, list_P4547456442757143711BT_int: $tType).
% 59.76/13.06  thf(type_def_72, type, produc4894624898956917775BT_int: $tType).
% 59.76/13.06  thf(type_def_73, type, list_P7037539587688870467BT_nat: $tType).
% 59.76/13.06  thf(type_def_74, type, produc9072475918466114483BT_nat: $tType).
% 59.76/13.06  thf(type_def_75, type, list_P7413028617227757229T_VEBT: $tType).
% 59.76/13.06  thf(type_def_76, type, produc8243902056947475879T_VEBT: $tType).
% 59.76/13.06  thf(type_def_77, type, list_P4002435161011370285od_o_o: $tType).
% 59.76/13.06  thf(type_def_78, type, list_P6285523579766656935_o_nat: $tType).
% 59.76/13.06  thf(type_def_79, type, list_P7333126701944960589_nat_o: $tType).
% 59.76/13.06  thf(type_def_80, type, list_P6011104703257516679at_nat: $tType).
% 59.76/13.06  thf(type_def_81, type, list_P5647936690300460905T_VEBT: $tType).
% 59.76/13.06  thf(type_def_82, type, option4927543243414619207at_nat: $tType).
% 59.76/13.06  thf(type_def_83, type, option6357759511663192854e_term: $tType).
% 59.76/13.06  thf(type_def_84, type, produc6241069584506657477e_term: $tType).
% 59.76/13.06  thf(type_def_85, type, produc8551481072490612790e_term: $tType).
% 59.76/13.06  thf(type_def_86, type, set_Pr8056137968301705908nteger: $tType).
% 59.76/13.06  thf(type_def_87, type, set_Pr1872883991513573699nt_int: $tType).
% 59.76/13.06  thf(type_def_88, type, set_Pr1281608226676607948nteger: $tType).
% 59.76/13.06  thf(type_def_89, type, set_Pr9222295170931077689nt_int: $tType).
% 59.76/13.06  thf(type_def_90, type, set_Code_integer: $tType).
% 59.76/13.06  thf(type_def_91, type, produc8763457246119570046nteger: $tType).
% 59.76/13.06  thf(type_def_92, type, produc7773217078559923341nt_int: $tType).
% 59.76/13.06  thf(type_def_93, type, produc4953844613479565601on_nat: $tType).
% 59.76/13.06  thf(type_def_94, type, produc2233624965454879586on_nat: $tType).
% 59.76/13.06  thf(type_def_95, type, produc7248412053542808358at_nat: $tType).
% 59.76/13.06  thf(type_def_96, type, produc4471711990508489141at_nat: $tType).
% 59.76/13.06  thf(type_def_97, type, produc8306885398267862888on_nat: $tType).
% 59.76/13.06  thf(type_def_98, type, produc3447558737645232053on_num: $tType).
% 59.76/13.06  thf(type_def_99, type, produc7036089656553540234on_num: $tType).
% 59.76/13.06  thf(type_def_100, type, produc1193250871479095198on_num: $tType).
% 59.76/13.06  thf(type_def_101, type, produc1908205239877642774nteger: $tType).
% 59.76/13.06  thf(type_def_102, type, produc2285326912895808259nt_int: $tType).
% 59.76/13.06  thf(type_def_103, type, produc6121120109295599847at_nat: $tType).
% 59.76/13.06  thf(type_def_104, type, produc5491161045314408544at_nat: $tType).
% 59.76/13.06  thf(type_def_105, type, produc5542196010084753463at_nat: $tType).
% 59.76/13.06  thf(type_def_106, type, product_prod_nat_num: $tType).
% 59.76/13.06  thf(type_def_107, type, produc8064648209034914857omplex: $tType).
% 59.76/13.06  thf(type_def_108, type, produc2115011035271226405et_int: $tType).
% 59.76/13.06  thf(type_def_109, type, produc7819656566062154093et_nat: $tType).
% 59.76/13.06  thf(type_def_110, type, produc4411394909380815293omplex: $tType).
% 59.76/13.06  thf(type_def_111, type, produc2422161461964618553l_real: $tType).
% 59.76/13.06  thf(type_def_112, type, set_Pr6308028481084910985omplex: $tType).
% 59.76/13.06  thf(type_def_113, type, set_Pr2522554150109002629et_int: $tType).
% 59.76/13.06  thf(type_def_114, type, set_Pr5488025237498180813et_nat: $tType).
% 59.76/13.06  thf(func_def_0, type, produc5542196010084753463at_nat: $tType).
% 59.76/13.06  thf(func_def_1, type, set_Pr1281608226676607948nteger: $tType).
% 59.76/13.06  thf(func_def_2, type, produc1908205239877642774nteger: $tType).
% 59.76/13.06  thf(func_def_3, type, produc5491161045314408544at_nat: $tType).
% 59.76/13.06  thf(func_def_4, type, set_Pr9222295170931077689nt_int: $tType).
% 59.76/13.06  thf(func_def_5, type, produc2285326912895808259nt_int: $tType).
% 59.76/13.06  thf(func_def_6, type, set_Pr8056137968301705908nteger: $tType).
% 59.76/13.06  thf(func_def_7, type, produc8763457246119570046nteger: $tType).
% 59.76/13.06  thf(func_def_8, type, set_Pr1872883991513573699nt_int: $tType).
% 59.76/13.06  thf(func_def_9, type, produc7773217078559923341nt_int: $tType).
% 59.76/13.06  thf(func_def_10, type, produc1193250871479095198on_num: $tType).
% 59.76/13.06  thf(func_def_11, type, produc8306885398267862888on_nat: $tType).
% 59.76/13.06  thf(func_def_12, type, produc6121120109295599847at_nat: $tType).
% 59.76/13.06  thf(func_def_13, type, produc4471711990508489141at_nat: $tType).
% 59.76/13.06  thf(func_def_14, type, produc7036089656553540234on_num: $tType).
% 59.76/13.06  thf(func_def_15, type, produc2233624965454879586on_nat: $tType).
% 59.76/13.06  thf(func_def_16, type, produc6241069584506657477e_term: $tType).
% 59.76/13.06  thf(func_def_17, type, set_Pr6308028481084910985omplex: $tType).
% 59.76/13.06  thf(func_def_18, type, set_fi4554929511873752355omplex: $tType).
% 59.76/13.06  thf(func_def_19, type, list_P7413028617227757229T_VEBT: $tType).
% 59.76/13.06  thf(func_def_20, type, produc8551481072490612790e_term: $tType).
% 59.76/13.06  thf(func_def_21, type, set_Pr5488025237498180813et_nat: $tType).
% 59.76/13.06  thf(func_def_22, type, set_Pr2522554150109002629et_int: $tType).
% 59.76/13.06  thf(func_def_23, type, produc8064648209034914857omplex: $tType).
% 59.76/13.06  thf(func_def_24, type, option6357759511663192854e_term: $tType).
% 59.76/13.06  thf(func_def_25, type, produc3447558737645232053on_num: $tType).
% 59.76/13.06  thf(func_def_26, type, produc4953844613479565601on_nat: $tType).
% 59.76/13.06  thf(func_def_27, type, produc7248412053542808358at_nat: $tType).
% 59.76/13.06  thf(func_def_28, type, set_fi7789364187291644575l_real: $tType).
% 59.76/13.06  thf(func_def_29, type, filter6041513312241820739omplex: $tType).
% 59.76/13.06  thf(func_def_30, type, list_P7037539587688870467BT_nat: $tType).
% 59.76/13.06  thf(func_def_31, type, list_P4547456442757143711BT_int: $tType).
% 59.76/13.06  thf(func_def_32, type, list_P5647936690300460905T_VEBT: $tType).
% 59.76/13.06  thf(func_def_33, type, list_P7524865323317820941T_VEBT: $tType).
% 59.76/13.06  thf(func_def_34, type, produc8243902056947475879T_VEBT: $tType).
% 59.76/13.06  thf(func_def_35, type, set_Pr5085853215250843933omplex: $tType).
% 59.76/13.06  thf(func_def_36, type, produc7819656566062154093et_nat: $tType).
% 59.76/13.06  thf(func_def_37, type, produc2115011035271226405et_int: $tType).
% 59.76/13.06  thf(func_def_38, type, produc8923325533196201883nteger: $tType).
% 59.76/13.06  thf(func_def_39, type, list_P3126845725202233233VEBT_o: $tType).
% 59.76/13.06  thf(func_def_40, type, list_P7495141550334521929T_VEBT: $tType).
% 59.76/13.06  thf(func_def_41, type, filter2146258269922977983l_real: $tType).
% 59.76/13.06  thf(func_def_42, type, option4927543243414619207at_nat: $tType).
% 59.76/13.06  thf(func_def_43, type, filter1242075044329608583at_nat: $tType).
% 59.76/13.06  thf(func_def_44, type, set_Pr6218003697084177305l_real: $tType).
% 59.76/13.06  thf(func_def_45, type, list_P6011104703257516679at_nat: $tType).
% 59.76/13.06  thf(func_def_46, type, list_P8198026277950538467nt_nat: $tType).
% 59.76/13.06  thf(func_def_47, type, list_P5707943133018811711nt_int: $tType).
% 59.76/13.06  thf(func_def_48, type, produc9072475918466114483BT_nat: $tType).
% 59.76/13.06  thf(func_def_49, type, produc4894624898956917775BT_int: $tType).
% 59.76/13.06  thf(func_def_50, type, produc1531783533982839933T_VEBT: $tType).
% 59.76/13.06  thf(func_def_51, type, set_Pr1261947904930325089at_nat: $tType).
% 59.76/13.06  thf(func_def_52, type, set_Pr958786334691620121nt_int: $tType).
% 59.76/13.06  thf(func_def_53, type, produc4411394909380815293omplex: $tType).
% 59.76/13.06  thf(func_def_54, type, list_P7333126701944960589_nat_o: $tType).
% 59.76/13.06  thf(func_def_55, type, list_P5087981734274514673_int_o: $tType).
% 59.76/13.06  thf(func_def_56, type, list_P6285523579766656935_o_nat: $tType).
% 59.76/13.06  thf(func_def_57, type, list_P3795440434834930179_o_int: $tType).
% 59.76/13.06  thf(func_def_58, type, set_list_VEBT_VEBT: $tType).
% 59.76/13.06  thf(func_def_59, type, produc334124729049499915VEBT_o: $tType).
% 59.76/13.06  thf(func_def_60, type, produc2504756804600209347T_VEBT: $tType).
% 59.76/13.06  thf(func_def_61, type, produc6271795597528267376eger_o: $tType).
% 59.76/13.06  thf(func_def_62, type, produc2422161461964618553l_real: $tType).
% 59.76/13.06  thf(func_def_63, type, product_prod_num_num: $tType).
% 59.76/13.06  thf(func_def_64, type, product_prod_nat_num: $tType).
% 59.76/13.06  thf(func_def_65, type, product_prod_nat_nat: $tType).
% 59.76/13.06  thf(func_def_66, type, product_prod_int_nat: $tType).
% 59.76/13.06  thf(func_def_67, type, product_prod_int_int: $tType).
% 59.76/13.06  thf(func_def_68, type, list_P4002435161011370285od_o_o: $tType).
% 59.76/13.06  thf(func_def_69, type, set_list_complex: $tType).
% 59.76/13.06  thf(func_def_70, type, set_set_complex: $tType).
% 59.76/13.06  thf(func_def_71, type, list_list_nat: $tType).
% 59.76/13.06  thf(func_def_72, type, list_VEBT_VEBT: $tType).
% 59.76/13.06  thf(func_def_73, type, set_list_nat: $tType).
% 59.76/13.06  thf(func_def_74, type, set_list_int: $tType).
% 59.76/13.06  thf(func_def_75, type, product_prod_int_o: $tType).
% 59.76/13.06  thf(func_def_76, type, product_prod_o_int: $tType).
% 59.76/13.06  thf(func_def_77, type, set_VEBT_VEBT: $tType).
% 59.76/13.06  thf(func_def_78, type, set_set_nat: $tType).
% 59.76/13.06  thf(func_def_79, type, set_set_int: $tType).
% 59.76/13.06  thf(func_def_80, type, set_Code_integer: $tType).
% 59.76/13.06  thf(func_def_81, type, list_complex: $tType).
% 59.76/13.06  thf(func_def_82, type, set_list_o: $tType).
% 59.76/13.06  thf(func_def_83, type, set_complex: $tType).
% 59.76/13.06  thf(func_def_84, type, filter_real: $tType).
% 59.76/13.06  thf(func_def_85, type, option_num: $tType).
% 59.76/13.06  thf(func_def_86, type, option_nat: $tType).
% 59.76/13.06  thf(func_def_87, type, filter_nat: $tType).
% 59.76/13.06  thf(func_def_88, type, set_char: $tType).
% 59.76/13.06  thf(func_def_89, type, list_real: $tType).
% 59.76/13.06  thf(func_def_90, type, set_real: $tType).
% 59.76/13.06  thf(func_def_91, type, list_nat: $tType).
% 59.76/13.06  thf(func_def_92, type, list_int: $tType).
% 59.76/13.06  thf(func_def_93, type, vEBT_VEBT: $tType).
% 59.76/13.06  thf(func_def_94, type, set_rat: $tType).
% 59.76/13.06  thf(func_def_95, type, set_num: $tType).
% 59.76/13.06  thf(func_def_96, type, set_nat: $tType).
% 59.76/13.06  thf(func_def_97, type, set_int: $tType).
% 59.76/13.06  thf(func_def_98, type, code_integer: $tType).
% 59.76/13.06  thf(func_def_99, type, extended_enat: $tType).
% 59.76/13.06  thf(func_def_100, type, list_o: $tType).
% 59.76/13.06  thf(func_def_101, type, complex: $tType).
% 59.76/13.06  thf(func_def_102, type, set_o: $tType).
% 59.76/13.06  thf(func_def_103, type, char: $tType).
% 59.76/13.06  thf(func_def_104, type, real: $tType).
% 59.76/13.06  thf(func_def_105, type, rat: $tType).
% 59.76/13.06  thf(func_def_106, type, num: $tType).
% 59.76/13.06  thf(func_def_107, type, nat: $tType).
% 59.76/13.06  thf(func_def_108, type, int: $tType).
% 59.76/13.06  thf(func_def_109, type, archim7802044766580827645g_real: real > int).
% 59.76/13.06  thf(func_def_110, type, archim3151403230148437115or_rat: rat > int).
% 59.76/13.06  thf(func_def_111, type, archim6058952711729229775r_real: real > int).
% 59.76/13.06  thf(func_def_112, type, bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat).
% 59.76/13.06  thf(func_def_113, type, bNF_re3403563459893282935_int_o: (int > int > $o) > ((int > $o) > (int > $o) > $o) > (int > int > $o) > (int > int > $o) > $o).
% 59.76/13.06  thf(func_def_114, type, bNF_re711492959462206631nt_int: (int > int > $o) > ((int > int) > (int > int) > $o) > (int > int > int) > (int > int > int) > $o).
% 59.76/13.06  thf(func_def_115, type, bNF_re157797125943740599nt_int: (int > int > $o) > ((int > product_prod_int_int) > (int > product_prod_int_int) > $o) > (int > int > product_prod_int_int) > (int > int > product_prod_int_int) > $o).
% 59.76/13.06  thf(func_def_116, type, bNF_re3461391660133120880nt_rat: (int > int > $o) > ((int > product_prod_int_int) > (int > rat) > $o) > (int > int > product_prod_int_int) > (int > int > rat) > $o).
% 59.76/13.06  thf(func_def_117, type, bNF_re5089333283451836215nt_o_o: (int > int > $o) > ($o > $o > $o) > (int > $o) > (int > $o) > $o).
% 59.76/13.06  thf(func_def_118, type, bNF_re4712519889275205905nt_int: (int > int > $o) > (int > int > $o) > (int > int) > (int > int) > $o).
% 59.76/13.06  thf(func_def_119, type, bNF_re6250860962936578807nt_int: (int > int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (int > product_prod_int_int) > (int > product_prod_int_int) > $o).
% 59.76/13.06  thf(func_def_120, type, bNF_re2214769303045360666nt_rat: (int > int > $o) > (product_prod_int_int > rat > $o) > (int > product_prod_int_int) > (int > rat) > $o).
% 59.76/13.06  thf(func_def_121, type, bNF_re578469030762574527_nat_o: (nat > nat > $o) > ((nat > $o) > (nat > $o) > $o) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 59.76/13.06  thf(func_def_122, type, bNF_re1345281282404953727at_nat: (nat > nat > $o) > ((nat > nat) > (nat > nat) > $o) > (nat > nat > nat) > (nat > nat > nat) > $o).
% 59.76/13.06  thf(func_def_123, type, bNF_re4705727531993890431at_o_o: (nat > nat > $o) > ($o > $o > $o) > (nat > $o) > (nat > $o) > $o).
% 59.76/13.06  thf(func_def_124, type, bNF_re5653821019739307937at_nat: (nat > nat > $o) > (nat > nat > $o) > (nat > nat) > (nat > nat) > $o).
% 59.76/13.06  thf(func_def_125, type, bNF_re6830278522597306478at_int: (nat > nat > $o) > (product_prod_nat_nat > int > $o) > (nat > product_prod_nat_nat) > (nat > int) > $o).
% 59.76/13.06  thf(func_def_126, type, bNF_re8402795839162346335um_int: (num > num > $o) > ((num > int) > (num > int) > $o) > (num > num > int) > (num > num > int) > $o).
% 59.76/13.06  thf(func_def_127, type, bNF_re1822329894187522285nt_int: (num > num > $o) > (int > int > $o) > (num > int) > (num > int) > $o).
% 59.76/13.06  thf(func_def_128, type, bNF_re5228765855967844073nt_int: (product_prod_int_int > product_prod_int_int > $o) > ((product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > $o).
% 59.76/13.06  thf(func_def_129, type, bNF_re8699439704749558557nt_o_o: (product_prod_int_int > product_prod_int_int > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 59.76/13.06  thf(func_def_130, type, bNF_re7145576690424134365nt_int: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int) > (product_prod_int_int > product_prod_int_int) > $o).
% 59.76/13.06  thf(func_def_131, type, bNF_re7627151682743391978at_rat: (product_prod_int_int > rat > $o) > ((product_prod_int_int > product_prod_int_int) > (rat > rat) > $o) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > (rat > rat > rat) > $o).
% 59.76/13.06  thf(func_def_132, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 59.76/13.06  thf(func_def_133, type, bNF_re8279943556446156061nt_rat: (product_prod_int_int > rat > $o) > (product_prod_int_int > rat > $o) > (product_prod_int_int > product_prod_int_int) > (rat > rat) > $o).
% 59.76/13.06  thf(func_def_134, type, bNF_re717283939379294677_int_o: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > $o) > (int > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (int > int > $o) > $o).
% 59.76/13.06  thf(func_def_135, type, bNF_re7408651293131936558nt_int: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (int > int > int) > $o).
% 59.76/13.06  thf(func_def_136, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 59.76/13.06  thf(func_def_137, type, bNF_re4555766996558763186at_nat: (product_prod_nat_nat > int > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (int > nat) > $o).
% 59.76/13.06  thf(func_def_138, type, bNF_re7400052026677387805at_int: (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > int > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (int > int) > $o).
% 59.76/13.06  thf(func_def_139, type, bNF_re4202695980764964119_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 59.76/13.06  thf(func_def_140, type, bNF_re3099431351363272937at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > $o).
% 59.76/13.06  thf(func_def_141, type, bNF_re3666534408544137501at_o_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 59.76/13.06  thf(func_def_142, type, bNF_re8246922863344978751at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (product_prod_nat_nat > nat) > $o).
% 59.76/13.06  thf(func_def_143, type, bNF_re2241393799969408733at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat) > (product_prod_nat_nat > product_prod_nat_nat) > $o).
% 59.76/13.06  thf(func_def_144, type, binomial: nat > nat > nat).
% 59.76/13.06  thf(func_def_145, type, gbinomial_complex: complex > nat > complex).
% 59.76/13.06  thf(func_def_146, type, gbinomial_int: int > nat > int).
% 59.76/13.06  thf(func_def_147, type, gbinomial_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_148, type, gbinomial_rat: rat > nat > rat).
% 59.76/13.06  thf(func_def_149, type, gbinomial_real: real > nat > real).
% 59.76/13.06  thf(func_def_150, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 59.76/13.06  thf(func_def_151, type, bit_and_not_num: num > num > option_num).
% 59.76/13.06  thf(func_def_152, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 59.76/13.06  thf(func_def_153, type, bit_concat_bit: nat > int > int > int).
% 59.76/13.06  thf(func_def_154, type, bit_or_not_num_neg: num > num > num).
% 59.76/13.06  thf(func_def_155, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 59.76/13.06  thf(func_def_156, type, bit_ri7919022796975470100ot_int: int > int).
% 59.76/13.06  thf(func_def_157, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 59.76/13.06  thf(func_def_158, type, bit_ri631733984087533419it_int: nat > int > int).
% 59.76/13.06  thf(func_def_159, type, bit_se725231765392027082nd_int: int > int > int).
% 59.76/13.06  thf(func_def_160, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_161, type, bit_se8568078237143864401it_int: nat > int > int).
% 59.76/13.06  thf(func_def_162, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_163, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 59.76/13.06  thf(func_def_164, type, bit_se2159334234014336723it_int: nat > int > int).
% 59.76/13.06  thf(func_def_165, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_166, type, bit_se2000444600071755411sk_int: nat > int).
% 59.76/13.06  thf(func_def_167, type, bit_se2002935070580805687sk_nat: nat > nat).
% 59.76/13.06  thf(func_def_168, type, bit_se1409905431419307370or_int: int > int > int).
% 59.76/13.06  thf(func_def_169, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_170, type, bit_se545348938243370406it_int: nat > int > int).
% 59.76/13.06  thf(func_def_171, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_172, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 59.76/13.06  thf(func_def_173, type, bit_se7879613467334960850it_int: nat > int > int).
% 59.76/13.06  thf(func_def_174, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_175, type, bit_se2923211474154528505it_int: nat > int > int).
% 59.76/13.06  thf(func_def_176, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_177, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 59.76/13.06  thf(func_def_178, type, bit_se4203085406695923979it_int: nat > int > int).
% 59.76/13.06  thf(func_def_179, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_180, type, bit_se6526347334894502574or_int: int > int > int).
% 59.76/13.06  thf(func_def_181, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_182, type, bit_se1146084159140164899it_int: int > nat > $o).
% 59.76/13.06  thf(func_def_183, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 59.76/13.06  thf(func_def_184, type, bit_take_bit_num: nat > num > option_num).
% 59.76/13.06  thf(func_def_185, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 59.76/13.06  thf(func_def_186, type, bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o).
% 59.76/13.06  thf(func_def_187, type, bit_un6178654185764691216or_num: num > num > option_num).
% 59.76/13.06  thf(func_def_188, type, bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o).
% 59.76/13.06  thf(func_def_189, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 59.76/13.06  thf(func_def_190, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 59.76/13.06  thf(func_def_191, type, bit_un2480387367778600638or_num: num > num > option_num).
% 59.76/13.06  thf(func_def_192, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 59.76/13.06  thf(func_def_193, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 59.76/13.06  thf(func_def_194, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_195, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_196, type, code_int_of_integer: code_integer > int).
% 59.76/13.06  thf(func_def_197, type, code_integer_of_int: int > code_integer).
% 59.76/13.06  thf(func_def_198, type, code_integer_of_num: num > code_integer).
% 59.76/13.06  thf(func_def_199, type, code_nat_of_integer: code_integer > nat).
% 59.76/13.06  thf(func_def_200, type, code_negative: num > code_integer).
% 59.76/13.06  thf(func_def_201, type, code_num_of_integer: code_integer > num).
% 59.76/13.06  thf(func_def_202, type, code_positive: num > code_integer).
% 59.76/13.06  thf(func_def_203, type, code_Target_negative: num > int).
% 59.76/13.06  thf(func_def_204, type, code_Target_positive: num > int).
% 59.76/13.06  thf(func_def_205, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 59.76/13.06  thf(func_def_206, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 59.76/13.06  thf(func_def_207, type, comple4887499456419720421f_real: set_real > real).
% 59.76/13.06  thf(func_def_208, type, complete_Sup_Sup_int: set_int > int).
% 59.76/13.06  thf(func_def_209, type, comple1385675409528146559p_real: set_real > real).
% 59.76/13.06  thf(func_def_210, type, arg: complex > real).
% 59.76/13.06  thf(func_def_211, type, cis: real > complex).
% 59.76/13.06  thf(func_def_212, type, cnj: complex > complex).
% 59.76/13.06  thf(func_def_213, type, complex2: real > real > complex).
% 59.76/13.06  thf(func_def_214, type, im: complex > real).
% 59.76/13.06  thf(func_def_215, type, re: complex > real).
% 59.76/13.06  thf(func_def_216, type, csqrt: complex > complex).
% 59.76/13.06  thf(func_def_217, type, imaginary_unit: complex).
% 59.76/13.06  thf(func_def_218, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 59.76/13.06  thf(func_def_219, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 59.76/13.06  thf(func_def_220, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 59.76/13.06  thf(func_def_221, type, adjust_div: product_prod_int_int > int).
% 59.76/13.06  thf(func_def_222, type, adjust_mod: int > int > int).
% 59.76/13.06  thf(func_def_223, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_224, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 59.76/13.06  thf(func_def_225, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 59.76/13.06  thf(func_def_226, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 59.76/13.06  thf(func_def_227, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 59.76/13.06  thf(func_def_228, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_229, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 59.76/13.06  thf(func_def_230, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_231, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_232, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 59.76/13.06  thf(func_def_233, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_234, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 59.76/13.06  thf(func_def_235, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 59.76/13.06  thf(func_def_236, type, comm_s4660882817536571857er_int: int > nat > int).
% 59.76/13.06  thf(func_def_237, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_238, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 59.76/13.06  thf(func_def_239, type, comm_s7457072308508201937r_real: real > nat > real).
% 59.76/13.06  thf(func_def_240, type, semiri3624122377584611663nteger: nat > code_integer).
% 59.76/13.06  thf(func_def_241, type, semiri5044797733671781792omplex: nat > complex).
% 59.76/13.06  thf(func_def_242, type, semiri1406184849735516958ct_int: nat > int).
% 59.76/13.06  thf(func_def_243, type, semiri1408675320244567234ct_nat: nat > nat).
% 59.76/13.06  thf(func_def_244, type, semiri773545260158071498ct_rat: nat > rat).
% 59.76/13.06  thf(func_def_245, type, semiri2265585572941072030t_real: nat > real).
% 59.76/13.06  thf(func_def_246, type, invers8013647133539491842omplex: complex > complex).
% 59.76/13.06  thf(func_def_247, type, inverse_inverse_rat: rat > rat).
% 59.76/13.06  thf(func_def_248, type, inverse_inverse_real: real > real).
% 59.76/13.06  thf(func_def_249, type, at_bot_real: filter_real).
% 59.76/13.06  thf(func_def_250, type, at_top_nat: filter_nat).
% 59.76/13.06  thf(func_def_251, type, at_top_real: filter_real).
% 59.76/13.06  thf(func_def_252, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 59.76/13.06  thf(func_def_253, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 59.76/13.06  thf(func_def_254, type, eventually_real: (real > $o) > filter_real > $o).
% 59.76/13.06  thf(func_def_255, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 59.76/13.06  thf(func_def_256, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 59.76/13.06  thf(func_def_257, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 59.76/13.06  thf(func_def_258, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 59.76/13.06  thf(func_def_259, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 59.76/13.06  thf(func_def_260, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 59.76/13.06  thf(func_def_261, type, finite_card_o: set_o > nat).
% 59.76/13.06  thf(func_def_262, type, finite_card_complex: set_complex > nat).
% 59.76/13.06  thf(func_def_263, type, finite_card_list_nat: set_list_nat > nat).
% 59.76/13.06  thf(func_def_264, type, finite_card_nat: set_nat > nat).
% 59.76/13.06  thf(func_def_265, type, finite_card_char: set_char > nat).
% 59.76/13.06  thf(func_def_266, type, finite_finite_o: set_o > $o).
% 59.76/13.06  thf(func_def_267, type, finite3207457112153483333omplex: set_complex > $o).
% 59.76/13.06  thf(func_def_268, type, finite_finite_int: set_int > $o).
% 59.76/13.06  thf(func_def_269, type, finite_finite_list_o: set_list_o > $o).
% 59.76/13.06  thf(func_def_270, type, finite8712137658972009173omplex: set_list_complex > $o).
% 59.76/13.06  thf(func_def_271, type, finite3922522038869484883st_int: set_list_int > $o).
% 59.76/13.06  thf(func_def_272, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 59.76/13.06  thf(func_def_273, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 59.76/13.06  thf(func_def_274, type, finite_finite_nat: set_nat > $o).
% 59.76/13.06  thf(func_def_275, type, finite_finite_num: set_num > $o).
% 59.76/13.06  thf(func_def_276, type, finite_finite_rat: set_rat > $o).
% 59.76/13.06  thf(func_def_277, type, finite_finite_real: set_real > $o).
% 59.76/13.06  thf(func_def_278, type, finite6551019134538273531omplex: set_set_complex > $o).
% 59.76/13.06  thf(func_def_279, type, finite6197958912794628473et_int: set_set_int > $o).
% 59.76/13.06  thf(func_def_280, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 59.76/13.06  thf(func_def_281, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 59.76/13.06  thf(func_def_282, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 59.76/13.06  thf(func_def_283, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 59.76/13.06  thf(func_def_284, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 59.76/13.06  thf(func_def_285, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_286, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_287, type, comp_C3531382070062128313er_num: (code_integer > code_integer) > (num > code_integer) > num > code_integer).
% 59.76/13.06  thf(func_def_288, type, comp_int_int_num: (int > int) > (num > int) > num > int).
% 59.76/13.06  thf(func_def_289, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 59.76/13.06  thf(func_def_290, type, id_o: $o > $o).
% 59.76/13.06  thf(func_def_291, type, id_nat: nat > nat).
% 59.76/13.06  thf(func_def_292, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 59.76/13.06  thf(func_def_293, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 59.76/13.06  thf(func_def_294, type, inj_on_real_real: (real > real) > set_real > $o).
% 59.76/13.06  thf(func_def_295, 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 > $o).
% 59.76/13.06  thf(func_def_296, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 59.76/13.06  thf(func_def_297, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 59.76/13.06  thf(func_def_298, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 59.76/13.06  thf(func_def_299, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 59.76/13.06  thf(func_def_300, type, map_fu4333342158222067775at_rat: (rat > product_prod_int_int) > ((product_prod_int_int > product_prod_int_int) > rat > rat) > (product_prod_int_int > product_prod_int_int > product_prod_int_int) > rat > rat > rat).
% 59.76/13.06  thf(func_def_301, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 59.76/13.06  thf(func_def_302, type, map_fu5673905371560938248nt_rat: (rat > product_prod_int_int) > (product_prod_int_int > rat) > (product_prod_int_int > product_prod_int_int) > rat > rat).
% 59.76/13.06  thf(func_def_303, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 59.76/13.06  thf(func_def_304, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 59.76/13.06  thf(func_def_305, type, gcd_Gcd_int: set_int > int).
% 59.76/13.06  thf(func_def_306, type, gcd_Gcd_nat: set_nat > nat).
% 59.76/13.06  thf(func_def_307, type, bezw: nat > nat > product_prod_int_int).
% 59.76/13.06  thf(func_def_308, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 59.76/13.06  thf(func_def_309, type, gcd_gcd_int: int > int > int).
% 59.76/13.06  thf(func_def_310, type, gcd_gcd_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_311, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 59.76/13.06  thf(func_def_312, type, abs_abs_Code_integer: code_integer > code_integer).
% 59.76/13.06  thf(func_def_313, type, abs_abs_complex: complex > complex).
% 59.76/13.06  thf(func_def_314, type, abs_abs_int: int > int).
% 59.76/13.06  thf(func_def_315, type, abs_abs_rat: rat > rat).
% 59.76/13.06  thf(func_def_316, type, abs_abs_real: real > real).
% 59.76/13.06  thf(func_def_317, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 59.76/13.06  thf(func_def_318, type, minus_minus_complex: complex > complex > complex).
% 59.76/13.06  thf(func_def_319, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_320, type, minus_minus_int: int > int > int).
% 59.76/13.06  thf(func_def_321, type, minus_minus_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_322, type, minus_minus_rat: rat > rat > rat).
% 59.76/13.06  thf(func_def_323, type, minus_minus_real: real > real > real).
% 59.76/13.06  thf(func_def_324, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 59.76/13.06  thf(func_def_325, type, minus_minus_set_int: set_int > set_int > set_int).
% 59.76/13.06  thf(func_def_326, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 59.76/13.06  thf(func_def_327, type, minus_minus_set_real: set_real > set_real > set_real).
% 59.76/13.06  thf(func_def_328, type, one_one_Code_integer: code_integer).
% 59.76/13.06  thf(func_def_329, type, one_one_complex: complex).
% 59.76/13.06  thf(func_def_330, type, one_on7984719198319812577d_enat: extended_enat).
% 59.76/13.06  thf(func_def_331, type, one_one_int: int).
% 59.76/13.06  thf(func_def_332, type, one_one_nat: nat).
% 59.76/13.06  thf(func_def_333, type, one_one_rat: rat).
% 59.76/13.06  thf(func_def_334, type, one_one_real: real).
% 59.76/13.06  thf(func_def_335, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 59.76/13.06  thf(func_def_336, type, plus_plus_complex: complex > complex > complex).
% 59.76/13.06  thf(func_def_337, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_338, type, plus_plus_int: int > int > int).
% 59.76/13.06  thf(func_def_339, type, plus_plus_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_340, type, plus_plus_num: num > num > num).
% 59.76/13.06  thf(func_def_341, type, plus_plus_rat: rat > rat > rat).
% 59.76/13.06  thf(func_def_342, type, plus_plus_real: real > real > real).
% 59.76/13.06  thf(func_def_343, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 59.76/13.06  thf(func_def_344, type, sgn_sgn_complex: complex > complex).
% 59.76/13.06  thf(func_def_345, type, sgn_sgn_int: int > int).
% 59.76/13.06  thf(func_def_346, type, sgn_sgn_rat: rat > rat).
% 59.76/13.06  thf(func_def_347, type, sgn_sgn_real: real > real).
% 59.76/13.06  thf(func_def_348, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 59.76/13.06  thf(func_def_349, type, times_times_complex: complex > complex > complex).
% 59.76/13.06  thf(func_def_350, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_351, type, times_times_int: int > int > int).
% 59.76/13.06  thf(func_def_352, type, times_times_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_353, type, times_times_num: num > num > num).
% 59.76/13.06  thf(func_def_354, type, times_times_rat: rat > rat > rat).
% 59.76/13.06  thf(func_def_355, type, times_times_real: real > real > real).
% 59.76/13.06  thf(func_def_356, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 59.76/13.06  thf(func_def_357, type, uminus1482373934393186551omplex: complex > complex).
% 59.76/13.06  thf(func_def_358, type, uminus_uminus_int: int > int).
% 59.76/13.06  thf(func_def_359, type, uminus_uminus_rat: rat > rat).
% 59.76/13.06  thf(func_def_360, type, uminus_uminus_real: real > real).
% 59.76/13.06  thf(func_def_361, type, uminus1532241313380277803et_int: set_int > set_int).
% 59.76/13.06  thf(func_def_362, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 59.76/13.06  thf(func_def_363, type, uminus612125837232591019t_real: set_real > set_real).
% 59.76/13.06  thf(func_def_364, type, zero_z3403309356797280102nteger: code_integer).
% 59.76/13.06  thf(func_def_365, type, zero_zero_complex: complex).
% 59.76/13.06  thf(func_def_366, type, zero_z5237406670263579293d_enat: extended_enat).
% 59.76/13.06  thf(func_def_367, type, zero_zero_int: int).
% 59.76/13.06  thf(func_def_368, type, zero_zero_nat: nat).
% 59.76/13.06  thf(func_def_369, type, zero_zero_rat: rat).
% 59.76/13.06  thf(func_def_370, type, zero_zero_real: real).
% 59.76/13.06  thf(func_def_371, type, groups6621422865394947399nteger: (complex > code_integer) > set_complex > code_integer).
% 59.76/13.06  thf(func_def_372, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 59.76/13.06  thf(func_def_373, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 59.76/13.06  thf(func_def_374, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 59.76/13.06  thf(func_def_375, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 59.76/13.06  thf(func_def_376, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 59.76/13.06  thf(func_def_377, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 59.76/13.06  thf(func_def_378, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 59.76/13.06  thf(func_def_379, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 59.76/13.06  thf(func_def_380, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 59.76/13.06  thf(func_def_381, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 59.76/13.06  thf(func_def_382, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 59.76/13.06  thf(func_def_383, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 59.76/13.06  thf(func_def_384, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 59.76/13.06  thf(func_def_385, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 59.76/13.06  thf(func_def_386, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 59.76/13.06  thf(func_def_387, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 59.76/13.06  thf(func_def_388, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 59.76/13.06  thf(func_def_389, type, groups977919841031483927at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 59.76/13.06  thf(func_def_390, type, groups4567486121110086003t_real: (product_prod_nat_nat > real) > set_Pr1261947904930325089at_nat > real).
% 59.76/13.06  thf(func_def_391, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 59.76/13.06  thf(func_def_392, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 59.76/13.06  thf(func_def_393, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 59.76/13.06  thf(func_def_394, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 59.76/13.06  thf(func_def_395, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 59.76/13.06  thf(func_def_396, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 59.76/13.06  thf(func_def_397, type, groups8682486955453173170nteger: (complex > code_integer) > set_complex > code_integer).
% 59.76/13.06  thf(func_def_398, type, groups3708469109370488835omplex: (complex > complex) > set_complex > complex).
% 59.76/13.06  thf(func_def_399, type, groups858564598930262913ex_int: (complex > int) > set_complex > int).
% 59.76/13.06  thf(func_def_400, type, groups861055069439313189ex_nat: (complex > nat) > set_complex > nat).
% 59.76/13.06  thf(func_def_401, type, groups225925009352817453ex_rat: (complex > rat) > set_complex > rat).
% 59.76/13.06  thf(func_def_402, type, groups766887009212190081x_real: (complex > real) > set_complex > real).
% 59.76/13.06  thf(func_def_403, type, groups3827104343326376752nteger: (int > code_integer) > set_int > code_integer).
% 59.76/13.06  thf(func_def_404, type, groups7440179247065528705omplex: (int > complex) > set_int > complex).
% 59.76/13.06  thf(func_def_405, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 59.76/13.06  thf(func_def_406, type, groups1707563613775114915nt_nat: (int > nat) > set_int > nat).
% 59.76/13.06  thf(func_def_407, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 59.76/13.06  thf(func_def_408, type, groups2316167850115554303t_real: (int > real) > set_int > real).
% 59.76/13.06  thf(func_def_409, type, groups3455450783089532116nteger: (nat > code_integer) > set_nat > code_integer).
% 59.76/13.06  thf(func_def_410, type, groups6464643781859351333omplex: (nat > complex) > set_nat > complex).
% 59.76/13.06  thf(func_def_411, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 59.76/13.06  thf(func_def_412, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 59.76/13.06  thf(func_def_413, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 59.76/13.06  thf(func_def_414, type, groups129246275422532515t_real: (nat > real) > set_nat > real).
% 59.76/13.06  thf(func_def_415, type, groups4075276357253098568at_int: (product_prod_nat_nat > int) > set_Pr1261947904930325089at_nat > int).
% 59.76/13.06  thf(func_def_416, type, groups4077766827762148844at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 59.76/13.06  thf(func_def_417, type, groups6225526099057966256nteger: (real > code_integer) > set_real > code_integer).
% 59.76/13.06  thf(func_def_418, type, groups713298508707869441omplex: (real > complex) > set_real > complex).
% 59.76/13.06  thf(func_def_419, type, groups4694064378042380927al_int: (real > int) > set_real > int).
% 59.76/13.06  thf(func_def_420, type, groups4696554848551431203al_nat: (real > nat) > set_real > nat).
% 59.76/13.06  thf(func_def_421, type, groups4061424788464935467al_rat: (real > rat) > set_real > rat).
% 59.76/13.06  thf(func_def_422, type, groups1681761925125756287l_real: (real > real) > set_real > real).
% 59.76/13.06  thf(func_def_423, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 59.76/13.06  thf(func_def_424, type, groups4561878855575611511st_nat: list_nat > nat).
% 59.76/13.06  thf(func_def_425, type, the_int: (int > $o) > int).
% 59.76/13.06  thf(func_def_426, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 59.76/13.06  thf(func_def_427, type, the_real: (real > $o) > real).
% 59.76/13.06  thf(func_def_428, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 59.76/13.06  thf(func_def_429, type, if_complex: $o > complex > complex > complex).
% 59.76/13.06  thf(func_def_430, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_431, type, if_int: $o > int > int > int).
% 59.76/13.06  thf(func_def_432, type, if_list_int: $o > list_int > list_int > list_int).
% 59.76/13.06  thf(func_def_433, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 59.76/13.06  thf(func_def_434, type, if_nat: $o > nat > nat > nat).
% 59.76/13.06  thf(func_def_435, type, if_num: $o > num > num > num).
% 59.76/13.06  thf(func_def_436, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 59.76/13.06  thf(func_def_437, type, if_option_num: $o > option_num > option_num > option_num).
% 59.76/13.06  thf(func_def_438, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 59.76/13.06  thf(func_def_439, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 59.76/13.06  thf(func_def_440, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 59.76/13.06  thf(func_def_441, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_442, type, if_rat: $o > rat > rat > rat).
% 59.76/13.06  thf(func_def_443, type, if_real: $o > real > real > real).
% 59.76/13.06  thf(func_def_444, type, if_set_int: $o > set_int > set_int > set_int).
% 59.76/13.06  thf(func_def_445, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 59.76/13.06  thf(func_def_446, type, abs_Integ: product_prod_nat_nat > int).
% 59.76/13.06  thf(func_def_447, type, rep_Integ: int > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_448, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 59.76/13.06  thf(func_def_449, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 59.76/13.06  thf(func_def_450, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 59.76/13.06  thf(func_def_451, type, nat2: int > nat).
% 59.76/13.06  thf(func_def_452, type, pcr_int: product_prod_nat_nat > int > $o).
% 59.76/13.06  thf(func_def_453, type, power_int_real: real > int > real).
% 59.76/13.06  thf(func_def_454, type, ring_1_Ints_real: set_real).
% 59.76/13.06  thf(func_def_455, type, ring_17405671764205052669omplex: int > complex).
% 59.76/13.06  thf(func_def_456, type, ring_1_of_int_int: int > int).
% 59.76/13.06  thf(func_def_457, type, ring_1_of_int_rat: int > rat).
% 59.76/13.06  thf(func_def_458, type, ring_1_of_int_real: int > real).
% 59.76/13.06  thf(func_def_459, type, inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_460, type, inf_inf_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_461, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 59.76/13.06  thf(func_def_462, type, sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_463, type, sup_sup_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_464, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 59.76/13.06  thf(func_def_465, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 59.76/13.06  thf(func_def_466, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 59.76/13.06  thf(func_def_467, type, at_infinity_real: filter_real).
% 59.76/13.06  thf(func_def_468, type, append_int: list_int > list_int > list_int).
% 59.76/13.06  thf(func_def_469, type, append_nat: list_nat > list_nat > list_nat).
% 59.76/13.06  thf(func_def_470, type, drop_nat: nat > list_nat > list_nat).
% 59.76/13.06  thf(func_def_471, type, fold_int_int: (int > int > int) > list_int > int > int).
% 59.76/13.06  thf(func_def_472, type, fold_nat_nat: (nat > nat > nat) > list_nat > nat > nat).
% 59.76/13.06  thf(func_def_473, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 59.76/13.06  thf(func_def_474, type, cons_int: int > list_int > list_int).
% 59.76/13.06  thf(func_def_475, type, cons_nat: nat > list_nat > list_nat).
% 59.76/13.06  thf(func_def_476, type, nil_int: list_int).
% 59.76/13.06  thf(func_def_477, type, nil_nat: list_nat).
% 59.76/13.06  thf(func_def_478, type, hd_nat: list_nat > nat).
% 59.76/13.06  thf(func_def_479, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 59.76/13.06  thf(func_def_480, type, set_o2: list_o > set_o).
% 59.76/13.06  thf(func_def_481, type, set_complex2: list_complex > set_complex).
% 59.76/13.06  thf(func_def_482, type, set_int2: list_int > set_int).
% 59.76/13.06  thf(func_def_483, type, set_list_nat2: list_list_nat > set_list_nat).
% 59.76/13.06  thf(func_def_484, type, set_nat2: list_nat > set_nat).
% 59.76/13.06  thf(func_def_485, type, set_real2: list_real > set_real).
% 59.76/13.06  thf(func_def_486, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 59.76/13.06  thf(func_def_487, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 59.76/13.06  thf(func_def_488, type, list_update_o: list_o > nat > $o > list_o).
% 59.76/13.06  thf(func_def_489, type, list_update_complex: list_complex > nat > complex > list_complex).
% 59.76/13.06  thf(func_def_490, type, list_update_int: list_int > nat > int > list_int).
% 59.76/13.06  thf(func_def_491, type, list_update_nat: list_nat > nat > nat > list_nat).
% 59.76/13.06  thf(func_def_492, type, list_update_real: list_real > nat > real > list_real).
% 59.76/13.06  thf(func_def_493, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.06  thf(func_def_494, type, nth_o: list_o > nat > $o).
% 59.76/13.06  thf(func_def_495, type, nth_complex: list_complex > nat > complex).
% 59.76/13.06  thf(func_def_496, type, nth_int: list_int > nat > int).
% 59.76/13.06  thf(func_def_497, type, nth_list_nat: list_list_nat > nat > list_nat).
% 59.76/13.06  thf(func_def_498, type, nth_nat: list_nat > nat > nat).
% 59.76/13.06  thf(func_def_499, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 59.76/13.06  thf(func_def_500, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 59.76/13.06  thf(func_def_501, type, nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o).
% 59.76/13.06  thf(func_def_502, type, nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int).
% 59.76/13.06  thf(func_def_503, type, nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat).
% 59.76/13.06  thf(func_def_504, type, nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT).
% 59.76/13.06  thf(func_def_505, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 59.76/13.06  thf(func_def_506, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 59.76/13.06  thf(func_def_507, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 59.76/13.06  thf(func_def_508, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 59.76/13.06  thf(func_def_509, type, nth_real: list_real > nat > real).
% 59.76/13.06  thf(func_def_510, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.06  thf(func_def_511, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 59.76/13.06  thf(func_def_512, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 59.76/13.06  thf(func_def_513, type, product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 59.76/13.06  thf(func_def_514, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 59.76/13.06  thf(func_def_515, type, product_int_o: list_int > list_o > list_P5087981734274514673_int_o).
% 59.76/13.06  thf(func_def_516, type, product_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 59.76/13.06  thf(func_def_517, type, product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat).
% 59.76/13.06  thf(func_def_518, type, produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 59.76/13.06  thf(func_def_519, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 59.76/13.06  thf(func_def_520, type, product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat).
% 59.76/13.06  thf(func_def_521, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 59.76/13.06  thf(func_def_522, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 59.76/13.06  thf(func_def_523, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 59.76/13.06  thf(func_def_524, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 59.76/13.06  thf(func_def_525, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 59.76/13.06  thf(func_def_526, type, replicate_o: nat > $o > list_o).
% 59.76/13.06  thf(func_def_527, type, replicate_complex: nat > complex > list_complex).
% 59.76/13.06  thf(func_def_528, type, replicate_int: nat > int > list_int).
% 59.76/13.06  thf(func_def_529, type, replicate_nat: nat > nat > list_nat).
% 59.76/13.06  thf(func_def_530, type, replicate_real: nat > real > list_real).
% 59.76/13.06  thf(func_def_531, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.06  thf(func_def_532, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 59.76/13.06  thf(func_def_533, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 59.76/13.06  thf(func_def_534, type, take_nat: nat > list_nat > list_nat).
% 59.76/13.06  thf(func_def_535, type, upt: nat > nat > list_nat).
% 59.76/13.06  thf(func_def_536, type, upto: int > int > list_int).
% 59.76/13.06  thf(func_def_537, type, upto_aux: int > int > list_int > list_int).
% 59.76/13.06  thf(func_def_538, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 59.76/13.06  thf(func_def_539, type, suc: nat > nat).
% 59.76/13.06  thf(func_def_540, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 59.76/13.06  thf(func_def_541, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 59.76/13.06  thf(func_def_542, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 59.76/13.06  thf(func_def_543, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 59.76/13.06  thf(func_def_544, type, pred: nat > nat).
% 59.76/13.06  thf(func_def_545, type, semiri4939895301339042750nteger: nat > code_integer).
% 59.76/13.06  thf(func_def_546, type, semiri8010041392384452111omplex: nat > complex).
% 59.76/13.06  thf(func_def_547, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 59.76/13.06  thf(func_def_548, type, semiri1314217659103216013at_int: nat > int).
% 59.76/13.06  thf(func_def_549, type, semiri1316708129612266289at_nat: nat > nat).
% 59.76/13.06  thf(func_def_550, type, semiri681578069525770553at_rat: nat > rat).
% 59.76/13.06  thf(func_def_551, type, semiri5074537144036343181t_real: nat > real).
% 59.76/13.06  thf(func_def_552, type, semiri2816024913162550771omplex: (complex > complex) > nat > complex > complex).
% 59.76/13.06  thf(func_def_553, type, semiri8420488043553186161ux_int: (int > int) > nat > int > int).
% 59.76/13.06  thf(func_def_554, type, semiri8422978514062236437ux_nat: (nat > nat) > nat > nat > nat).
% 59.76/13.06  thf(func_def_555, type, semiri7787848453975740701ux_rat: (rat > rat) > nat > rat > rat).
% 59.76/13.06  thf(func_def_556, type, semiri7260567687927622513x_real: (real > real) > nat > real > real).
% 59.76/13.06  thf(func_def_557, type, size_size_list_o: list_o > nat).
% 59.76/13.06  thf(func_def_558, type, size_s3451745648224563538omplex: list_complex > nat).
% 59.76/13.06  thf(func_def_559, type, size_size_list_int: list_int > nat).
% 59.76/13.06  thf(func_def_560, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 59.76/13.06  thf(func_def_561, type, size_size_list_nat: list_nat > nat).
% 59.76/13.06  thf(func_def_562, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 59.76/13.06  thf(func_def_563, type, size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat).
% 59.76/13.06  thf(func_def_564, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 59.76/13.06  thf(func_def_565, type, size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat).
% 59.76/13.06  thf(func_def_566, type, size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat).
% 59.76/13.06  thf(func_def_567, type, size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat).
% 59.76/13.06  thf(func_def_568, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 59.76/13.06  thf(func_def_569, type, size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat).
% 59.76/13.06  thf(func_def_570, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 59.76/13.06  thf(func_def_571, type, size_size_list_real: list_real > nat).
% 59.76/13.06  thf(func_def_572, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 59.76/13.06  thf(func_def_573, type, size_size_num: num > nat).
% 59.76/13.06  thf(func_def_574, type, size_size_option_nat: option_nat > nat).
% 59.76/13.06  thf(func_def_575, type, size_size_option_num: option_num > nat).
% 59.76/13.06  thf(func_def_576, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 59.76/13.06  thf(func_def_577, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 59.76/13.06  thf(func_def_578, type, nat_list_encode: list_nat > nat).
% 59.76/13.06  thf(func_def_579, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 59.76/13.06  thf(func_def_580, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_581, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 59.76/13.06  thf(func_def_582, type, nat_prod_encode: product_prod_nat_nat > nat).
% 59.76/13.06  thf(func_def_583, type, nat_set_decode: nat > set_nat).
% 59.76/13.06  thf(func_def_584, type, nat_set_encode: set_nat > nat).
% 59.76/13.06  thf(func_def_585, type, nat_triangle: nat > nat).
% 59.76/13.06  thf(func_def_586, type, root: nat > real > real).
% 59.76/13.06  thf(func_def_587, type, sqrt: real > real).
% 59.76/13.06  thf(func_def_588, type, bitM: num > num).
% 59.76/13.06  thf(func_def_589, type, inc: num > num).
% 59.76/13.06  thf(func_def_590, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 59.76/13.06  thf(func_def_591, type, neg_nu7009210354673126013omplex: complex > complex).
% 59.76/13.06  thf(func_def_592, type, neg_numeral_dbl_int: int > int).
% 59.76/13.06  thf(func_def_593, type, neg_numeral_dbl_rat: rat > rat).
% 59.76/13.06  thf(func_def_594, type, neg_numeral_dbl_real: real > real).
% 59.76/13.06  thf(func_def_595, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 59.76/13.06  thf(func_def_596, type, neg_nu6511756317524482435omplex: complex > complex).
% 59.76/13.06  thf(func_def_597, type, neg_nu3811975205180677377ec_int: int > int).
% 59.76/13.06  thf(func_def_598, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 59.76/13.06  thf(func_def_599, type, neg_nu6075765906172075777c_real: real > real).
% 59.76/13.06  thf(func_def_600, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 59.76/13.06  thf(func_def_601, type, neg_nu8557863876264182079omplex: complex > complex).
% 59.76/13.06  thf(func_def_602, type, neg_nu5851722552734809277nc_int: int > int).
% 59.76/13.06  thf(func_def_603, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 59.76/13.06  thf(func_def_604, type, neg_nu8295874005876285629c_real: real > real).
% 59.76/13.06  thf(func_def_605, type, neg_numeral_sub_int: num > num > int).
% 59.76/13.06  thf(func_def_606, type, bit0: num > num).
% 59.76/13.06  thf(func_def_607, type, bit1: num > num).
% 59.76/13.06  thf(func_def_608, type, one: num).
% 59.76/13.06  thf(func_def_609, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 59.76/13.06  thf(func_def_610, type, size_num: num > nat).
% 59.76/13.06  thf(func_def_611, type, num_of_nat: nat > num).
% 59.76/13.06  thf(func_def_612, type, numera6620942414471956472nteger: num > code_integer).
% 59.76/13.06  thf(func_def_613, type, numera6690914467698888265omplex: num > complex).
% 59.76/13.06  thf(func_def_614, type, numera1916890842035813515d_enat: num > extended_enat).
% 59.76/13.06  thf(func_def_615, type, numeral_numeral_int: num > int).
% 59.76/13.06  thf(func_def_616, type, numeral_numeral_nat: num > nat).
% 59.76/13.06  thf(func_def_617, type, numeral_numeral_rat: num > rat).
% 59.76/13.06  thf(func_def_618, type, numeral_numeral_real: num > real).
% 59.76/13.06  thf(func_def_619, type, pow: num > num > num).
% 59.76/13.06  thf(func_def_620, type, pred_numeral: num > nat).
% 59.76/13.06  thf(func_def_621, type, sqr: num > num).
% 59.76/13.06  thf(func_def_622, type, none_nat: option_nat).
% 59.76/13.06  thf(func_def_623, type, none_num: option_num).
% 59.76/13.06  thf(func_def_624, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 59.76/13.06  thf(func_def_625, type, some_nat: nat > option_nat).
% 59.76/13.06  thf(func_def_626, type, some_num: num > option_num).
% 59.76/13.06  thf(func_def_627, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 59.76/13.06  thf(func_def_628, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 59.76/13.06  thf(func_def_629, type, case_option_int_num: int > (num > int) > option_num > int).
% 59.76/13.06  thf(func_def_630, type, case_option_num_num: num > (num > num) > option_num > num).
% 59.76/13.06  thf(func_def_631, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 59.76/13.06  thf(func_def_632, type, map_option_num_num: (num > num) > option_num > option_num).
% 59.76/13.06  thf(func_def_633, type, size_option_nat: (nat > nat) > option_nat > nat).
% 59.76/13.06  thf(func_def_634, type, size_option_num: (num > nat) > option_num > nat).
% 59.76/13.06  thf(func_def_635, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 59.76/13.06  thf(func_def_636, type, the_nat: option_nat > nat).
% 59.76/13.06  thf(func_def_637, type, the_num: option_num > num).
% 59.76/13.06  thf(func_def_638, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.06  thf(func_def_639, type, bot_bo5358457235160185703eger_o: (code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o).
% 59.76/13.06  thf(func_def_640, type, bot_bo1403522918969695512_int_o: (int > option6357759511663192854e_term) > product_prod_int_int > $o).
% 59.76/13.06  thf(func_def_641, type, bot_bo3000040243691356879eger_o: (produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o).
% 59.76/13.06  thf(func_def_642, type, bot_bo8662317086119403298_int_o: (produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o).
% 59.76/13.06  thf(func_def_643, type, bot_bot_int_int_o: int > int > $o).
% 59.76/13.06  thf(func_def_644, type, bot_bo4199563552545308370d_enat: extended_enat).
% 59.76/13.06  thf(func_def_645, type, bot_bot_nat: nat).
% 59.76/13.06  thf(func_def_646, type, bot_bot_set_complex: set_complex).
% 59.76/13.06  thf(func_def_647, type, bot_bot_set_int: set_int).
% 59.76/13.06  thf(func_def_648, type, bot_bot_set_nat: set_nat).
% 59.76/13.06  thf(func_def_649, type, bot_bot_set_num: set_num).
% 59.76/13.06  thf(func_def_650, type, bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger).
% 59.76/13.06  thf(func_def_651, type, bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int).
% 59.76/13.06  thf(func_def_652, type, bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger).
% 59.76/13.06  thf(func_def_653, type, bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int).
% 59.76/13.06  thf(func_def_654, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 59.76/13.06  thf(func_def_655, type, bot_bot_set_rat: set_rat).
% 59.76/13.06  thf(func_def_656, type, bot_bot_set_real: set_real).
% 59.76/13.06  thf(func_def_657, type, bot_bot_set_set_int: set_set_int).
% 59.76/13.06  thf(func_def_658, type, ord_less_complex_o: (complex > $o) > (complex > $o) > $o).
% 59.76/13.06  thf(func_def_659, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 59.76/13.06  thf(func_def_660, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 59.76/13.06  thf(func_def_661, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 59.76/13.06  thf(func_def_662, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 59.76/13.06  thf(func_def_663, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 59.76/13.06  thf(func_def_664, type, ord_less_int: int > int > $o).
% 59.76/13.06  thf(func_def_665, type, ord_less_nat: nat > nat > $o).
% 59.76/13.06  thf(func_def_666, type, ord_less_num: num > num > $o).
% 59.76/13.06  thf(func_def_667, type, ord_less_rat: rat > rat > $o).
% 59.76/13.06  thf(func_def_668, type, ord_less_real: real > real > $o).
% 59.76/13.06  thf(func_def_669, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 59.76/13.06  thf(func_def_670, type, ord_less_set_complex: set_complex > set_complex > $o).
% 59.76/13.06  thf(func_def_671, type, ord_less_set_int: set_int > set_int > $o).
% 59.76/13.06  thf(func_def_672, type, ord_less_set_nat: set_nat > set_nat > $o).
% 59.76/13.06  thf(func_def_673, type, ord_less_set_num: set_num > set_num > $o).
% 59.76/13.06  thf(func_def_674, type, ord_less_set_rat: set_rat > set_rat > $o).
% 59.76/13.06  thf(func_def_675, type, ord_less_set_real: set_real > set_real > $o).
% 59.76/13.06  thf(func_def_676, type, ord_less_set_set_int: set_set_int > set_set_int > $o).
% 59.76/13.06  thf(func_def_677, type, ord_le3636971675376928563eger_o: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > $o).
% 59.76/13.06  thf(func_def_678, type, ord_le2124322318746777828_int_o: ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > $o).
% 59.76/13.06  thf(func_def_679, type, ord_le4340812435750786203eger_o: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > $o).
% 59.76/13.06  thf(func_def_680, type, ord_le5643404153117327598_int_o: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > $o).
% 59.76/13.06  thf(func_def_681, type, ord_le4573692005234683329plex_o: (complex > $o) > (complex > $o) > $o).
% 59.76/13.06  thf(func_def_682, type, ord_le6741204236512500942_int_o: (int > int > $o) > (int > int > $o) > $o).
% 59.76/13.06  thf(func_def_683, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 59.76/13.06  thf(func_def_684, type, ord_le6558929396352911974_nat_o: (list_nat > list_nat > $o) > (list_nat > list_nat > $o) > $o).
% 59.76/13.06  thf(func_def_685, type, ord_le1520216061033275535_nat_o: (list_nat > $o) > (list_nat > $o) > $o).
% 59.76/13.06  thf(func_def_686, type, ord_le2646555220125990790_nat_o: (nat > nat > $o) > (nat > nat > $o) > $o).
% 59.76/13.06  thf(func_def_687, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 59.76/13.06  thf(func_def_688, type, ord_le1598226405681992910_int_o: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > $o).
% 59.76/13.06  thf(func_def_689, type, ord_le8369615600986905444_int_o: (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 59.76/13.06  thf(func_def_690, type, ord_le5604493270027003598_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 59.76/13.06  thf(func_def_691, type, ord_le704812498762024988_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 59.76/13.06  thf(func_def_692, type, ord_le2556027599737686990_num_o: (product_prod_num_num > product_prod_num_num > $o) > (product_prod_num_num > product_prod_num_num > $o) > $o).
% 59.76/13.06  thf(func_def_693, type, ord_le2239182809043710856_num_o: (product_prod_num_num > $o) > (product_prod_num_num > $o) > $o).
% 59.76/13.06  thf(func_def_694, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 59.76/13.06  thf(func_def_695, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 59.76/13.06  thf(func_def_696, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 59.76/13.06  thf(func_def_697, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 59.76/13.06  thf(func_def_698, type, ord_le4104064031414453916r_real: filter_real > filter_real > $o).
% 59.76/13.06  thf(func_def_699, type, ord_less_eq_int: int > int > $o).
% 59.76/13.06  thf(func_def_700, type, ord_less_eq_nat: nat > nat > $o).
% 59.76/13.06  thf(func_def_701, type, ord_less_eq_num: num > num > $o).
% 59.76/13.06  thf(func_def_702, type, ord_less_eq_rat: rat > rat > $o).
% 59.76/13.06  thf(func_def_703, type, ord_less_eq_real: real > real > $o).
% 59.76/13.06  thf(func_def_704, type, ord_less_eq_set_o: set_o > set_o > $o).
% 59.76/13.06  thf(func_def_705, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 59.76/13.06  thf(func_def_706, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 59.76/13.06  thf(func_def_707, type, ord_less_eq_set_int: set_int > set_int > $o).
% 59.76/13.06  thf(func_def_708, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 59.76/13.06  thf(func_def_709, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 59.76/13.06  thf(func_def_710, type, ord_less_eq_set_num: set_num > set_num > $o).
% 59.76/13.06  thf(func_def_711, type, ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o).
% 59.76/13.06  thf(func_def_712, type, ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o).
% 59.76/13.06  thf(func_def_713, type, ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o).
% 59.76/13.06  thf(func_def_714, type, ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o).
% 59.76/13.06  thf(func_def_715, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 59.76/13.06  thf(func_def_716, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 59.76/13.06  thf(func_def_717, type, ord_less_eq_set_real: set_real > set_real > $o).
% 59.76/13.06  thf(func_def_718, type, ord_le4403425263959731960et_int: set_set_int > set_set_int > $o).
% 59.76/13.06  thf(func_def_719, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 59.76/13.06  thf(func_def_720, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 59.76/13.06  thf(func_def_721, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_722, type, ord_max_int: int > int > int).
% 59.76/13.06  thf(func_def_723, type, ord_max_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_724, type, ord_max_num: num > num > num).
% 59.76/13.06  thf(func_def_725, type, ord_max_rat: rat > rat > rat).
% 59.76/13.06  thf(func_def_726, type, ord_max_real: real > real > real).
% 59.76/13.06  thf(func_def_727, type, ord_max_set_int: set_int > set_int > set_int).
% 59.76/13.06  thf(func_def_728, type, ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat).
% 59.76/13.06  thf(func_def_729, type, ord_min_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_730, type, order_Greatest_nat: (nat > $o) > nat).
% 59.76/13.06  thf(func_def_731, type, order_9091379641038594480t_real: (nat > real) > $o).
% 59.76/13.06  thf(func_def_732, type, order_mono_nat_nat: (nat > nat) > $o).
% 59.76/13.06  thf(func_def_733, type, order_mono_nat_real: (nat > real) > $o).
% 59.76/13.06  thf(func_def_734, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 59.76/13.06  thf(func_def_735, type, order_7092887310737990675l_real: (real > real) > $o).
% 59.76/13.06  thf(func_def_736, type, top_top_set_o: set_o).
% 59.76/13.06  thf(func_def_737, type, top_top_set_nat: set_nat).
% 59.76/13.06  thf(func_def_738, type, top_top_set_real: set_real).
% 59.76/13.06  thf(func_def_739, type, top_top_set_char: set_char).
% 59.76/13.06  thf(func_def_740, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 59.76/13.06  thf(func_def_741, type, power_power_complex: complex > nat > complex).
% 59.76/13.06  thf(func_def_742, type, power_power_int: int > nat > int).
% 59.76/13.06  thf(func_def_743, type, power_power_nat: nat > nat > nat).
% 59.76/13.06  thf(func_def_744, type, power_power_rat: rat > nat > rat).
% 59.76/13.06  thf(func_def_745, type, power_power_real: real > nat > real).
% 59.76/13.06  thf(func_def_746, type, produc6137756002093451184nteger: (code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > produc8763457246119570046nteger).
% 59.76/13.06  thf(func_def_747, type, produc4305682042979456191nt_int: (int > option6357759511663192854e_term) > product_prod_int_int > produc7773217078559923341nt_int).
% 59.76/13.06  thf(func_def_748, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 59.76/13.06  thf(func_def_749, type, produc3209952032786966637at_nat: (nat > nat > nat) > produc7248412053542808358at_nat > produc4471711990508489141at_nat).
% 59.76/13.06  thf(func_def_750, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 59.76/13.06  thf(func_def_751, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 59.76/13.06  thf(func_def_752, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 59.76/13.06  thf(func_def_753, type, produc8603105652947943368nteger: (produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > produc1908205239877642774nteger).
% 59.76/13.06  thf(func_def_754, type, produc5700946648718959541nt_int: (produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > produc2285326912895808259nt_int).
% 59.76/13.06  thf(func_def_755, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 59.76/13.06  thf(func_def_756, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 59.76/13.07  thf(func_def_757, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 59.76/13.07  thf(func_def_758, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 59.76/13.07  thf(func_def_759, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 59.76/13.07  thf(func_def_760, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 59.76/13.07  thf(func_def_761, type, product_Pair_int_o: int > $o > product_prod_int_o).
% 59.76/13.07  thf(func_def_762, type, product_Pair_int_int: int > int > product_prod_int_int).
% 59.76/13.07  thf(func_def_763, type, product_Pair_int_nat: int > nat > product_prod_int_nat).
% 59.76/13.07  thf(func_def_764, type, produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT).
% 59.76/13.07  thf(func_def_765, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_766, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 59.76/13.07  thf(func_def_767, type, produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat).
% 59.76/13.07  thf(func_def_768, type, product_Pair_num_num: num > num > product_prod_num_num).
% 59.76/13.07  thf(func_def_769, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 59.76/13.07  thf(func_def_770, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 59.76/13.07  thf(func_def_771, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 59.76/13.07  thf(func_def_772, type, produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex).
% 59.76/13.07  thf(func_def_773, type, produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int).
% 59.76/13.07  thf(func_def_774, type, produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat).
% 59.76/13.07  thf(func_def_775, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 59.76/13.07  thf(func_def_776, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 59.76/13.07  thf(func_def_777, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 59.76/13.07  thf(func_def_778, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 59.76/13.07  thf(func_def_779, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 59.76/13.07  thf(func_def_780, type, produc127349428274296955eger_o: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > produc8763457246119570046nteger > $o).
% 59.76/13.07  thf(func_def_781, type, produc2592262431452330817omplex: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_complex) > produc8763457246119570046nteger > set_complex).
% 59.76/13.07  thf(func_def_782, type, produc8604463032469472703et_int: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_int) > produc8763457246119570046nteger > set_int).
% 59.76/13.07  thf(func_def_783, type, produc3558942015123893603et_nat: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_nat) > produc8763457246119570046nteger > set_nat).
% 59.76/13.07  thf(func_def_784, type, produc815715089573277247t_real: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_real) > produc8763457246119570046nteger > set_real).
% 59.76/13.07  thf(func_def_785, type, produc2558449545302689196_int_o: ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > produc7773217078559923341nt_int > $o).
% 59.76/13.07  thf(func_def_786, type, produc8289552606927098482et_nat: ((int > option6357759511663192854e_term) > product_prod_int_int > set_nat) > produc7773217078559923341nt_int > set_nat).
% 59.76/13.07  thf(func_def_787, type, produc8709739885379107790t_real: ((int > option6357759511663192854e_term) > product_prod_int_int > set_real) > produc7773217078559923341nt_int > set_real).
% 59.76/13.07  thf(func_def_788, type, produc6253627499356882019eger_o: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > produc1908205239877642774nteger > $o).
% 59.76/13.07  thf(func_def_789, type, produc1573362020775583542_int_o: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > produc2285326912895808259nt_int > $o).
% 59.76/13.07  thf(func_def_790, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 59.76/13.07  thf(func_def_791, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 59.76/13.07  thf(func_def_792, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 59.76/13.07  thf(func_def_793, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 59.76/13.07  thf(func_def_794, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 59.76/13.07  thf(func_def_795, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 59.76/13.07  thf(func_def_796, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 59.76/13.07  thf(func_def_797, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 59.76/13.07  thf(func_def_798, type, produc8580519160106071146omplex: (int > int > set_complex) > product_prod_int_int > set_complex).
% 59.76/13.07  thf(func_def_799, type, produc73460835934605544et_int: (int > int > set_int) > product_prod_int_int > set_int).
% 59.76/13.07  thf(func_def_800, type, produc4251311855443802252et_nat: (int > int > set_nat) > product_prod_int_int > set_nat).
% 59.76/13.07  thf(func_def_801, type, produc6452406959799940328t_real: (int > int > set_real) > product_prod_int_int > set_real).
% 59.76/13.07  thf(func_def_802, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 59.76/13.07  thf(func_def_803, 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).
% 59.76/13.07  thf(func_def_804, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 59.76/13.07  thf(func_def_805, type, produc1917071388513777916omplex: (nat > nat > complex) > product_prod_nat_nat > complex).
% 59.76/13.07  thf(func_def_806, type, produc6840382203811409530at_int: (nat > nat > int) > product_prod_nat_nat > int).
% 59.76/13.07  thf(func_def_807, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 59.76/13.07  thf(func_def_808, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_809, type, produc6207742614233964070at_rat: (nat > nat > rat) > product_prod_nat_nat > rat).
% 59.76/13.07  thf(func_def_810, type, produc1703576794950452218t_real: (nat > nat > real) > product_prod_nat_nat > real).
% 59.76/13.07  thf(func_def_811, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 59.76/13.07  thf(func_def_812, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 59.76/13.07  thf(func_def_813, type, produc3914248068834153634plex_o: (set_complex > set_complex > $o) > produc8064648209034914857omplex > $o).
% 59.76/13.07  thf(func_def_814, type, produc4109468873575309990_int_o: (set_int > set_int > $o) > produc2115011035271226405et_int > $o).
% 59.76/13.07  thf(func_def_815, type, produc6247414631856714078_nat_o: (set_nat > set_nat > $o) > produc7819656566062154093et_nat > $o).
% 59.76/13.07  thf(func_def_816, type, product_fst_int_int: product_prod_int_int > int).
% 59.76/13.07  thf(func_def_817, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 59.76/13.07  thf(func_def_818, type, product_snd_int_int: product_prod_int_int > int).
% 59.76/13.07  thf(func_def_819, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 59.76/13.07  thf(func_def_820, type, abs_Rat: product_prod_int_int > rat).
% 59.76/13.07  thf(func_def_821, type, fract: int > int > rat).
% 59.76/13.07  thf(func_def_822, type, frct: product_prod_int_int > rat).
% 59.76/13.07  thf(func_def_823, type, rep_Rat: rat > product_prod_int_int).
% 59.76/13.07  thf(func_def_824, type, field_5140801741446780682s_real: set_real).
% 59.76/13.07  thf(func_def_825, type, field_7254667332652039916t_real: rat > real).
% 59.76/13.07  thf(func_def_826, type, normalize: product_prod_int_int > product_prod_int_int).
% 59.76/13.07  thf(func_def_827, type, of_int: int > rat).
% 59.76/13.07  thf(func_def_828, type, pcr_rat: product_prod_int_int > rat > $o).
% 59.76/13.07  thf(func_def_829, type, positive: rat > $o).
% 59.76/13.07  thf(func_def_830, type, quotient_of: rat > product_prod_int_int).
% 59.76/13.07  thf(func_def_831, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 59.76/13.07  thf(func_def_832, type, vanishes: (nat > rat) > $o).
% 59.76/13.07  thf(func_def_833, type, real_V2521375963428798218omplex: set_complex).
% 59.76/13.07  thf(func_def_834, type, real_V5970128139526366754l_real: (real > real) > $o).
% 59.76/13.07  thf(func_def_835, type, real_V3694042436643373181omplex: complex > complex > real).
% 59.76/13.07  thf(func_def_836, type, real_V975177566351809787t_real: real > real > real).
% 59.76/13.07  thf(func_def_837, type, real_V1022390504157884413omplex: complex > real).
% 59.76/13.07  thf(func_def_838, type, real_V7735802525324610683m_real: real > real).
% 59.76/13.07  thf(func_def_839, type, real_V4546457046886955230omplex: real > complex).
% 59.76/13.07  thf(func_def_840, type, real_V2046097035970521341omplex: real > complex > complex).
% 59.76/13.07  thf(func_def_841, type, real_V1485227260804924795R_real: real > real > real).
% 59.76/13.07  thf(func_def_842, type, field_nat: set_Pr1261947904930325089at_nat > set_nat).
% 59.76/13.07  thf(func_def_843, type, algebr932160517623751201me_int: int > int > $o).
% 59.76/13.07  thf(func_def_844, type, algebr934650988132801477me_nat: nat > nat > $o).
% 59.76/13.07  thf(func_def_845, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 59.76/13.07  thf(func_def_846, type, divide1717551699836669952omplex: complex > complex > complex).
% 59.76/13.07  thf(func_def_847, type, divide_divide_int: int > int > int).
% 59.76/13.07  thf(func_def_848, type, divide_divide_nat: nat > nat > nat).
% 59.76/13.07  thf(func_def_849, type, divide_divide_rat: rat > rat > rat).
% 59.76/13.07  thf(func_def_850, type, divide_divide_real: real > real > real).
% 59.76/13.07  thf(func_def_851, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 59.76/13.07  thf(func_def_852, type, dvd_dvd_complex: complex > complex > $o).
% 59.76/13.07  thf(func_def_853, type, dvd_dvd_int: int > int > $o).
% 59.76/13.07  thf(func_def_854, type, dvd_dvd_nat: nat > nat > $o).
% 59.76/13.07  thf(func_def_855, type, dvd_dvd_rat: rat > rat > $o).
% 59.76/13.07  thf(func_def_856, type, dvd_dvd_real: real > real > $o).
% 59.76/13.07  thf(func_def_857, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 59.76/13.07  thf(func_def_858, type, modulo_modulo_int: int > int > int).
% 59.76/13.07  thf(func_def_859, type, modulo_modulo_nat: nat > nat > nat).
% 59.76/13.07  thf(func_def_860, type, zero_n356916108424825756nteger: $o > code_integer).
% 59.76/13.07  thf(func_def_861, type, zero_n1201886186963655149omplex: $o > complex).
% 59.76/13.07  thf(func_def_862, type, zero_n2684676970156552555ol_int: $o > int).
% 59.76/13.07  thf(func_def_863, type, zero_n2687167440665602831ol_nat: $o > nat).
% 59.76/13.07  thf(func_def_864, type, zero_n2052037380579107095ol_rat: $o > rat).
% 59.76/13.07  thf(func_def_865, type, zero_n3304061248610475627l_real: $o > real).
% 59.76/13.07  thf(func_def_866, type, suminf_complex: (nat > complex) > complex).
% 59.76/13.07  thf(func_def_867, type, suminf_int: (nat > int) > int).
% 59.76/13.07  thf(func_def_868, type, suminf_nat: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_869, type, suminf_real: (nat > real) > real).
% 59.76/13.07  thf(func_def_870, type, summable_complex: (nat > complex) > $o).
% 59.76/13.07  thf(func_def_871, type, summable_int: (nat > int) > $o).
% 59.76/13.07  thf(func_def_872, type, summable_nat: (nat > nat) > $o).
% 59.76/13.07  thf(func_def_873, type, summable_real: (nat > real) > $o).
% 59.76/13.07  thf(func_def_874, type, sums_complex: (nat > complex) > complex > $o).
% 59.76/13.07  thf(func_def_875, type, sums_int: (nat > int) > int > $o).
% 59.76/13.07  thf(func_def_876, type, sums_nat: (nat > nat) > nat > $o).
% 59.76/13.07  thf(func_def_877, type, sums_real: (nat > real) > real > $o).
% 59.76/13.07  thf(func_def_878, type, collect_o: ($o > $o) > set_o).
% 59.76/13.07  thf(func_def_879, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 59.76/13.07  thf(func_def_880, type, collect_complex: (complex > $o) > set_complex).
% 59.76/13.07  thf(func_def_881, type, collect_int: (int > $o) > set_int).
% 59.76/13.07  thf(func_def_882, type, collect_list_o: (list_o > $o) > set_list_o).
% 59.76/13.07  thf(func_def_883, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 59.76/13.07  thf(func_def_884, type, collect_list_int: (list_int > $o) > set_list_int).
% 59.76/13.07  thf(func_def_885, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 59.76/13.07  thf(func_def_886, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_887, type, collect_nat: (nat > $o) > set_nat).
% 59.76/13.07  thf(func_def_888, type, collect_num: (num > $o) > set_num).
% 59.76/13.07  thf(func_def_889, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 59.76/13.07  thf(func_def_890, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 59.76/13.07  thf(func_def_891, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 59.76/13.07  thf(func_def_892, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 59.76/13.07  thf(func_def_893, type, collec5108298041176329748omplex: (produc8064648209034914857omplex > $o) > set_Pr6308028481084910985omplex).
% 59.76/13.07  thf(func_def_894, type, collec957716948307931664et_int: (produc2115011035271226405et_int > $o) > set_Pr2522554150109002629et_int).
% 59.76/13.07  thf(func_def_895, type, collec6662362479098859352et_nat: (produc7819656566062154093et_nat > $o) > set_Pr5488025237498180813et_nat).
% 59.76/13.07  thf(func_def_896, type, collect_rat: (rat > $o) > set_rat).
% 59.76/13.07  thf(func_def_897, type, collect_real: (real > $o) > set_real).
% 59.76/13.07  thf(func_def_898, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 59.76/13.07  thf(func_def_899, type, collect_set_int: (set_int > $o) > set_set_int).
% 59.76/13.07  thf(func_def_900, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 59.76/13.07  thf(func_def_901, type, collect_VEBT_VEBT: (vEBT_VEBT > $o) > set_VEBT_VEBT).
% 59.76/13.07  thf(func_def_902, type, image_int_int: (int > int) > set_int > set_int).
% 59.76/13.07  thf(func_def_903, type, image_nat_int: (nat > int) > set_nat > set_int).
% 59.76/13.07  thf(func_def_904, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 59.76/13.07  thf(func_def_905, type, image_nat_real: (nat > real) > set_nat > set_real).
% 59.76/13.07  thf(func_def_906, type, image_nat_char: (nat > char) > set_nat > set_char).
% 59.76/13.07  thf(func_def_907, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 59.76/13.07  thf(func_def_908, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 59.76/13.07  thf(func_def_909, type, image_real_real: (real > real) > set_real > set_real).
% 59.76/13.07  thf(func_def_910, type, image_char_nat: (char > nat) > set_char > set_nat).
% 59.76/13.07  thf(func_def_911, type, insert_int: int > set_int > set_int).
% 59.76/13.07  thf(func_def_912, type, insert_nat: nat > set_nat > set_nat).
% 59.76/13.07  thf(func_def_913, type, insert_real: real > set_real > set_real).
% 59.76/13.07  thf(func_def_914, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 59.76/13.07  thf(func_def_915, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 59.76/13.07  thf(func_def_916, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_917, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 59.76/13.07  thf(func_def_918, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 59.76/13.07  thf(func_def_919, type, set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o).
% 59.76/13.07  thf(func_def_920, type, set_or1266510415728281911st_int: int > int > set_int).
% 59.76/13.07  thf(func_def_921, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 59.76/13.07  thf(func_def_922, type, set_or7049704709247886629st_num: num > num > set_num).
% 59.76/13.07  thf(func_def_923, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 59.76/13.07  thf(func_def_924, type, set_or1222579329274155063t_real: real > real > set_real).
% 59.76/13.07  thf(func_def_925, type, set_or370866239135849197et_int: set_int > set_int > set_set_int).
% 59.76/13.07  thf(func_def_926, type, set_or4662586982721622107an_int: int > int > set_int).
% 59.76/13.07  thf(func_def_927, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 59.76/13.07  thf(func_def_928, type, set_ord_atLeast_real: real > set_real).
% 59.76/13.07  thf(func_def_929, type, set_ord_atMost_int: int > set_int).
% 59.76/13.07  thf(func_def_930, type, set_ord_atMost_nat: nat > set_nat).
% 59.76/13.07  thf(func_def_931, type, set_ord_atMost_num: num > set_num).
% 59.76/13.07  thf(func_def_932, type, set_ord_atMost_rat: rat > set_rat).
% 59.76/13.07  thf(func_def_933, type, set_ord_atMost_real: real > set_real).
% 59.76/13.07  thf(func_def_934, type, set_or58775011639299419et_int: set_int > set_set_int).
% 59.76/13.07  thf(func_def_935, type, set_or6656581121297822940st_int: int > int > set_int).
% 59.76/13.07  thf(func_def_936, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 59.76/13.07  thf(func_def_937, type, set_or5832277885323065728an_int: int > int > set_int).
% 59.76/13.07  thf(func_def_938, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 59.76/13.07  thf(func_def_939, type, set_or1633881224788618240n_real: real > real > set_real).
% 59.76/13.07  thf(func_def_940, type, set_or5849166863359141190n_real: real > set_real).
% 59.76/13.07  thf(func_def_941, type, set_ord_lessThan_int: int > set_int).
% 59.76/13.07  thf(func_def_942, type, set_ord_lessThan_nat: nat > set_nat).
% 59.76/13.07  thf(func_def_943, type, set_ord_lessThan_num: num > set_num).
% 59.76/13.07  thf(func_def_944, type, set_ord_lessThan_rat: rat > set_rat).
% 59.76/13.07  thf(func_def_945, type, set_or5984915006950818249n_real: real > set_real).
% 59.76/13.07  thf(func_def_946, type, ascii_of: char > char).
% 59.76/13.07  thf(func_def_947, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 59.76/13.07  thf(func_def_948, type, comm_s629917340098488124ar_nat: char > nat).
% 59.76/13.07  thf(func_def_949, type, integer_of_char: char > code_integer).
% 59.76/13.07  thf(func_def_950, type, unique3096191561947761185of_nat: nat > char).
% 59.76/13.07  thf(func_def_951, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 59.76/13.07  thf(func_def_952, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 59.76/13.07  thf(func_def_953, type, topolo4899668324122417113eq_int: (nat > int) > $o).
% 59.76/13.07  thf(func_def_954, type, topolo4902158794631467389eq_nat: (nat > nat) > $o).
% 59.76/13.07  thf(func_def_955, type, topolo1459490580787246023eq_num: (nat > num) > $o).
% 59.76/13.07  thf(func_def_956, type, topolo4267028734544971653eq_rat: (nat > rat) > $o).
% 59.76/13.07  thf(func_def_957, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 59.76/13.07  thf(func_def_958, type, topolo3100542954746470799et_int: (nat > set_int) > $o).
% 59.76/13.07  thf(func_def_959, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 59.76/13.07  thf(func_def_960, type, topolo2815343760600316023s_real: real > filter_real).
% 59.76/13.07  thf(func_def_961, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 59.76/13.07  thf(func_def_962, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 59.76/13.07  thf(func_def_963, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 59.76/13.07  thf(func_def_964, type, arccos: real > real).
% 59.76/13.07  thf(func_def_965, type, arcosh_real: real > real).
% 59.76/13.07  thf(func_def_966, type, arcsin: real > real).
% 59.76/13.07  thf(func_def_967, type, arctan: real > real).
% 59.76/13.07  thf(func_def_968, type, arsinh_real: real > real).
% 59.76/13.07  thf(func_def_969, type, artanh_real: real > real).
% 59.76/13.07  thf(func_def_970, type, cos_complex: complex > complex).
% 59.76/13.07  thf(func_def_971, type, cos_real: real > real).
% 59.76/13.07  thf(func_def_972, type, cos_coeff: nat > real).
% 59.76/13.07  thf(func_def_973, type, cosh_real: real > real).
% 59.76/13.07  thf(func_def_974, type, cot_real: real > real).
% 59.76/13.07  thf(func_def_975, type, diffs_complex: (nat > complex) > nat > complex).
% 59.76/13.07  thf(func_def_976, type, diffs_int: (nat > int) > nat > int).
% 59.76/13.07  thf(func_def_977, type, diffs_rat: (nat > rat) > nat > rat).
% 59.76/13.07  thf(func_def_978, type, diffs_real: (nat > real) > nat > real).
% 59.76/13.07  thf(func_def_979, type, exp_complex: complex > complex).
% 59.76/13.07  thf(func_def_980, type, exp_real: real > real).
% 59.76/13.07  thf(func_def_981, type, ln_ln_real: real > real).
% 59.76/13.07  thf(func_def_982, type, log: real > real > real).
% 59.76/13.07  thf(func_def_983, type, pi: real).
% 59.76/13.07  thf(func_def_984, type, powr_real: real > real > real).
% 59.76/13.07  thf(func_def_985, type, sin_complex: complex > complex).
% 59.76/13.07  thf(func_def_986, type, sin_real: real > real).
% 59.76/13.07  thf(func_def_987, type, sin_coeff: nat > real).
% 59.76/13.07  thf(func_def_988, type, sinh_real: real > real).
% 59.76/13.07  thf(func_def_989, type, tan_complex: complex > complex).
% 59.76/13.07  thf(func_def_990, type, tan_real: real > real).
% 59.76/13.07  thf(func_def_991, type, tanh_complex: complex > complex).
% 59.76/13.07  thf(func_def_992, type, tanh_real: real > real).
% 59.76/13.07  thf(func_def_993, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 59.76/13.07  thf(func_def_994, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_995, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_996, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_997, type, vEBT_VEBT_high: nat > nat > nat).
% 59.76/13.07  thf(func_def_998, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_999, type, vEBT_VEBT_low: nat > nat > nat).
% 59.76/13.07  thf(func_def_1000, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1001, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1002, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1003, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1004, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1005, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1006, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1007, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 59.76/13.07  thf(func_def_1008, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1009, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 59.76/13.07  thf(func_def_1010, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1011, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1012, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1013, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1014, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1015, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 59.76/13.07  thf(func_def_1016, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1017, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1018, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 59.76/13.07  thf(func_def_1019, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 59.76/13.07  thf(func_def_1020, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 59.76/13.07  thf(func_def_1021, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 59.76/13.07  thf(func_def_1022, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 59.76/13.07  thf(func_def_1023, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 59.76/13.07  thf(func_def_1024, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 59.76/13.07  thf(func_def_1025, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 59.76/13.07  thf(func_def_1026, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 59.76/13.07  thf(func_def_1027, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 59.76/13.07  thf(func_def_1028, type, vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o).
% 59.76/13.07  thf(func_def_1029, type, vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o).
% 59.76/13.07  thf(func_def_1030, type, vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o).
% 59.76/13.07  thf(func_def_1031, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 59.76/13.07  thf(func_def_1032, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 59.76/13.07  thf(func_def_1033, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1034, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 59.76/13.07  thf(func_def_1035, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1036, type, vEBT_is_pred_in_set: set_nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1037, type, vEBT_vebt_pred: vEBT_VEBT > nat > option_nat).
% 59.76/13.07  thf(func_def_1038, type, vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1039, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 59.76/13.07  thf(func_def_1040, type, accp_nat: (nat > nat > $o) > nat > $o).
% 59.76/13.07  thf(func_def_1041, type, accp_P6019419558468335806at_nat: (produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o) > produc4471711990508489141at_nat > $o).
% 59.76/13.07  thf(func_def_1042, type, accp_P5496254298877145759on_nat: (produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o) > produc8306885398267862888on_nat > $o).
% 59.76/13.07  thf(func_def_1043, type, accp_P7605991808943153877on_num: (produc1193250871479095198on_num > produc1193250871479095198on_num > $o) > produc1193250871479095198on_num > $o).
% 59.76/13.07  thf(func_def_1044, type, accp_P3267385326087170368at_nat: (produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o) > produc5542196010084753463at_nat > $o).
% 59.76/13.07  thf(func_def_1045, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 59.76/13.07  thf(func_def_1046, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 59.76/13.07  thf(func_def_1047, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 59.76/13.07  thf(func_def_1048, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 59.76/13.07  thf(func_def_1049, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1050, type, finite8643634255014194347omplex: set_Pr6308028481084910985omplex).
% 59.76/13.07  thf(func_def_1051, type, finite_psubset_int: set_Pr2522554150109002629et_int).
% 59.76/13.07  thf(func_def_1052, type, finite_psubset_nat: set_Pr5488025237498180813et_nat).
% 59.76/13.07  thf(func_def_1053, type, measure_int: (int > nat) > set_Pr958786334691620121nt_int).
% 59.76/13.07  thf(func_def_1054, type, fChoice_real: (real > $o) > real).
% 59.76/13.07  thf(func_def_1055, type, member_o: $o > set_o > $o).
% 59.76/13.07  thf(func_def_1056, type, member_complex: complex > set_complex > $o).
% 59.76/13.07  thf(func_def_1057, type, member_int: int > set_int > $o).
% 59.76/13.07  thf(func_def_1058, type, member_list_o: list_o > set_list_o > $o).
% 59.76/13.07  thf(func_def_1059, type, member_list_nat: list_nat > set_list_nat > $o).
% 59.76/13.07  thf(func_def_1060, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1061, type, member_nat: nat > set_nat > $o).
% 59.76/13.07  thf(func_def_1062, type, member_num: num > set_num > $o).
% 59.76/13.07  thf(func_def_1063, type, member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o).
% 59.76/13.07  thf(func_def_1064, type, member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o).
% 59.76/13.07  thf(func_def_1065, type, member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o).
% 59.76/13.07  thf(func_def_1066, type, member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o).
% 59.76/13.07  thf(func_def_1067, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 59.76/13.07  thf(func_def_1068, type, member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o).
% 59.76/13.07  thf(func_def_1069, type, member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o).
% 59.76/13.07  thf(func_def_1070, type, member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o).
% 59.76/13.07  thf(func_def_1071, type, member_rat: rat > set_rat > $o).
% 59.76/13.07  thf(func_def_1072, type, member_real: real > set_real > $o).
% 59.76/13.07  thf(func_def_1073, type, member_set_int: set_int > set_set_int > $o).
% 59.76/13.07  thf(func_def_1074, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1075, type, deg: nat).
% 59.76/13.07  thf(func_def_1076, type, m: nat).
% 59.76/13.07  thf(func_def_1077, type, ma: nat).
% 59.76/13.07  thf(func_def_1078, type, mi: nat).
% 59.76/13.07  thf(func_def_1079, type, na: nat).
% 59.76/13.07  thf(func_def_1080, type, summary: vEBT_VEBT).
% 59.76/13.07  thf(func_def_1082, type, treeList: list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1083, type, xa: nat).
% 59.76/13.07  thf(func_def_1102, type, sP0: (option4927543243414619207at_nat > option4927543243414619207at_nat > $o) > option4927543243414619207at_nat > option4927543243414619207at_nat > $o).
% 59.76/13.07  thf(func_def_1103, type, sP1: real > (real > real > $o) > real > $o).
% 59.76/13.07  thf(func_def_1104, type, sP2: rat > rat > rat > $o).
% 59.76/13.07  thf(func_def_1105, type, sP3: rat > rat > rat > $o).
% 59.76/13.07  thf(func_def_1106, type, sP4: rat > rat > rat > $o).
% 59.76/13.07  thf(func_def_1107, type, sP5: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1108, type, sP6: int > int > int > $o).
% 59.76/13.07  thf(func_def_1109, type, sP7: int > int > int > $o).
% 59.76/13.07  thf(func_def_1110, type, sP8: int > int > int > $o).
% 59.76/13.07  thf(func_def_1111, type, sP9: (option4927543243414619207at_nat > option_nat > $o) > option4927543243414619207at_nat > option_nat > $o).
% 59.76/13.07  thf(func_def_1112, type, sP10: int > (int > $o) > $o).
% 59.76/13.07  thf(func_def_1113, type, sP11: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1114, type, sP12: (option_nat > option4927543243414619207at_nat > $o) > option_nat > option4927543243414619207at_nat > $o).
% 59.76/13.07  thf(func_def_1115, type, sP13: (int > $o) > $o).
% 59.76/13.07  thf(func_def_1116, type, sP14: (option_nat > option_nat > $o) > option_nat > option_nat > $o).
% 59.76/13.07  thf(func_def_1117, type, sP15: nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1118, type, sP16: nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1119, type, sP17: nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1120, type, sP18: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1121, type, sP19: nat > nat > list_VEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1122, type, sP20: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1123, type, sP21: list_VEBT_VEBT > nat > nat > $o).
% 59.76/13.07  thf(func_def_1124, type, sP22: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 59.76/13.07  thf(func_def_1125, type, sP23: nat > nat > nat > list_VEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1126, type, sP24: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1127, type, sP25: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1128, type, sP26: vEBT_VEBT > nat > $o).
% 59.76/13.07  thf(func_def_1129, type, sP27: nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1130, type, sP28: option_nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1131, type, sP29: vEBT_VEBT > $o > $o).
% 59.76/13.07  thf(func_def_1132, type, sP30: vEBT_VEBT > $o > $o).
% 59.76/13.07  thf(func_def_1133, type, sK31: real > real > real > real).
% 59.76/13.07  thf(func_def_1134, type, sK32: (option4927543243414619207at_nat > option4927543243414619207at_nat > $o) > option4927543243414619207at_nat > option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1135, type, sK33: (option4927543243414619207at_nat > option4927543243414619207at_nat > $o) > option4927543243414619207at_nat > option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1136, type, sK34: list_VEBT_VEBT > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1137, type, sK35: (real > rat) > real).
% 59.76/13.07  thf(func_def_1138, type, sK36: (real > rat) > real).
% 59.76/13.07  thf(func_def_1139, type, sK37: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1140, type, sK38: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1141, type, sK39: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1142, type, sK40: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1143, type, sK41: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1144, type, sK42: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1145, type, sK43: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1146, type, sK44: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1147, type, sK45: int > int).
% 59.76/13.07  thf(func_def_1148, type, sK46: (real > nat) > real).
% 59.76/13.07  thf(func_def_1149, type, sK47: (real > nat) > real).
% 59.76/13.07  thf(func_def_1150, type, sK48: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1151, type, sK49: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1152, type, sK50: int > int > int).
% 59.76/13.07  thf(func_def_1153, type, sK51: int > int).
% 59.76/13.07  thf(func_def_1154, type, sK52: nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1155, type, sK53: int > int).
% 59.76/13.07  thf(func_def_1156, type, sK54: real > (real > real > $o) > real > real).
% 59.76/13.07  thf(func_def_1157, type, sK55: real > real > (real > real > $o) > real > real).
% 59.76/13.07  thf(func_def_1158, type, sK56: real > real > (real > real > $o) > real > real).
% 59.76/13.07  thf(func_def_1159, type, sK57: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1160, type, sK58: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1161, type, sK59: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1162, type, sK60: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1163, type, sK61: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1164, type, sK62: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1165, type, sK63: (int > $o) > int).
% 59.76/13.07  thf(func_def_1166, type, sK64: int > nat).
% 59.76/13.07  thf(func_def_1167, type, sK65: (real > real) > real).
% 59.76/13.07  thf(func_def_1168, type, sK66: (real > real) > real).
% 59.76/13.07  thf(func_def_1169, type, sK67: nat > nat > nat).
% 59.76/13.07  thf(func_def_1170, type, sK68: set_int > set_int > int).
% 59.76/13.07  thf(func_def_1171, type, sK69: int > int).
% 59.76/13.07  thf(func_def_1172, type, sK70: real > real > real).
% 59.76/13.07  thf(func_def_1173, type, sK71: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1174, type, sK72: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1175, type, sK73: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1176, type, sK74: int > int > int).
% 59.76/13.07  thf(func_def_1177, type, sK75: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1178, type, sK76: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1179, type, sK77: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1180, type, sK78: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1181, type, sK79: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1182, type, sK80: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1183, type, sK81: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1184, type, sK82: (list_VEBT_VEBT > $o) > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1185, type, sK83: real > nat > (nat > real) > nat > real).
% 59.76/13.07  thf(func_def_1186, type, sK84: (real > nat) > real).
% 59.76/13.07  thf(func_def_1187, type, sK85: (real > nat) > real).
% 59.76/13.07  thf(func_def_1188, type, sK86: nat > nat).
% 59.76/13.07  thf(func_def_1189, type, sK87: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1190, type, sK88: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1191, type, sK89: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1192, type, sK90: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1193, type, sK91: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1194, type, sK92: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1195, type, sK93: set_nat > nat > nat).
% 59.76/13.07  thf(func_def_1196, type, sK94: real > real > real).
% 59.76/13.07  thf(func_def_1197, type, sK95: int > nat).
% 59.76/13.07  thf(func_def_1198, type, sK96: rat > rat).
% 59.76/13.07  thf(func_def_1199, type, sK97: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1200, type, sK98: (int > real) > int).
% 59.76/13.07  thf(func_def_1201, type, sK99: (int > real) > int).
% 59.76/13.07  thf(func_def_1202, type, sK100: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1203, type, sK101: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1204, type, sK102: real > nat).
% 59.76/13.07  thf(func_def_1205, type, sK103: nat > nat).
% 59.76/13.07  thf(func_def_1206, type, sK104: int > nat).
% 59.76/13.07  thf(func_def_1207, type, sK105: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1208, type, sK106: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1209, type, sK107: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1210, type, sK108: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1211, type, sK109: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1212, type, sK110: rat > rat).
% 59.76/13.07  thf(func_def_1213, type, sK111: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1214, type, sK112: code_integer > code_integer > code_integer).
% 59.76/13.07  thf(func_def_1215, type, sK113: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1216, type, sK114: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1217, type, sK115: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1218, type, sK116: rat > rat).
% 59.76/13.07  thf(func_def_1219, type, sK117: real > real > nat).
% 59.76/13.07  thf(func_def_1220, type, sK118: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1221, type, sK119: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1222, type, sK120: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1223, type, sK121: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1224, type, sK122: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1225, type, sK123: (real > rat) > real).
% 59.76/13.07  thf(func_def_1226, type, sK124: (real > rat) > real).
% 59.76/13.07  thf(func_def_1227, type, sK125: nat > nat > nat).
% 59.76/13.07  thf(func_def_1228, type, sK126: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1229, type, sK127: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1230, type, sK128: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1231, type, sK129: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1232, type, sK130: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1233, type, sK131: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1234, type, sK132: vEBT_VEBT > option4927543243414619207at_nat).
% 59.76/13.07  thf(func_def_1235, type, sK133: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1236, type, sK134: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1237, type, sK135: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1238, type, sK136: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1239, type, sK137: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1240, type, sK138: list_VEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1241, type, sK139: list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1242, type, sK140: list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1243, type, sK141: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1244, type, sK142: nat > nat).
% 59.76/13.07  thf(func_def_1245, type, sK143: real > real > nat).
% 59.76/13.07  thf(func_def_1246, type, sK144: rat > rat).
% 59.76/13.07  thf(func_def_1247, type, sK145: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1248, type, sK146: real > real > real).
% 59.76/13.07  thf(func_def_1249, type, sK147: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1250, type, sK148: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1251, type, sK149: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1252, type, sK150: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1253, type, sK151: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1254, type, sK152: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1255, type, sK153: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1256, type, sK154: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1257, type, sK155: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1258, type, sK156: real > real).
% 59.76/13.07  thf(func_def_1259, type, sK157: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1260, type, sK158: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1261, type, sK159: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1262, type, sK160: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1263, type, sK161: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1264, type, sK162: (real > real) > real).
% 59.76/13.07  thf(func_def_1265, type, sK163: (real > real) > real).
% 59.76/13.07  thf(func_def_1266, type, sK164: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1267, type, sK165: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1268, type, sK166: real > real).
% 59.76/13.07  thf(func_def_1269, type, sK167: rat > rat).
% 59.76/13.07  thf(func_def_1270, type, sK168: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1271, type, sK169: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1272, type, sK170: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1273, type, sK171: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1274, type, sK172: int > int).
% 59.76/13.07  thf(func_def_1275, type, sK173: real > nat > real).
% 59.76/13.07  thf(func_def_1276, type, sK174: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1277, type, sK175: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1278, type, sK176: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1279, type, sK177: nat > (nat > real) > nat).
% 59.76/13.07  thf(func_def_1280, type, sK178: (nat > real) > nat > nat).
% 59.76/13.07  thf(func_def_1281, type, sK179: nat > nat > nat).
% 59.76/13.07  thf(func_def_1282, type, sK180: real > real).
% 59.76/13.07  thf(func_def_1283, type, sK181: (int > rat) > int).
% 59.76/13.07  thf(func_def_1284, type, sK182: (int > rat) > int).
% 59.76/13.07  thf(func_def_1285, type, sK183: real > real).
% 59.76/13.07  thf(func_def_1286, type, sK184: rat > rat).
% 59.76/13.07  thf(func_def_1287, type, sK185: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1288, type, sK186: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1289, type, sK187: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1290, type, sK188: rat > rat > rat).
% 59.76/13.07  thf(func_def_1291, type, sK189: nat > nat).
% 59.76/13.07  thf(func_def_1292, type, sK190: nat > nat).
% 59.76/13.07  thf(func_def_1293, type, sK191: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1294, type, sK192: real > nat).
% 59.76/13.07  thf(func_def_1295, type, sK193: nat > nat > nat).
% 59.76/13.07  thf(func_def_1296, type, sK194: real > real > real > real).
% 59.76/13.07  thf(func_def_1297, type, sK195: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1298, type, sK196: int > int > int).
% 59.76/13.07  thf(func_def_1299, type, sK197: real > nat).
% 59.76/13.07  thf(func_def_1300, type, sK198: int > int > int).
% 59.76/13.07  thf(func_def_1301, type, sK199: (nat > int) > nat).
% 59.76/13.07  thf(func_def_1302, type, sK200: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1303, type, sK201: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1304, type, sK202: rat > rat > rat).
% 59.76/13.07  thf(func_def_1305, type, sK203: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1306, type, sK204: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1307, type, sK205: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1308, type, sK206: vEBT_VEBT > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1309, type, sK207: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1310, type, sK208: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1311, type, sK209: vEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1312, type, sK210: nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1313, type, sK211: nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1314, type, sK212: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1315, type, sK213: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1316, type, sK214: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1317, type, sK215: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1318, type, sK216: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1319, type, sK217: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1320, type, sK218: list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1321, type, sK219: list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1322, type, sK220: list_VEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1323, type, sK221: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1324, type, sK222: nat > (nat > real) > real).
% 59.76/13.07  thf(func_def_1325, type, sK223: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1326, type, sK224: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1327, type, sK225: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1328, type, sK226: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1329, type, sK227: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1330, type, sK228: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1331, type, sK229: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1332, type, sK230: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1333, type, sK231: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1334, type, sK232: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1335, type, sK233: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1336, type, sK234: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1337, type, sK235: nat > nat).
% 59.76/13.07  thf(func_def_1338, type, sK236: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1339, type, sK237: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1340, type, sK238: int > int).
% 59.76/13.07  thf(func_def_1341, type, sK239: (real > int) > real).
% 59.76/13.07  thf(func_def_1342, type, sK240: (real > int) > real).
% 59.76/13.07  thf(func_def_1343, type, sK241: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1344, type, sK242: int > int).
% 59.76/13.07  thf(func_def_1345, type, sK243: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1346, type, sK244: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1347, type, sK245: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1348, type, sK246: nat > nat).
% 59.76/13.07  thf(func_def_1349, type, sK247: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1350, type, sK248: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1351, type, sK249: nat > nat > nat).
% 59.76/13.07  thf(func_def_1352, type, sK250: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1353, type, sK251: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1354, type, sK252: real > real).
% 59.76/13.07  thf(func_def_1355, type, sK253: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1356, type, sK254: (real > real) > real).
% 59.76/13.07  thf(func_def_1357, type, sK255: (real > real) > real).
% 59.76/13.07  thf(func_def_1358, type, sK256: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1359, type, sK257: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1360, type, sK258: rat > rat).
% 59.76/13.07  thf(func_def_1361, type, sK259: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1362, type, sK260: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1363, type, sK261: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1364, type, sK262: rat > rat).
% 59.76/13.07  thf(func_def_1365, type, sK263: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1366, type, sK264: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1367, type, sK265: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1368, type, sK266: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1369, type, sK267: (real > rat) > real).
% 59.76/13.07  thf(func_def_1370, type, sK268: (real > rat) > real).
% 59.76/13.07  thf(func_def_1371, type, sK269: nat > nat).
% 59.76/13.07  thf(func_def_1372, type, sK270: (option4927543243414619207at_nat > option_nat > $o) > option4927543243414619207at_nat > option_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1373, type, sK271: (option4927543243414619207at_nat > option_nat > $o) > option4927543243414619207at_nat > option_nat > nat).
% 59.76/13.07  thf(func_def_1374, type, sK272: (real > int) > real).
% 59.76/13.07  thf(func_def_1375, type, sK273: (real > int) > real).
% 59.76/13.07  thf(func_def_1376, type, sK274: real > real > nat).
% 59.76/13.07  thf(func_def_1377, type, sK275: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1378, type, sK276: real > real).
% 59.76/13.07  thf(func_def_1379, type, sK277: (int > real) > int).
% 59.76/13.07  thf(func_def_1380, type, sK278: (int > real) > int).
% 59.76/13.07  thf(func_def_1381, type, sK279: int > int).
% 59.76/13.07  thf(func_def_1382, type, sK280: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1383, type, sK281: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1384, type, sK282: real > nat > real).
% 59.76/13.07  thf(func_def_1385, type, sK283: real > real).
% 59.76/13.07  thf(func_def_1386, type, sK284: rat > rat > rat > rat).
% 59.76/13.07  thf(func_def_1387, type, sK285: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1388, type, sK286: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1389, type, sK287: nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1390, type, sK288: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1391, type, sK289: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1392, type, sK290: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1393, type, sK291: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1394, type, sK292: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1395, type, sK293: code_integer > code_integer > code_integer).
% 59.76/13.07  thf(func_def_1396, type, sK294: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1397, type, sK295: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1398, type, sK296: int > int).
% 59.76/13.07  thf(func_def_1399, type, sK297: (nat > int) > nat).
% 59.76/13.07  thf(func_def_1400, type, sK298: nat > nat > nat).
% 59.76/13.07  thf(func_def_1401, type, sK299: nat > nat > nat).
% 59.76/13.07  thf(func_def_1402, type, sK300: nat > nat > nat).
% 59.76/13.07  thf(func_def_1403, type, sK301: int > int).
% 59.76/13.07  thf(func_def_1404, type, sK302: nat > nat > nat).
% 59.76/13.07  thf(func_def_1405, type, sK303: (real > real) > real).
% 59.76/13.07  thf(func_def_1406, type, sK304: (real > real) > real).
% 59.76/13.07  thf(func_def_1407, type, sK305: nat > nat > nat).
% 59.76/13.07  thf(func_def_1408, type, sK306: rat > rat > rat > rat).
% 59.76/13.07  thf(func_def_1409, type, sK307: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1410, type, sK308: (rat > nat) > rat).
% 59.76/13.07  thf(func_def_1411, type, sK309: nat > nat > nat).
% 59.76/13.07  thf(func_def_1412, type, sK310: nat > nat > nat).
% 59.76/13.07  thf(func_def_1413, type, sK311: nat > nat > nat).
% 59.76/13.07  thf(func_def_1414, type, sK312: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1415, type, sK313: real > real).
% 59.76/13.07  thf(func_def_1416, type, sK314: (nat > real) > nat > real > nat).
% 59.76/13.07  thf(func_def_1417, type, sK315: (real > rat) > real).
% 59.76/13.07  thf(func_def_1418, type, sK316: (real > rat) > real).
% 59.76/13.07  thf(func_def_1419, type, sK317: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1420, type, sK318: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1421, type, sK319: int > nat).
% 59.76/13.07  thf(func_def_1422, type, sK320: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1423, type, sK321: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1424, type, sK322: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1425, type, sK323: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1426, type, sK324: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1427, type, sK325: nat > vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1428, type, sK326: nat > vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1429, type, sK327: nat > vEBT_VEBT > option4927543243414619207at_nat).
% 59.76/13.07  thf(func_def_1430, type, sK328: real > real > real > real).
% 59.76/13.07  thf(func_def_1431, type, sK329: real > real).
% 59.76/13.07  thf(func_def_1432, type, sK330: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1433, type, sK331: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1434, type, sK332: (nat > int) > nat).
% 59.76/13.07  thf(func_def_1435, type, sK333: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1436, type, sK334: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1437, type, sK335: int > int).
% 59.76/13.07  thf(func_def_1438, type, sK336: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1439, type, sK337: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1440, type, sK338: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1441, type, sK339: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1442, type, sK340: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1443, type, sK341: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1444, type, sK342: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1445, type, sK343: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1446, type, sK344: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1447, type, sK345: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1448, type, sK346: (int > rat) > int).
% 59.76/13.07  thf(func_def_1449, type, sK347: (int > rat) > int).
% 59.76/13.07  thf(func_def_1450, type, sK348: (real > real) > real).
% 59.76/13.07  thf(func_def_1451, type, sK349: (real > real) > real).
% 59.76/13.07  thf(func_def_1452, type, sK350: (nat > set_int) > nat).
% 59.76/13.07  thf(func_def_1453, type, sK351: rat > rat > rat).
% 59.76/13.07  thf(func_def_1454, type, sK352: nat > set_nat > nat).
% 59.76/13.07  thf(func_def_1455, type, sK353: real > real > real > real).
% 59.76/13.07  thf(func_def_1456, type, sK354: int > int).
% 59.76/13.07  thf(func_def_1457, type, sK355: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1458, type, sK356: vEBT_VEBT > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1459, type, sK357: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1460, type, sK358: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1461, type, sK359: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1462, type, sK360: vEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1463, type, sK361: nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1464, type, sK362: nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1465, type, sK363: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1466, type, sK364: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1467, type, sK365: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1468, type, sK366: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1469, type, sK367: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1470, type, sK368: vEBT_VEBT > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1471, type, sK369: vEBT_VEBT > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1472, type, sK370: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1473, type, sK371: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1474, type, sK372: nat > nat).
% 59.76/13.07  thf(func_def_1475, type, sK373: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1476, type, sK374: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1477, type, sK375: real > real > real).
% 59.76/13.07  thf(func_def_1478, type, sK376: (option_nat > option4927543243414619207at_nat > $o) > option_nat > option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1479, type, sK377: (option_nat > option4927543243414619207at_nat > $o) > option_nat > option4927543243414619207at_nat > nat).
% 59.76/13.07  thf(func_def_1480, type, sK378: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1481, type, sK379: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1482, type, sK380: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1483, type, sK381: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1484, type, sK382: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1485, type, sK383: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1486, type, sK384: nat > nat).
% 59.76/13.07  thf(func_def_1487, type, sK385: nat > nat > nat).
% 59.76/13.07  thf(func_def_1488, type, sK386: (int > $o) > int).
% 59.76/13.07  thf(func_def_1489, type, sK387: (int > $o) > int).
% 59.76/13.07  thf(func_def_1490, type, sK388: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1491, type, sK389: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1492, type, sK390: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1493, type, sK391: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1494, type, sK392: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1495, type, sK393: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1496, type, sK394: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1497, type, sK395: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1498, type, sK396: int > int > int).
% 59.76/13.07  thf(func_def_1499, type, sK397: option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1500, type, sK398: real > real > nat).
% 59.76/13.07  thf(func_def_1501, type, sK399: rat > rat > rat).
% 59.76/13.07  thf(func_def_1502, type, sK400: nat > nat).
% 59.76/13.07  thf(func_def_1503, type, sK401: rat > rat).
% 59.76/13.07  thf(func_def_1504, type, sK402: real > real).
% 59.76/13.07  thf(func_def_1505, type, sK403: int > nat).
% 59.76/13.07  thf(func_def_1506, type, sK404: int > nat).
% 59.76/13.07  thf(func_def_1507, type, sK405: num > num).
% 59.76/13.07  thf(func_def_1508, type, sK406: num > num).
% 59.76/13.07  thf(func_def_1509, type, sK407: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1510, type, sK408: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1511, type, sK409: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1512, type, sK410: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1513, type, sK411: nat > (nat > real) > real > nat > real).
% 59.76/13.07  thf(func_def_1514, type, sK412: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1515, type, sK413: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1516, type, sK414: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1517, type, sK415: (rat > rat > $o) > rat).
% 59.76/13.07  thf(func_def_1518, type, sK416: nat > nat).
% 59.76/13.07  thf(func_def_1519, type, sK417: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1520, type, sK418: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1521, type, sK419: (real > real) > real).
% 59.76/13.07  thf(func_def_1522, type, sK420: (real > real) > real).
% 59.76/13.07  thf(func_def_1523, type, sK421: (option_nat > option_nat > $o) > option_nat > option_nat > nat).
% 59.76/13.07  thf(func_def_1524, type, sK422: (option_nat > option_nat > $o) > option_nat > option_nat > nat).
% 59.76/13.07  thf(func_def_1525, type, sK423: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1526, type, sK424: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1527, type, sK425: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1528, type, sK426: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1529, type, sK427: int > int).
% 59.76/13.07  thf(func_def_1530, type, sK428: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1531, type, sK429: rat > rat).
% 59.76/13.07  thf(func_def_1532, type, sK430: real > real).
% 59.76/13.07  thf(func_def_1533, type, sK431: real > real > real).
% 59.76/13.07  thf(func_def_1534, type, sK432: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1535, type, sK433: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1536, type, sK434: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1537, type, sK435: int > int > int).
% 59.76/13.07  thf(func_def_1538, type, sK436: nat > nat > nat).
% 59.76/13.07  thf(func_def_1539, type, sK437: nat > nat > nat).
% 59.76/13.07  thf(func_def_1540, type, sK438: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1541, type, sK439: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1542, type, sK440: nat > nat).
% 59.76/13.07  thf(func_def_1543, type, sK441: option_nat > nat).
% 59.76/13.07  thf(func_def_1544, type, sK442: nat > nat > nat).
% 59.76/13.07  thf(func_def_1545, type, sK443: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1546, type, sK444: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1547, type, sK445: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1548, type, sK446: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1549, type, sK447: (real > $o) > (real > $o) > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1550, type, sK448: real > (real > $o) > (real > $o) > real).
% 59.76/13.07  thf(func_def_1551, type, sK449: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1552, type, sK450: nat > nat).
% 59.76/13.07  thf(func_def_1553, type, sK451: nat > nat).
% 59.76/13.07  thf(func_def_1554, type, sK452: nat > nat > nat).
% 59.76/13.07  thf(func_def_1555, type, sK453: nat > nat > nat).
% 59.76/13.07  thf(func_def_1556, type, sK454: nat > nat > nat).
% 59.76/13.07  thf(func_def_1557, type, sK455: nat > nat).
% 59.76/13.07  thf(func_def_1558, type, sK456: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1559, type, sK457: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1560, type, sK458: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1561, type, sK459: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1562, type, sK460: rat > rat).
% 59.76/13.07  thf(func_def_1563, type, sK461: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1564, type, sK462: nat > nat > nat).
% 59.76/13.07  thf(func_def_1565, type, sK463: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1566, type, sK464: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1567, type, sK465: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1568, type, sK466: real > (real > $o) > real).
% 59.76/13.07  thf(func_def_1569, type, sK467: real > real).
% 59.76/13.07  thf(func_def_1570, type, sK468: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1571, type, sK469: int > nat).
% 59.76/13.07  thf(func_def_1572, type, sK470: int > nat).
% 59.76/13.07  thf(func_def_1573, type, sK471: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1574, type, sK472: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1575, type, sK473: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1576, type, sK474: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1577, type, sK475: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1578, type, sK476: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1579, type, sK477: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1580, type, sK478: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1581, type, sK479: list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1582, type, sK480: list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1583, type, sK481: list_VEBT_VEBT > nat > nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1584, type, sK482: list_VEBT_VEBT > nat > nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1585, type, sK483: list_VEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1586, type, sK484: vEBT_VEBT > nat > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1587, type, sK485: code_integer > code_integer).
% 59.76/13.07  thf(func_def_1588, type, sK486: rat > rat).
% 59.76/13.07  thf(func_def_1589, type, sK487: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1590, type, sK488: nat > nat).
% 59.76/13.07  thf(func_def_1591, type, sK489: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1592, type, sK490: int > int > int > int).
% 59.76/13.07  thf(func_def_1593, type, sK491: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1594, type, sK492: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1595, type, sK493: (real > real > $o) > real).
% 59.76/13.07  thf(func_def_1596, type, sK494: real > real).
% 59.76/13.07  thf(func_def_1597, type, sK495: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1598, type, sK496: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1599, type, sK497: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1600, type, sK498: nat > nat).
% 59.76/13.07  thf(func_def_1601, type, sK499: int > int).
% 59.76/13.07  thf(func_def_1602, type, sK500: rat > rat > rat).
% 59.76/13.07  thf(func_def_1603, type, sK501: real > nat > real).
% 59.76/13.07  thf(func_def_1604, type, sK502: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1605, type, sK503: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1606, type, sK504: int > int > int > int).
% 59.76/13.07  thf(func_def_1607, type, sK505: int > int > int > int).
% 59.76/13.07  thf(func_def_1608, type, sK506: (real > rat) > real).
% 59.76/13.07  thf(func_def_1609, type, sK507: (real > rat) > real).
% 59.76/13.07  thf(func_def_1610, type, sK508: int > nat).
% 59.76/13.07  thf(func_def_1611, type, sK509: list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1612, type, sK510: list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1613, type, sK511: list_VEBT_VEBT > nat > nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1614, type, sK512: list_VEBT_VEBT > nat > nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1615, type, sK513: list_VEBT_VEBT > vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1616, type, sK514: list_VEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1617, type, sK515: (nat > int) > nat).
% 59.76/13.07  thf(func_def_1618, type, sK516: nat > nat > nat).
% 59.76/13.07  thf(func_def_1619, type, sK517: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1620, type, sK518: (rat > real) > rat).
% 59.76/13.07  thf(func_def_1621, type, sK519: rat > rat).
% 59.76/13.07  thf(func_def_1622, type, sK520: nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1623, type, sK521: nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1624, type, sK522: nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1625, type, sK523: nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1626, type, sK524: nat > vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1627, type, sK525: nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1628, type, sK526: nat > vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1629, type, sK527: vEBT_VEBT > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1630, type, sK528: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1631, type, sK529: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1632, type, sK530: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1633, type, sK531: vEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1634, type, sK532: vEBT_VEBT > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1635, type, sK533: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1636, type, sK534: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1637, type, sK535: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1638, type, sK536: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1639, type, sK537: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1640, type, sK538: vEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1641, type, sK539: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1642, type, sK540: vEBT_VEBT > nat > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1643, type, sK541: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1644, type, sK542: vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1645, type, sK543: vEBT_VEBT > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1646, type, sK544: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1647, type, sK545: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1648, type, sK546: nat > nat).
% 59.76/13.07  thf(func_def_1649, type, sK547: (vEBT_VEBT > $o) > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1650, type, sK548: (int > real) > int).
% 59.76/13.07  thf(func_def_1651, type, sK549: (int > real) > int).
% 59.76/13.07  thf(func_def_1652, type, sK550: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1653, type, sK551: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1654, type, sK552: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1655, type, sK553: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1656, type, sK554: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1657, type, sK555: real > real).
% 59.76/13.07  thf(func_def_1658, type, sK556: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1659, type, sK557: (real > rat) > real).
% 59.76/13.07  thf(func_def_1660, type, sK558: (real > rat) > real).
% 59.76/13.07  thf(func_def_1661, type, sK559: nat > nat).
% 59.76/13.07  thf(func_def_1662, type, sK560: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1663, type, sK561: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1664, type, sK562: (int > rat) > int).
% 59.76/13.07  thf(func_def_1665, type, sK563: (int > rat) > int).
% 59.76/13.07  thf(func_def_1666, type, sK564: product_prod_nat_nat > nat).
% 59.76/13.07  thf(func_def_1667, type, sK565: product_prod_nat_nat > nat).
% 59.76/13.07  thf(func_def_1668, type, sK566: option_nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1669, type, sK567: option_nat > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1670, type, sK568: option_nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1671, type, sK569: option_nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1672, type, sK570: option_nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1673, type, sK571: option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1674, type, sK572: option_nat > vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1675, type, sK573: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1676, type, sK574: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1677, type, sK575: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1678, type, sK576: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1679, type, sK577: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1680, type, sK578: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1681, type, sK579: int > nat).
% 59.76/13.07  thf(func_def_1682, type, sK580: nat > nat > nat).
% 59.76/13.07  thf(func_def_1683, type, sK581: nat > nat).
% 59.76/13.07  thf(func_def_1684, type, sK582: (int > $o) > (int > $o) > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1685, type, sK583: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1686, type, sK584: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1687, type, sK585: real > real).
% 59.76/13.07  thf(func_def_1688, type, sK586: int > int > int > int).
% 59.76/13.07  thf(func_def_1689, type, sK587: int > int > int > int).
% 59.76/13.07  thf(func_def_1690, type, sK588: real > real > nat).
% 59.76/13.07  thf(func_def_1691, type, sK589: rat > rat).
% 59.76/13.07  thf(func_def_1692, type, sK590: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1693, type, sK591: (real > int) > real).
% 59.76/13.07  thf(func_def_1694, type, sK592: (real > int) > real).
% 59.76/13.07  thf(func_def_1695, type, sK593: (nat > set_int) > nat).
% 59.76/13.07  thf(func_def_1696, type, sK594: int > int).
% 59.76/13.07  thf(func_def_1697, type, sK595: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1698, type, sK596: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1699, type, sK597: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1700, type, sK598: (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1701, type, sK599: nat > nat > nat).
% 59.76/13.07  thf(func_def_1702, type, sK600: int > int > int).
% 59.76/13.07  thf(func_def_1703, type, sK601: int > int > int).
% 59.76/13.07  thf(func_def_1704, type, sK602: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1705, type, sK603: set_nat > nat > nat).
% 59.76/13.07  thf(func_def_1706, type, sK604: set_nat > nat > nat).
% 59.76/13.07  thf(func_def_1707, type, sK605: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1708, type, sK606: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1709, type, sK607: nat > nat > nat).
% 59.76/13.07  thf(func_def_1710, type, sK608: (int > $o) > int).
% 59.76/13.07  thf(func_def_1711, type, sK609: int > (int > $o) > (int > $o) > int).
% 59.76/13.07  thf(func_def_1712, type, sK610: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1713, type, sK611: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1714, type, sK612: int > int).
% 59.76/13.07  thf(func_def_1715, type, sK613: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1716, type, sK614: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1717, type, sK615: rat > rat).
% 59.76/13.07  thf(func_def_1718, type, sK616: real > real).
% 59.76/13.07  thf(func_def_1719, type, sK617: (int > rat) > int).
% 59.76/13.07  thf(func_def_1720, type, sK618: (int > rat) > int).
% 59.76/13.07  thf(func_def_1721, type, sK619: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1722, type, sK620: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1723, type, sK621: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1724, type, sK622: nat > nat > nat).
% 59.76/13.07  thf(func_def_1725, type, sK623: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1726, type, sK624: (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1727, type, sK625: real > real > real).
% 59.76/13.07  thf(func_def_1728, type, sK626: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1729, type, sK627: (nat > rat) > nat).
% 59.76/13.07  thf(func_def_1730, type, sK628: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1731, type, sK629: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1732, type, sK630: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1733, type, sK631: (nat > real) > nat).
% 59.76/13.07  thf(func_def_1734, type, sK632: nat > nat > nat).
% 59.76/13.07  thf(func_def_1735, type, sK633: vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1736, type, sK634: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1737, type, sK635: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1738, type, sK636: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1739, type, sK637: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1740, type, sK638: vEBT_VEBT > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1741, type, sK639: nat > $o > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1742, type, sK640: nat > $o > vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1743, type, sK641: nat > $o > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1744, type, sK642: nat > $o > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1745, type, sK643: nat > $o > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1746, type, sK644: nat > $o > vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1747, type, sK645: nat > $o > vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1748, type, sK646: vEBT_VEBT > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1749, type, sK647: vEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1750, type, sK648: vEBT_VEBT > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1751, type, sK649: nat > list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1752, type, sK650: int > nat).
% 59.76/13.07  thf(func_def_1753, type, sK651: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1754, type, sK652: (rat > rat) > rat).
% 59.76/13.07  thf(func_def_1755, type, sK653: int > (int > $o) > int).
% 59.76/13.07  thf(func_def_1756, type, sK654: (nat > nat) > nat).
% 59.76/13.07  thf(func_def_1757, type, sK655: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1758, type, sK656: nat > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1759, type, sK657: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1760, type, sK658: rat > rat).
% 59.76/13.07  thf(func_def_1761, type, sK659: (real > nat) > real).
% 59.76/13.07  thf(func_def_1762, type, sK660: (real > nat) > real).
% 59.76/13.07  thf(func_def_1763, type, sK661: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1764, type, sK662: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1765, type, sK663: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1766, type, sK664: (nat > nat > $o) > nat).
% 59.76/13.07  thf(func_def_1767, type, sK665: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1768, type, sK666: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1769, type, sK667: real > real).
% 59.76/13.07  thf(func_def_1770, type, sK668: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1771, type, sK669: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1772, type, sK670: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1773, type, sK671: (rat > $o) > (rat > $o) > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1774, type, sK672: rat > (rat > $o) > (rat > $o) > rat).
% 59.76/13.07  thf(func_def_1775, type, sK673: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1776, type, sK674: (rat > int) > rat).
% 59.76/13.07  thf(func_def_1777, type, sK675: nat > set_nat > nat).
% 59.76/13.07  thf(func_def_1778, type, sK676: set_nat > nat).
% 59.76/13.07  thf(func_def_1779, type, sK677: (nat > $o) > nat > nat > nat).
% 59.76/13.07  thf(func_def_1780, type, sK678: vEBT_VEBT > nat > nat > nat).
% 59.76/13.07  thf(func_def_1781, type, sK679: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1782, type, sK680: (int > $o) > int > int).
% 59.76/13.07  thf(func_def_1783, type, sK681: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1784, type, sK682: (vEBT_VEBT > $o) > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1785, type, sK683: option_nat > nat).
% 59.76/13.07  thf(func_def_1786, type, sK684: int > int > int).
% 59.76/13.07  thf(func_def_1787, type, sK685: nat > (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1788, type, sK686: nat > (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1789, type, sK687: list_VEBT_VEBT > nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1790, type, sK688: vEBT_VEBT > nat > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1791, type, sK689: (nat > vEBT_VEBT > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1792, type, sK690: (nat > vEBT_VEBT > $o) > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1793, type, sK691: (nat > vEBT_VEBT > $o) > nat > list_VEBT_VEBT).
% 59.76/13.07  thf(func_def_1794, type, sK692: (nat > vEBT_VEBT > $o) > nat > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1795, type, sK693: nat > (nat > vEBT_VEBT > $o) > vEBT_VEBT).
% 59.76/13.07  thf(func_def_1796, type, sK694: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1797, type, sK695: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1798, type, sK696: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1799, type, sK697: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1800, type, sK698: list_VEBT_VEBT > nat > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1801, type, sK699: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1802, type, sK700: nat > nat > nat > nat).
% 59.76/13.07  thf(func_def_1803, type, sK701: nat > nat > nat).
% 59.76/13.07  thf(func_def_1804, type, sK702: option_nat > nat).
% 59.76/13.07  thf(func_def_1805, type, sK703: (nat > real) > real > nat > real).
% 59.76/13.07  thf(func_def_1806, type, sK704: (nat > real) > real > nat > real).
% 59.76/13.07  thf(func_def_1807, type, sK705: (nat > real) > nat > nat).
% 59.76/13.07  thf(func_def_1808, type, sK706: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1809, type, sK707: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1810, type, sK708: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1811, type, sK709: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1812, type, sK710: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1813, type, sK711: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1814, type, sK712: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1815, type, sK713: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1816, type, sK714: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1817, type, sK715: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1818, type, sK716: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1819, type, sK717: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1820, type, sK718: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1821, type, sK719: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1822, type, sK720: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1823, type, sK721: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1824, type, sK722: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1825, type, sK723: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1826, type, sK724: (nat > $o) > nat > nat > nat).
% 59.76/13.07  thf(func_def_1827, type, sK725: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1828, type, sK726: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1829, type, sK727: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1830, type, sK728: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1831, type, sK729: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1832, type, sK730: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1833, type, sK731: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1834, type, sK732: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1835, type, sK733: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1836, type, sK734: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1837, type, sK735: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1838, type, sK736: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1839, type, sK737: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1840, type, sK738: (int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1841, type, sK739: nat > nat).
% 59.76/13.07  thf(func_def_1842, type, sK740: vEBT_VEBT > nat > list_VEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1843, type, sK741: (int > int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1844, type, sK742: (int > int > $o) > int > int > int).
% 59.76/13.07  thf(func_def_1845, type, sK743: nat > vEBT_VEBT > nat > nat).
% 59.76/13.07  thf(func_def_1846, type, sK744: option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1847, type, sK745: int > int > (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1848, type, sK746: int > int > (int > int > $o) > int).
% 59.76/13.07  thf(func_def_1849, type, sK747: option4927543243414619207at_nat > product_prod_nat_nat).
% 59.76/13.07  thf(func_def_1850, type, sK748: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1851, type, sK749: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1852, type, sK750: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1853, type, sK751: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1854, type, sK752: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1855, type, sK753: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1856, type, sK754: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1857, type, sK755: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1858, type, sK756: set_nat > nat).
% 59.76/13.07  thf(func_def_1859, type, sK757: nat > set_nat > nat).
% 59.76/13.07  thf(func_def_1860, type, sK758: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1861, type, sK759: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1862, type, sK760: (nat > real) > nat > (nat > real) > real).
% 59.76/13.07  thf(func_def_1863, type, sK761: (nat > real) > nat > (nat > real) > nat).
% 59.76/13.07  thf(func_def_1864, type, sK762: nat > (nat > real) > real).
% 59.76/13.07  thf(func_def_1865, type, sK763: nat > (nat > real) > nat).
% 59.76/13.07  thf(func_def_1866, type, sK764: nat > nat > nat).
% 59.76/13.07  thf(func_def_1867, type, sK765: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1868, type, sK766: nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1869, type, sK767: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1870, type, sK768: vEBT_VEBT > $o).
% 59.76/13.07  thf(func_def_1871, type, sK769: list_VEBT_VEBT > vEBT_VEBT > nat).
% 59.76/13.07  thf(func_def_1872, type, sK770: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1873, type, sK771: nat > nat > (nat > $o) > nat).
% 59.76/13.07  thf(func_def_1874, type, sK772: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1875, type, sK773: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1876, type, sK774: (nat > $o) > nat > nat).
% 59.76/13.07  thf(func_def_1877, type, ph775: !>[X0: $tType]:(X0)).
% 59.76/13.07  thf(f43982,plain,(
% 59.76/13.07    $false),
% 59.76/13.07    inference(subsumption_resolution,[status(thm)],[f43981,f30560])).
% 59.76/13.07  thf(f30560,plain,(
% 59.76/13.07    ((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) != none_nat)),
% 59.76/13.07    inference(cnf_transformation,[status(thm)],[f1])).
% 59.76/13.07  thf(f1,axiom,(
% 59.76/13.07    ((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) != none_nat)),
% 59.76/13.07    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',fact_0_False)).
% 59.76/13.07  thf(f43981,plain,(
% 59.76/13.07    ((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) = none_nat)),
% 59.76/13.07    inference(equality_resolution,[status(thm)],[f43980])).
% 59.76/13.07  thf(f43980,plain,(
% 59.76/13.07    ( ! [X0 : option_nat] : (((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) != X0) | (none_nat = X0)) )),
% 59.76/13.07    inference(superposition,[status(thm)],[f42972,f31771])).
% 59.76/13.07  thf(f31771,plain,(
% 59.76/13.07    ( ! [X0 : option_nat] : (((some_nat @ (sK441 @ X0)) = X0) | (none_nat = X0)) )),
% 59.76/13.07    inference(cnf_transformation,[status(thm)],[f28764])).
% 59.76/13.07  thf(f28764,plain,(
% 59.76/13.07    ! [X0 : option_nat] : (((some_nat @ (sK441 @ X0)) = X0) | (none_nat = X0))),
% 59.76/13.07    inference(skolemisation,[status(esa),new_symbols(skolem,[sK441])],[f26553,f28763])).
% 59.76/13.07  thf(f28763,plain,(
% 59.76/13.07    ! [X0 : option_nat] : (? [X1 : nat] : ((some_nat @ X1) = X0) => ((some_nat @ (sK441 @ X0)) = X0))),
% 59.76/13.07    introduced(definition,[],[choice_axiom])).
% 59.76/13.07  thf(f26553,plain,(
% 59.76/13.07    ! [X0 : option_nat] : (? [X1 : nat] : ((some_nat @ X1) = X0) | (none_nat = X0))),
% 59.76/13.07    inference(ennf_transformation,[status(thm)],[f24578])).
% 59.76/13.07  thf(f24578,plain,(
% 59.76/13.07    ! [X0 : option_nat] : ((none_nat != X0) => ~! [X1 : nat] : ((some_nat @ X1) != X0))),
% 59.76/13.07    inference(rectify,[status(thm)],[f86])).
% 59.76/13.07  thf(f86,axiom,(
% 59.76/13.07    ! [X10 : option_nat] : ((none_nat != X10) => ~! [X23 : nat] : ((some_nat @ X23) != X10))),
% 59.76/13.07    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',fact_85_option_Oexhaust)).
% 59.76/13.07  thf(f42972,plain,(
% 59.76/13.07    ( ! [X0 : nat] : (((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) != (some_nat @ X0))) )),
% 59.76/13.07    inference(subsumption_resolution,[status(thm)],[f29704,f30824])).
% 59.76/13.07  thf(f30824,plain,(
% 59.76/13.07    (thesis != $true)),
% 59.76/13.07    inference(cnf_transformation,[status(thm)],[f24320])).
% 59.76/13.07  thf(f24320,plain,(
% 59.76/13.07    (thesis != $true)),
% 59.76/13.07    inference(flattening,[status(thm)],[f16769])).
% 59.76/13.07  thf(f16769,plain,(
% 59.76/13.07    ~(thesis = $true)),
% 59.76/13.07    inference(fool_elimination,[status(thm)],[f16768])).
% 59.76/13.07  thf(f16768,plain,(
% 59.76/13.07    ~thesis),
% 59.76/13.07    inference(rectify,[status(thm)],[f10250])).
% 59.76/13.07  thf(f10250,negated_conjecture,(
% 59.76/13.07    ~thesis),
% 59.76/13.07    inference(negated_conjecture,[status(cth)],[f10249])).
% 59.76/13.07  thf(f10249,conjecture,(
% 59.76/13.07    thesis),
% 59.76/13.07    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',conj_1)).
% 59.76/13.07  thf(f29704,plain,(
% 59.76/13.07    ( ! [X0 : nat] : ((thesis = $true) | ((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) != (some_nat @ X0))) )),
% 59.76/13.07    inference(cnf_transformation,[status(thm)],[f26591])).
% 59.76/13.07  thf(f26591,plain,(
% 59.76/13.07    ! [X0 : nat] : (((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) != (some_nat @ X0)) | (thesis = $true))),
% 59.76/13.07    inference(ennf_transformation,[status(thm)],[f21395])).
% 59.76/13.07  thf(f21395,plain,(
% 59.76/13.07    ! [X0 : nat] : (((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) = (some_nat @ X0)) => (thesis = $true))),
% 59.76/13.07    inference(fool_elimination,[status(thm)],[f21394])).
% 59.76/13.07  thf(f21394,plain,(
% 59.76/13.07    ! [X0 : nat] : (((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) = (some_nat @ X0)) => thesis)),
% 59.76/13.07    inference(rectify,[status(thm)],[f10248])).
% 59.76/13.07  thf(f10248,axiom,(
% 59.76/13.07    ! [X292 : nat] : (((vEBT_vebt_mint @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) = (some_nat @ X292)) => thesis)),
% 59.76/13.07    file('/export/starexec/sandbox2/tmp/tmp.JavhEzEqM4/DTF2THF_22586.p',conj_0)).
% 59.76/13.07  % SZS output end Proof for DTF2THF_22586
% 59.76/13.07  % (22889)------------------------------
% 59.76/13.07  % (22889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.76/13.07  % (22889)Termination reason: Refutation
% 59.76/13.07  
% 59.76/13.07  % (22889)Memory used [KB]: 33389
% 59.76/13.07  % (22889)Time elapsed: 1.081 s
% 59.76/13.07  % (22889)Instructions burned: 3433 (million)
% 59.76/13.07  % (22889)------------------------------
% 59.76/13.07  % (22889)------------------------------
% 59.76/13.07  % (22693)Success in time 7.725 s
%------------------------------------------------------------------------------